Te History of Mathematical Proof in Ancient raditions
Tis radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the rst mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers, and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the rst comprehensive, textually based history of proof.
Jeremy Gray, Professor of the History of Mathematics, Open University ‘Each of the papers in this volume, starting with the amazing “Prologue” by the editor, Karine Chemla, contributes to nothing less than a revolution in the way we need to think about both the substance and the historiography of ancient non-Western mathematics, as well as a reconception of the problems that need to be addressed if we are to get beyond myth-eaten ideas of “unique Western rationality” and “the Greek miracle”. I found reading this volume a thrilling intellectual adventure. It deserves a very wide audience.’
Hilary Putnam, Cogan University Professor Emeritus, Harvard University is Senior Researcher at the CNRS (Research Unit SPHERE, University Paris Diderot, France), and a Senior Fellow at the Institute for the Study of the Ancient World at New York University. She is also Professor on a Guest Chair at Northwestern University, Xi‘an, as well as at Shanghai Jiaotong University and Hebei Normal University, China. She was awarded a Chinese Academy of Sciences Visiting Professorship for Senior Foreign Scientists in 2009.
Te History of Mathematical Proof In Ancient raditions Edited by
Cambridge, New York, Melbourne, Madrid, Cape own, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press Te Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107012219 © Cambridge University Press 2012 Tis publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library ISBN 9781107012219 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
List of gures [ix] List of contributors [xii] Note on references [xiv] Acknowledgements [xv] Prologue Historiography and history of mathematical proof: a research programme [1]
Shaping ancient Greek mathematics: the critical editions of Greek texts in the nineteenth century 1 Te Euclidean ideal of proof in Te Elements and philological uncertainties of Heiberg’s edition of the text [69] 2 Diagrams and arguments in ancient Greek mathematics: lessons drawn from comparisons of the manuscript diagrams with those in modern critical editions [135] 3 Te texture of Archimedes’ writings: through Heiberg’s veil [163]
Shaping ancient Greek mathematics: the philosophers’ contribution 4 John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations [206]
Forming views on the ‘Others’ on the basis of mathematical proof 5 Contextualizing Playfair and Colebrooke on proof and demonstration in the Indian mathematical tradition (1780–1820) [228] v
vi
Contents
6 Overlooking mathematical justi cations in the Sanskrit tradition: the nuanced case of G. F. W. Tibaut [260] 7 Te logical Greek versus the imaginative Oriental: on the historiography of ‘non-Western’ mathematics during the period 1820–1920 [274]
:
Critical approaches to Greek practices of proof 8 Te pluralism of Greek ‘mathematics’ . . .
[294]
Proving with numbers: in Greece 9 Generalizing about polygonal numbers in ancient Greek mathematics [311] 10 Reasoning and symbolism in Diophantus: preliminary observations
[327]
Proving with numbers: establishing the correctness of algorithms 11 Mathematical justi cation as non-conceptualized practice: the Babylonian example [362] 12 Interpretation of reverse algorithms in several Mesopotamian texts [384] 13 Reading proofs in Chinese commentaries: algebraic proofs in an algorithmic context [423] 14 Dispelling mathematical doubts: assessing mathematical correctness of algorithms in Bhāskara’s commentary on the mathematical chapter of the Āryabhatīya [487]
˙
Contents
Te later persistence of traditions of proving in Asia: late evidence of traditions of proof 15 Argumentation for state examinations: demonstration in traditional Chinese and Vietnamese mathematics [509]
Te later persistence of traditions of proving in Asia: interactions of various traditions 16 A formal system of the Gougu method: a study on Li Rui’s
Detailed Outline of Mathematical Procedures for the Right-Angled riangle [552]
Index [574]
vii
Figures
1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
2.10 2.11
2.12 2.13 2.14 2.15 2.16 2.17 3.1
3.2
3.3
3.4
3.5
extual history: the philological approach. Euclid’s Elements. ypology of deliberate structural alterations. Euclid’s Elements. Proposition XII.15. Diagrams for Euclid’s Elements, Book XI, Proposition 12. Diagrams for Euclid’s Elements, Book I, Proposition 13. Diagrams for Euclid’s Elements, Book I, Proposition 7. Diagrams for Euclid’s Elements, Book I, Proposition 35. Diagrams for Euclid’s Elements, Book VI, Proposition 20. Diagrams for Euclid’s Elements, Book I, Proposition 44. Diagrams for Euclid’s Elements, Book II, Proposition 7. Diagrams for Apollonius’ Conica, Book I, Proposition 16. Diagrams for Euclid’s Elements, Book IV, Proposition 16. Dashed lines were drawn in and later erased. Grey lines were drawn in a different ink or with a different instrument. Diagrams for Archimedes’ Method, Proposition 12. Diagrams for Euclid’s Elements, Book XI, Proposition 33 and Apollonius’ Conica, Book I, Proposition 13. Diagrams for Teodosius’ Spherics, Book II, Proposition 6. Diagrams for Teodosius’ Spherics, Book II, Proposition 15. Diagrams for Euclid’s Elements, Book III, Proposition 36. Diagrams for Euclid’s Elements, Book III, Proposition 21. Diagrams for Euclid’s Elements, Book I, Proposition 44. Diagrams for Euclid’s Elements, Book I, Proposition 22. Heiberg’s diagrams for Sphere and Cylinder I.16 and the reconstruction of Archimedes’ diagrams. A reconstruction of Archimedes’ diagram for Sphere and Cylinder I.15. Heiberg’s diagram for Sphere and Cylinder I.9 and the reconstruction of Archimedes’ diagram. Heiberg’s diagram for Sphere and Cylinder I.12 and the reconstruction of Archimedes’ diagram. Heiberg’s diagram for Sphere and Cylinder I.33 and the reconstruction of Archimedes’ diagram.
ix
x
List of figures
3.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7
5.8 9.1 9.2 9.3 9.4 9.5 9.6 11.1 11.2 11.3 11.4 11.5 11.6 11.7 13.1 13.2 13.3
13.4
13.5
13.6
13.7 13.8
13.9
13.10 14.1
Te general case of a division of the sphere. Te square a2. Te square a2 minus the square b2. Te rectangle of sides a + b and b a. Te square a2. Te square b2. Te square (a + b)2. Te area (a + b)2 minus the squares a2 and b2 equals twice the product ab. A right-angled triangle ABC and its height BD. Geometric representation of polygonal numbers. Te generation of square numbers. Te generation of the rst three pentagonal numbers. Te graphic representation of the fourth pentagonal number. Diophantus’ diagram, Polygonal Numbers, Proposition 4. Diophantus’ diagram, Polygonal Numbers. Te con guration of VA 8390 #1. Te procedure of BM 13901 #1, in slightly distorted proportions. Te con guration discussed in MS #1. Te con guration of MS #2. Te situation of MS #1. Te transformations of MS #1. Te procedure of YBC 6967. Te truncated pyramid with circular base. Te truncated pyramid with square base. Te layout of the algorithm up to the point of the multiplication of fractions. Te execution of the multiplication of fractions on the surface for computing. Te basic structure of algorithms 1 and 2, for the truncated pyramid with square base. Te basic structure of algorithm 2 , which begins the computation of the volume sought for.
Algorithm 5: cancelling opposed multiplication and division. Te division between quantities with fractions on the surface for computing. Te multiplication between quantities with fractions on the surface for computing. Te layout of a division or a fraction on the surface for computing. Names of the sides of a right-angled triangle.
List of gures
14.2 14.3 14.4 14.5 14.6 14.7 14.8 16.1 16.2
A schematized gnomon and light. Proportional astronomical triangles. Altitude and zenith. Latitude and co-latitude on an equinoctial day. Inner segments and elds in a trapezoid. An equilateral pyramid with a triangular base. Te proportional properties of similar triangles. Te gougu shape (right-angled triangle). Li Rui’s diagram for his explanation for the fourth problem in
Detailed Outline of Mathematical Procedures for the Right-Angled riangle. 16.3
Li Rui’s diagram for his explanation for the eighth problem in
Detailed Outline of Mathematical Procedures for the Right-Angled riangle.
xi
Contributors
Independent scholar (retire d), Gärtringen, Germany Directrice de recherche, REHSEIS, UMR SPHERE, CNRS and University Paris Diderot, PRES Sorbonne Paris Cité, France Department of Philosophy and Department of Classics, el Aviv University, Israel Emeritus Professor, Section for Philosophy and Science Studies, Roskilde University, Roskilde, Denmark Chargée de recherche, REHSEIS, UMR SPHERE, CNRS and University Paris Diderot, PRES Sorbonne Paris Cité, France . . .
Professor, Needham Research Institute, Cambridge, UK
Emeritus Professor, Philosophy and Conceptual Foundations of Science, University of Chicago, USA (deceased 2010) Professor, Department of Classics, Stanford University, Palo Alto, USA Directrice de recherche, REHSEIS, UMR SPHERE, CNRS and University Paris Diderot, PRES Sorbonne Paris Cité, Paris, France Professor, School of Social Sciences, Jawaharlal Nehru University, New Delhi, India Professor, Department of Human Sciences, Osaka Prefecture University, Japan Assistant Professor, School of International Liberal Studies, Waseda University, okyo, Japan
xii
Senior Researcher, IHNS, Chinese Academy of Science, Beijing, China
List of contributors
Directeur de recherche, ANHIMA, CNRS UMR 8210, Paris, France Assistant Professor, Center for General Education and Institute of History, National sing-Hua University, Hsinchu, R.O.C., aiwan
xiii
Note on references
Te following books are frequently referred to in the notes. We use the following abbreviations to refer to them. CG2004 Chemla, K. and Guo Shuchun (2004) Les Neuf Chapitres: le classique mathématique de la Chine ancienne et ses commentaires . Paris. (1817) Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhāscara. ranslated by H. . Colebrooke. London.
C1817
Colebrooke,
H. .
H1995
Hayashi, . (1995) Te Bakhshali Manuscript: An Ancient Indian Mathematical reatise. Groningen.
H2002
Høyrup, J. (2002) Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. New York.
LD1987
Li Yan, Du Shiran ([1963] 1987) Mathematics in Ancient China: A Concise History (Zhongguo gudai shuxue jianshi). Beijing. Updated and translated in English by J. N. Crossley and A. W. C. Lun, Chinese Mathematics: A Concise History. Oxford.
N1999
Netz, R. (1999) Te Shaping of Deduction in Greek Mathematics. Cambridge.
1893/5 annery, P. (1893–5) Diophanti Alexandrini opera omnia cum graecis commentariis, edidit et latine interpretatus, vol. : 1893; vol. : 1895. Leipzig.
xiv
Acknowledgements
Te book that the reader has in his or her hands is based on the research carried out within the context of a working group that convened in Paris for three months during the spring of 2002. Te core members of the group were: Geoffrey Lloyd, Ian Mueller, Dhruv Raina, Reviel Netz and myself. Other colleagues took part in some or all of the weekly discussions: Alain Bernard, Armelle Debru, Marie-José Durand-Richard, Pierre-Sylvain Filliozat, Catherine Jami, Agathe Keller, François Patte, Christine Proust, ian Miao, Bernard Vitrac and Alexei Volkov. As a complement to its work, this group organized a workshop to tackle questions for which no specialist could be found within the srcinal set of participants ( www.pieaipas.msh-paris.fr/IMG/pdf/RAPPOR_groupe_Chemla.pdf). Te whole endeavour has been made possible thanks to the International Advanced Study Program set up by the Maison des sciences de l’homme, Paris, in collaboration with Reid Hall, Columbia University at Paris. It is my pleasure to express to these institutions my deepest gratitude. I completed the writing of the introduction at the Dibner Institute, MI, to which I am pleased to address my heartfelt thanks. Stays at the Max Planck Institute, Berlin, in 2007, and at Le Mas Pascal, Cavillargues, in 2008 and 2009, have provided the quietness needed to complete the project. Tanks for that to Hans-Jörg Rheinberger, Jean-Pascal Jullien and Gilles Vandenbroeck. For the preparation of this volume, the core members of the group acted as an editorial board. I express my deepest gratitude to those who accepted the anonymous work of being referees. Micah Ross, Guo Yuanyuan, Wang Xiaofei, Leonid Zhmud and Zhu Yiwen have played a key role in the elaboration of this book. I have pleasure here in expressing my deepest thanks to them as well as to those who read versions of this introduction: Bruno Belhoste, Evelyn Fox Keller, Ramon Guardans and Jacques Virbel.
xv
Prologue
Historiography and history o mathematical proo: a research programme
Pour Oriane, ces raisonnements sur les raisonnements
I Introduction: a standard view Te standard history o mathematical proo in ancient traditions at the present day is disturbingly simple. Tis perspective can be represented by the ollowing assertions. (1) Mathematical proo emerged in ancient Greece and achieved a mature orm in the geometrical works o Euclid, Archimedes and Apollonius. (2) Te ull- edged theory underpinning mathematical proo was ormulated in Aristotle’s Posterior Analytics, which describes the model o demonstration rom which any piece o knowledge adequately known should derive. (3) Beore these developments took place in classical Greece, there was no evidence o proo worth mentioning, a act which has contributed to the promotion o the concept o ‘Greek miracle’. Tis account also implies that mathematical proo is distinctive o Europe, or it would appear that no other mathematical tradition has ever shown interest in establishing the truth o statements.1 Finally, it is assumed that mathematical proo, as it is practised today, is inherited exclusively rom these Greek ancestors. Are things so simple? Tis book argues that they are not. But we shall see that some preliminary analysis is required to avoid alling into the old, amiliar pitalls. Powerul rhetorical devices have been constructed which perpetuate this simple view, and they need to be identi ed beore any meaningul discussion can take place. Tis should not surprise us. As Geoffrey Lloyd has repeatedly stressed, some o these devices were shaped in the context o erce debates among competing ‘masters o truth’ in ancient Greece, and these devices continue to have effective orce.2 1
2
See, or example, M. Kline’s crude evaluation o what a procedure was in Mesopotamia and how it was derived, quoted in J. Høyrup’s chapter, p. 363. Te rst lay sinologist to work on ancient Chinese texts related to mathematics, Edouard Biot, does not ormulate a higher assessment – see the statement quoted in A. Volkov’s chapter, p. 512. On Biot’s special emphasis on the lack o proos in Chinese mathematical texts, compare Martija-Ochoa 2001 –2: 61. See chapter 3 in Lloyd 1990: 73–97, Lloyd 1996a. Lloyd has also regularly emphasized how ‘Te concentration on the model o demonstration in the Organon and in Euclid, the one that
1
2
Studies o mathematical proo as an aspect o the intellectual history o the ancient world have echoed the belies summarized above – in part, by concentrating mainly on Euclid’sElements and Archimedes’ writings, the subtleties o which seem to be in nite. Te practice o proo to which these writings bear witness has impressed many minds, well beyond the strict domain o mathematics. Since antiquity, versions o Euclid’sElements, in Greek, in Arabic, in Latin, in Hebrew and later in the various vernacular languages o Europe, have regularly constituted a central piece o mathematical education, even though they were by no means the only element o mathematical education. Te proos in these editions were widely emulated by those interested in the value o incontrovertibility attached to them and they inspired the discussions o many philosophers. However, some versions o Euclid’s Elements have also been used since early modern times – in Europe and elsewhere – in ways that show how mathematical proo has been enrolled or unexpected purposes. One stunning example will suffice to illustrate this point. At the end o the sixteenth century, European missionaries arrived at the southern door o China. As a result o the difficulties encountered in entering China and capturing the interest o Chinese literati, the Jesuit Matteo Ricci devised a strategy o evangelism in which the science and technology available in Europe would play a key part. One o the rst steps taken in this programme was the publication o a Chinese version o Euclid’s Elements in 1607. Prepared by Ricci himsel in collaboration with the Chinese convert and high official Xu Guangqi, this translation was based on Clavius’ edition o the Elements, which Ricci had studied in Rome, while he was a student at the Collegio Romano. Te purpose o the translation was maniold. wo aspects are important or us here. First, the purportedly superior value o the type o geometrical knowledge introduced, when compared to the mathematical knowledge available to Chinese literati at that time, was expected to plead in avour o those who possessed that knowledge, namely, European missionaries. Additionally, the kind o certainty such a type o proo was prized or securing in mathematics could also be claimed or the theological teachings which the missionaries introduced simultaneously and which made use o reasoning similar to the proo o Euclidean geometry.3 Tus, in the rst large-scale intellectual contact between Europe
3
proceeds via valid deductive argument rom premises that are themselves indemonstrable but necessary and sel-evident, that concentration is liable to distort the Greek materials already – let alone the interpretation o Chinese texts.’ (Lloyd 1992: 196.) On Ricci’s background and evangelization strategy, see Martzloff 1984. Martzloff 1995 is devoted more generally to the translations o Clavius’s textbooks on the mathematical sciences
Mathematical proo: a research programme
and China mediated by the missionaries, mathematical proo played a role having little to do with mathematics stricto sensu. It is difficult to imagine that such a use and such a context had no impact on its reception in China.4 Tis topic will be revisited later. Te example outlined is ar rom unique in showing the role o mathematical proo outside mathematics. In an article signi cantly titled ‘What mathematics has done to some and only some philosophers’, Ian Hacking (2000) stresses the strange uses that mathematical proo inspired in philosophy as well as in theological arguments. In it, he diagnoses how mathematics, that is, in act, the experience o mathematical proo, has ‘inected’
4
into Chinese at the time. Engelriet 1993 discusses the relationship between Euclid’s Elements and teachings on Christianity in Ricci’s European context. More generally, this article outlines the role which Clavius allotted to mathematical sciences in Jesuit schools and in the wider Jesuit strategy or Europe. For a general and excellent introduction to the circumstances o the translation o Euclid’sElements into Chinese, an analysis and a complete bibliography, see Engelriet 1998. Xu Guangqi’s biography and main scholarly works were the object o a collective endeavour: Jami, Engelriet and Blue 2001. Martzloff 1981, Martzloff 1993 are devoted to the reception o this type o geometry in China, showing the variety o reactions that the translation o the Elements aroused among Chinese literati. On the other hand, the process o introduction o Clavius’ textbook or arithmetic was strikingly different. See Chemla 1996, Chemla 1997a. Leibniz appears to have been the rst scholar in uErope who, one century afer the Jesuits had arrived in China, became interested in the question o knowing whether ‘the Chinese’ ever developed mathematical proos in their past. In his letter to Joachim Bouvet sent rom Braunschweig on 15 February 1701, Leibniz asked whether the Jesuit, who was in evangelistic mission in China, could give him any inormation about geometrical proos in China: ‘J’ay souhaité aussi de sçavoir si ce que les Chinois ont eu anciennement de Geometrie, a esté accompagné de quelques demonstrations, et particulièrement s’ils ont sçû il y a long temps l’égalité du quarré de l’hypotenuse aux deux quarrés des costés, ou quelque autre telle proposition de la Geometrie non populaire.’ (Widmaier 2006 : 320; my emphasis.) In act, Leibniz had already expressed this interest ew years earlier, in a letter written in Hanover on 2 December 1697, to the same correspondent: ‘Outre l’Histoire des dynasties chinoises . . ., il audroit avoir soin de l’Histoire des inventions [,] des arts, des loix, des religions, et d’autres établissements[.] Je voudrois bien sçavoir par exemple s’il[s] n’ont eu il y a long temps quelque chose d’approchant de nostre Geometrie, et si l’egalité du quarré de l’Hypotenuse à ceux des costés du triangle rectangle leur a esté connue, et s’ils ont eu cette proposition par tradition ou commerce des autres peuples, ou par l’experience, ou en n par demonstration, soit trouvée chez eux ou apportée d’ailleurs.’ (Widmaier 2006: 142–4, my emphasis.) o this, Bouvet replied on 28 February 1698: ‘Le point au quel on pretend s’appliquer davantage comme le plus important est leur chronologie . . . Apres quoy on travaillera sur leur histoire naturelle et civile[,] sur leur physique, leur morale, leurs loix, leur politique, leurs Arts, leurs mathematiques et leur medecine, qui est une des matieres sur quoy je suis persuadé que la Chine peut nous ournir de[s] plus belles connaissances.’ (Widmaier 2006 : 168.) In his letter rom 1697 (Widmaier 2006 : 144–6), Leibniz expressed the conviction that, even though ‘their speculative mathematics’ could not hold the comparison with what he called ‘our mathematics’, one could still learn rom them. o this, in a sequel to the preceding letter, Bouvet expressed a strong agreement (Widmaier 2006: 232). Mathematics, especially proo, was already a ‘measure’ used or comparative purposes.
3
4
‘some central parts o [the] philosophy [o some philosophers], parts that have nothing intrinsically to do with mathematics’ (p. 98). What is important or us to note or the moment is that through such non-mathematical uses o mathematical proo the actors’ perception o proo has been colored by implications that were oreign to mathematics itsel. Tis observation may help to account or the astonishing emotion that ofen permeates debates on mathematical proo – ordinary ones as well as more academic ones – while other mathematical issues meet with indifference.5 On the other hand, these historical uses o proo in non-mathematical domains, as well as uses still ofen ound in contemporary societies, led to overvaluation o some values attached to proo (most importantly the incontrovertibility o its conclusion and hence the rigour o its conduct) and the undervaluing and overshadowing o other values that persist to the present. In this sense, these uses contributed to biases in the historical and philosophical discussion about mathematical proo, in that the values on which the discussion mainly ocused were brought to the ore by agendas most meaningul outside the eld o mathematics. Te resulting distortion is, in my view and as I shall argue in greater detail below, one o the main reasons why the historical analysis o mathematical proo has become mired down 6 and has ailed to accommodate new evidence discovered in the last decades. Moreover, it also imposed restrictions on the philosophical inquiry into proo. Accordingly, the challenge conronting us is to reinstate some autonomy in our thinking about mathematical proo. o meet this challenge effectively, a critical awareness derived rom a historical outlook is essential.
II Remarks on the historiography of mathematical proof Te historical episode just invoked illustrates how the type o mathematical proo epitomized by Euclid’sElements (notwithstanding the differences between the various orms the book has taken) has been used by some (European) practitioners to claim superiority o their learning over that o other practitioners. In the practice o mathematics as such, proo became a means o distinction among practices and consequently among social groups. In the nineteenth century, the same divide was projected back into history. In parallel with the proessionalization o science and the shaping o 5
6
Te same argument holds with respect to ‘science’. For example, the social and political uses o the discourses on ‘methodology’ within the milieus o practitioners, as well as vis-à-vis wider circles, were at the ocus o Schuster and Yeo 1986. However, previous attempts paid little attention to the uses o these discourses outside Europe. I was led to the same diagnosis through a different approach in Chemla 1997b.
Mathematical proo: a research programme
a scienti c community, history and philosophy o science emerged during that century as domains o inquiry in their own right.7 Euclid’s Elements thus became an object o the past, to be studied as such, along with other Greek, Arabic, Indian, Chinese and soon Babylonian and Egyptian sources that were progressively discovered.8 By the end o the nineteenth century, as François Charette shows in his contribution, mathematical proo had again become the weapon with which some Greek sources were evaluated and ound superior to all the others: a pattern similar to the one outlined above was in place, but had now been projected back in history. Te standard history o mathematical proo, the outline o which was recalled at the beginning o this introduction, had taken shape. In this respect, the dismissive assertion ormulated in 1841 by Jean-Baptiste Biot – Edouard Biot’s ather – was characteristic and premonitory, when he exposed this peculiar habit o mind, ollowing which the Arabs, as the Chinese and Hindus, limited their scienti c writings to the statement o a series o rules, which, once given, ought only to be veri ed by their applications, without requiring any logical demonstration or connections between them: this gives those Oriental nations a remarkable character o dissimilarity, I would even add o intellectual ineriority, comparatively to the Greeks, with whom any proposition is established by reasoning, and generates logically deduced consequences.9
Tis book challenges the historical validity o this thesis. Te issue at hand is not merely to determine whether this representation o a worldwide history o mathematical proo holds true or not. We shall also question whether the idea that this quotation conveys is relevant with respect to 7 8
See or example Laudan 1968, Yeo 1981, Yeo 1993, especially chapter 6. Between 1814 and 1818, Peyrard, who had been librarian at the Ecole Polytechnique, translated Euclid’sElements as well as his other writings on the basis o a manuscript in Greek that Napoleon had brought back rom the Vatican. He had also published a translation o Archimedes’ books (Langins 1989.) Many o those active in developing history and philosophy o science in France (Carnot, Brianchon, Poncelet, Comte, Chasles), especially mathematics, had connections to the Ecole Polytechnique. More generally, on the history o the historiography o mathematics, including the account o Greek texts, compare Dauben and Scriba 2002.
9
Tis is a quotation with At which F. Charette begins (p. 274). See the srcinal ormulation on p. 274. roughly the same time,hiswechapter nd under William Whewell’s pen the ollowing assessment: ‘Te Arabs are in the habit o giving conclusions without demonstrations, precepts without the investigations by which they are obtained; as i their main object were practical rather than speculative, – the calculation o results rather than the exposition o theory. Delambre [here, Whewell adds a ootnote with the reerence] has been obliged to exercise great ingenuity, in order to discover the method in which Ibn Iounis proved his solution o certain difficult problems.’ (Whewell 1837: 249.) Compare Yeo 1993: 157. Te distinction which ‘science’ enables Whewell to draw between Europe and the rest o the world in his History o the Inductive Scienceswould be worth analysing urther but alls outside the scope o this book.
5
6
proo. As we shall see, comparable debates on the practice o proo have developed within the eld o mathematics at the present day too.
First lessons from historiography, or: how sources have disappeared from the historical account of proof Several reasons suggest that we should be wary regarding the standard narrative. o begin with, some historiographical re ection is helpul here. As some o the contributions in this volume indicate, the end o the eighteenth century and the rst three-quarters o the nineteenth century by no means witnessed a consensus in the historical discourse about proo comparable to the one that was to become so pervasive later. In the chapter devoted to the development o British interest in the Indian mathematical tradition, Dhruv Raina shows how in the rst hal o the nineteenth century, Colebrooke, the rst translator o Sanskrit mathematical writings into a European language, interpreted these texts as containing a kind o algebraic analysis orming a well arranged science with a method aided by devices, among which symbols and literal signs are conspicuous. wo acts are worth stressing here. On the one hand, Colebrooke compared what he translated to D’Alembert’s conception o analysis. Tis comparison indicates that he positioned the Indian algebra he discovered with respect to the mathematics developed slightly beore him and, let me emphasize, speci cally with respect to ‘analysis’. When Colebrooke wrote, analysis was a eld in whi ch rigour had not yet become a central concern. Hal a century later in his biography o his ather, Colebrooke’s son would assess the same acts in an entirely different way, stressing the practical character o the mathematics written in Sanskrit and its lack o rigour. As Raina emphasizes, a general evolution can be perceived here. We shall come back to this evolution shortly. On the other hand, Colebrooke read in the Sanskrit texts the use o ‘algebraic methods’, the rules o which were proved in turn by geometric means. In act, Colebrooke discussed ‘geometrical and algebraic demonstrations’ o algebraic rules, using these expressions to translate Sanskrit terms. He showed how the geometrical demonstrations ‘illustrated’ the rules with diagrams having particular dimensions. We shall also come back later to this detail. Later in the century, as Charette indicates, the visual character o these demonstrations was opposed to Greek proos and assessed positively or negatively according to the historian. As or ‘algebraic proos’ , Colebrooke compared some o the proos developed by Indian authors to those o Wallis,
Mathematical proo: a research programme
or example, thereby leaving little doubt as to Colebrooke’s estimation o these sources: namely,that Indian scholars had carried out genuine algebraic proos. I we recapitulate the previous argument, we see that Colebrooke read in the Sanskrit texts a rather elaborate system o proo in which the algebraic rules used in the application o algebra were themselves proved. Moreover, he pointed resolutely to the use in these writings o ‘algebraic proos’. It is striking that these remarks were not taken up in later histori10 ography. Why did this evidence disappear rom subsequent accounts? Tis rst observation raises doubts about the completeness o the record on which the standard narrative examined is based. But there is more. Reading Colebrooke’s account leads us to a much more general observation: algebraic proo as a kind o proo essential to mathematical practice today is, in act, absent rom the standard account o the early history o mathematical proo. Te early processes by which algebraic proo was constituted are still terra incognitatoday. In act, there appears to be a correlation between the evidence that vanished rom the standard historical narrative and segments missing in the early history o proo. We can interpret this state o the historiography as a symptom o the bias in the historical approach to proo that I described above. Various chapters in this book will have a contribution to make to this page in the early history o mathematical proo. Let us or now return to our critical examination o the standard view rom a historiographical perspective. Charette’s chapter, which sketches the evolution o the appreciation o Indian, Chinese, Egyptian and Arabic source material during the nineteenth century with respect to mathematical proo, also provides ample evidence that many historians o that time discussed what they considered proos in writings which they quali ed as ‘Oriental’. For some, these proos were inerior to those ound in Euclid’s Elements. For others, these proos represented alternatives to Greek ones, the rigour characteristic o the latter being regularly assessed as a burden or even verging on rigidity. Te de cit in rigour o Indian proos was thus not systematically deemed an impediment to their consideration as proos, even interesting ones. It is true that historians had not yet lost their awareness that this distinctive eature made them comparable to early modern proos. One characteristic o these early historical works is even more telling when we contrast it with attitudes towards ‘non-Western’ texts today: when conronted with Indian writings in which assertions were not 10
Te same question is raised in Srinivas 2005: 213–14. Te author also emphasizes that Colebrooke and his contemporary C. M. Whish both noted that there were proos in ancient mathematical writings in Sanskrit.
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accompanied by proos, we nd more than one historian in thenineteenth century expressing his conviction that the assertion had once been derived on the basis o a proo. As late as the 1870s, this characteristic held true o, or instance, G. F. W. Tibaut in his approach to the geometry o the Sulbasutras, described below by Agathe Keller. It is true that Tibaut criticized the dogmatic attitude he attributed to Sanskrit writings dealing with science, in which he saw opinions different rom those expounded by the author treated with contempt – a act that he related to how proos were presented. It is also true that the practical religious motivations driving the Indian developments in geometry he studied diminished their value to him. In his view, these motivations betrayed the lack o ree inquiry that should characterize scienti c endeavour. Note here how these judgements projected the values attached to science in Tibaut’s scholarly circles back into history.11 Yet he never doubted that proos were at the basis o the statements contained in the ancient texts. For example, or the general case o ‘Pythagorean theorem’, he was convinced that the authors used some means to ‘satisy themselves o the general truth’ o the proposition. And he judged it a necessary task or the historian to restore these reasonings. Tis is how, or the speci c case when the two sides o the right-angled triangle have equal length, Tibaut unhesitatingly attributed the reasoning recorded in Plato’s Meno to the authors o the Sulbasutras. As the reader will nd out in the historiographical chapters o this book, he was not the only one to hold such views. On the other hand, it is revealing that while he was looking or geometrical proos rom which the statements o the Sulbasutras were derived, Tibaut discarded evidence o arithmetical reasoning contained in ancient commentaries on these texts. He preerred to attribute to the authors rom antiquity a geometrical proo that he would reely restore. In other words, he did not consider commentators o the past worth attending to and, in particular, did not describe how they proceeded in their proos. o sum up the preceding remarks, even i, in the nineteenth century, ‘the Greeks’ were thought to have carried out proos that were quite speci c, there were historians who recognized that other types o proos could be ound in other kinds o sources. Even when proos were not recorded, historians might grant that the achievements recorded in the writings had been obtained by proos that they thus strove to restore. However, as Charette concludes with respect to the once-known ‘non-Western’ source material, ‘much o the twentieth-century historiography simply disre11
Te moral, political and religious dimensions o the discourse on methodology have begun to be explored. See, or example, the introduction and various chapters in Schuster and Yeo 1986. More remains to be done.
Mathematical proo: a research programme
garded the evidence already available’. One could add that the assumption that outside the ew Greek geometrical texts listed above, there were no proos at all in ancient mathematical sources has become predominant today. It is clearly a central issue or our project to understand the processes which marginalized some o the known sources to such an extent that they were eventually erased rom the early history o mathematical proo. In any event, the elements just recalled again suggest caution regarding the standard narrative.
Other lessons from historiography, or: nineteenth-century ideas on computing Raina and Charette highlight another process that gained momentum in the nineteenth century and that will prove quite meaningul or our purpose. Tey show how mathematics provided a venue or progressive development o an opposition between styles soon understood to characterize distinct ‘civilizations’. In act, as a result o this development, by the end o the century ‘the Greeks’ were more generally contrasted with all the other ‘Orientals’, because they privileged geometry over any other branch o mathematics, while ‘the others’ were thought o as having stressed com12
putations and rules,the that is, algorithms, and algebra, instead. Charette discusses various means byarithmetic which historians accommodated the somewhat abundant evidence that challenged this division. Tis remark simultaneously reveals and explains a wide lacuna in the standard account o the early history o proo: this account is mute with respect to proos relating to arithmetical statements or addressing the correctness o algorithms. From this perspective, Colebrooke’s remarks on ‘algebraic analysis’ take on a new signi cance, since they pertain precisely to proos o that kind. In addition, the absence o algebraic proo rom the standard early history, noted above, appears to be one aspect o a systematic gap. I we exclude the quite peculiar kind o number theory to be ound in the ‘arithmetic books’ o Euclid’sElements, or in Diophantus’ Arithmetics, the standard history has little to say about how practitioners developed proos or statements related to numbers and computations. Yet there is no doubt that all societies had number systems and developed means o 12
From the statement by J.B. Biot in 1841 (quoted by F. Charette) to the statement by M. Kline in 1972 (quoted by Høyrup) – both cited above – there is a remarkable stability in the arguments by which algorithms are trivialized: they are interpreted as verbal instructions to be ollowed without any concern or justi cation. An analysis o the historiography o computation would certainly be quite helpul in situating such approaches within a broader context. Tis point will be taken up later.
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computing with them. Can we believe that proving the correctness o these algorithms was not a key issue or Athenian public accounts or or the Chinese bureaucracy?13 Could these rely on checks lef to trial and error? Clearly, there is a whole section missing in the early history o proo as it took shape in the last centuries.14 In act, there appear two correlated absences in the narrative we are analysing: on the one hand, most traditions are missing,15 while on the other hand, proos o a certain type are lacking. Is it because we have no evidence or this kind o proo? Such is not the case, and it will come as no surprise to discover that most o the chapters on proo that ollow address precisely those theorems dealing with numbers or algorithms. From a historiographic perspective, again, it would be quite interesting to understand better the historical circumstances that account or this lacuna.
Creating the standard history As Charette recalls in the conclusion o his chapter, the standard early history o mathematical proo took shape and became dominant in relation to the political context o the European imperialist enterprise. As was the case with the European missionaries in China a ew centuries earlier, mathematical proo played a key role in the process o shaping ‘European civilization’ as superior to the others – a process to which not only science, but also history o science, more generally contributed at that time. Te analysis developed above still holds, and I shall not repeat it. Te role that was allotted to proo in this ramework tied it to issues that extended ar beyond the domain o mathematics. Tese ties explain, in my view, why mathematical proo has meant so much to so many people – a point that still holds true today. Tese uses o proo have also badly constrained its historical and philosophical analysis, placing emphasis on some values rather than others or reasons that lay outside mathematics. 13 14
15
What is at stake today in the trustworthiness o computing is discussed in MacKenzie 2001. Te ailure that results rom not having yet systematically developed the portion o the history o mathematical proo has unortunate consequences in how some philosophers o mathematics deal with ‘calculations’, as opposed to ‘proos’. o take an example among those to whom I reer in this introduction, however insightul Hacking 2000 may be, the paragraph entitled ‘Te unpuzzling character o calculation’ (pp. 101–3) records some common misconceptions about computing that call or rethinking. See n. 45. As is ofen the case, when ‘non-Western traditions’ – as they are sometimes called – are missing, other traditions in the West have been marginalized in, or even lef out rom, the historiography. Lloyd directly addresses this act in his own contribution to this volume.
Mathematical proo: a research programme
Understanding what other elements played a part in the shaping o our narrative is another way o developing our critical awareness o the narrative. As R. Yeo has argued regarding the case o early Victorian Britain in the publications mentioned above, the proessionalization o science and the development o the sense o a ‘scienti c community’, as well as the need o the practitioners to reinorce the unity o ‘science’ or themselves and its value in the eyes o the public, can be correlated with an increase in the size and number o publications devoted to the ‘scienti c method’. Te distinctive eatures o the method enabled it to maintain the cohesion o the community and enhance the value o the social group in the eyes o the public. It shaped the social and proessional status o those who were soon to be called ‘scientists’. Philosophy o science and history o science emerged and developed as disciplines through this historical process and were instrumental in the pursuit o the question o method. How were the understanding and discussion o mathematical proo in uenced by this global trend? In my view, this is a key issue or our topic, to which we shall come back below but which awaits urther research.16 A consideration o the mainstream development o academic mathematics during the nineteenth century casts more light on our narrative rom yet another perspective. It also allows the perception o other elements that may have played a part in constructing the narrative. Indeed, the approach to proos o the past at different time periods correlates with more general trends in the mathematics o the time. On the one hand, as we saw, in the rst decades o the nineteenth century, Colebrooke was reading his Indian
16
Clearly, proo was a topic o explicit discussion within disciplinary writings, as the rst edition o George Peacock’sreatise o Algebra(1830) shows. Te pages starting rom paragraph 142, on p. 109, were devoted to the question: ‘What constitutes a demonstration?’ Further, John Stuart Mill’s discussion o methodology, in hisA System o Logic, Ratiocinative and Inductive, rst published in 1843, encompassed ananalysis o mathematical proo and led him to offer an interpretation o Euclidean proos as reliant on an inductive oundation and their certainty as an illusion (p. 296). Tis example shows how re ections o mathematical proos were in uenced by wider discussion o methodology. By comparison, Auguste Comte’s considerations on demonstrations were less systematic. Conversely, another question is worth exploring: what role did ideas about and practices o mathematical proos play in shaping the various discourses about methodology? Even though considerations about demonstration are pervasive in the methodological books o that period, it seems to me that this eature has received little attention. An exception is the discussion o Whewell’s ideas regarding the various practices o proo in the context o his wider concern or the teaching o mathematics and physics in Yeo 1993: 218–22. In this case, questions o method relate to pedagogic efficiency and tie mathematics to natural science. Hacking 1980 (reprinted as chapter 13 in Hacking 2002: 200–13) sheds interesting light on the question o the emergence o methodology in the seventeenth century. On the issue o mathematical proo as such, this article is updated in Hacking 2000.
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sources with mathematical analysis in mind. His comparisons were with Wallis or D’Alembert. On the other hand, at the end o the nineteenth century, when Greek geometry overshadowed all other evidence or the early history o proo, the value o rigour had been growing in signi cance or some decades, and academic mathematics was witnessing the beginning o a new practice o axiomatic systems which would soon become the dominant trend in the twentieth century.17 Tese arguments suggest that different actors brought about the shif in historiography outlined above and could account or the outline o the now-standard narrative o the early history o proo. Some o these actors clearly relate to the state o mathematics at a given time, both institutionally and intellectually, but others are not directly related to it. Te in uence o some o these actors may be elt at the present day and could explain the lingering belie in this narrative as well as the signi cance widely attached to it. However, the same arguments invite us to look at this narrative with critical eyes: the narrative belongs to its time and the time may have come that we need to replace it.
Dissatisfactions: overemphasizing certainty For more than three decades now, some historians o mathematics have published articles and books arguing that the Chinese, Babylonian and Indian sources on which they were working contained mathematical proos .18 17
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It would be interesting to document these correlations in greater detail. See e.g. I. oth’s work on the history o axiomatization. Other changes in the mathematics o the nineteenth century also probably had an impact on the historiography in exactly the same way such as the increasing marginalization o computing and the division between pure and applied mathematics, which were soon perceived as two distinct pursuits and to be carried out in separate institutions. Tibaut’s critical remarks, mentioned above, on the practical orientation o the mathematics in the Sulbasutras are probably an echo o the latter trend and illustrate a typical moti o nineteenth- and twentieth-century historical publications. Regarding the marginalization o computing and its impact on historiography, I reer to the orthcoming joint publication by Marie-José Durand-Richard, Agathe Keller and Dhruv Raina. For the Chinese case, let us mention the rst research works no the topic published in English: Wagner 1975, Wagner 1978, Wagner 1979. One must also mention the rst works in Chinese systematically addressing issue: and the 8th issue o the Kejishi wenjio the (Collection o papers on the history othe science technology), in journal 1982; the 11th issue journal Kexueshi jikan(Collected papers in history o science); Wu Wenjun 1982. Since then, the publications are too numerous to be listed here. Te reader can nd amore complete bibliography in CG2004. Te rst publication on thetopic o proos that could be read in the Mesopotamian sources is Høyrup 1990. Since then, Høyrup has continued exploring this issue, and other specialists o the eld have joined him to support and develop this thesis. Asynthesis o the outcomes o this research programme, the results o which were widely adopted by the narrow circle o specialists o Mesopotamian mathematics, was published: H2002. As or the Indian case, we can reer the reader to H1995: 75–7, Jain 1995. Tese were ollowed more recently by Patte 2004, Srinivas 2005, Keller 2006, among others.
Mathematical proo: a research programme
Tey worked independently o each other and the proos they discussed were quite different in nature. Moreover, their interpretation o the acts conronting them was not uniorm. However, they brought orward extensive evidence, partly new, partly old, which challenged the received view o the early history o mathematical proo. It is interesting to note that, in a way, they were partly returning to a past historiography. A puzzling act is that, beyond the strict circle o specialists in the same domain, these results were at best ignored, but, more requently, were rejected outright. Clearly, these publications have so ar not managed to bring about any change in the view o the early history o mathematical proo held by historians and philosophers o science at large, or the wider population. Tis sustained ailed reception needed to be analysed. Tus, this book is not only devoted to the history but also contains a section on the historiography o mathematical proo. Needless to say, much more remains to be done in this domain. Tese circumstances also explain why I chose to begin this introduction with historiographical remarks. Some urther actors are at play in how mathematical proo is approached in our societies at large, and we need to recognize these actors in order to restore some reedom to the discussion and come to grips with the new evidence. On the basis o the analysis outlined above, we see two types o obstacles which could hinder the development o the discussion. Firstly, the whole question o mathematical proo is entangled with extrascholarly uses in which it plays an important part – among these uses are those o the issues addressed earlier which are related to claims o identity.19 Additionally, and in relation to this point, an image o what a mathematical proo endeavours has crystallized and blurs the analysis. My claim is that this image is biased and that dealing with the new evidence mentioned above presents an opportunity or us to locate this distortion and to think about mathematical proo anew. We have reached the crux o the argument. Let me explain in greater detail. Te essential value usually attached to mathematical proo – topmost or its wide cultivation and esteem outside the sphere o mathematics – is that, as the word ‘proo’ itsel indicates, it yields certainty: the conclusion which has been proved can (hopeully) be accepted as true. 20 Securing the 19
20
How social groups construct identity through science or history o science is more generally a key issue, on which much more research ought to be done. Grabiner 1988 argues that certainty and applicability were the two eatures through which mathematics was most in uential to ‘Western thought’. Certainly, these two eatures occupy a prominent position in Xu Guangqi’s preace to the Chinese translation o Euclid’s Elements (Engelriet 1998: 291–7). Grabiner’s analysis o how the certainty yielded by proo was in uential, especially in theology, reveals dimensions o the importance regularly attached
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truth o a piece o knowledge and convincing an opponent o the incontrovertibility o an assertion seem to be what mathematical proo offers and the ideal it embodies. Clearly, i we adopt this viewo proo, we are immediately orced to admit that starting points (de nitions, axioms) are mandatory or the activity o proo, i we are to achieve these goals. Moreover, the validity o these starting points must be agreed upon, regardless o how this agreement is reached. In his chapter, Geoffrey Lloyd treats at length the variety o terms used to designate these starting points in ancient Greece and the intensity o interest in, and debate about, them that this variety re ects. On this basis, and this is where requirements such as rigour appear to come in, valid arguments are required to derive assertions rom the starting points in a trustworthy way, and new assertions depend on the rst ones or the starting points, and so on. In other words, as soon as one has granted the premise that the goal o mathematical proo is to prove in an indisputable way, then the conclusion ollows unavoidably that this aim can be only achieved within the ramework o an axiomatic–deductive system o one sort or another. In the context o this assumption, Euclid’sElements is the rst known mathematical writing that contains proos, and any claim that a given source contains proos has to be judged accordingly. And such claims have actually been judged by that very standard. Tis is, in my view, the simple device by which Greek geometricalwritings have become so central to the discussion o proo that they cannot possibly be challenged, and this position lies at the core o the recent rejection o the claim that Babylonian, Chinese or Indian sources contained proos by some part o the community o history and philosophy o science (among others). Te reasoning will look simplistic to many. However, I claim that this is precisely the core o the matter.21 I I am right, this is the point on which critical analysis must be exercised or us to open our historical inquiry into proo wider. Te eature o mathematical proo just examined is certainly quite meaningul, and was indeed deemed so outside mathematics. However, on what basis do we grant ‘incontrovertibility’ asthe essential value and goal o mathematical proo within mathematics itsel ?
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to this value. Hacking 2000 is a bright analysis o what certainty and its cognate values have meant or some philosophers. I ormulated the reasoning relying on present-day perception o what yields certainty. Although certainty, starting points and modes o reasonings based on the latter to secure the ormer remained a stable constellation o elements in the history o discussions about mathematical proo, the meanings and contents attached to them displayed variation in history. As Orna Harari shows in her chapter in this book, earlier views were quite different rom present-day ones. Compare Mancosu 1996, especially chapter 1.
Mathematical proo: a research programme
o examine this question, let us restrict the discussion to mathematical proo as such, as carried out within the context o mathematics. Te recollection o a simple act will prove useul here: many mathematical proos produced throughout history by duly acknowledged scholars were not presented within axiomatic–deductive systems.22 In act, the periods during which advanced mathematical writings were predominantly composed in such a way are much shorter than the periods when they were not. In tandem with the lack o interest in an axiomatic–deductive organization o mathematical knowledge, the authors ofen did not place much emphasis on rigour. Yet they reerred to what they wrote as proos.23 One may argue that these practitioners o mathematics overlooked some diffi culties and made errors. But these objections cannot possibly obliterate the innumerable theories proposed and results obtained with precisely such types o proo. Tese remarks have an inescapable consequence: it reveals that or a air number o practitioners o mathematics the goals o proo cannot have been only ascertaining incontrovertibility and assuring certainty through achieving conviction, i such was ever their goal at all. Nevertheless, they considered it worthwhile to look or proos, and their practice o proo was no less productive rom a mathematical point o view. In my view, this perception o proo still holds true today. Even though, in their discourse on the contemporary practices o proo, mathematicians may stress the axiomatic–deductive ramework within which they work and emphasize the certainty yielded by proos as well as the rigour necessary in their production,24 the unctions they ascribe to proo in their 22
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Ironically enough, the proo that lies at the core o Plato’s Meno and that has exerted a huge in uence in the history o philosophy (Hacking 2000) is not ormulated within an axiomatic– deductive system. Philosophers o the present day such as Lakatos 1970 held ‘a no-oundation view o mathematics’ (Hacking 2000: 124). Unortunately such views have not yet shown any clear impact on the history o ancient mathematics. Rav 1999: 15–19 lists several examples o major domains o mathematics o the present day, or which axioms have not been proposed and that are nevertheless elt to be rigorous. He urther emphasizes the various meanings o ‘axioms’ as used in modern practice. I am not aware o any historical publication which denies that Leibniz, Euler, Poncelet, Poincaré or others o their ilk wrote down actual proos and suggests that these men should be erased rom the history o mathematical proo: whatever the evaluation may be, it is without contest that they contributed to shaping practices o proo. More revealing examples are discussed in Jaffe and Quinn 1993: 7–8. Te act that Jaffe and Quinn reer to cases o ‘weak standards o proo ’ and suggest that, in some cases, ‘expressions such as “motivation” or “supporting argument” should replace “proo ”’ in actors’ language indicates that in the contemporary mathematical literature the label ‘proo ’ reers to a great variety o types o arguments (Jaffe and Quinn 1993: 7, 10). Tis topic recurs below. See the very different and lucid account in Turston 1994: 10–11. Among other rereshing insights into the activity o proo, Turston rejects the ‘hidden assumption that there is
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actual work seem quite different and multiaceted, in act. Some insight on this point can be gained rom the contributions to a debate that broke out in the pages o the Bulletin o the American Mathematical Societyabout a decade ago.25 Te paper by Jaffe and Quinn that launched the discussion recognized the importance o ‘speculating’ – which they called ‘theoretical mathematics’ – or the development o mathematics, in addition to proos which secure certainty. However, the authors expressed concerns regarding the conusion that could arise rom conounding rigorous proos (ones that bring certainty), insights, arguments and so on. As a consequence, they suggested norms o publication that would distinguish explicitly between, on the one hand, ‘theorem’, ‘show’, ‘construct’, ‘proo’ and, on the other 26 hand, ‘conjecture’, ‘predict’, ‘motivation’, and ‘supporting argument’. One may venture to recognize in this opposition a divide o the type we are examining with respect to history. It is impossible to review the debate in detail here. However, or our purposes, it is interesting to observe the intensity o reaction that this suggestion elicited in the mathematical community. From the responses published in the Bulletin, a much more complex image o the activity o proo emerges, in which rigorous proos appear to arouse mixed eelings and cohabit with all kinds o other modalities o proo.27 Moreover, the relation o proo to other aspects o mathematical activity appears to be quite intricate and calls or urther analysis. In relation to our topic, I interpret the act that, ironically, many mathematicians do not nd it iffi d cult to recognize as proos arguments rom Chinese or Indian texts although other scholars deny them this quality as an additional sign o this coexistence o motley practices o proo in the mathematical community. Were urther evidence still necessary, these acts indicate that there are con icting ideas among mathematicians about what a proo is or should be. Why, in such circumstances, should historians or philosophers opt or one idea as the correct one and civilize the past, let alone the present, on this basis?
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uniorm, objective and rmly established theory and practice o proo’(p. 1.) A comparable, yet different, account o proo, which is quite critical o standard views, is provided by Rav 1999. Some o the pieces written or this debate were already mentioned above. Here are the reerences to the entire core exchange: Jaffe and Quinn 1993, Atiyah, Borel, Chaitin, Friedan, Glimm, Gray, Hirsch, Lane, Mandelbrot, Ruelle, Schwarz, Uhlenbeck, Tom, Witten and Zeeman 1994, Jaffe and Quinn 1994, Turston 1994. Jaffe and Quinn 1993: 10. Te relationship between the written text o the proo and the collective oral activity related to proo that emerges rom these testimonies presents a potentially worrying complexity to the historian, whose only sources are written vestiges with a aint relation to real processes o proo production.
Mathematical proo: a research programme
In connection with this issue, and to return to the question whether certainty is the main motivation or looking or proos today, it is interesting to note that many responses to the srcinal paper by Jaffe and Quinn maniest a concern that too strict a control in order to assure certainty could entail losses or the discipline. By contrast, the debate also allows one to observe how many different unctions and expectations mathematicians attach to proo: bringing ‘clarity and reliability’; providing ‘eedback and corrections’, ‘new insights and unexpected new data’ (Jaffe et al. 1993), ‘clues to new and unexpected phenomena’ (Jaffeet al. 1994), ‘ideas and techniques’ (Atiyah et al. 1994), ‘understanding’,28 ‘mathematical concepts which are quite interesting in themselves, and lead to urther mathematics’; ‘helping support o certain vision or the structure o ’ a mathematical object (Turston 1994).29 Only with this variety o objectives in mind can we account or some otherwise mysterious practices. For instance, how else could we explain why rewriting a proo or already well-established statements can be ruitul? 30 Restricting ourselves to consideration o proo in the more limited domain o mathematics brought to light a wealth o reasons which motivated the writing o proos or mathematicians.31 Moreover, it suggests the great loss or the historical inquiry on mathematical proo i these proos, the values attached to them, and the motivations to ormulate them and write them down were not considered. 28
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A comment by Martin Davis on the our-colour theorem nicely illustrates this point: the problem with the computer proo, in his view, is not so much the lack o certainty it entails, but the act that it does not put us in a position to understand where the ‘4’ comes rom, and whether it is accidental or not (Martin Davis, 2 October 2007, personal communication). As I alluded to it above, rigour is a contested value in these pages (see the contributions by Mandelbrot, Tom). What is more, it must be stressed that in contemporary mathematics, as it may have been the case or the Aristotle o the Posterior Analytics, the value attached to rigour is perhaps linked more to the understanding and additional insights it provides than to the increased certainty it yields. Hilbert 1900, or example, testi es to the idea that rigour yields ruitulness and provides a guide to determine the importance o a problem (in the English translation: Hilbert 1902: 441). However, as Rav 1999 stresses, even when proos are wrong or inadequate, they remain the main source rom which new concepts emerge and new theories are developed. He urther suggests that it is in proos, rather than in theorems, that mathematicians look or mathematical knowledge and understanding: ‘Conceptual and methodological innovations are inextricably bound to the search or and the discovery o proos, thereby establishing links between theories, systematizing knowledge and spurring urther developments.’ (Rav 1999:6). Tis point was stressed in Chemla 1992, which relies on how Rota 1990 had discussed the issue. Some historians have attempted to widen the history o proo by suggesting that the actors o the past used various means to convince their peers o the truth o a statement. In this vein attention has been paid to the rhetorical means that the actors employed. Te preceding remarks show why this does not help rame a wide enough perspective on the activity o proo.
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New perspectives, or: the project of the book From this vantage point, two conclusions can be discerned. Firstly, we see how a history o proo limited to inquiry into how practitioners devised the means o establishing a statement in an incontrovertible way runs the risk o being truncated. Tis, in my view, is what happens when the Babylonian, Chinese and Indian evidence is lef out. Secondly, and conversely, the outline sketched above suggests another kind o programme or a history o mathematical proo, one likely to be more open and allow us to derive bene ts rom the multiplicity o our sources. We may be interested in understanding the aims pursued by different collectives o practitioners in the past when they maniested an interest in the reasons why a statement was true or an algorithm was correct. We may also wonder how they shaped the practices o proo in relation to the aims they pursued and how they lef written evidence o these practices. 32 In act, some o these other unctions associated with proo were explicitly identi ed in the past and they were at times perceived as more important than assuring certainty. In relation to this, epistemological values distinct rom that o incontrovertibility have been used to assess proos. In this respect, one can recall the seventeenth-century debates about how to secure increased clarity through mathematical proos, thereby achieving conviction and understanding. Seen in this light, the versions o Euclid’s Elements o the past were not much prized, and new kinds o Elements were composed to ul l more adequately the new requirements demanded rom mathematical proo.33 Tis example illustrates how different types o proo were created in relation to different agendas or proving. How would such a programme translate with respect to ancient traditions? Tis is the inquiry o the present book, as one step towards opening a wider space or a historical and epistemological investigation into mathematical proo. Te book is mainly – we shall see why ‘mainly’ shortly – devoted to the earliest known proos in mathematics. By the term ‘proo ’, it should be now clear why we simply mean texts in which the ambition o accounting or the truth o an assertion or the correctness o an algorithm can be identi ed as one o the actors’ intentions. In other words, we do not restrict our corpus 32
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Tis analysis and this programme develop the suggestion I ormulated in Chemla 1997b: 229–31. On this question, see chapter 4, ‘L’interprétation d’Euclide chez Pascal et Arnauld’, in Gardies 1984: 85–108.
Mathematical proo: a research programme
a priori by reerence to norms and values t hat would appear to us as characterizing proos in an essential way. From this basis, the various chapters aim at identiying the variety o goals and unctions that were assigned to proo in different times and places as well as the variety o practices that were constructed accordingly. In brie, the authors seek to analyse why and how practitioners o the past chose to execute proos. Moreover, they attempt to understand how the activity o proving was tied to other dimensions o mathematical activity and, when possible, to determine the social or proessional environments within which these developments took place. Beyond such an agenda, several more general questions remain on our horizon. From a historical point o view, we need to question whether the history o mathematical proo presents the linear pattern which today seems to be implicitly assumed. How did the various practices o proo clearly distinguished in present day mathematical practice inherit rom and draw on earlier equally distinct practices? In more concrete terms, we seek to understand how the various practices o proo identi ed in ancient traditions or their components (like ways o proceeding or motivations), developed, circulated and interacted with one another. Tese are some o the questions that arise when attempting to account or the construction o proo as a central but multiaceted mathematical endeavour that unolded in history in a less straightorward way than it was once believed. From an epistemological point o view, on the other hand, we are interested in the understanding about mathematical proo in general that can be derived rom studying these early sources rom this perspective.
Further lessons from historiography, or: the historical analysis of critical editions Te analysis developed so ar was needed to raise an awareness o the various meanings that have overloaded – and still overload – the term ‘proo ’ in the historiography o mathematics. We brought to light how agendas involved in this issue ettered the development o a broader programme which would consider proo as a practice and analyse it in all its dimensions. Beore we outline how the present book contributes to this larger programme, urther preliminary remarks o another type are still needed. Our approach to proos rom the past is mediated by written texts. In his contribution to the debate evoked above, wherein he described the
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collective work involved in the making o a proo eventually produced and written down by an individual, W. Turston makes us ully aware o the bias that such an approach represents. In act, there are urther difficulties linked to the nature o the sources with which the historian works. Some o these sources, like Babylonian tablets, were discovered in archaeological excavations, on a spot where they were used by actors. Others came down to us through the written tradition. In most cases, the physical medium has travelled.34 In the end o the best-case scenario, those that can be read are available to us through critical editions. Trough the various processes o transmission and reshaping o the primary sources, the agendas related to proo described earlier may have lef an imprint. In such cases, our analysis o the source material would be biased at its root. We shall illustrate this problem with a undamental example, which will bring us back to nineteenth-century historiography o proo and a dimension o its ormation that we have not yet contemplated. Above, we outlined the contribution that this book makes to analysing the evolution o European historiography o science with respect to ‘non-Western’ proos. As a complementary account, the rst section o Part I in the book ocuses on the approach to Greek geometrical texts that developed in the late nineteenth century and the beginning o the twentieth century. Tree chapters examine how the critical editions o Euclid’sElements and Archimedes’ writings produced by the philologist Johann Heiberg, on which we still depend or our access to these texts, re ect, and hence convey, his own vision o the mathematics o ancient Greece. Tese chapters illustrate a new element involved in the historiographic turn described above: the production o critical editions. Let us sketch why they invite us to maintain a critical distance rom the way sources have come down to us, lest we unconsciously absorb the agendas that shaped these editions. Te problem affecting these critical editions was rst exposed by Wilbur Knorr, in an article published in 1996, the title o which was quite explicit: ‘Te wrong text o Euclid: on Heiberg’s text and its alternatives’.35 In it, Knorr explained why in his view, Heiberg shaped Euclid’s text on the basis o his own assumptions regarding the practice o axiomatic–deductive systems in ancient Greece. Knorr’s article began with a critical examination o a debate which at the end o the nineteenth century opposed Heiberg to 34
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Te research programme entitled ‘Looking at it rom Asia and Arica: a critical analysis o the processes that shaped the sources o history o science’, led by Florence Bretelle-Establet and to which A. Bréard, C. Jami, A. Keller, C. Proust and mysel contributed helped me clariy my views on these questions. Knorr 1996. A paper that appeared posthumously took up this issue once again: Knorr 2001.
Mathematical proo: a research programme
Klamroth, a historian who specialized in Arabic mathematics. Te debate concerned the role ascribed to the editions and translations into Arabic and Latin carried out between the eighth and the thirteenth centuries – the so-called ‘indirect tradition’ – in the making o the critical edition o the Elements. Heiberg’s position was that the Greek manuscripts dating rom the ninth century onwards – the ‘direct tradition’ – were closer to Euclid’s srcinal text. In contrast, Klamroth argued that the Arabic and Latin witnesses, less complete rom a logical point o view, bore testimony to earlier states o the text, whereas the Greek documents had already been contaminated by the various uses to which the text had been put in the centuries between its composition by Euclid and the transliteration into minuscule that took place in Byzantium. In brie, Heiberg was committed to the view that Euclid’s Elements contained a minimum o logical gaps in the mathematical composition which it delineated. Tis supposition dictated the choice o sources on which he based his edition and motivated his rejection o other documents as derivative. Tis is how his selective treatment o the written evidence contributed to reshaping Euclid closer to his own vision. aking up Klamroth’s thesis, Knorr held the opposite view: or him, the Arabic and Latin witnesses were closer to the srcinal Euclid, and the additions o logical steps were carried out by later editors o the Elements. Te consequence o the resurgence o the debate was clear: some textual doubts were thereby raised regarding Euclid’s srcinal ormulation o his proos. In articulating a critical analysis o this kind regarding the nineteenthcentury edition o the Elements still widely used today or the rst time since the publication o Heiberg’s volumes, Knorr launched a research programme o tremendous importance to our topic. How much does our perception o the practice o proo in the Elements depend on the choices carried out by Heiberg? In other words, how ar is his vision o Euclidean proo, ormed at the end o the nineteenth century, conveyed through the text o his critical edition? Such are the undamental questions raised. Te example illustrates clearly, I believe, a much more general problem, which can be ormulated as ollows: how do critical editions affect the theses held by historians o science and the transmission o this inheritance to the next generations o scholars? Tis general issue is to be kept in mind with respect to all the sources mentioned in this volume. However, beyond providing the illustration o a general difficulty, the example o the Elements is in itsel o speci c importance or our topic. In act, the problem it raises extends beyond the case o the Elements, since soon afer the publication o Knorr’s rst paper, a diculty o the same kind became maniest with respect to Heiberg’s critical
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edition o Archimedes’ writings.36 What can we learn about the issue o proo by examining the philologist’s impact on our present-day vision o Euclid and Archimedes? Te three chapters o this book that are devoted to the analysis o the nineteenth-century editions o Greek geometrical texts rom antiquity – the rst one dealing with the Elements, the second one with the general issue o the critical edition o diagrams and the third one with Archimedes’ texts – represent three critical approaches to Heiberg’s philological choices and their impact on the editing o the proos. Teir argumentation bene ts rom the wealth o twentieth-century publications on the Arabic and Latin translations and editions o the Greek geometrical texts. Let us outline here brie y the distinct textual problems on which these chapters ocus. Each chapter represents one way in which our understanding o the proos preserved in the geometrical writings o ancient Greece is affected by their representation developed in the editions commonly employed. In his contribution to the volume, Bernard Vitrac examines different types o divergences between proos, to which the various manuscripts that bear witness to Euclid’sElements testiy. More speci cally, Vitrac ocuses on a corpus o differences that were caused by deliberate intervention. Since these transormations were most certainly carried out by an author in the past who wanted to manipulate the logical or mathematical nature o the text, they indicate clearly the points at which we are in danger o attributing to Euclid reworking o the Elements undertaken afer him. Tree types o divergences are examined. Te rst one, about which the debate described above broke out, relates to the terseness o the text o proos: some proos are ound to be more complete rom a logical point o view in some manuscripts than in others. Vitrac brings to light that the interpretation made by the two opponents in the debate relied on divergent views o the possible evolution o such a book as the Elements. Klamroth’s thesis presupposed that the evolution o the text could only be a progressive expansion, motivated by the desire to make the deductive system more and more complete rom a logical or a mathematical point o view. In contrast, Heiberg suggested that the Arabic and Latin versions were based on an epitome o the Euclidean text, on which account he could marginalize their use in restoring the Elements. Vitrac provides an analysis o the various logical gaps and concludes that the later additions to the Greek text that the indirect tradition allows us to perceive in the Greek manuscripts are linked to a logical concern regarding the mathematical content o the text. 36
Chemla 1999.
Mathematical proo: a research programme
Tis also holds true or most o the material added (propositions, lemmas, porisms). Tese remarks seem to support Klamroth’s view. In this respect, Vitrac considers the indirect tradition as more authentic, a act which calls or a re-examination o proos in the various versions o Euclid’sElements. Vitrac suggests, however, that the enlargement and ‘improvement’ o the Elements could have started in Greek and continued in Arabic and Latin. Te extant versions all seem to bear signs o corruption by such activity. Te second type o divergence between the sources Vitrac examines relates to the order in which propositions are arranged. Tis order constitutes a key ingredient in an axiomatic–deductive structure. In act, the order does vary according to the version o the text. Te decisions implemented by any critical edition hence represent an interpretation o Euclid’s srcinal deductive structure. However, on this count, Vitrac suggests the provisional conclusion that the indirect tradition more requently bears witness to modi cations o this type. Te third kind o divergence which he analyses has perhaps the greatest impact on our perception o Euclid’s proos, since it relates to major dierences between the sources: substitution o proos, integration o these substitutions in a set o related proos, addition or subtraction o cases, and double proos, o which Heiberg kept only one according to criteria that need to be examined. Such cases indicate that proos and their modi cation were the subject o a continuous effort, part o which was integrated into the editions o the Elements available to us today. In conclusion, beore we consult the critical editions o Arabic, AraboHebrew or Arabo-Latin versions o Euclid’sElements, it may be difficult to go substantially urther in the analysis o the proos or the deductive system attributed to Euclid. Most probably, this goal may remain orever out o reach. However, we can already appreciate the extent to which the textual decisions made by the philologist affect the discourse on the practice o proo in ancient Greece. Tis remark shows that the discourse on the practice o proo in ancient Greece may not be as solidly ounded as was previously thought. As Vitrac suggests in his conclusion, rather than holding to the romantic ideal o some day retrieving the srcinal Elements, it may be ar more reasonable and interesting to consider the various versions o Euclid’s Elements or which we have evidence. Tis new perspective would provide us with a better grasp o the various orms that the text took in history – namely, the orms through which different generations o scholars read and used the Elements. Ken Saito and Nathan Sidoli critically examine the work o the philologist rom an entirely different perspective. Te purpose o their chapter is
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to draw attention to the act that the diagrams inserted by Heiberg in his edition o Euclid’s Elements, among others, are quite different rom those actually and stably contained in the manuscripts. Te sources indicate that the diagrams were more ofen than not quite particular, representing the general case not by means o a generic gure, but rather by means o a remarkable and singular con guration. By contrast, Saito and Sidoli show how Heiberg tacitly altered the diagrams, modernizing them and thereby conspicuously making them look more aithul to the situation under study and more generic than they actually were. Tese operations inserted diagrams in the nineteenth-century edition o the Elements which displayed an arti cial continuity between past practices and mathematical practices at the time, not only with respect to their appearance, but also with respect to their way o expressing the general. Furthermore, the Greek diagrams were thereby shown as being demonstrably more different rom the diagrams having speci c dimensions contained in the Sanskrit or Chinese sources than the manuscripts actually indicated. Such issues may look minor, but they are not. In act, Saito (2006) discusses a case in which the option chosen by the philologist in the restoration o the gure has had a crucial impact on the restored text. His conclusion is that, on both counts, Heiberg’s choice seems to admit the results o a later intervention 37
as genuine. It is important to notice that, in modernizing the diagrams in this way, Heiberg removed any hint o the actors’ ways o drawing and using gures, thereby impeding through his edition any study o the ancient practices with geometrical gures. Saito’s and Sidoli’s critical analysis o the gures that Heiberg included in his editions such as the Elements is in ull agreement with what Reviel Netz shows in the ollowing chapter about Heiberg’s edition o Archimedes’ writings. In this chapter, Netz analyses more generally by which kinds o operation Heiberg’s philological interventions lef a lingering imprint on Greek mathematical texts o antiquity as we read them today. However, concentrating on the Danish philologist’s critical edition o Archimedes’ writings, particularly the second edition published between 1910 and 1915, Netz demonstrates urther the speci cs o Heiberg’s editorial operations with respect to the Syracusan’sOpera Omnia. Netz’s analysis distinguishes three types o intervention that, in his words ‘produce[d] an Archimedes who was textually explicit, consistent, rigorous and yet opaque’. In particular, Netz’s overall broader argument reveals how Heiberg shaped Archimedes’ 37
Saito 2006 : 97–144 compares Heiberg’s diagrams in Book o theElements with those o the Greek manuscripts which ormed the basis o his critical edition.
Mathematical proo: a research programme proos according to his vision. In conclusion, we understand better how we were mistaken, when we took Heiberg’s words or Archimedes’ writings as the manuscripts bear witness to them. o start with, Netz examines the diagrams o the critical edition. Clearly, like cases analysed by Saito and Sidoli, the diagrams used by Heiberg differ markedly rom the evidence contained in the manuscripts, and Heiberg drew the diagrams according to his own understanding o what the srcinal diagrams might have looked like. Yet Netz argues that the manuscripts represent a coherent and perectly valid practice with diagrams. Further, three criteria allow him to discern how the ancient diagrams, drawn within the context o this practice, systematically differ rom those which Heiberg substituted. Note that one o Netz’s criteria relates to a eature already discussed by Saito and Sidoli: Heiberg tended to picture elements o the diagram as unequal that the manuscripts, in contrast to the discourse, drew as equal. Interestingly, the two chapters suggest slightly different interpretations o this ancient element o practice. Te broader analysis developed by Netz urther leads him to restore an ancient and consistent regime o conceiving and using diagrams which Heiberg’s critical edition concealed and replaced with another more modern usage, or which there exists no ancient evidence. In addition, Netz argues that, in relation to this transormation, the role o the diagrams in the text underwent a dramatic shif: although the ancient evidence preserves diagrams that were an integral component o the argumentative text, Heiberg turned the diagrams into mere ‘aids’, dispensable elements or reading a discursive text that was ‘logically sel enclosed’. Tis rst conclusion thus identi es one way in which the critical edition distorted the texts o Archimedes’ proos with respect to the extant manuscripts. Te second systematic intervention by Heiberg which Netz analyses is the bracketing o words, sentences and passages in Archimedes’ writings, despite the act that the manuscripts all agree on the wording o these passages. In other words, by rejecting portions as belonging to the srcinal text, Heiberg modi ed the received text o Archimedes’ writings in conormity with the representation that he had ormed or Archimedes as a sharp contrast to Euclid. While, or Heiberg, Euclid was characterized by the careul expression o the ull- edged argument, Archimedes’ style was, in his view, to ocus on the main line o the proo, leaving aside ‘obvious’ details. Accordingly, Heiberg designated many passages o the received text as possible interpolations. Heiberg thus made Archimedes’ style more coherent than what the manuscript evidence shows. Netz brings to light Heiberg’s uneven pattern o bracketing and suggests actors which account or it. What is important or us here are the conclusions that Netz’s analysis allows
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him to derive with respect to the text o proos: Heiberg’s bracketing occurs mainly in the texts or proos, at precisely those points which suggest that Heiberg elt that overly simple arguments in the course o a proo could not be due to Archimedes. Te more elementary the treatises, the more bracketing Heiberg carried out. In conclusion, Heiberg imposed on the text o the proo his expectation regarding Archimedes’ way o proo. Lastly, Netz brings to light the subtle ways at the global level o the corpus o texts in which Heiberg established Archimedes as a mathematician who adopted a uniorm style and wrote down his treatises according to the same systematic pattern. By contrast, Netz suggests that Archimedes’ writings maniested variety in several ways and at different levels. What matters most or us, again, is how the philologist’s operations have a bearing on our perception o proos and the sequence o them in ‘axiomatic–deductive organizations’. And, here, the description o the editorial situation that Netz offers us is quite striking. He reveals how Heiberg orced divisions between propositions, types o propositions and components o propositions onto texts that did not lend themselves equally well to the operations and thus arti cially created the sense o a standardized mathematical text, in conormity to modern expectations. In addition, Netz reveals Heiberg himsel decided to give some propositions the status o postulate and others that o de nition, with the manuscripts containing nothing o the sort. In that way, beyond the Archimedean corpus, the whole corpus o Greek geometrical texts acquired more coherence than what the written evidence records. ogether, these three chapters bring to light various respects by which the critical editions tacitly convey nineteenth-century or early-twentiethcentury representations in place o Greek mathematical proos to inattentive readers. Still more will be developed on this point in relation to Diophantus below. Tese conclusions provide impetus or developing urther research on these topics, in order to understand how representations o ancient mathematics were ormed in the nineteenth century and how they adhered to other representations and uses o Greek antiquity. Another chapter o the book inquires urther in this direction o research. It complements our critical analysis o the historical ormation o our understanding o Greek ideas o proo and shows how ruitul urther research o that kind could be or sharpening our critical awareness. In this chapter, Orna Harari draws on the hindsight o history and questions the conviction widely shared today that Aristotle’s theory o demonstration in the Posterior Analytics can be interpreted in reerence to its presumptive illustration, that is, Euclid’sElements. In act, she establishes that this use o these two pieces o evidence in relation to each other became
Mathematical proo: a research programme
commonplace only in modern times. Tis brings us back to the issue o the part played in our story by the philosophy o science as it took shape as a discipline in the nineteenth century. o make this point, Harari digs into the history o the discussions that bore on the question o the conormity o mathematical proos – particularly, those contained in Euclid’sElements – to Aristotle’s theory o demonstration. Her historical inquiry highlights that the present-day discussions o the issue are at odds with how the question was understood and tackled rom late antiquity until the Renaissance. In contrast to the discussions by John Philoponus and Proclus which took Aristotle’s theory as their oundation and inquired into whether and how mathematical proos, and which mathematical proos conormed to the Aristotelian theory, the contemporary view reversed the perspective. It took Euclid’s Elements as a basis on which Aristotle’s theory o demonstration had to be interpreted and understood. Tis repositioning reveals a undamental shif in the interpretation o Aristotle’s Posterior Analytics. By analysing how Philoponus and Proclus discussed the issue, she emphasizes that, despite essential differences between their approaches, they both understood the key problems to be whether proos established mathematical attributes that belong to their subjects essentially and whether the middle term o a syllogism could serve as the cause o the conclusion. Tus, or these authors, the problem o the applicability o Aristotle’s theory o demonstration related to the non-ormal requirements o the theory. Te same criterion holds true or the discussion until the Renaissance. By contrast, whatever conclusions they reach, contemporary interpretations o the question only consider the ormal requirements. Te main point o the discussion has hence become whether an interpretation o the syllogism could be offered that could accommodate what is to be ound in, say, Euclid’sElements. Harari’s contribution thereby exposes the anachronism underpinning the common, present-day reading o the relationship o Euclid and Aristotle to each other and highlights how much stranger they might become – both to us and to each other – i we attempted to restore them back to the context o the discussions and problems rom which they emerged, so ar as this is possible. Can we establish a correlation between the modern readings o Euclid and Aristotle and the way in which the critical editions discussed above were carried out? Such questions are interesting to keep in mind generally when analysing the various editions o Euclid’s Elements produced th roughout history. Tese remarks conclude our analysis o past historiographies o proo and our identi cation o the actors at play in shaping and maintaining them. Among these actors, we identi ed elements o the contexts in which
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historiographic ideas were ormed, the values which past historians had deemed central, the agendas they adopted and the critical editions they produced. We are now brought back to the new agenda we suggested above: what can be gained by widening our perspective on the practices o proo while considering a richer collection o sources?
III Broader perspectives on the history of proof Widening our perspective on Greek texts: epistemological values and goals attached to proof Te biases in the history o proo on which the oregoing analysis shed light rst coloured the treatment o the source material written in Greek. Historical approaches to proo in ancient Greece have so ar concentrated mainly on a restricted corpus o texts and have limited the issues addressed. Te ensuing account was accordingly con ned in its scope and lef wide ranges o evidence overlooked. Some o the chapters in this book deal precisely with part o this evidence. o begin with, Geoffrey Lloyd’s chapter indicates some o the bene ts that could be derived rom a radical broadening o the corpus o Greek proos under consideration. In particular, he discusses some o the new questions that emerge rom this extended context, with regard to the practices o proo in ancient Greece. Lloyd rst reminds us o the act that, despite the importance historiography granted to Euclid’s Elements and cognate geometrical texts, mathematical arguments in ancient Greece were by no means restricted to proos o the type that this corpus embodies. As Lloyd illustrates by means o examples, Greek sources on mathematical sciences provide ample evidence o other orms o argument as well as discussions on the relative value o proos. 38 Enlarging the set o sources under consideration thus opens a space in which the various practices o proo and the values attached to them become an object o historical inquiry. Some o these sources bear witness to the act that some authors ound it important to use various modes o reasoning. Lloyd recalls the case o Archimedes, who expounds in Te Method why it is ruitul to consider a gure as composed o indivisibles and to interpret it in a mechanical way in order to yield the result sought or. However, as Lloyd insists, although Archimedes deemed such reasoning essential to the discovery o the result to be proved, in 38
Lloyd has made this point on other domains o inquiry; compare or instance Lloyd 1996b.
Mathematical proo: a research programme
Archimedes’ view, this type o argument could not be conclusive and had to be ollowed by another purely geometrical proo. Our explorations into matters o proo will allow us to come back to this example below, rom a new perspective. Let us stress or now that different kinds o reasoning have different kinds o value. Furthermore, Lloyd stresses that in numerous domains o inquiry in ancient Greece, there were debates about the value o their starting points or the proper methods to ollow, and securing conviction was a key issue. Keeping too narrow a ocus on mathematics in this respect conceals important phenomena. Here two points are worth emphasizing. Firstly, within this extended ramework, it appears that proos carried out according to an axiomatic–deductive pattern were developed in several areas and were by no means con ned to mathematics, although even in antiquity, geometry came to be perceived as a singular eld in this respect. Te recurring use o such a practice o proo echoes the variety o terms used throughout the sources to demand ‘irreutable’ arguments. One is hence led to wonder how ar, as regards ancient Greece, the history o an axiomatic–deductive practice can be conducted while remaining within the history o mathematics, or to what extent the interpretation o this practice can be based only on mathematical sources. Here too, we encounter the impact o a orm o anachronism. Since this kind o proo is at the present day deemed to be essential to, and even characteristic o, mathematics, historiography has approached the question o axiomatic–deductive proo mainly rom within the eld o mathematics, disregarding the act that it was employed much more widely in antiquity. What greater understanding o such a practice o proo would a broad historical inquiry o proo more geometrico yield? Tis is the issue at stake here. Secondly, such an importance granted to one type o method and organization o knowledge cannot hide a much wider phenomenon which Lloyd wants to emphasize: the numerous debates on the correct way o conducting an inquiry. We seem to have here an idiosyncrasy o ancient Greek writings, or at least among the writings that have been handed down to us. Te unique multiplicity o ‘second-order disputes’ evidenced in ‘most areas o inquiry’ leads Lloyd to suggest a third expansion. Lloyd grants that disputes between practitioners o mathematics or other domains o inquiry are a widespread phenomenon worldwide in the ancient world. However, his comparison o such debates, in ancient Greece and elsewhere, leads him to an important observation, namely, that the modes o settling debates in various collectives appear to differ. Lloyd thus invites us to consider engaging in a discussion on the standards o proo in
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order to conclusively resolve debates as a social phenomenon. Te new and important research programme which he proposes intends to account or the development o such attempts to adjudicate debates in social terms. In other words, Lloyd calls or developing a social account o the emergence o second-order discussions on proo. Tese suggestions show how, by concentrating on a set o texts wider than the usual geometrical writings, one can de ne new horizons or research on proo in Greek sources. In recent years, though, new approaches to proos in the writings that provided the standard historiography with its basis have taken shape. Tese approaches are interesting or us, since they have brought to the ore epistemological values other than being conducive to truth which, as ar as our sources can tell, may have been attached to proo, thereby sidelining the issue o certainty that has dominated the discussion on ancient proos. o mention but one example, I shall show how, in my view, Ken Saito has advocated a new way o interpreting proos in the core corpus. Saito takes as his starting point the thesis that, when one considers this collection o texts as a whole, there emerges rom the corpus a set o ‘elementary techniques’ that orm a ‘tool-box’ on which Greek geometers relied.39 Moreover, he argues that the practitioners developed knowledge o how to combine the elements in the tool-box in standard and locally valid methods – combinations which he also calls ‘techniques’, or ‘patterns o argument’. In Saito’s view, the ‘method o exhaustion’, which was named and discussed as such only in the seventeenth century, constitutes an example o such a method. His approach not only yields an analysis o the method as a speci c sequence o elements taken rom the tool-box, but it also embeds a technique that has been long recognized into a larger set o similar techniques which recur in proos. What is worth stressing is his remark that, or reasons yet unknown, these methods do not seem to have been described at a meta-mathematical level or even named at the time. Nevertheless, the sources bear witness to patterns o proo which circulate between proos and to the stabilization o a orm o knowledge about them. An initial hypothesis can be ormulated with respect to these methods: it is by reading the text o a proo per se and not merely as establishing the 39
Te insight about the ‘tool-box’ was introduced and worked out by Saito rom 1994 on (see Saito and assora 1998 andwww.greekmath.org/diagram/). It was urther developed in N1999: 216–35. Te latter book guresprominently among the publications that opened new perspectives in the approach to deduction in the Greek mathematical texts o what I called the ‘core corpus’. I develop here re ections on a tiny part o the new ideas that were introduced in this wider context. Saito’s project on the Greek mathematical tool-box has not yet come to completion. o present his ideas here, I rely on personal communication and on drafs that he sent me in 2005 and which contain abstracts o part o his project.
Mathematical proo: a research programme
truth o a proposition that such techniques could be grasped. Te hypothesis accounts or how the techniques brought to light took shape. It may also account or one o the motivations at play in making proos explicit and writing them down. One can go one step urther and speculate about why, as ar as we know, in ancient Greece the methods in question were neither named, nor analysed in any second-order discussion. Tis point leads me to a second hypothesis with respect to the text o a proo: were not some o the proos written down with the purpose o displaying a given technique which they put into play? In that case, general techniques would have been expressed through the proos o particular propositions and thereby also motivated the expression o these proos in writing. In other words, some proos were to be read as a kind o paradigm, making a statement o more general validity than a rstreading would indicate. Te interpretation o the texts o these proos would be comparable in that respect to how a problem and the procedure or solving it made sense in the Babylonian or Chinese writings.40 Whatever the case, the essential point here is that the text o a proo was not read only as establishing a proposition, but also as a possible source or working techniques. Moreover, the generality and importance o a textual unit in these books would not lie only in the proposition itsel, but also in the technique brought into play in its proo. Let us consider these various points one by one to grasp what is more generally at stake here. o begin with, the rst hypothesis ormulated above suggests that readers were likely to read a proo or itsel and not merely or its capacity to establish the statement proved. Tere is nothing surprising about this assumption. Te recent debate on which we commented in Section bore witness to such uses o the text o proos: some o these mathematicians testi ed to the act that they read proos, seeking, among other things, techniques and also concepts. Tis constitutes a challenge or us: how are we, as historians, to gather evidence in order to take this dimension o the interest in proo into account more generally and rigorously? Interestingly enough, the hypothesis on the practice o proo prompted by Saito’s suggestion echoes with how, as we shall see, proos o the correctness o algorithms were conducted in the earliest extant Chinese sources attesting to practices o proo.41 In all thes e contexts, the proos appear not to have been only means 40
41
On the latter, a discussion and bibliography can be ound in Chemla 2009. Note that I use paradigm in the sense that grammarians use this word. Also note that the text o a proo could either state a general technique or document its existence by the act o bringing it into play. See below and Chemla 1992, Chemla 1997b.
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or an end, but their texts were also read as conveying other meanings. Te ‘techniques’ read in the proos do not have the same nature in different contexts. However, what is important to note here is that in all these cases the epistemological value o the proo cannot be exhausted by the question o determining whether it duly establishes the statement to be proved. According to the rst hypothesis too, the reader looked or something general in a proo – a method, the use o which could extend beyond the limits o a proposition. Te act that, as Saito showed, some techniques o somewhat general validity were actually composed indicates the possible outcome o such a search. A straightorward interpretation o the text o each proo taken separately would miss this eature o the practice o proo. Te virtue o the techniques thereby identi ed was their potential useulness in other contexts: i we ollow this interpretation, a certain ruitulness was recognized in it. Tese concerns indicate epistemological values that actors may have attached to proos and that too narrow a ocus on certainty could hide rom our view. Te preceding remarks illustrate what kind o bene ts could be derived rom re-examining standard texts with wider expectations in mind. Tey also bring to light an issue that will prove essential in what ollows. Te way in which actors have read proos or have written them down, the
motivation driving the composition o explicit proos, cannot be taken or
granted. As I have indicated, reading meanings into proos is apparently a widely shared practice. However, this does not mean that practitioners belonging to different scholarly cultures read the meanings in texts in the same way or that the texts intended the meanings to be read in the same way. Whether one accepts only the rst hypothesis or both hypotheses as ormulated, the perception o the various dimensions o the Greek texts to which I have just alluded requires an unusual and speci c reading o the text. I one admits the second hypothesis, texts o proos were to be read as paradigms. Interestingly enough, as we saw in Section o this introduction, the diagrams in Greek texts seem to have required the same kind o reading, at least i we agree on the act that the srcinal gures resembled those in the manuscripts and not those which Heiberg drew. Interpretation o the sources appears more generally to be a delicate procedure, on which our ability to perceive the various dimensions o the proos examined will depend. As I shall argue below, this problem is intrinsic to our endeavour: it is, in my view, tied to the act that shaping a practice o proo has always involved designing a kind o text to work out and deliver the proos. Te task o interpreting the texts thus cannot be separated rom the job o describing the practice o proo to which they adhere.
Mathematical proo: a research programme
Te lines o inquiry just outlined illustrate some o the issues that more generally have imposed themselves as central issues in the ollowing chapters o the book. o begin with, these issues are taken up rom different perspectives in the next two chapters o the book, both also devoted to Greek sources. Te issue o generality in relation to proo directs Ian Mueller’s analysis o marginalized Greek writings dealing with numbers, albeit rom a different perspective. Because they have been overshadowed by the treatment o arithmetic in Books to o Euclid’s Elements, the techniques o proo used by Nicomachus in his Introduction to Arithmeticand by Diophantus in On Polygonal Numbers have not yet been the object o detailed analysis. Ian Mueller chooses to ocus on them because they deal with numbers – polygonal numbers – in a singular way, approaching them through the prism o con guration and procedure o generation. Tese eatures raise the problem o de ning the polygonal numbers as general objects, making general statements about them, and proving such statements in a general way. Te challenge is to reach generality not only with respect to all polygonal numbers o a speci c type, such as triangular or square numbers, but also to de ne and work with n-agonal numbers. Both Nicomachus and Diophantus attempted to meet with this challenge, by composing treatments o these numbers in general, stating propositions about them, and accounting or the validity o these statements. In particular, both authors set themselves the task o establishing the value o the nth j-agonal number. Te conclusion o Mueller’s analysis is that both attempts equally ail to establish the conclusion aimed at with ull generality. Nonetheless, the differences between the ways the two authors shape textual elements to approach polygonal numbers, ormulate statements about them and design modes o proving to deal with the topic raise considerable interest. Tis is what emerges rom Mueller’s detailed description o the different techniques o reasoning by which both authors address these numbers and try to establish their properties. Nicomachus makes use o speci c diagrams that iconically represent the numbers as con gurations o units. In addition, Nicomachus introduces a key tool – sequences o numbers – in a way that will be characteristic o his approach. o begin with, he constructs arithmetical ways o generating these sequences. He then strives to establish relationships between these sequences and the rst equences s o polygonal numbers (triangular, square, pentagonal and so on). It is or this task that Nicomachus’ diagrams are brought into play. Because o their eatures, these diagrams can be used to indicate the reason o the correctness o the relationship only or the
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rst terms o the sequences. However, this is how Nicomachus argues in avour o the general statement, whereby his establishment o this general statement differs rom modern standards. In the second step, Nicomachus urther brings to light patterns in the modes o generation o the rst sequences, thereby indicating the general structure o the set o sequences o polygonal numbers and pointing to urther relationships between these sequences. Again, Nicomachus indicates the general pattern and argues or it by highlighting the pattern or the rst equences. s And again this is where his approach alls short o modern standards. Te most general statement by which Nicomachus summarizes his procedure o proo consists o a table o numbers. It collects in its rows the sequences introduced and more. Since it displays the pattern o relationship between the lines, the table allows Nicomachus to generate subsequent lines by deploying the same pattern urther and thereby determining the value o any polygonal number. Te textual elements brought into play (diagrams, sequences and table o numbers) and the ways o using and articulating them by modes o reasoning contrast sharply with how Diophantus approaches the same topic. Te core o Diophantus’ treatise On Polygonal Numbers consists o purely arithmetical and general propositions. Tese propositions state arithmetical properties in the orm o relationships holding between numbers. Diophantus proves these relationships through representations o numbers as lines, using the representations in a way that is speci c to this branch o inquiry. Diophantus attempts to ormulate the value o the nth j-agonal number as a proposition o this kind. Te diagrams used and the propositions stated thus exhibit a style completely different rom Nicomachus’. However, their kind o generality is precisely what constitutes the problem or concluding the proo. It is in Diophantus’ attempt to connect these general statements to polygonal numbers with ull generality that Mueller identi es the gap in Diophantus’ proo. Te tools Diophantus uses here are too general to allow him to recapture the details o the general objects that polygonal numbers represent. He manages to establish the link only or the rst n-agonal numbers. Tese two texts devoted to the same topic illustrate quite vividly the plurality o practices in Greek mathematics, the study o which Lloyd advocated. Mueller highlights differences in the ways o making diagrams and relating them to the mathematical objects being studied. He shows the distinct ways in which diagrams are employed in the arguments being developed, thereby bringing to light two distinct kinds o arithmetical methods. Additional interest in this case study derives rom what is revealed when the proos are considered rom the viewpoint o generality. Clearly, both texts
Mathematical proo: a research programme
betray an ambition to reach a high level o generality. Mueller’s contrastive analysis discloses how distinct means are constructed and combined or the proos to ul l this ambition. Despite their ailure in modern eyes to achieve their goal, the two sets o proos in the texts appear to orm two strikingly different, but careully designed, architectures o arguments inspired by the task that the authors had set or themselves. aking the value o generality into account in his interpretation allows Mueller to use ner tools and describe with greater accuracy the argumentative structures and the differences between them. Mueller thus highlights how the conduct o proos can bear the hallmark o epistemological values prized by the actors. More generally, Mueller’s analysis indicates how much more there can be to the study o a practice o proo than simply assessing whether proos adequately establish their conclusions or not. Te kinds o elements the practitioners design or their proos, the ways in which they use them, and other questions, all essential or a historical inquiry into the activity o proo, will appear quite ruitul in the ollowing chapters. In particular, the question o how a kind o text has been designed or a certain practice o proo – a question that the multiplicity o the proos examined brings to the ore – appears relevant again or the urther analysis o the sources. Its undamental character will soon become even clearer.
Further widening: the text of a proof In his Arithmetics, Diophantus opts or a completely different style o composition and presents solutions or hundreds o problems relating to integers. Each problem is ollowed by the reasoning that leads to the determination o a solution. o ormulate the problems and the kind o proo ollowing them, both o which involve statements relating to numbers and unknowns, Diophantus regularly makes use o symbols. In his chapter, Reviel Netz ocuses on the question o determining the role played by this symbolism in the development o the reasonings Diophantus proposed to establish the solutions to the problems. Te act that the symbols introduced are essential to Diophantus’ project is made clear by the act that they are the main topic o the introduction to his book. On the other hand, Diophantus stands in contrast to his known predecessors in that he makes explicit the reasoning by which he establishes the solutions to problems. Tereore, the question o how the ormer are linked to the latter is not only natural, but also essential to an analysis o Diophantus’ activity o proving. Such is the main question o the chapter. It pertains, as one can see, to the text with which an argument is conducted.
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Aware that the symbols o Diophantus must be distinguished rom those o Vieta, Netz rst studies the speci c historicalcontext in which they were designed and describes them in detail, on which basis he examines how the editors o the nineteenth century transcribed them in their critical editions and translations. His conclusions are twoold. On the one hand, Netz shows that the symbols are located at the level o noun-phrases, but are not used or either the relations or the structural terms speci c to a problem. Moreover, he establishes their nature o being ‘allographs’ o the words they stand or, that is, they write these words in another way. On the other hand, Netz reveals that the use o these symbols is nowhere as systematic in the manuscripts as Paul annery presented them in his 1893–5 edition.42 annery designed the proos, rather than the statement o problems, as the locus or the use o symbols, a act which does not correspond to what is ound in the manuscripts. Moreover, annery introduced a distinction between some terms which he systematically rendered as symbols and other terms which he always wrote down in ull, thereby establishing two different kinds o terms, in contrast to the manuscripts which use abbreviations or both kinds in comparable ways. We meet again with the necessity o a critical awareness regarding the critical editions carried out in the nineteenth century. Tis preliminary analysis provides a sound basis on which Netz can address the main question raised by his chapter: what is the correlation between Diophantus’ use o such symbols and the speci c kind o proo he systematically presented? In Netz’s view, Diophantus undertook to gather problems he had received and complete their collection in a systematic way. Moreover, his ambition was to present them or a literate, elite readership. In relation to this goal, Diophantus opted or a solution o each problem in the orm o ‘analysis’. Hence Netz also addresses a part o the history o proo that alls outside the scope o Euclid’s Elements. Tis holds true not only because these proos proceed through analysis. In addition, the point in Diophantus’ Arithmetics is not to establish the truth o a statement, but rather to ul l a task correctly. In a context in which the procedure o the solution provided or problems was also a topic or debate, Netz argues, writing down the reasoning which establishes how the task was correctly ul lled contributed to showing the suitability o the mode o solution adopted. In other words, or Netz, the proo here intended to highlight the natural and rational character o the method chosen to solve a given 42
Compare 1893/5. Te 1974 reprint o the book is reely available on Gallica: http://gallica.bn.r/.
Mathematical proo: a research programme
problem. o this end, Diophantus shaped, primarily thanks to his symbols, a kind o text that would enable the reader to survey in the best way possible the method ollowed. Tis is how Netz argues in support o his thesis that the expressions ormed with the speci c symbols introduced are consubstantial with the project and the kind o proo speci c to Diophantus’ Arithmetics. Note that here again, as in Mueller’s chapter, the examined proos proceed through operating with statements o equality between numbers. However, in the Arithmetics, the symbols developed helped carry out such operations in a speci c way, linked to the peculiar eatures o Diophantus’ reasonings. Since they were allographs, they allowed the reader to keep the meaning o the computations in mind. On that count, these symbols differ rom modern symbolism. Tis difference in nature possibly echoes a difference in use: Diophantus’ symbols do not seem to have been used or proving through blind computations, as is the case with modern symbolism. Instead, they helped orm a kind o writing transparent with respect to the meaning o the statement. Since the symbols were abbreviations, they enhanced the surveyability o the expressions, in the same way as the technical writing o a number helps understand the structure o the number.43 Tis conclusion raises a general question. Te surveyability o a procedure or a proo depends on the kind o text constructed to write down and work with the proo or procedure. Which resources did various groups o practitioners create, or borrow, to this end? Netz’s contribution can be viewed as a step towards a systematic inquiry in that direction. We shall soon meet with urther evidence that can be ruitully analysed rom the s ame perspective. o create this orm o writing, Diophantus made use o possibilities available in the written culture o his time, but used them in a way speci c to his project. As Netz stresses, Euclid’sElements also exhibits evidence o creating a speci c language, characterized by distinctive ormulaic expressions. Tus we meet with the same phenomenon already emphasized on several occasions above rom yet another perspective: the kind o text used is correlated to the type o proo developed. Given the act that the kinds o proo and the project embodied by the Elements differ rom the objectives o Diophantus, the kind o writing employed in the Elements differs rom those used by Diophantus. 43
Neugebauer also interpreted some eatures o Mesopotamian ways o writing mathematics as making statements surveyable. Høyrup 2006 quotes at length the passages by Neugebauer on this point and discusses them, with respect to Mesopotamian, Greek, Latin, Arabic and Indian sources as well as sources written in vernacular European languages.
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From another viewpoint, Diophantus’ text can be contrasted with other types o problem texts, which also attest to mathematical work on and with operations or computations. Several o the ollowing chapters consider types o writing o the latter kind. Both the use o operations on statements o equality and the introduction o symbols to carry out these operations ound in the Arithmetics contrast with what other traditions ormatted as algorithmic solutions to problems, or which the correctness needed to be, and was, established. Even i these other writings bear witness to other means o proo, via other techniques and in pursuit o other goals, many parallels can be drawn between the Arithmetics and these other texts. Tese texts all deal with operations and operations on operations, illustrating how different modes o manipulating mathematical operations were devised in history. Most o these texts reveal an ideal o writing sequences and combinations o operations in such a way that the meaning becomes transparent. However, despite the act that they shared a common eature, in what ollows we shall see that how this ideal was achieved depended on the context and the type o text constructed. Lastly, these writings all raise the question o what was meant by a problem and the procedure attached to it. Was a particular problem representative only o itsel, or was it read more generally as a paradigm or all problems in the same class? Netz develops an interpretation o the way in which Diophantus conceived o generality. Whether or not this interpretation is accepted, it stresses an essential point: the symbols used by Diophantus were not abstract. Tis eature sheds an additional light on how these symbols differed rom modern ones. Moreover, it implies that i they had a general meaning, it was conveyed in a speci c way, requiring again a speci c reading. Tis chapter thus leads to two general conclusions, essential or our purposes. Firstly, Netz’s article analyses how different groups o mathematicians created different kinds o text in relation to the practice o proo they adopted. Note that this approach offers one o the ways in which one could systematically develop the programme suggested by Lloyd and account or the variety o practices o proo in ancient Greece. More generally, Netz oregrounds the act that proos have been conducted in history with various kinds o texts, each being shaped in relation to the operations speci c to a given kind o proo. Te text o the proo is correlated to the act o proving. Te general question raised by Netz in his approach to Greek sources may be phrased as ‘What types o text were shaped or the conduct o which kind o proos?’ and has already proved relevant above. Clearly, this question opens a eld o inquiry into proo that could be – and will prove so below – extremely ruitul. In particular, we can expect that the
Mathematical proo: a research programme
development o this inquiry provides means or interpreting these texts more accurately. In conjunction with the rst point, Netz’s treatment yields insight into how different the purposes or developing proos may have been. Tis brings us back to the programme suggested above or our historical inquiry into proo, namely, the restoration o the motivations behind the development o proos and the description o the diversity in their conduct accordingly. However, beore we go urther in widening the set o sources to be considered with these issues in mind, a last point must be emphasized. Netz’s discussion illustrates how the resources Diophantus introduced or a given type o proo were adopted to design the text o another kind o proo, i.e. algebraic proos, in modern times. More precisely, Netz’s analysis highlights why Diophantus’ proos are not algebraic in nature. Nonetheless, the shaping o the modern algebraic proo made use, or the conduct o a reasoning, o symbolic resources similar to those designed within the ramework o the Arithmetics. Tis conclusion offers our rst insight into the history o algebraic proos. What are its other components and how did they take shape? Tese are some o the questions to which we shall come back below.
Proving the correctness of algorithms Te ideal o transparency, which Netz interprets as inorming the symbolism used by Diophantus, is also the main orce driving the way Babylonian practitioners o mathematics composed the text o algorithms, according to the interpretation o Jens Høyrup. Beore explaining this point urther, let us rst recall some basic eatures o the writings to which we now turn. Tese documents are, or the most part, composed o problems ollowed by algorithms which solve them. Te act that the algorithm correctly solves the problem is the statement to be proved, in contrast to what we nd in Euclid’s Elements, where proos mainly deal with the truth o theorems.44 In such contexts, proving means establishing that the procedure carries 44
Te claim here takes into account the act that the statement o a problem in the Elements does not include the statement o how to carry out a task. Interestingly enough, except or some speci c cases, the scholarship devoted to Euclid’s Elements has paid much less attention to problems than to theorems. Tere are exceptions like Harari 2003. However, the problems still await urther study qua problems. How was the solution written down as text and how did the proo relate to the ormulation o the solution as such? Tese are questions that seem to me to be promising or uture research. It may well be that afer these problems have been studied more in depth, the statement contrasting proos in Euclid’s Elements with those o algorithms may have to be made more precise.
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out correctly the task or which it is given, that is, that the algorithm yields the desired result. In this ramework, the ideal o transparency that the Mesopotamian tablets embody consists o the act established by Høyrup that the texts or procedures were simultaneously prescribing computations and indicating the reasons underlying their correctness.45 Since we have no second-order comments by Babylonians explaining how these texts should be interpreted, it took some time beore this property was recognized. Once again, like the previous examples, this case shows how given collectives o practitioners shaped speci c kinds o text to work with operations and establish their correctness. It also highlights how this ormation and standardization o texts invited problems o interpretation. Te technical character o the texts hindered their interpretation by historians, who ailed to identiy how proos were expressed and hence drew derogatory conclusions, such as M. Kline’s. In this case, however, recognizing the proo in the text required understanding something with respect to proo as well, that is, that the rationale o a procedure can, at times, be given in the description o the procedure itsel and not as a separate text – this is precisely the maniestation o the ideal o transparency in this context, which demonstrates that the same ideal can appear in various ways. More accurately, when we examine Mesopotamian texts such as those with which Høyrup establishes his point rom this perspective, we observe that the texts o algorithms do not only contain speci c prescriptions or operations that achieve transparency, but also contain elements o the reasoning that develops along the statement o the algorithm. Again, widening the corpus o proos under consideration leads us to deeper insights into how a proo can be ormulated. Tis expansion o the corpus also broadens our understanding o the motivations or writing down proos in the ancient traditions. In Høyrup’s 45
One speaks o the ‘correctness’ o the algorithm. On this theme, it may be helpul to clariy two points about which I ofen read misleading comments. Firstly, the text o an algorithm is the statement to be proved and not its proo. It is on the basis o this distinction that one can make the point that in Mesopotamian tablets, the two texts (the statement o the algorithm and the ormulation o its proo) merged with each other. Moreover, to perceive this requires a speci c reading, whereby two layers o meaning are discerned in the statement o the algorithm. Secondly, the aim in proving the correctness o an algorithm is not only to show that the algorithm yields an exact value – or to establish how accurate or inaccurate the value is – but also to establish that the sequence o operations prescribed yields the desired magnitude. So the depiction o algorithms only in association to approximations is doubly misleading. Tese basic misconceptions lie at the root o what most commentators who have been discounting computation have claimed. Te section entitled ‘Te unpuzzling character o calculation’ in Hacking 2000: 101–3 comments on the text o an algorithm, overlooking the act that this is the statement to be proved and not the proo. Te same pages make other claims that are contradicted by the conclusions reached here.
Mathematical proo: a research programme
view, the proos he reads in the ormulation o the procedure intend to guarantee an understanding o the reasons why the operations should be carried out. He even suggests they are proos precisely because they have this goal. We see how the exclusive ocus on the unction o proo as yielding certainty would leave out these sources as irrelevant or a history o proo. However, these texts demonstrate that one motivating interest in proos and their transcription in one way or another may have been not only – or perhaps not at all – to convince someone o the truth o a statement but to make one understand the statement. Tis is still a strong motivation or mathematicians today, as is evidenced, or example, by the debate analysed in Section and it has been so all through the history o mathematics. Let us pause or a while to consider the goal o ‘understanding’ within the context o a practice o proo intended to establish the correctness o algorithms. Far rom being the nal point o the analysis, it is in act only its beginning. Te possibility that some proos aim at providing an ‘understanding’ raises an essential question, or which the Babylonian case allows urther inquiry: what techniques or dispositis were devised to provide an ‘understanding’ o the algorithms in the milieus o scribes? By Høyrup’s restoration, geometrical diagrams seem to have supported the procedure. Moreover, these diagrams were made in a way which allowed material transormations o their shape. Te speci c terms which prescribed the operations designated such material transormations which helped make sense o the computations. Te arguments supporting this hypothesis come rom a close analysis o the terms used to write down the algorithm. However, this conclusion would have remained only speculation, had not Høyrup discovered some texts rom Susa that make explicit the kind o training required by such a mode o understanding. Tese texts are revealing or several reasons. Te explanations in them that produce the ‘understanding’ are developed very speci cally within the ramework o paradigmatic situations similar to those described in the outline o some geometrical problems. We hold that these explanations reveal how the context o geometrical problems may have provided situations as well as numerical values with which the understanding o the effect o operations could be grasped. Te texts rom Susa also reveal how diagrams with highly particular dimensions were used in the same way. Tis parallel between geometrical gures and problems, as wellas this way o using geometrical problems, compellingly evokes the case o some Chinese mathematical sources, about which two points can be established. Firstly, the problems were not only a question to be addressed, but, as the
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paradigms that they were, they also provided a semantic eld or interpreting the operations o the algorithm or the operations required or the proo o correctness. Secondly, just as problems provided particular numerical data, geometrical gures displayed simple dimensions, and they were used in the same way to make explicit the meaning o operations.46 In a moment, we shall come back to this comparison, but note that exactly the same situation holds true or Sanskrit sources analysed by Colebrooke.47 In addition, the Susa texts that Høyrup analyses ormulate the explanations by describing the result o each operation in two ways: on the one hand, a numerical value is provided and, on the other hand, an interpretation o the magnitude which is determined is made explicit in the geometrical terms o the eld o interpretation. Such a kind o ‘meaning’ or the effect o operations recalls what is ound in Chinese texts. In the latter sources, a speci c concept (yi) is reserved to designate that ‘meaning’, and the meaning is made explicit by reerence to the situations introduced in the statements o problem. In my own chapter on early China, I discuss the interpretation o this concept and provide cases where it is used in Chinese sources. In correlation with this parallel, in early Chinese mathematical writings we also nd algorithms that are transparent regarding the reasons o their correctness: the successive operations are prescribed in such a way as to simultaneously indicate their ‘meaning’, which can be exhibited directly in the context o the situation described by the problem.48 Tis parallel shows that the early mathematical cultures which worked with algorithms developed partially similar techniques or ‘understanding’, even though they did so in different ways, as we shall make clearer below. More broadly, these remarks raise a general issue. Tey invite us to study systematically the devices, or dispositis, that various human collectives constructed or ‘understanding’ and interpreting the ‘meaning’ o operations, or conversely, the kind o ‘interpretation’ that was rejected. Interestingly enough, this question enables a perspective rom which we may cast a new light on the ‘Method’ described by Archimedes in the text devoted to this topic, which Lloyd discusses in his chapter. Indeed, what Archimedes offers with his ‘mechanical method’ is a way o ‘interpreting a gure’ in terms o weight – speci cally, an interpretation rom which he can 46
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See chapter A, in CG2004: 28–38 and Chemla 2009, which presents a ully developed analysis o these issues. See p. 6. Chemla 1991. Chemla 2010 analyses more generally the two undamental ways in which the text o an algorithm can reer to the reasons or its correctness. Both can be recognized in the way in which texts or algorithms were recorded in the tablets discussed by Høyrup.
Mathematical proo: a research programme
derive a result. Even though he discards this method as inappropriate or a proo, as did a tradition o scholars who developed comparable proos, the question remains open or us to understand what this kind o interpretation actually achieved or him.49 Getting back to our Mesopotamian documents, I am aware that some historians may question whether such modes o establishing the correctness o algorithms ought to be considered proos. Te Babylonian source material allows us to shed light on the difficulty that this division would entail in the history o mathematics. In act, the techniques that the scribes used to provide an ‘understanding’ o the type discussed above could be, and appear to have been later on, taken up in other practices o proo – where the quali cation as ‘proo’ is less disputed. As Høyrup has suggested in previous publications, there is a strong historical continuity between the modes o argumentation alluded to above, which appear to have been developed in Babylonian scribal milieus on the one hand and what are explicitly recorded as proos in Arabic algebraic texts rom the ninth century onwards on the other hand.50 I only or this reason, these techniques o ‘understanding’ do belong, in my view, to the history o mathematical proo. Te continuity evoked is o the same kind as that mentioned above with respect to the textual techniques devised by Diophantus to develop his arguments. As a provisional conclusion, one may suggest that the text o a proo is a technical text, the shaping o which may have bene ted rom all kinds o resources available. Conversely, in some cases, the ormation o a technical text or working out a kind o proo led to developing techniques that could be brought to bear in other mathematical activities. In the case o Babylonian tablets, not only the operations used in a procedure, but – as is clearly shown by the Susa texts – also the transormations o algorithms, 49
50
In the same way, Krob 1991 has developed a proo o a combinatorial theorem based on an interpretation involving a plate, beads and pebbles. Such eatures are unusual in mathematical publications. Tey occur more requently in some elds, like combinatorics, than in others. Te reasons why it is so are worth exploring, since they shed light on social aspects o proving. It is clear that precisely because o these eatures, not all mathematicians o the present day will accept the proo Krob 1991 gives as a proo. Tis approach to the question, however, leaves unanswered the questions which I nd quite ascinating: what does the interpretation do or the reasoning? And why do practitioners nd it appealing to make use o such devices or dispositis o interpretation within proos? Approaching the problem through the controversies among mathematicians would yield interesting results. See, or instance, Høyrup 1986. In his recent edition and translation into French o al-Khwarizmi’s book on algebra, Rashed 2007 puts orward a different hypothesis or the history o these proos, interpreting them rather as composed within the ramework o Euclidean geometry.
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such as embedding them into other algorithms or modiying their lists o operations, could be established on the basis o problems and/or gures. In the case o Diophantus, equalities were transormed qua equalities. Both techniques were adopted in Arabic algebraic texts. In addition to providing insights into how actors carried out interpretation or the algorithms recorded on Babylonian tablets, Høyrup suggests that the need or understanding perhaps developed in relation to teaching. In other words, he links the proessional context o training scribes in Mesopotamia to the development o certain kinds o proo. Interestingly enough, as we shall see below, such a hypothesis nicely ts withA. Volkov’s thesis regarding the use o proos or a teaching context in East Asia. Christine Proust’s chapter suggests capturing an interest in the correctness o algorithms in another kind o Mesopotamian tablet, which contain texts consisting o only numbers. Note that, here, the work o the exegete is particularly challenging, since she has to argue or an interpretation o texts that contain no words, only numbers. Te method Proust uses to read these traces is deeply subtle but o particular interest or us. At rst sight, the tablets at the ocus o Proust’s attention appear merely to betray an interest in ‘checking’ the numerical results yielded by an algorithm. Seen in that light, they recall some o the texts discussed by Høyrup, in which a similar concern can be identi ed. However, as we shall see, the two types o text call or different modes o interpretation. Te tablet VA 8390, discussed by Høyrup, contains a ‘veri cation’ part, comparable in some sense to the ‘synthesis’ ollowing the ‘analysis’ in Diophantus’ Arithmetics. Tis part o the text relies on the values produced by the algorithm as well as on the procedure described by the statement o the problem to show that the values obtained satisy the relationships stated in the problem. However, the actual unction o this section in the text should not be interpreted too hastily: as Høyrup emphasizes, it does not merely yield a ‘numerical control’ o the solution, since the way in which the ‘veri cation’ procedure is written down requires the same kind o ‘understanding’ rom the reader as that attached to the text o the direct algorithm. Te nature and practice o the ‘veri cation’ must thus be considered somewhat urther, without being taken or granted a priori. extual structures o this type are characteristic o other tablets, in which, once an algorithm has yielded a result, this result is then subjected to another procedure, immediately appended to the srcinal one and which has ofen been interpreted as a veri cation o it. Te tablets on which Proust ocuses in her chapter display such a structure. Te main algorithm she examines is the one used to compute reciprocals o regular
Mathematical proo: a research programme numbers.51 Once the computation o a reciprocal has been recorded on the tablet, the same algorithm is applied to the result and shows that one thereby returns to the starting point o the srcinal algorithm. In act, more accurately, this structure is typical o only one type o tablet devoted to the algorithm computing reciprocals, precisely those tablets that contain only numbers. Tese tablets record successive numbers produced through the ow o computations according to a determined and highly speci c layout until the result is yielded, and then record numbers obtained through applying the algorithm to the result. However, as Proust emphasizes, another type o text also reers to the same algorithm. In these other tablets, the algorithm is expressed in words and the text prescribes the operations to be carried out in succession. Among all the tablets containing either ormulation (the two never occur on the same tablet), Proust chooses to concentrate on two tablets (ablet A and ablet B), one or each kind o expression. In act, she selects the two texts that repeat the same pattern in a signi cant number o sections. Te verbal expression o the algorithm had been essential or Sachs to interpret the purely numerical expression or it. However, once he had established that the two tablets relate to the same algorithm, a key question remained, which Proust addresses: why do we have two expressions o the same algorithm? What are the speci c meanings conveyed by each o them? And, especially in her case, what does the numerical tablet say? o answer these questions, Proust combines several methods. She restores the practices o computation to which both tablets adhere, bringing to light that they relate to the ow o computations in different ways. She also compares the tablets to other parallel specimens. Lastly, she examines every detail o the numerical tablet (ablet A): the layout, the numbers chosen, the way o conducting the algorithm in the direct and the reverse computations. Trough sophisticated reasoning, Proust can establish that the second part o each section – the one containing the computations in the reverse direction – did most probablynot play the part o checking the results o the direct algorithm. She urther demonstrates that the layout designed to record the numbers, as evidenced in ablet A, was created or such kind o texts and introduces a way o managing the space o the tablet that wasarti cial. Tis conclusion leads her to suggest that the spatial elements o the layout, like columns, are precisely those which convey the meanings expressed by ablet A. We see here at its closest how the composition o a kind o text relates to the work carried out with a text. In her view, the columns may be 51
For greater technical detail, I reer the reader to Proust’ s chapter.
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interpreted as related to the statement o rules which ground the correctness o the algorithm. Proust thereby accounts or the meaning behind the numerical display as ound in ablet A, suggesting that it made sense or its readers in a way comparable to how an algebraic ormula makes sense or us today. For her, the numerical text enjoys a kind o transparency in regard to the algorithm treated, making the operation o the procedure explicit. Te reader could thereby see why and how the algorithm worked. Tis is how Proust argues that the numerical text bears witness to an interest in the correctness o the algorithm or computing reciprocals.Note that Proust’s argument is in agreement with what Høyrup has shown. Although they operate in different ways, they both highlight that a speci c kind o inscription has been designed to note down an algorithm while pointing out the reasons or its correctness. In some sense, Proust’s thesis with respect to these tablets concurs with Netz’s conclusions on theArithmetics. In her view, ablet A bears witness to the development o an arti cial kind o text designed to make the algorithm surveyable. Yet, in both cases, different aspects o the working o the computations are made surveyable. Note urther that, once more, the act that actors constructed a speci c kind o text to make speci c statements with respect to algorithms means that historians have to design sophisticated methods to argue how such texts should be interpreted and what they mean. Here an interest in the correctness o the procedure can only be perceived through lengthy consideration o the text itsel. Let us pause here or a while and consider what we have accomplished in this subsection so ar. We have entered the world o proving the correctness o algorithms. As was stressed in Section o this Introduction, this was precisely a part missing rom the standard account o the history o proo in the ancient world. By enlarging the set o sources and the issues about proos considered, we began to see the emergence o a new continent. But there is more. We saw above that an operation – take multiplication, or example – computes two things: a number and a meaning. A multiplication can produce the value which is claimed to be the product o two numbers. Or it can be interpreted as computing the area o a rectangle. On this basis, we see that Proust analyses texts addressing the ormer eature o the operation, whereas Høyrup considers texts that deal with the latter eature. In what ollows, we shall proceed in the development o this segment o the history o proo, showing how various groups o actors have established the correctness o algorithms. Proust’s nal point about ablet A relates to its speci c structure, namely, the display o an application o the algorithm ollowed by the display o its
Mathematical proo: a research programme
application to the result. On that count, her conclusion is that the overall structure o the text makes a statement regarding the act that the algorithm or computing reciprocals is its own reverse algorithm. Similar tablets can be ound or square-root extractions, displaying that squaring and squareroot extraction are in the same way the reverse o one another. A similar interest in algorithms that are the reverse o one another – where one algorithm cancels the effect o the other – emerges as central to a type o proo to which Chinese early mathematical sources bear witness.52 It is to this type o proo that my own chapter is devoted.
Algebraic proofs in an algorithmic context Like some o the Babylonian tablets analysed above, the earliest Chinese writings attesting to mathematical activity stricto sensu are composed o problems and algorithms solving them. Te practice o proo to which they bear witness also aims at establishing the correctness o algorithms. Among these writings, those that were handed down through the written tradition are o a type quite different rom that o the Babylonian tablets just examined.53 Te most important one or our purpose, Te Nine Chapters on Mathematical Procedures (Jiuzhang suanshu), was probably completed in the rst century and considered a ‘classic’ soon thereafer. In correlation with this adoption, commentaries on it were composed, some o which were elt to be so essential to the reading o Te Nine Chapters that they were handed down with it. Tese are the commentary composed by Liu Hui and completed in 263 as well as the one written under the supervision o Li Chuneng and presented to the throne in 656. wo key acts regarding the commentaries prove essential or us in relation to mathematical proo. First o all, the commentaries attest to how ancient readers approached the classic as such. Tis highlights why, as historians, when we interpret Te Nine Chapters, we are in quite a different situation rom that conronting historians who deal with sources or which no ancient commentary
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Chemla 1997–8. In addition to the source material handed down through the written tradition, we now have recourse to writings that archaeologists excavated rom tombs. Te most important o them, the Book o Mathematical Procedures(Suanshushu), ound in a tomb sealed in c. 186 , is useul or, but not central to, our purpose. Such sources can be compared to the Babylonian tablets with respect to the way in which they were ound and the conditions in which we can interpret them. However, it is not yet clear within which milieus and how they were used.
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exists.54 As alluded to above, Te Nine Chapters, like Babylonian tablets, describes some o the algorithms in such a way that they are transparent in regard to the reasons accounting or the correctness o the computations they prescribe. At this stage in the reasoning, ‘transparency’ is an observer’s category. However, it is crucial that, with respect to this Chinese document, the commentators did read the text o the algorithm as transparent and made precisely these reasons explicit in their exegesis. ‘ransparency’ can thus also be shown to correspond to an actor’s category. It is in this context that the commentators bring to light exactly the same type o ‘meaning’ that Høyrup suggests reading in the transparent algorithms ound in Babylonian tablets. In the Chinese case, we can thus demonstrate that this is the way in which the earliest observable readers actually did ‘interpret’ the texts. Such evidence supports the hypothesis that the practitioners o mathematics in ancient China designed a kind o text to ormulate algorithms, similar to that shaped in Mesopotamia to express algorithms transparent about the reasons o their correctness. Te proo expressed in this way was read as such by ancient readers.55 From the point o view o the reception, afer all, the historical continuity between Babylonian and Arabic sources also indicates that Babylonian proos were read in this way by subsequent practitioners. On the other hand, rom the point o view o the text itsel, it is remarkable that in different contexts, the mode o expression chosen or indicating the reasons o the correctness was the same. In my view, this remark indirectly reinorces Høyrup’s argument, in that it shows the useulness o this property o the statements or practitioners. Te important point here is that or the Chinese commentators, in my interpretation, such a reading was a way o making the ‘meaning’ o the classic explicit. It is in order to designate that ‘meaning’ that they used the concept o yi, which I introduced above.56 Tis brings us back to the question, or which we now have plenty o evidence, o how the commentators made use o the context o a problem, or the geometrical analysis o a body, to ormulate the ‘meaning’ they read 54
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Using ancient commentaries to interpret an ancient text does not mean that we attribute anything ound in the commentaries to the text commented upon without caution. Chemla 1997–8 constitutes an example o how the two kinds o sources are treated separately and only thereafer articulated with each other. Te commentators read the expression o the reasons or the correctness in various elements o the classic. Te structure o the text is one o them; compare Chemla 1991. Te terms used in Te Nine Chaptersto prescribe an operation is another one – s ee, or instance, Chemla 1997–8. Chemla 2010 attempts to give a systematic treatment o this question and to highlight elements o a history o these kinds o text. On the act that commentators assumed that the classic indicated the ‘meaning’ or ‘reasoning’, see Chemla 2003, Chemla 2008a, Chemla 2008b.
Mathematical proo: a research programme
in the classic. Further analysis reveals that, beyond the similarity which we suggested above with the Babylonian case as interpreted by Høyrup, the understanding made explicit in the Chinese commentaries was not only provided by ‘geometrical’ interpretations, but could also be achieved, more generally, by recourse to the situation described in the statement o any kind o problem.57 In this sense, the way o generating a semantic analysis o operations differed. A landscape o similarities and differences starts emerging in our world history o mathematical proo in ancient traditions. Secondly, the act that the commentators made explicit the reasons underlying the correctness o the algorithms in such cases is one aspect o a much more general phenomenon. In effect, the commentaries systematically established the correctness o the algorithms contained in Te Nine Chapters, thereby bearing witness to a considered practice o proo or such kinds o statements. My own chapter ocuses on one dimension o this practice, which, as ar as I know, appears to be speci c to ancient China. Tis dimension, which reveals another undamental operation used to establish the correctness o procedures, sheds light on why the texts o algorithms are not all transparent about the reasons or their correctness. As I show, in some cases, to establish that an algorithm correctly ullled the task or which it was given, the commentators, on the one hand, established another algorithm ul lling the same task and, on the other hand, carried out operations on the text o this algorithm to transorm it into the proper algorithm, the correctness o which was to be established. Moreover, in such cases the commentary usually made explicit the reasons they adduced or explaining why, although the ormer algorithm was transparent, the classic substituted the latter algorithm or it. My chapter mainly ocuses on the section o such a proo in which the algorithm is reworked by means o transormations carried out on the list o operations directly. My claim is that, within a context in which mathematics was worked out on the basis o algorithms, this section o the proo represents a practice o algebraic proo. By algebraic proo, I mean, in this context, a proo that starts rom a statement o equality, rst established in a given way that is not o interest here and then transorms this srcinal equality as such and in a valid way into other equalities, until the desired equality is obtained. Te rst part o my claim is thus that the commentaries record proos o precisely this kind, with the only difference being that algorithms, and not equalities, are 57
Compare chapter A in CG2004, Chemla 2009.
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transormed. We would then have a orm o algebraic proo in an algorithmic ormat. Te second part o my claim relates to the concern or the validity o such a method o proo. In act, an analysis o the commentaries reveals that the exegetes considered the question o the validity o the operations applied to an algorithm as such. A close inspection shows that the exegetes linked the validity o their operations to the set o numbers introduced in Te Nine Chapters, which includes not only integers but also ractions and quadratic irrationals. Te key point, in their eyes, was that these quantities allowed the expression o the results o divisions and root extraction exactly, thereby allowing the inverse operation o multiplication to cancel the effect o these operations and restore the srcinal number. Tis point recalls the Mesopotamian tablets described by Proust, which demonstrate the same concern. Why was this act important or practitioners o mathematics in Mesopotamia? Further inquiry into that question could prove interesting or our topic. At the same time, I argue that it is when the commentators establish the correctness o algorithms or carrying out the arithmetic with ractions that they address the validity o applying some o the operations to lists o operations. Several points must be stressed here. Firstly, the analysis o this dimension o the practice o proo preserved rom ancient China brings to light an essential point, which allows us to capture a key eature o algebraic proo: the validity o such kinds o proo is essentially linked to the set o numbers with which one operates and how one operates with them. Tis point, I argue, was understood in ancient China, but it is a point o general validity regarding algebraic proo. Secondly, the question arises whether dimensions o algebraic proos as we practise them today may have historically taken shape within practices o proving the correctness o algorithms. Tis brings me back to a point raised at the beginning o this introduction. I insisted on the act that the standard account othe history o mathematical proo had nothing to say about the history o how the correctness o algorithms was established in the past. At this point, I am in a position to summarize our ndings on this question. We now see even more clearly that this was a lacuna which contributed to the marginalization o sources that were ‘non-Western’ and sources that bore witness to practices o proo related to computations. In addition, we also see that this lacuna may also prevent us rom providing a historical account o the emergence o algebraic proo. Last, but not least, i the answer to the previous question proves positive, a new historical question presents itsel quite naturally: one may urther
Mathematical proo: a research programme
wonder whether the algebraic proo in an algorithmic context as demonstrated in ancient China could not have played a part in the actual emergence o the algebraic proo as we practise it today. Tis set o issues demonstrates the ways in which the broadening o our corpus leads to the ormulation o new directions o research in the early history o mathematical proo.
Proving as an element of the interpretation of a classic Te Chinese case just examined is not the only historical instance in which the ormulation o mathematical proos took place within the ramework o commentaries on a classic. Agathe Keller’s chapter is devoted to the earliest known Indian source in which an interest or mathematical proo can be identi ed: it turns out to be the seventh-century commentary by Bhaskara I on the mathematical chapter o the fh-century astronomical treatise Aryabhatiya. As in the Chinese case, Keller shows how the development o arguments to establish the correctness o procedures is part o the activity o an exegete who comments on a classic.58 Te proo is part o Bhaskara’s way o justiying the classic, unless it justi es his own interpretation o the classic. A Sanskrit classic is composed o sutras, the interpretation o which requires skills. It is within this context that, when the classic deals with mathematics, proo – together with grammar – seems to be a means or a commentator to inquire into the meaning o the classic and to advance his interpretation. Despite the act that commenting on a classic provided the impetus or making proos explicit in both Sanskrit and Chinese, the way in which proos relate to the interpretation seems to present differences between the two contexts. In the case discussed by Keller, the classic, i.e. the Aryabhatiya, indicates algorithms. Te commentator Bhaskara states them ully, showing by means o Paninian grammar how the sutras mean the suggested algorithms and then accounting or why the suggested algorithms are correct. Bhaskara maniests his expectation that the classic does not provide explanations. By contrast, he introduces a set o terms (explaining, veriying, proving) that indicate how he understands the epistemological status o parts o his commentary. Keller provides evidence to support an interpretation o what ‘explaining’, ‘veriying’ and ‘proving’ meant or him, in terms o actual intellectual 58
Srinivas 2005 insists more generally on the act that in Indian writings proos occur in commentaries, and in Appendix A he provides a list o these commentaries.
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acts. Moreover, she delineates the techniques used by the commentator to account or an algorithm. Here, similarities with the Chinese sources appear. One o the key techniques Bhaskara uses is to highlight how a given procedure is in act supported by a undamental, general procedure, in the terms o which the srcinal procedure can be rewritten. Such a technique also appears in Chinese commentaries, where a technical term ‘meaning (yi )’ is used exclusively to reer to the kind o meaning o the procedure that a proo brings to light in this way.59 Tis similarity between the two contexts possibly derives rom the act that the activity o interpreting a classic inspired similar conceptions o the ‘higher meaning’ o an algorithm. Showing that different procedures can in act be explained in the terms o the same undamental, general procedure is one way in which proos highlight relationships between algorithms which at rst glance might appear unrelated. In such cases, something circulates among the proos, and thanks to the proos, in a way that can be compared to the techniques brought to light by Saito in the core corpus o Greek geometrical texts. Tis circulation again requires a reading o the proo in and o itsel, and not merely as a means to prove the correctness o a procedure. Moreover, what circulates between the proos differs depending on the context. In Sanskrit ′
and Chinese sources, a procedure circulates, that is, a statement o the same kind as the proposition to be proved. In other terms, the technique o proo is at the same time a new statement. Again, this echoes present-day mathematicians’ claim that proos are a source o knowledge or them. However, the procedure in question is not ordinary, since the mere act that it can be put to such uses indicates that it is more undamental and more general than others. One may hypothesize that the identi cation o procedures o this kind ormed one o the goals that motivated the interest in proving in these contexts.60 In this case, the historian would miss one o the epistemological expectations with respect to proving, were he to analyse it only rom the viewpoint o its ability to establish the statement to be proved. It is also interesting that, in the context o Bhaskara’s commentary as well as in the Chinese commentaries, gures were introduced or types o 59 60
On this ‘meaning’ yi , see the glossary I compiled, CG2004: 1022–3. For Chinese sources, there is evidence supporting the claim. Compare Chemla 1992, Chemla 1997b. We reach a conclusion that was already an outcome o Lakatos’ analysis o the activity o proving in Lakatos 1970. Tis convergence is not surprising: we share with Lakatos’ enterprise a starting point, that is, that there is more to proo than mere deduction. However, the nature o the statements produced in the contexts Lakatos studied and those we studied differs, showing that one could go deeper in the analysis o how proos yield mathematical knowledge (concepts, statements and techniques). ′
Mathematical proo: a research programme
‘explanation’ which are reerenced with speci c terms. Once the ‘explanation’ is given in the orm o such a diagram, it comes to a close. Is it that the argument is lef or the reader to develop or is it that it was developed orally? It is difficult to tell. However, we recognize a eature o proos that was requently mentioned in nineteenth-century accounts o ‘Indian’ mathematical reasoning but was subject to divergent assessments, as Charette’s chapter shows. Seen rom another angle, we may note that the written ormulation o a proo carried out in relation to a diagram took quite different orms in history. Further development o a comparative analysis o such texts arises as a possible venue or uture research. On the other hand, the commentator used the term ‘explanation’ (pratipadita) to reer speci cally to another component that he introduced: problems solved by means o the algorithm described. In which ways did the problems contribute to providing an explanation o the algorithms? Here too, the source material calls or a comparative analysis o the part allotted by different traditions to problems or establishing an algorithm. Te evidence discussed so ar illustrates the variety o contexts that may have prompted an interest in writing down proos. Te sources analysed by Keller and mysel show how commenting upon a canonical text has been an activity by which proos were made explicit. In addition Høyrup suggests the hypothesis that teaching could have motivated an interest in ormulating proos. In act, the two explanations are not mutually exclusive, i we embrace Volkov’s hypothesis that Chinese commentaries were composed within the context o mathematical education. We come back to this hypothesis below. In addition, the evidence discussed so ar also shows the variety o motivations that led to the ormulation o proos in the ancient traditions. What they contribute to our historical approach and understanding o mathematical proo is an issue to be taken up in the conclusion. Beore we can address our conclusions, however, one more dimension o our world history is worth considering.
Te persistence of traditions of proof in Asia One may be tempted to believe that it is relevant to adopt the perspective o a world history to deal with mathematical proo in ancient traditions, but that afer the seventeenth century, the story to be told is that o the ‘Western’ practices and their adoption worldwide. Te naltwo chapters o the book illustrate two ways in which such a view must be quali ed. Tey constitute the only incursions o this book into later traditions o proo. Te
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main reason or including them in the book is that they reveal historically interesting modes o continuity with what was analysed above. Alexei Volkov devotes his chapter to an apparently discrete topic: mathematical examinations in China and the sphere o Chinese in uence in East Asia. However, the link to our questions appears immediately. Te issue at hand or him is that o the relation between the practices o examination in mathematics and mathematical proo as evidenced by the commentaries on Chinese classics. Tis question leads him to ocus on the extant evidence regarding the teaching o mathematics in this part o the world. Among all the channels through which mathematical knowledge was taught throughout Chinese history, the channel o state institutions is the least poorly documented. Relying on the extant Chinese administrative sources, Volkov describes the textbooks used or mathematics in the state educational system rom the seventh century onwards and the way in which they were used. It is important or us that among these textbooks, one nds precisely Te Nine Chapters on Mathematical Proceduresalong with the commentaries by Liu Hui and Li Chuneng introduced above.61 Moreover, Volkov discusses in great detail how the terse description o the kind o examinations the students had to take by the administrative sources can be interpreted concretely. Te interpretation o the extant administrative sources would have remained a matter o speculation, had not Volkov discovered a piece o evidence in nineteenth-century Vietnamese sources. Some elements o context are needed to understand this point better. As in Japan and Korea, Vietnamese state institutions had a history closely linked to that o their parallel institutions in China. In particular, rom the ang dynasty onwards, Chinese state institutions or teaching were imitated in East Asia and the textbooks used by these institutions were transmitted in this process. Moreover, state examinations in mathematics were held in all other contexts, including in Vietnam, as Volkov shows. Tis explains how Vietnamese sources clariy practices carried out in China: the margins ofen keep alive traditions that are modi ed in the centre. In Vietnam, an additional actor played a decisive role: at the beginning o the nineteenth century, Western books had not yet become in uential there. Te extant mathematical writings composed in Vietnam until that time consequently appear to belong mainly to a tradition on which Chinese 61
Please note that Volkov opts or another interpretation o the title o the Chinese book, translating it in a different way. Appendix 2 in his chapter presents various transcriptions and translations or the title o Chinese mathematical texts.
Mathematical proo: a research programme
books exerted a strong in uence. It is in such a Vietnamese source that Volkov ound a model or mathematical examination which he translates and analyses in his chapter. Tis piece o evidence leads him to put orward the hypothesis that the shape taken in China by the mathematical classics and the seventhcentury commentaries may re ect precisely the requirements o the teaching institution. On this basis, one can shed light on the connection between these texts and the examination system rom another angle. It is quite striking, indeed, that the administrative texts analysed by Volkov describe the tasks to be carried out by students in the seventh-century Chinese state institutions with technical terms that can be ound inter alia in Te Nine Chapters and the commentaries on the mathematical classics that were mentioned above. Tis holds true, as Volkov stresses, or words like wen ‘problem’, orda ‘answer’, which reer to components o texts like Te Nine Chapters. However, most importantly, this also holds true or terms like yi’ ‘meaning’, which is the second type o meaning given above or a procedure, a meaning that is intimately connected to the activity o proving. Such a link between the two types o sources supports Volkov’s thesis that commentaries played a key part in the training o students, since terms like yi are not to be ound in the texts o the mathematical ′
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classics themselves, but only in commentaries. In conclusion, Volkov suggests a social context or the interest in the proos o the correctness o algorithms in ancient China. wo points are worth emphasizing or our main argument here. Firstly, let me stress again what was said above: i Volkov’s hypothesis holds true, we would have at least two cases – East Asia and Babylon – in which the proessional context o teaching was instrumental or composing proos, even though the proos actually written down differed in the two contexts. Secondly, it is worth noting that this piece o evidence con rms the longevity o practices o proo in East Asia. Tis is but one example which shows 62
One may even go astep urther. We mentioned above two commentaries onTe Nine Chapters: the one completed by Liu Hui in 263, and the one presented to the throne by Li Chuneng in 656. In act, several scholars have produced clues which indicate that the text o the two commentaries may have been commingled during the process o transmission (in CG2004: 472–3, I have summarized the current contributions treating this difficult issue which awaits urther research). For the question discussed here, it may be relevant to note that many clues suggest that the concept o yi’, when used in relation to procedures, may belong to the layer o commentary rom the seventh century. I this is con rmed, the connection between the administrative sources and the seventh-century commentary would be even more striking. Te correlation between the terms used in both types o documents should invite us, in my view, to take the occurrence o the terms in the commentaries on the classics into account when interpreting the administrative prescriptions.
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that the late history o practices o proo bears witness to circulations and preservations that challenge the standard account sketched above.63 At the beginning o his chapter, Volkov recalls how the sinologist Edouard Biot, in his 1847 Essai sur l’histoire de l’instruction publique en Chine, dismissively belittled the ormat o problems in Chinese mathematical texts, their absence o proo and the elementary level o state education. With respect to what was discussed above, the additional denigration o everything classed as educational in the historiography o mathematics may be partly responsible or the lack o discernment regarding sources that could have modi ed Biot’s assessment at least to a certain extent. In China, the approach to mathematics o the past was strikingly different, i we judge it on the basis o Li Rui’sDetailed Outline o Mathematical Procedures or the Right-Angled rianglecompleted in 1806, which ian Miao analyses in her chapter. Tis text illustrates a second orm o prolonged relevance o ancient practices o proo, which reveals several interesting eatures. o be more precise, Li Rui’s practice o proo exempli es a revival o past Chinese practices o proo and shows how they were at that time transormed mathematically while simultaneously reshaped under the in uence o – or rather as an alternative to – practices o proo identi ed as ‘Western’. Te topic on which Li Rui chose to write his book, the right-angled triangle, was one in which, as he knew, an interest was documented in both Chinese and Greek antiquity. Te ninth o Te Nine Chapters is devoted to the right-angled triangle and it is the subject o theorems in the part o Clavius’ edition o Euclid’sElements that Ricci and Xu Guangqi translated into Chinese in 1607. Li Rui approached the right-angled triangle as was done in the tradition which descends rom Te Nine Chapters. Among the various identi able traces o this approach, one notes that his book takes the orm o problems or which solutions are provided in the orm o algorithms. In addition, Li Rui makes use o the traditional terminology developed throughout Chinese history and completed in the Song dynasty to designate the quantities attached to a triangle. On the other hand, ian argues, the in uence o the Elements can be perceived in the act that Li Rui provided a systematic set o solutions to all the problems that can be encountered. Moreover, he organized this set according to the dependencies o its elements. In the system produced, the 63
A similar kind o continuity in the practice o proo is described by François Patte, in his work on sixteenth-century Sanskrit commentaries; see Patte 2004.
Mathematical proo: a research programme
solution to any problem depended only on those beore it. Te proos o the correctness o the algorithms were thus a key element or deciding over the structure o the system. ian highlights several mathematical innovations in the book. o begin with, Li Rui invoked combinatorial methods to state and solve any problem that could be asked about a right-angled triangle. Moreover, Li Rui innovatively employed the ancient ‘heavenly unknown (tianyuan)’64 method to establish the correctness o the algorithms which solve each o the problems in the most uniorm way possible. Te earliest surviving evidence or this method, which is equivalent to the modern practice o using polynomial algebra to set up an equation which solves a problem, dates to 1248, the year that Li Ye completed his Sea-Mirror o the Circle Measurements (Ceyuan haijing). Afer having been orgotten in China, the method had been recovered by Mei Juecheng in the rst hal o the eighteenth century, thanks to the understanding Mei gained through his acquaintance with European books o algebra.65 In particular, Mei deciphered the meaning o the algebraic symbolisms or writing down polynomials and equations that had been developed in China a ew centuries earlier and had since been lost. Li Rui could thus rely on the method and its related symbolisms that had been rescued rom oblivion only a ew decades beore he wrote his book. When using the symbolism to establish algebraically the correctness o the algorithms he stated, Li Rui was using symbols that differed in orm rom those o Diophantus, but which had played a similar part in the past. Like Li Ye, Li Rui used these symbolic notations to account or the correctness o the equation – the ‘procedure’ – yielded to solve a general problem. However, the way in which Li Rui was now using them modi ed the status o the proos carried out with them. Te main point that ian highlights in this respect is that, when considering given quantities attached to a triangle as data, Li Rui discriminated among the different categories o triangles according to the relative size o the data in them. More precisely, in contrast to Li Ye beore him, Li Rui ormulated as many problems as there were distinguishable cases so that he could prove the correctness o the general equation in a way that would be valid or each case and that would establish 64
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Te literal interpretation o the expression tianyuan is ‘celestial srcin’. Tis interpretation permits the identi cation o occurrences o the concept beore the thirteenth century in the set o mathematical classics gathered in the seventh century; see above. I shall come back to this point in a uture publication. On this episode, compare Needham and W ang Ling 1959: 53, Horng Wann-sheng 1993: 175–6, Yabuuti Kiyosi 2000: 141–3.
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the equation with ull generality. Tis distinction between cases relates to a concern about the validity o the operations in the proo and the generality o the proven statement. Li Rui distinguished cases in such a way that the proo carried out through polynomial computations would be valid or all triangles o the same case. Tis step ensured the correctness o the kind o algebraic proo which he conducted in a way that Li Ye’s proos beore him did not. As a consequence, Li Rui established general equations through polynomial computations – the proo o their correctness – that were valid or the particular category o triangles delineated and he probably developed this structure o proo intentionally. Otherwise, there would be no reason or him to differentiate different cases o a given type. Yet, even though this eature reveals that Li Rui was interested in the generality o procedures, like the ancient Chinese mathematical texts, he expressed this property or each case within the context o a particular problem, which he thus used as a paradigm. We see here again that the search or generality and the ways in which generality is expressed both account or speci c eatures o the practice o proo that was constructed. Several other elements maniest how, through his mathematical practice, Li Rui simultaneously presented himsel as continuing the tradition o his Chinese predecessors o the past and yet changed it. His deployment o geometrical diagrams to provide yet another (geometric) proo o the correctness o the equation is one o these elements. However, although the diagrams clearly call to mind Li Ye’s own diagrams in hisYigu yanduan, completed in 1259, or Yang Hui’s 1261 commentary onTe Nine Chapters, they betray differences, due not least o all to an in uence o Western practices with geometrical gures. Further, like Xu Guangqi beore him, Li Rui seems to be using the concept o ‘meaning (yi )’ in a way that displays affinity with how the commentators on Te Nine Chapters used the same term. Tis reveals a continuity o mathematical theory that has not yet been addressed adequately. In addition, ian surmises that Li Rui was also interested in showing the power o the ‘procedure o the right-angled triangle (gougushu)’ – the ′
ancient name and ormulation or Pythagoras’ theorem – to solve any problem in a uniorm way. Li’s book can be interpreted as having explicitly developed the system covered by this older procedure, even i it had been presented in the past in relation to a particular problem. In conclusion, the Detailed Outline o Mathematical Procedures or the Right-Angled riangle demonstrates a synthesis o goals or and techniques o proo, which take their srcins rom both East and West. Te book
Mathematical proo: a research programme
composes a new type o text with which to carry out proos, one that integrates different agendas. Most importantly, however, i we ollow ian’s interpretation o it, we can read Li Rui’s discourse and practice as illustrating the politics o the proo, in that they attempt to embody the ideal o proving in the ‘Chinese way’, andnot in the ‘Western’ way. Some decades later, the politics o the historiography o mathematical proo would become by ar more visible.
IV Conclusion: a research programme on mathematical proofs It is time to gather the various threads that we have ollowed and conclude, by considering our ndings with respect to ancient mathematics and the research programme that they open or us. Let us begin with acts. What we have seen emerging in Section is the outline o a history o proving the correctness o algorithms in the ancient world. Mesopotamian, Chinese and Indian sources bear witness to the act that practitioners have attended to the correctness o the algorithms with which they have practised mathematics. An analysis o their attempts helps us identiy some o the undamental operations involved in such proos. We have seen that these practitioners have striven to establish how an algorithm correctly yields the desired magnitude and the value that can be attached to it. o do so, they have designed devices or dispositis that have allowed them to ormulate the ‘meaning’ o operations. Te proos they constructed share common eatures. Tey also demonstrate speci cities in the way in which proo was practised. Among the speci cities noted in the way o approaching the correctness o algorithms, one act proved o special relevance or a history o proving. Chinese sources demonstrate the act that operations – meta-operations, i one wishes – were sometimes applied to the sequence o operations that an algorithm constitutes. Tese meta-operations were used to transorm an algorithm known to be true, qua algorithm, into another algorithm, the correctness o which was to be established. Moreover, these sources bear witness to the act that a connection was established between the validity o these meta-operations and the numbers with which one worked. I suggested the conclusion that we have here a kind o ‘algebraic proo within an algorithmic context’. Tis remark leads to several questions. What kind o understanding can practices developed speci cally to prove the correctness o algorithms yield
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into the nature o algebraic proos, on the one hand, and the process o their emergence, on the other? I a historical link can be established between the two, what evidence can we nd regarding the historical process by which both kinds o proo were connected? Tis question opens onto another one, much more general: through what concrete historical processes did algebraic proo take shape and develop? Te analyses developed in this introduction have brought to light several elements inherent to that kind o proo as we experience it: textual techniques, re ections on numbers and problems o generality. What other elements constitute algebraic proo and how did this cluster crystallize? What type o historicity is attached to it? Tis book offers a contribution to this agenda by identiying elements essential to algebraic proo and hypothesizing a historical scenario regarding the kinds o practice in which these elements took shape. Clearly, much more remains to be done. Tese rst results show the bene ts that broadening o the scope o sources taken into consideration can produce through the change o perspective we advocated in the approach to proos and their history. A scarcely considered branch o the history o proo thus emerges: namely, the history o proving the correctness o algorithms. And as it takes shape, it elucidates parts o the history o proo that still await better understanding. In correlation with opening new pages in the history o proo, we have been naturally led to approach the topic o proo more comprehensively. From this global perspective, we understand more clearly the link between the devaluation o computation as a mathematical activity, which was and still is quite widespread, and the exclusive ocus on only some proos, written in ancient Greece, that has dominated the history o mathematics. Now, what changes will this outline o the history o proving the correctness o algorithms bring in the history o proo? How ar will these tools o analysis allow historians to examine anew other proos, or instance proos written in Greek? Tese remain open questions. Our exploration o ancient practices o proo has met with another important issue, which is worth pondering urther. As suggested by Lloyd, Høyrup, Keller and Volkov among others, the interest in proo and, more speci cally, in writing proos down has been stimulated by distinct activities and social contexts. Among those activities and contexts we have seen, let us mention the rivalry between competing schools o thought or the development and promotion o one tradition as opposed to another, teaching mathematics or interpreting a classic, all activities that need not have been exclusive o each other. Te list is by no means exhaustive. Still, this remark brings to the ore two points that are important much more
Mathematical proo: a research programme
generally. On the one hand, proving is an activity that takes place in speci c social and proessional groups which have speci c agendas. On the other hand, as we saw, the practices o proo betray a variety o modalities which one can attempt to correlate to the social groups which sustain them. Tis leaves us with two tasks: nding the means to describe the practices in their variety and identiying the social and proessional contexts that are relevant to account or their ormation and relative stability. Such a research programme is quite meaningul to inquire into the history o proo in the ancient world. Indeed, only along these lines can we hope to bring to light and accommodate the variety o practices in a way more satisactory than the old model o competing civilizations which has been pre-eminent rom the nineteenth century onwards. However, the research programme is laden with difficulties. Te evidence available with respect to ancient time periods is in general so scanty that rigorously reconstructing the social environment in which proos were actually composed is an ideal or the most part out o reach. One can only put orward hypotheses. In that context, concentrating on the description o the varying practices appears to be an initial means o overcoming the difficulties and perhaps discerning rom mathematical sources different social groups that carried out the practice o proo. Tis is the project on which we ocus in the book and what our explorations into matters o proo open to re ections o wider relevance. Te conclusions which we propose bring orth some suggestions or the task o describing practices o proo whose value appears to me to exceed the scope o the ancient world to which we have restricted ourselves. Let me comment on some o these suggestions by way o conclusion. Among the various sets o sources which they treat, the chapters in this book identiy different goals ascribed to proo, different values attached to proving and different qualities required rom a proo. In this Introduction, I have outlined some o them. We have seen that some proos seem to be conducted in order to understand the statement proved or the text which states it. In other cases, proos have appeared to have had as one o their goals the identi cation o undamental operations or the display o a technique. We have also seen that in some contexts, proos were expected to be general or to comply with an ideal o generality. In others, they should bring clarity, yield ruitulness or maniest simplicity. Much more remains to be done in identiying goals and values practitioners have attached – and still attach today – to proo and the constraints they imposed on themselves. What is important is that in each o these cases the identi cation o these elements, ar rom being the end o the inquiry, constitutes only its
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beginning. Indeed, the main question then raised is to identiy how the way in which the proo is conducted or written down helps practitioners to reach the goals, achieve the values or implement the qualities they value. Tis is where the issue o the practices o proo is inextricably linked to the issue o the expectations actors have with respect to proos. In relation to this issue, I introduced the notion o devices or dispositis that actors have created in various contexts to carry out key operations with respect to the proo. We have seen that the dispositis constructed by Mesopotamian scribes or Chinese scholars to make explicit the ‘meaning’ o operations in an algorithm had commonalities as well as speci c differences. Te differences between the two Greek texts dealing with polygonal numbers that Mueller described can also be approached in these terms: the dispositis used by the authors to treat their topic show two distinct attempts at achieving generality. Seen in this light, axiomatic–deductive systems appear to be a dispositi designed to yield certainty. Describing these dispositis appears to me as a method to attend more closely to differences between the various practices o proo, thereby breaking down what is all too ofen presented collectively as ‘the mathematical practice’. Can we spot transormations in the modalities o proo that demonstrate a change o values or a combination o a larger set o values? Which o these goals, o these values, o these qualities were held together? Which combinations can we identiy and how have these various constraints been held together? Which o them seem to have been in tension with each other, because they were difficult to ul l simultaneously? Archimedes’ practices o proo offer a case study that can be approached rom this perspective. All the questions that arise in this context now explain, I hope, how an overly strict ocus on the value o certainty would yield an essentially truncated account o mathematical proo. Clearly, such an approach does not do justice to the variety o agendas that were ascribed to proo and to the variety o practices that were developed accordingly. When describing the diverse practices o proo exhibited in ancient sources, the various chapters o the book collectively bring to the ore another act that is, in my view, both important and o general relevance. Tey converge on the conclusion that various types o technical texts have been designed or the conduct o proos, depending on the context in which these proos have been written down and the constraints bearing on them. Let me gather various hints that support this conclusion. Te texts o proos we have mentioned consist o distinct basic components. Among them, one can list equalities, proportions or lists o
Mathematical proo: a research programme
operations. Moreover, within the context o distinct practices o proo, these basic components appear to have been composed in various ways and to have been combined in distinct kinds o technical texts. Among the kinds o texts and inscriptions we encountered, let me recall a ew: texts or algorithms transparent with respect to the reasons o their correctness; the material dispositis by means o which their meaning was made explicit; symbolic inscriptions o different sorts (including those o Diophantus, those which Colebrooke rst described in the Sanskrit texts, and those o the Chinese past revivi ed by Li Rui); and texts composed with ormulaic languages. In addition, it regularly appeared that paradigms in the orm o particular gures or mathematicalproblems were used to ormulate general proos. Tis variety o texts developed or proos merely re ects the variety o contexts within which proos were carried out. Tis means that the design o texts is, in an important sense, an indicator o the context in which they were composed. Moreover, the shaping o kinds o texts to carry out proos is an aspect o the practice o proo as such which has been little studied so ar. Tis shaping demands study, even i only as a limited component o the practice o proo. However, there is another equally undamental reason to study this range o phenomena. Te examples just summarized remind us o the act that the interpretation o the text o a proo is a thorny issue, and it is so in relation to the effort involved in the construction o a kind o text adequate or the execution o proos o a certain type. In other words, it is because each human collective which carried out mathematical proos deliberately designed texts or this activity that these texts cannot be interpreted straightorwardly.66 Tis claim can be illustrated easily with the example o the recently mentioned transparent algorithms. In order to read a proo in the statement o the algorithm itsel, the historian has to establish the way in which the texts made sense. Te interpretation o paradigms as paradigms would constitute another example. Tese remarks explain why the relation between the type o text used and the kind o proo developed is an essential topic or uture research. It is essential not only because the shaping o texts to carry out proos is an aspect o the practice o proo in itsel, but also because inquiring into this issue yields better tools to interpret the texts in question. 66
In this respect, we return to the conclusions that emerged rom the collective effort published in Chemla 2004.
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Such are some o the general issues that emerged rom our historical analysis o ancient practices o proos. As such, they appear to me to provide useul directions o research i we are to develop more generally a genuinely historical approach to the activity o proving and understand the motley practices o mathematical proos as such. What results can these issues yield or the study o modern proos? Let this task constitute our uture endeavour.
Bibliography Atiyah, M., Borel, A., Chaitin, G. J., et al. (1994) ‘Responses to “Teoretical mathematics: toward a cultural synthesis o mathematics and theoretical physics”, by A. Jaffe and F. Quinn ’, Bulletin o the American Mathematical Society 30: 178–207. Chemla, K. (1991) ‘Teoretical aspects o the Chinese algorithmic tradition ( rst to third century)’, Historia Scientiarum42: 75–98 (+errata in the ollowing issue). (1992) ‘Résonances entre démonstration et procédure: remarques sur le commentaire de Liu Hui ( e siècle) aux Neu chapitres sur les procédures mathématiques (1er siècle)’, in Regards Obliques sur l’Argumentation en Chine , ed. K. Chemla, Extrême-Orient, Extrême-Occident 14: 91–129. (1996) ‘Que signi e l’expression “mathématiques européennes” vue de Chine?’, in L’Europe mathématique: Histoires, mythes, identités (Mathematical Europe: History, Myth, Identity), ed. C. Goldstein, J. Gray and J. Ritter. Paris: 219–45. (1997a) ‘Re ections on the worldwide history o the rule o alse double position, or: how a loop was closed’, Centaurus, an International Journal o the History o Mathematics, Science and echnology39: 97–120. (1997b) ‘What is at stake in mathematical proos rom third-century China?’, Science in Context 10: 227–51. (1997–8) ‘Fractions and irrationals between algorithm and proo in ancient China’, Studies in History o Medicine and Science , N.S. 15: 31–54. (1999) ‘Commentaires, éditions et autres textes seconds: quel enjeu pour l’histoire des mathématiques? Ré exions inspirées par la note de Reviel Netz’, 5
: 127–48. Revue d’histoire destextuelles mathématiques (2003) ‘Les catégories de “Classique” et de “Commentaire” dans leur mise en oeuvre mathématique en Chine ancienne’, in Figures du texte scienti que, ed. J.-M. Berthelot. Paris: 55–79. (ed.) (2004) History o Science, History o ext. Dordrecht. (2008a) ‘Antiquity in the shape o a canon: views on antiquity rom the outlook o mathematics’, in Perceptions o Antiquity in Chinese Civilization , ed. D. Kuhn and H. Stahl. Heidelberg: 191–208.
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(2008b) Classic and Commentary: An Outlook Based on Mathematical Sources . Preprint MPING, vol. 344. Berlin. (2009) ‘On mathematical problems as historically determined artiacts: re ections inspired by sources rom ancient China’, Historia Mathematica 36: 213–46. (2010) ‘Proo in the wording: two modalities rom ancient Chinese algorithms’, in Explanation and Proo in Mathematics: Philosophical and Educational Perspectives, ed. G. Hanna, H. N. Jahnke and H. Pulte. Dordrecht: 253–85. Dauben, J., and Scriba, C. J. (eds.) (2002) Writing the History o Mathematics: Its Historical Development. Basel. Engelriet, P. (1993) ‘Te Chinese Euclid and its European context’, in L’Europe e en Chine: Interactions scienti ques, religieuses et culturelles aux e et siècles, ed. C. Jami and H. Delahaye. Paris: 111–35. (1998) Euclid in China: Te Genesis o the First ranslation o Euclid’s Elements in 1607 and Its Reception up to 1723. Leiden. Gardies, J.-L. (1984) Pascal entre Eudoxe et Cantor. Paris. Grabiner, J. (1988) ‘Te centrality o mathematics in the history o Western thought’, Mathematics Magazine 61: 220–30. Hacking, I. (1980) ‘Proo and eternal truths: Descartes and Leibniz’, in Descartes: Philosophy, Mathematics and Physics, ed. S. Gaukroger. Brighton: 169–80. (2000) ‘What mathematics has done to some and only some philosophers’, in Mathematics and Necessity: Essays in the History o Philosophy, ed. . Smiley. Oxord: 83–138. (2002) Historical Ontology. Cambridge, Mass. Harari, O. (2003) ‘Existence and constructions in Euclid’s Elements’, Archive or History o Exact Sciences 57: 1–23. Hilbert, D. (1900) ‘Mathematische Probleme’, Nachrichten von der Königlichen Gesellschaf der Wissenschafen zu Göttingen, Mathematisch–Physikalische Klasse : 253–97. (1902) ‘Mathematical Problems’, Bulletin o the American Mathematical Society 8: 437–79. Horng Wann-sheng (1993) ‘Chinese mathematics at the turn o the nineteenth century: Jiao Xun, Wang Lai and Li Rui’, in Philosophy and Conceptual History o Science in aiwan, ed. C. H. Lin and D. Fu. Dordrecht: 167–208. Høyrup, J. (1986) ‘Al-Khwarizmi, Ibn urk, and the “Liber mensurationum”: on the srcins o Islamic algebra ’, Erdem 2: 445–84. (1990) ‘Algebra and naive geometry: an investigation o some basic aspects o Old Babylonian mathematical thought’, Altorientalische Forschungen 17: 27–69, 262–324. (2006) ‘Arti cial languages in Ancient Mesopotamia: a dubious and a less dubious case’, Journal o Indian Philosophy 34: 57–88.
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Jaffe, A., and Quinn, F. (1993) ‘“Teoretical mathematics”: toward a cultural synthesis o mathematics and theoretical physics’, Bulletin o the American Mathematical Society 29: 1–13. (1994) ‘Responses to comments on “Teoretical mathematics”’, Bulletin o the American Mathematical Society 30: 208–11. Jain, P. K. (1995) ‘A critical edition, English translation and commentary o the upodghata, savidhaprakarana and kuttakadhikara o the Suryaprakasa o Suryadasa (A commentary on Bhaskaracarya’s Bijaganita)’, PhD thesis, Simon Fraser University, Canada. Jami, C., Engelriet, P., and Blue, G. (eds.) (2001) Statecraf and Intellectual Renewal in Late Ming China: Te Cross-Cultural Synthesis o Xu Guangqi (1562–1633). Leiden. Keller, A. (2006) Expounding the Mathematical Seed: A ranslation o Bhaskara I on the Mathematical Chapter o the Aryabhatiya, 2 vols. Basel. Knorr, W. R. (1996) ‘Te wrong text o Euclid: on Heiberg’s text and its alternatives’, Centaurus 38: 208–76. (2001) ‘On Heiberg’s Euclid’, Science in Context 14: 133–43. Krob, D. (1991) ‘Algèbres de Lie partiellement commutatives libres’, in Séminaire de Mathématique, Rouen, 1989/90. Rouen: 81–90. Lakatos, I. (1970) Conjectures and Reutations. Cambridge. Langins, J. (1989) ‘Histoire de la vie et des ureurs de François Peyrard, bibliothécaire de l’Ecole polytechnique de 1795 à 1804 et traducteur renommé d’Euclide et d’Archimède’, Bulletin de la SABIX3: 2–12. Laudan, L. (1968) ‘Teories o scienti c method rom Plato to Mach: a bibliographical review’, History o Science 7: 1–63. Lloyd, G. E. R. (1990) Demystiying Mentalities. Cambridge. (1992) ‘Te Agora perspective’, in Regards obliques sur l’argumentation en Chine, ed. K. Chemla, Extrême-Orient, Extrême-Occident 14: 185–98. (1996a) Adversaries and Authorities: Investigations into Ancient Greek and Chinese Science. Cambridge. (1996b) ‘Te theories and practices o demonstration’, in Aristotelian Explorations, ed. G. E. R. Lloyd. Cambridge: 7–37. MacKenzie, D. A. (2001) Mechanizing Proo: Computing, Risk, and rust. Cambridge, Mass. Mancosu, P. (1996) Philosophy o Mathematics and Mathematical Practice in the . Oxord. SeventeenthI.Century Martija-Ochoa, (2001–2) ‘Edouard et Jean-Baptiste Biot: L’astronomie e siècle’, Diplôme d’études approondies, chinoise en France au University Paris Diderot, Département d’histoire et de philosophie des sciences. Martzloff, J.-C. (1981) ‘La géométrie euclidienne selon Mei Wending’, Historia Scientiarum 21: 27–42.
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(1984) ‘Sciences et techniques dans l’œuvre de Ricci’, Recherches de science religieuse 72: 37–49. (1993) ‘Eléments de ré exion sur les réactions chinoises à la géométrie e siècle: Le Jihe lunyue de Du Zhigeng euclidienne à la n du vu principalement à partir de la préace de l’auteur et deux notices bibliographiques rédigées par des lettrés illustres’, Historia Mathematica 20: 160–79. (1995) ‘Clavius traduit en chinois’, in Les jésuites à la Renaissance: Système
éducati et production du savoir, ed. L. Giard. Paris: 309–22. Needham, J., and Wang Ling (1959) ‘Section 19: Mathematics’, in Science and Civilisation in China, vol. , ed. J. Needham. Cambridge: 1–168. Patte, F. (2004) BHĀSKARĀCĀRYA, Le Siddhāntaśiroman . i, I–II : L’oeuvre mathématique et astronomique de Bhāskarācārya, 2 vols. Geneva. Rashed, R. (2007) Al-Khwarizmi: Le commencement de l’algèbre. Paris. Rav, Y. (1999) ‘Why do we prove theorems?’, Philosophia Mathematica7: 5–41. Rota, G.-C. (1990) ‘Les ambiguïtés de la pensée mathématique’, Gazette des mathématiciens 45: 54–64. Saito, K. (2006) ‘A preliminary study in the critical assessment o diagrams in Greek mathematical works’, SCIAMVS 7: 81–144. Saito, K., and assora, R. (1998) Restoration o the ool-Box o Greek Mathematics: Summary o Research Project, Grant-in-Aid or Scienti c Researches (C2) in 1995–98. Osaka. Schuster, J. A., and Yeo, R. R. (eds.) (1986) Te Politics and Rhetoric o Scienti c Method: Historical Studies. Dordrecht. Srinivas, M. D. (2005) ‘Proos in Indian mathematics’, in Contributions to the History o Indian Mathematics, ed. G. G. Emch, R. Sridharan and M. D. Srinivas. New Delhi: 209–48. Turston, W. P. 1( 994) ‘On proo and progress in mathematics’, Bulletin o the American Mathematical Society 30: 161–77. Wagner, D. B. (1975) ‘Proo in ancient Chinese mathematics: Liu Hui on the volumes o rectilinear solids’, Candidatus magisterii thesis, University o Copenhagen. (1978) ‘Liu Hui and su Keng-Chih on the volume o a sphere ’, Chinese Science 3: 59–79. (1979) ‘An early Chinese derivation o the volume o a pyramid: Liu Hui, 3rd 6
century ’, Historia : 164–88. Mathematica Whewell, W. (1837) History o the Inductive Sciences rom the Earliest to the Present imes, vol. 1. Cambridge. Widmaier, R. (2006) Gottried Wilhelm Leibniz: Der Briewechsel mit den Jesuiten in China (1689–1714). Hamburg. Wu Wenjun (ed.) (1982) Jiuzhang suanshu yu Liu Hui (Te Nine Chapters on Mathematical Procedures and Liu Hui). Beijing.
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Yabuuti Kiyosi (K. Baba and C. Jami, trans.) (2000) Une Histoire des mathématiques chinoises. Paris. Yeo, R. (1981) ‘Scienti c method and the image o science: 1831–1890’, in Te Parliament o Science, ed. R. MacLeod and P. Collins. London: 65–88. (1993) De ning Science: William Whewell, Natural Knowledge, and Public Debate in Early Victorian Britain. Cambridge.
1
Te Euclidean ideal o proo in Te Elements and philological uncertainties o Heiberg’s edition o the text ,
Introduction One o the last literary successors o Euclid, Nicolas Bourbaki, wrote at the beginning o his Éléments d’histoire des mathématiques: L’srcinalité essentielle des Grecs consiste précisément en un effort conscient pour ranger les démonstrations mathématiques en une succession telle que le passage d’un chaînon au suivant ne laisse aucune place au doute et contraigne l’assentiment universel … Mais, dès les premiers textes détaillés qui nous soient connus (et qui datent du milieu du e siècle), le « canon » idéal d’un texte mathématique est bien xé. Il trouvera sa réalisation la plus achevée chez les grands classiques, Euclide, Archimède, Apollonius; la notion de démonstration, chez ces auteurs, ne diffère en rien de la nôtre.1
I am unsure what was intended by the last possessive, whether it acts as the royal or editorialwe designating the ‘author’, or i it ought to be understood in a more general way: ‘la nôtre’ could mean that o the Modernists, o the twentieth-century mathematicians, o the French, or ormalists. All jokes aside, the affirmation supposes a well-de ned and universally accepted conception o what constitutes a mathematical proo. Te aorementioned conception, the citation or which is ound in a chapter titled ‘Fondements des mathématiques, Logique, Téorie des ensembles’, is at once logical, psychological (through a rejection o doubt), and ‘sociological’ (based on universal consensus). Perhaps this assertion ought to be considered nothing more than a distant echo o the Aristotelian affirmation that all scienti c assertions (not just mathematical statements) are necessary and universal. Te ollowing list o Greek geometers is also interesting. It contains the classics, and the triumvirate was probably intended to ollow chronological order. Here, then, Euclid is not simply a convenient label, sometimes used to designate one or several o the many adaptations o Euclid’s amous work, as when one speaks about the Euclid o Campanus c(. 1260–70), the Arab 1
Bourbaki 1974: 10.
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Euclid or the Euclid o the sixteenth century. Rather, thisEuclid indicates the third-century Hellenistic geometer and author o theElements. o speak about the Hellenistic Euclid, to describe the contents o his composition with precision – which certainly implies the act that it quali es as a ‘classic’ – and to adopt or reject its approach towards proo presumes a reasonably certain knowledge o the text o theElements. Precisely this knowledge, however, is in doubt. o examine these assumptions, in the rst part I revisit some inormation (or hypotheses) concerning the transmission o ancient Greek texts, particularly the text o the Elements. I emphasize there the indirect character o our knowledge about this subject, and I review the history o the text proposed by the Danish philologist J. L. Heiberg, at the time when he produced, in the 1880s, the critical edition o the Greek text to which the majority o modern studies on Euclid still reer.2 I raise some uncertainties and mention the recent criticism o W. Knorr.3 In the second part, I give examples o differences between preserved versions o the text, illustrating the uncertainties which dismantle our knowledge about the Euclidean text, notably the texts o certain proos.
Re ections on the History of the ext of the Elements A brief history of the ancient Greek texts
Lest the present study become too complicated,4 let us admit that there existed in thirteen books a Hellenistic edition (ἔκδοσις) o the Elements (τὰ Στοικεῖα), corresponding, at least in rough outline, to that which has come down to us and produced by Euclid or one o his closest students.5 In 2
3 4
5
Heiberg and Menge, 1883–1916. It has been partially re-edited and (seemingly) revised by E. S. Stamatis: Heiberg and Stamatis, 1969–77. In the ollowing, I will designate these editions by the EHM and EHS respectively. Knorr 1996. Te literature on this subject is immense. I have consulted Pasquali 1952, Dain 1975, Reynolds and 1988,(aDorandi 2000 about papyri) and Wilson Irigoin 2003 collection o (which articles contains publishedextensive betweeninormation 1954 and 2001, plus several unpublished studies). At least two other possibilities are conceivable, by analogy with some known cases o ancient scholarly editions: • Euclid had produced two versions o his text: the rst, a provisional copy , or a restricted circle o students, correspondents or riends; the other, revised and authorized. Tis corresponds with the composition o the Conics o Apollonius, as described by the author himsel in the introduction o Book (o hisrevised version). Consequently, this hypothesis
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Greek antiquity, when there existed neither printing press nor any orm o copyright, edition signi ed ‘the introduction o a text into circulation among a circle o readers larger than the school, riends and students o the author’ – in other words, a ‘publication’ in the minimal sense o having been ‘rendered public’ and o having been reproduced rom a manuscript revised and corrected by the author (or a collaborator).6 Te books o the Hellenistic era (third to rst century beore our era) were written in majuscule and, in theory, on only one side o papyrus scrolls o a modest and relatively standardized size. Tus, they were rather limited in contents. 7 In the case o the Elements, this tradition implies a likely division into feen rolls, each containing one book, with the exception o the lengthy Book .8 O course, like practically any other text rom Greek antiquity, the ‘srcinal’ (which was not necessarily an autograph copy)9 has not come down to us. Te rather limited liespan o such papyrus scrolls required that they be periodically recopied, with each copy capable o introducing new aults and, even more importantly, alterations. Certainly chance played a role in the preservation o particular papyri, but, in the long run, because o the ragility o the writing material, a text could come to us only i certain communities ound enough interest in it to reproduce it requently. In the course o these recopyings, two particularly important technical operations occurred in the history o the ancient Greek book: • the change rom papyrus scrolls (volumina) rst to papyrus codices but later to parchment codices, and • the Byzantine transliteration. allows the possibility o variations by the author rom the beginning o the textual tradition. Nonetheless, there is no evidence o this process or the Elements. • Euclid had not gone to the trouble o producing an ἔκδοσις in the technical sense o the term. His writings had been circulated in his ‘school’ (in a orm that we evidently do not know), and the edition was made some time later, such as at the beginning o the Roman era in the circle o Heron o Alexandria. Tis scenario is traced in the history o the body o ‘scholarly’ works o Aristotle, officially edited only afer the rst centur y beore our era, by Andronicos o Rhodes, among others.
6 7 8 9
In order to be able to dismiss such a (completely speculative) hypothesis, ully detailed testaments about the role o the Elements in the course o the three centuries beore our era must be in evidence, and this is not the case. On the contrary, we are nearly certain that Heron had made an important contribution to the Elements – in particular rom a textual point o view – but the epoch in which he lived (traditionally, afer the work o Neugebauer, the second hal o the rst century is named) is not ree rom dispute. Tis second hypothesis has been suggested to me by A. Jones. I thank him or it. Te most amous case is that o the edition o the works o Plotinus byorphyry. P See Reynolds and Wilson 1988: 2–3. Dorandi 1986. See Dorandi 2000: 51–75.
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Te rst operation, apparently begun in Rome at the beginning o our era, is nothing more than the adoption o the book with pages, written on both sides and with contents de nitely more important than the volumen. Tis shif allows the composition o textual collections and the development o marginal commentaries which previously appeared in a separate scroll. Writings that were not converted into this ormat had a relatively small chance o being transmitted down to us. Te texts known only through papyrus scrolls are small in number and requently nothing more than ragments. In other words, in the case o the Elements, the creation o (at least one) archetypal codex must be postulated. We know nothing o when this abrication occurred or who (whether a mathematician or an institution similar to a library with a centre or copying) undertook this labour. However, the adoption o the codex was a rather slow operation which spanned rom the rst through to the ourth centuries o our era, and beyond. Te act that this adoption was applied in wholesale to the texts rom previous eras probably ought to be attributed to the revival o the study o classical texts under the Antonines (second century). 10 Te other operation, the Byzantine transliteration, was more limited than the change rom scrolls to codices. It was done in the Byzantine empire rom the end o the eighth century. Te Byzantine transliteration consisted o using a orm o cursive minuscule or the edition o texts in place o the majuscule writing termed uncial. Previously, cursive minuscule had been limited to the drafing o administrative documents, but uncial had proven too large and thus ‘costly’ or use with parchment. Here, too, the success and systematization o the process were certainly linked with a renewed interest in ancient texts during the course o the ‘Byzantine Renaissance’, which began in the 850s and was associated with individuals like Leo the Wise (or the Philosopher), the patriarch Photius and Arethas o Cesarea. Such transliteration was a rather delicate technical operation composed o two phases – the rst (and the largest) o which ell in the ninth and tenth centuries, the second in the years 1150–1300. 11 Here, again, translation acted as a lter. Non-transliterated texts progressively ceased to be read. Save or some ortunate circumstances, they disappeared. For the ancient writings which survived these two transormations, we may, i we are reasonably optimistic, emphasize on the one hand the act that on occasions in these two situations, the editors intervened in important ways, and the specimens were produced according to particularly 10
11
On the change rom scroll to codex, see the accessible summary by Reynolds and Wilson 1988: 23–6. C. also Blanchard 1989. C. Irigoin 2003: 6–7.
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‘authorized’ manners which played a decisive role in the transmission. Tese two circumstances produced the archetypal codex (or codices) o the Roman era and the transliterated example or examples in minuscule beginning in the ninth century. On the other hand, on these occasions there was the risk and opportunity that the substance or presentation o these texts would be radically modi ed. Te oldest preserved complete examples o the Elements in thirteen books were produced immediately afer the transliteration into minuscule which has just been called into question. Tey are: • one manuscript rom the Vatican Library,Vaticanus gr. 190, assigned to the years 830–50 according to palaeographic and codicological considerations;12 • one manuscript rom the Bodleian Library at Oxord, D’Orville 301, which, other than its exceptional state o conservation, has the advantage o having been explicitly dated, since its copying, ordered rom the cleric Stephanos by Arethas, who was then deacon, was completed in September 888. wo remarks are in order: (1) Tese pieces o evidence are rom more than a thousand years afer the hypothetical srcinal o Euclid. (2) Te case o the Elements is, however, one o the most avourable (or, perhaps, least unavourable?) in the collection o proane Greek texts. Other than these two precious copies, about eighty manuscripts containing the text (either complete or in part) are known; o these roughly thirty predate the feenth century. Likewise, a palimpsest, dated to the end o the seventh or the beginning o the eighth century and written in uncial, contains extracts rom Books and .13 It thus seems assured that the study o the Elements had not completely ceased during the so-called Dark Ages o Byzantine history (650–850). Also known are several papyrus ragments,14 the oldest o which are ascribed to the rst century and the most 12
13 14
C. Irigoin 2003: 215 (srcinal publication, 1962). C. Follieri 1977, particularly 144; Mogenet and ihon 1985, 23–4 (Vatican r. 190 = ms probably rom the rst hal o theninth century) and 80–1. At the time o Heiberg, this copy was assigned to the tenth century, and the manuscript in the Bodleian was considered the oldest. One sometimes still nds this debatable assertion. See Heiberg 1885. C. EHS: I: 187–9 and Fowler 1987: 204–14 and Plates 1–3.
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recent to the third century. In contrast to the manuscripts, the papyri have the privileged position o being documents rom Antiquity. An author represented among the papyri is likely to have been used in teaching. In the mathematical realm, the bulk o papyri preserved or us represent two categories: (1) very elementary school documents, and (2) astronomical texts. It is thereore signi cant that Euclid is the only geometer o the ‘scholarly’ tradition who appears in this type o text. Direct and indirect traditions
Nicolas Bourbaki probably did not consult the manuscripts o the Elements to determine his opinion about the subject o the Euclidean ideal o proo, and it is the same or the majority o Euclid’s modern readers. Generally, they rely on a translation, or i they know ancient Greek, on a critical edition produced by a modern philologist. In the case o the Greek texts o the Elements, the critical edition was produced by J. L. Heiberg. I he reads the work in Greek, the reader labours under the illusion that he has read what Euclid has written. In this respect, the philological terminology and its label ‘direct tradition’ can be misleading. Te ‘direct’ tradition designates the set o Greek manuscripts and papyri which contain the text either in its totality or in part. Despite this label, we must not orget the considerable number o intermediaries that came between us and the author, even in the direct tradition. Tese intermediaries include not only the copyists, who we would like to believe did nothing more than passively reproduce the text, but also, more importantly, those who took an active part in the transmission o the text – in particular ancient and medieval re-editors and, last o all, the philologists who, beginning with the collection o the available inormation, have constructed the critical edition that we read today. I have thus reported, too brie y, the several elements o the history o the preceding ancient Greek texts to make the point that our knowledge about the text o the Elements, like that o the majority o other ancient Greek texts, is essentially indirect. Classical philology is not without resources. It has developed methods to ‘reverse’ the course o time. Tese methods make it possible to trace the relationships between manuscripts, to detect the mistakes o the copyists, and in the ‘good’ cases to reconstitute an ancestor o the tradition, ofen immediately beore the transliteration, sometimes an ancient prototype rom late antiquity or rom the Roman era. In the case o a Hellenistic author, this result is still rather removed rom the ‘original’ and thus necessitates appeals to other sources. Tese sources constitute the so-called indirect tradition
Te Elements
Indirect Tradition
Direct Tradition
y r o t n e v n I l a u t x e T
Greek Manuscripts, Papyri
and uncertainties in Heiberg’s edition
Translations Ancient Medieval Latin, Arabic, Syriac (?) Persian, Latin, Hebrew, Syriac,
Quotes by Greek Authors (Non-Mathematical Authors, Commentators on Euclid. Other Geometers)
Citations by Authors in a Language other than Greek
Armenian ?
Establishment of Text
Reconstruction of the History of the Text
Work of the Editor of the Greek Text
Figure 1.1 extual history: the philological approach.
(see Figure 1.1). Generally, it is used to decide between variant manuscripts or as con rmation in the testing o conjectures about the state o the text beore the production o the oldest preserved manuscripts. In brie, the work o the editor comprises two dimensions: (1) the establishment o the text, and (2) the reconstruction o what philologists call the ‘textual history’, that is to ollow the avatars o the manuscripts, but also the commentaries and translations through which we have access to the text, to review the evidence about the use o the work in education, in controversies, or its presence in libraries. Although the one dimension is certainly articulated with respect to the other, it is nonetheless convenient to distinguish between them. For the reconstruction o the textual history, all inormation ought to be taken into account. Because the collected sources will probably be contradictory (variants among manuscripts, incompatible quotations, etc.), it is necessary to classiy the inormation and search or plausible explanations (accidents in copying, editorial action by a re-editor, in uence o a commentary through marginal notations, decisions o the translator, in uence o pedagogical, philosophical or mathematical context) in order to provide an account o the development o the manuscript. Since the history o the text serves to justiy the choices made in its establishment (see the owchart, in Figure 1.1 above), it must be understood how the two aspects o the philological work are articulated.
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In the case o the Elements, the group o sources which constitutes the indirect tradition is rich. First o all, in the case o citations by ancient authors, the Elements received commentaries on several occasions (namely, by Heron 15 Te o Alexandria, Pappus o Alexandria, Proclus o Lycia, Simplicius (?)). Elements were also used abundantly by the authors o late antiquity. Some extracts o several o these commentaries are ound in the thousands o marginal annotations contained in the manuscripts o the text. Moreover, tracing the indirect tradition o the translations, quotations and commentaries in languages other than Greek is practically unmanageable, even when the task is limited to ancient and medieval periods. Consequently, it is impossible to imagine an exhaustive textual history undertaken by a single individual. Te rst task or whoever wants to edit the text will be to limit the pertinent inormation, in a way that is not only selective enough to be operational, but also wide-ranging enough that no essential elements are lef behind. In the matter o editing a Greek text, in Greek, it is reasonable that the philologists privilege the direct tradition o manuscripts and papyri or the establishment o the text. Tey also emphasize the obvious limits o the different elements o the indirect tradition. Whether the quotations are in Greek or not, philologists note that the citations were sometimes made rom memory. As or the translations, they introduce into the process o transmission not only the passage rom one language to another in which the linguistic structures may be somewhat different, but also the preliminary operation o the comprehension o the text, which is not necessarily implied or a proessional copy. Indeed, there is even something about which to be happy when the Greek text no longer exists. Hellenists are generally grateul to the Latin, Syriac, Arabic, Persian, Armenian and Hebrew translators or having preserved whole elds o ancient literature. In the case o mathematics, the medieval Arabic translations have had great importance or our knowledge o Apollonius, Diocles, Heron, Menelaus, Ptolemy and Diophantus, to mention only the best-known cases. Tese examples suggest not only that the savants o the Arab world had assiduously sought out Greek manuscripts – indeed, they have borne requent witness to this subject – but also that they had some skill in nding them in ormerly Hellenized areas. Te decline o Greek as a scienti c language and the ascendancy o Syriac and then Arabic made translation necessary. Te possibility is thus oreseen that, in so doing, these translations had preserved an earlier state o the text than that transmitted by the 15
Te rst and last are accessible indirectly, thanks to the Persian commenta tor an-Nayrîzî. Heron is also cited several times by Proclus.
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manuscripts elaborated in the Byzantine world. Consequently, important decisions must be made about instances in which the medieval translations show important textual divergences rom the version o the same work preserved in Greek. As we will see, it is exactly this situation which occurs in the case o the Elements o Euclid. In the case o such divergences, at least two explanations may be imagined: (1) Te medieval translators took great liberties with the text, and they did not hesitate to adapt it to their own ends. (2) Teir versions were based on Greek models appreciably different rom those which we know. Tus, we can imagine that these models were (i) more authentic, or, (ii) on the contrary, more corrupt, than our manuscripts. In either case, it will be necessary to make an account o the history o the text, to establish the innovative inormality or rigorous delity othe translators, to account or the methods and the context o the transmission. It is clear that, within the ramework o hypotheses 1 or 2(ii), translations will not be taken into account in the establishment o the text. But i we prove that the translators scrupulously respected their models (non 1), which were less corrupted (2(i)) – let us remain realistic, though – what then? Te textual inventory in the case of the Elements
In order to produce his critical edition (1883–8), Heiberg had (partially) collated about twenty manuscripts. He continued this task or feen years afer the publication o the aorementioned edition, extending the scope to nearly thirty other manuscripts. He compared his edition with papyrus ragments, as they were discovered.16 In order to establish his text, he used seven o the eight manuscripts rom beore the thirteenth century. He systematically explored the indirect tradition o quotations by Greek authors and the tradition o ragments o ancient Latin translation. As or the medieval versions, they were not particularly well known. Heiberg used several previous works and, as ar as the phase o Arabic translations o the ninth century was concerned, he accepted the description published by M. Klamroth in 1881,17 at which time he inventoried the materials useul 16 17
See Heiberg 1885 and Heiberg 1903. At the time when he edited the chapter devoted to the medieval Arabic history o the text o the Elements in Heiberg 1882, he seems not to know Klamroth 1881, which he later criticized in his 1884 article.
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or the establishment o his edition. A debate – but not to say a polemic18 – between the two scholars ollowed on the subject o the obligation o recognizing the value o the indirect tradition rom the medieval era. At any rate, Heiberg knew that there had been at least two Arabic translations, that o al-Hajjâj (produced beore 805 and modi ed by the author or the Kali al-Ma’mun between 813 and 833), then that o Ishâq ibn Hunayn (†910–11) revised by Tâbit ibn Qurra (†901). Klamroth believed himsel to have the al-Hajjâj version or Books – and – and that o Ishâq or Books – . Te Hebrew and Arabo-Latin translations likewise began to be studied. Heiberg also knew (especially) about several recensions (alsely) attributed to Nâsir ad-Dîn at-ûsî (1201–73) and that o Campanus (†1296).19 From the comparison o Greek manuscripts produced by Heiberg and rom the statement that Klamroth had urnished concerning the Arabic Euclid emerges an assessment o the situation which I will describe roughly in the ollowing way: • For the ‘direct’ Greek tradition, it is necessary to distinguish two versions o the text in the collection o the thirteen Books o the Elements, and even three or .36– .17. A simple structural comparison o the manuscripts is sufficient to establish this point. Te two divergent versions o the complete text20 are represented on the one hand by the manuscript Vaticanus gr. 190 ( P) – the oldest complete manuscript – and, on the other, by the strongly connected BFVpqS manuscripts,21 as well as the Bologna manuscript (denoted as b),22 or the whole o the text, save the section .36– .17. In these twenty-one Propositions, the Bologna manuscript presents a structure completely different rom that o P and BFVpqS , which on the whole are less divergent rom each other than they are with respect to b. • For the indirect tradition o the Arabic translations, the report o Klamroth was that there was a considerable difference between the Greek and Arabic traditions. Tis difference went beyond the scope o the 18
19
20 21
22
I allow mysel to recall the rst part o Ro mmevaux, Djebbar and Vitrac 2001: 227–33 and 235–44, in which I analyse the arguments o the two parties. For a synthesized presentation o the Arabic, Arabo-Latin and Arabo-Hebrew traditions as they are known today, see Brentjes 2001a: 39–51 and De Young 2004: 313–23. Tis is what I have termed ‘dichotomy3’ (see Appendix, able 3). Codex Bodleianus, D’Orville, 301 (B), Codex Florentinus, Bibl. Laurentienne, , 3 ( F), Codex Vindobonensis, philos. Gr. 103 ( V); Codex Parisinus gr. 2466 (p); Codex Parisinus gr. 2344 (q); Codex Scolariensis gr. 221, F, , 5 (S). Te sigla used here are the same as those used by Heiberg. Codex Bononiensis, Bibl. communale, n°. 18–19.
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unavoidable variations between manuscripts. Klamroth urther declared that the Arabic tradition was characterized by a particular ‘thinness’ and several structural alterations in presentation (speci cally, in modi cation o order, division or regrouping).23 Te history of the text of the Elements in antiquity
Let us consider now the history o the text o the Elements. Starting with these inventories, let us examine the interpretation o the different pieces o evidence which our two scholars proposed. Te interpretation o Klamroth is simple: the ‘thinness’ o the Arabic (and Arabo-Latin) tradition is an indication o its greater purity. Te textual destiny o the Elements has been the ampli cation o its contents, particularly or pedagogical reasons. Te medieval evidence about the translators’ methods and the context in which they worked shows that the medieval translators had a real concern about the completeness o translated texts. Te gaps (with respect to the Greek text) cannot be ascribed to negligence on the part o these translators. Te additions are interpolations in the Greek tradition. Consequently, or Klamroth, it is necessary to take the indirect tradition into account, not only or the history o the text, but also in the establishment o the text.24 Te history o the text proposed by Heiberg is completely different. Tis history is clearly dependent on the way in which the transmission o the Elements was conceptualized by Hellenists since the Renaissance, particularly since the Latin translation produced by Zamberti, taken directly rom the Greek and published at Venice in 1505.25 Te presentation o this last work raised two essential questions: (1) For Zamberti, the ‘return’ to the Greek text was a remedy or the abuses to which the text had been subjected in medieval editions. Te ocus o his concern was the then highly renowned Latin recension o Campanus. Tis edition had just been printed at Venice in 1482 and was itsel composed rom an Arabo-Latin translation. A debate arose about the (linguistic and mathematical) competence o the translators and the quality o the models which would establish or quite some time the idea that the indirect medieval tradition could be discarded. (2) Zamberti presented his Elements as i the de nitions and the statements o the propositions were due to Euclid, while the proos were 23
24 25
He thus identi ed a well-established line o demarcation between the direct tradition and the indirect tradition. I have named this distinction ‘dichotomy 1’ (see Appendix, able 1). Generally, this position has been taken up by Knorr in his powerul 1996 study. See Weissenborn 1882.
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attributable to Teon o Alexandria. In act, we have a (single) example o this authorial division. Teon indicates explicitly, in his Commentary to the Almagest, that he had been given an edition o the Elements and that he had modi ed the last Proposition o Book ( .33 Heib.) in order to append an assertion concerning proportionality o sectors and arcs upon which they stand in equal circles. Zamberti’s attribution o proos to Teon was undoubtedly inerred rom the glosses ‘o the edition o Teon (ἐκ τῆς Θέωνος ἐκδοσεως)’ marked on the Greek manuscripts used by him. Consequently, since it was understood that Teon had re-edited the Elements in the second hal o the ourth century o our era, the question arose o what ought to be ascribed to Euclid and what ought to ascribed to the editorial actions o Teon. For someone like R. Simson (1756), the answers were particularly clear. All that was worthy o admiration srcinated with Euclid; all the de ciencies were due to the incompetence o the re-editor. Tus, the debate on the subject was open. When F. Peyrard, around 1808, undertook to check the Greek text or his new French translation o Elements which was based on the Oxord edition o 1703, he discovered among the manuscripts which had been brought back rom Italy by Gaspard Monge (afer the Napoleonic campaigns) a copy belonging to the Vatican Library (Vaticanus gr. 190), which contained neither mention ‘o the edition o Teon’ nor the additional portion at .33 and which diered considerably rom the twenty-two other manuscripts known to him. From this divergence, he deduced that this manuscript, unlike the others, preceded the re-edition o Teon and that it moreover contained the text o Euclid!26 He at once decided to make a new edition o the Greek text. Heiberg accepted (with some reworking) the interpretations o Peyrard, particularly the idea that all the manuscripts with the exception o Vaticanus 27 gr. 190 were derived rom Teon’s edition. He called these the ‘Teonine’. As or the Vatican copy, he was more careul. Heiberg noted that the copyist admits in the margins o Proposition .38 vulgo28 and Proposition .6 to have consulted two editions, one ‘ancient’ and the other ‘new’. Proposition .6 existed in the rst but was missing in the other. Exactly the oppo26 27
28
Peyrard 1814: xiii, xxv. Consequently, in the ollowing, I will use the abbreviation T to designate the aorementioned amily o manuscripts. Several Propositions appearing in the editio princeps(and reproduced in the ollowing editions) were discarded by Heiberg who designated them in this way lest there be some conusion in numbering. .38 vulgo was No. 38 in the preceding editions. It was rejected by Heiberg in the Appendix. His Proposition 38 was thus No. 39 in the previous editions.
Te Elements
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site was the case or Proposition .38 vulgo. Heiberg considered that the manuscript – which he would call P in homage to Peyrard – had been produced beginning with at least two models, one o which was pre-Teonian, and the other was Teonian. His edition was thus ounded on the comparison o P with T and on an examination o the total or partial agreement or disagreement between the two amilies. 29 From there, he claimed he had determined the editorial actions o Teon o Alexandria, and passed severe judgement on the changes. Teon’s re-edition o theElements did not compare avourably with the editions o the great poetical texts produced by the Alexandrian philologists o the second and third centuries beore the modern era.30 I we return to the terms o our previous line o reasoning and i we accept this history o the text, we ought to distinguish two textual archetypal manuscripts: the rst representing the re-edition o Teon and realized in the 370s, and the second corresponding to the pre-Teonine model called P. However, the alterations which Teon is supposed to have effected on the text, as deduced by a comparison with the manuscript P, are so limited that with a ew exceptions (which are listed in the Appendices), Heiberg believed he could combine the two versions in one text with a single apparatus criticus. For the divergent Greek text (b .36– .17), his solution was somewhat different. It seems that the discovery o this manuscript must be attributed to Heiberg in the context o the previously mentioned debate. In an 1884 article, he presented this new Greek evidence, taking the opportunity to respond to the arguments presented by Klamroth. Te reason or his approach was that this ‘dissenting’ Greek text and the Arabic translations are incontestably related in this portion o the text. Precisely this incomplete but incontestable structural agreement in opposition to the tradition in P + T constitutes the principal argument in the article by W. Knorr. However, noting that the text o b, copied in the eleventh century and also Teonine, is particularly de cient in section .36– .17, Heiberg introduced into the history o the text a Byzantine redactor, the author o an abridged version o the Elements, in order to explain the difference. From this abbreviated work was derived b .36– .17 and the models used by the Arabic translators. Te consequences or the edition o the text were clear. Aside rom some speci c reerences to the Latin recension o Campanus, the indirect medieval tradition which had been connected rom 29 30
See EHS: , 1, xxv–xxxvi. lviii. Te comparison is irrelevant: see Rommevaux, Djebbar and Vitrac 2001: 246–7.
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the beginning to a lower-quality model was not taken into account by the Danish editor. Te portion b .36– .17 was relegated to Appendix II o Volume 4 o the edition, together with portions o the text which Heiberg deemed inauthentic. In other words, his decisions (or rather his non-decisions) resulted in a critical edition that can be described as ‘conservative’. In order to clariy the meaning o this term, let us recall that the Greek text had undergone ve editions in recent times: the editio princeps by S. Grynée (Bâle, 1533), the edition by D. Gregory (Oxord, 1703), the edition by F. Peyrard (Paris, 1814–18), that o E. F. August (Berlin, 1826–29) and nally Heiberg’s own edition. I do not intend to examine in detail their respective merits, but two or three acts are clear. Te rst two editions were produced rom manuscripts belonging to the amily later characterized as ‘Teonine’. Despite the many discussions o the sixteenth century, 170 years had passed beore the appearance o a new edition, which Peyrard judged to be no better than the preceding! At any rate, Peyrard’s edition scarcely agrees with his history o the text. Afer he affirmed that the Vatican manuscript contained the text o Euclid, he continued to ollow the text o the editio princeps o 1533 (and thus the Teonine amily o texts) in several passages where the divergences are especially well-marked. Te quest or authenticity was not o primary importance. It was more important to present a mathematically correct Euclid. We may suppose that it is or this reason that Peyrard continued to ollow the Teonine amily which is more correct in the case o .19 and more general in the case o .38, but privileged P which is (apparently) less aulty in the case o .24 and more complete in the case o .6. Peyrard also wanted his edition to be easy to use. Quite bluntly, Peyrard admits to having retained what is now designated as Proposition .13 vulgo lest he introduce a shif in the enumeration o the Propositions o the book with respect to the previous editions – even though this proposition is omitted in P and is clearly an interpolation! More generally, he preserves most o the additional material (various additions, lemmas, alternate proos) which P would have been able to dismiss as inauthentic had it been taken into account. It was not until the edition o Heiberg that the primacy o manuscript P was truly assumed. A large part (but not all!) o the material thereafer considered additional was added to the Appendices inserted at the end o each o the our volumes. Whenever the textual divergence is marked and the result (in T) is identi ed as the product o a voluntary modi cation, the reading o P is retained, even i this destroys the mathematical coherence,
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as in the previously mentioned example o .19.31 Contrary to Peyrard, Heiberg does not admit that Euclid could have provided several proos or the same result, which would constitute what I have called above an ‘authorial variation’. We will return to this important topic later. For now, let us say simply that the criteria o Heiberg are simple. In the case o double proos, he retains as the sole, authentic proo that which occurs rst in P, whether it is better than the other or not. Te limitations o this edition thus result rom the adopted history o the text and the resulting principles o selection, while the merits o the edition derive rom a more coherent observation o these choices than Peyrard managed. Another (and not the least) o its merits is that the text as published corresponds rather well with something which had existed, namely manuscript P o the Vatican,32 whereas the archetypal texts reconstituted by the modern editors o ancient texts are sometimes nothing more than ctions or philological monsters. What it represents with respect to the ancient text is more uncertain. Te incidental remarks o the copyist o P already suggest a certain contamination between (at least) two branches o the tradition. Until the 1970s it was believed that the manuscripts resulting rom the transliteration were aithul copies o ancient models, with the only change being the replacement o one type o writing with another. Nowadays belie in this practice is not so sure, and there are even a number o cases in which it may be rankly doubted.33 We will see an argument (see below, p. 111) which casts doubts on the two oldest witnesses o the Elements (P and B). Let us assume that the copyist o P ollowed what was termed the ‘ancient edition’, and that he compared the ‘ancient edition’ with the ‘new edition’ only afer the copying. (Indeed, there is a good probability that this was the case.) Even so, our aith in the antiquity o the text produced in this way depends entirely on the con dence accorded to the history o the text proposed by Heiberg. In particular, the strength o the argument rests on the validity o the interpretation he proposes or the distinction between P and T in connection with the re-edition by Teon o Alexandria, around 370. Tis history was accepted by . L. Heath and J. Murdoch – who have signi cantly contributed to its diffusion – and thus by the majority o specialists. Disconnectedly and periodically challenged, this history was 31 32
33
See Vitrac 2004: 10–12. In a certain number o passages, and more generally or minor variants, Heiberg preserved the text o the Teonian amily. C. the list that he gives in EHS: , 1, xxxiv–xxxv. See Irigoin 2003: 37–53. Te (very illuminating) example rom the Hippocratic corpus is the object o the article reproduced on pp. 251–69 (srcinal publication 1975).
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thoroughly called into question by W. Knorr in his article o 1996. In particular, our late colleague there affi rms that all the preserved Greek manuscripts depend on the edition o Teon, that the differences between the Vatican manuscript and theT amily are microscopic, and that these differences are not characteristic o a re-edition. Stated differently, i the opinion o Knorr is adopted, the Euclid edited by Heiberg ought to correspond, at best, to the text in circulation at Alexandria in the second hal o the ourth century o our era. Te arguments o Knorr are not all o the same value – ar rom it.34 Te difference between P and T is real. It is not a question only o divergences attributable to errors by the copyist which philologists try to dismiss. Te reader can convince himsel o the extent o differences between P and T by consulting the list which I give in able 3 o the Appendix. However, it should also be emphasized that there is not, in this internal dichotomy in the Greek, any substitution o proos (!), any change in the order o the Propositions, or any Lemma which exists in one o the two versions but not in the other. When there are double proos, the order is always the same as in P and in T. At the present stage o my work, I see only two solutions: (i) to adopt Knorr’s opinion, or (ii) to conclude that the goal o Teon’s re-edition was not a large-scale alteration. About Teon’s motivations, we know next to nothing. He presents us with a single indication relating to the contents (the addition at .33). It is possible, or example, to conceive o the hypothesis that Teon’s re-edition was in act the transcription o the edition(s) written on scrolls into a version in the orm o a codex or codices. I the text o the previous vulgata appeared satisactory to him, the goal would not have been to propose a different mathematical composition, but to revitalize the treatise by adopting a new ormat or the old book. Te second hal o the ourth century represents a relatively late date, but it is known that the pagan circles sometimes resisted innovations which seemed to meet with their rst successes in Christian quarters.35 And, what is known, i not about Teon himsel, then at least about his daughter Hypatia, suggests that he was connected with pagan, neo-Platonic intellectual circles. Moreover, even i this explanation is adopted, nothing guarantees that he was the rst to unold this way, nor that he was the only one. On the other hand, it is certain that this version played an important role in the transmission o the Elements, as is proven by the statements contained in the amily o manuscripts titled T. 34 35
See Rommevaux, Djebbar and Vitrac 2001: 233–5 and 244–50. See van Haelst 1989: 14, 26–35.
Te Elements
and uncertainties in Heiberg’s edition
Te second scenario which might account or the limited but real variation shown between P and T satis es me more than Knorr’s reconstruction. We have only two criteria external to the text by which we can understand the aorementioned re-edition: the glosses ‘o the edition o Teon ( ἐκ τῆς Θέωνος ἐκδοσεως)’ and the presence or absence o the addition at .33. We have so little inormation about the history o the text36 that it is a little too daring to throw out some part o our inormation without external support or the decision. As or the problem discussed here, I do not believe that my hypotheses change anything regarding the state o the texts that the Greek manuscripts enable us to establish. It is probably approximately the text as it circulated around the turn o the third and ourth centuries o our era. Is it possible to advance rom here? With regard to the edition o a minimally coherent Greek text, I am not sure. However, other sources clariying the history o the text are provided to us, thanks to the indirect tradition and, in this arena, our situation is a little more avourable than the time-rame o the Klamroth–Heiberg debate. New contributions to the textual inventory
With regard to the indirect tradition o the quotations by Greek authors, we have two more valuable sources: • Te Persian commentator an-Nayrîzî has transmitted to us a certain number o testimonies about the commentaries o Heron and Simplicius, whose srcinal Greek texts are now lost. Some o them provide interesting inormation about the history o the text.37 Heiberg had taken note o this evidence. He had even taken part in the edition o Codex Leidensis 399 through which the commentary was rst known, although this edition was produced afer Heiberg’s edition o the Elements. He gives an analysis o these new materials, among other things, in an important 1903 article. • In the same vein, he had nothing except a very ragmentary knowledge about the commentary on Book , attributed to Pappus and preserved in an Arabic translation by al-Dimashqî, rom which Woepcke, around
36
37
In this regard, the indirect medieval tradition, so rich in new textual variants, teaches us nothing about the history o the text during antiquity, particularly about the existence or not o several editions o the Elements. In the case o Heron, see Brentjes 1997–8: 71–7; in this article Brentjes suggests that other Arabic authors knew about the commentary by Heron independently o an-Nayrîzî, in particular Ibn al-Haytham. In Brentjes 2000: 44–7, she shows that it is probably true or al-Karâbîsî, also. Heron proposed a number o textual emendations, among other things. See Vitrac 2004: 30–4.
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1855, published only extracts. Tenceorth, the text was edited and translated into multiple languages.38 In the course o the two decades during which Heiberg worked on the tradition o the text o Euclid, new inormation, accessible thanks to the indirect tradition,39 could have led him to alter certain editorial decisions made in the years 1883–6 at the time when he argued with Klamroth. Tese alterations might have stemmed notably rom taking into account manuscript b (in the portion where it diverges) and the indirect medieval tradition. Te works which he published in the years 1888–1903 are indispensable to those who use his critical edition. Regrettably, Heiberg did not produce a second revised edition, as he did or Archimedes, afer the discovery o the so-called Archimedes Palimpsest. 40 Tis text gave access to the previously unavailable Greek texts o On Floating Bodies and Te Method o Mechanical Teorems. o his eyes, the necessity o a revised edition was probably much smaller in the case o the Elements o Euclid, but the resumption o such a work would perhaps have led him to revise his position concerning the indirect medieval tradition. We know this tradition somewhat better than Klamroth or Heiberg, thanks to a more developed textual inventory. At least a score o manuscripts o the version called Ishâq–Tâbit have been identi ed today, 41 whereas Klamroth knew only two! Multiple works on the methods and contexts o medieval translations rom Greek into Syriac or Arabic, or rom Arabic into Latin or Hebrew, either in general or more directed toward mathematical texts, including the Elements, have been undertaken. Busard has published seven Arabo-Latin versions rom the twelfh and thirteenth centuries42 as well as a Greco-Latin version rom the twelfh century discovered by J. Murdoch.43 We even have partial editions o the Books and 38
39
40 41
42
43
See notably Tomson and Junge 1930. It might be argued that this partial knowledge led Heiberg to some debatable conclusions concerning the collection o the ‘Vatican’ scholia (see Vitrac 2003: 288–92) and the pre-Teonine state o the text o Book (see Euclid/Vitrac, 1998: 381–99). Let us add that the integrity o the text attributed to Pappus and the uniqueness o the author (pace Tomson and Junge 1930) are not at all certain (see Euclid/Vitrac, 1998: : 418–19). It ought to include the new inormation contained by the scholia ound in the margins o the Greek manuscripts and we once again know about these sources thanks to the monumental work o Heiberg. See EHS, , 1–2 and Heiberg 1888, to which should be added Heiberg 1903. Regrettably, in his ‘revision’(EHS), Stamatis did not supplement ‘Heiberg with Heiberg’. See Folkerts 1989 (with the corrections o Brentjes 2001: 52, n. 13). Some o these manuscripts contain ragments attributed to the translation by al-Hajjâj. Respectively Busard 1967–1972–1977 (HC), 1983 (Ad. I), 1984 (GC); Busard and Folkerts 1992 (RC); Busard 1996, 2001 (J), 2005 (Campanus). Complete reerences are provided in the bibliography. Busard 1987.
Te Elements
and uncertainties in Heiberg’s edition
– rom the so-called Ishâq–Tâbit version.44 A second manuscript o the commentary by an-Nayrîzî made it possible to complete the evidence rom the (mutilated) Codex Leidensis regarding the principles in Book .45 Several other commentaries (al-Mahânî,46 al-Farâbî,47 Ibn al-Haytham,48 al-Jayyâni,49 ‘Umar al-Khayyâm50) have also been edited, translated and analysed. Te wealth o materials since made available is exceptional. It is obvious that the history o the text o the Elements during the Middle Ages and perhaps even rom the beginning o the Renaissance ought to be entirely rewritten. Tis is clearly not what I propose to do in the remainder o this chapter, as this task surpasses my competence. I will adopt a more limited perspective and ocus on more modest aims. What does this renewed knowledge about the indirect tradition teach us about the history o the text in antiquity, more particularly about the redaction o mathematical proos? What are the limits? In so doing, I attempt to explore the consequences o the hypotheses put orth by Knorr. In his striking 1996 study, knowing that I was in the process o carrying out an annotated French translation (which was then partially published), he suggested that I compare the Greek text established by Heiberg with that o two Arabo-Latin translations, the rst attributed to Adelard o Bath and the second ascribed to Gerard o Cremona, the ormer composed around 1140, and the latter about 1180. Knorr was convinced that these versions transmitted to us a text less altered than the one contained in the Greek manuscripts. He believed that it was possible to reconstitute a Greek archetype rom the group o medieval 44
Engroff 1980; De Young 1981.
45
See Arnzen 2002. See also the new partial edition of the Latin translation by Gerard of Cremona, initially published as vol. ofEHM: ummers 1994. Te preserved Arab and Latin versions of the text of an-Nayrîzî may be described as passably divergent. See Brentjes 2001b: 17–55.
46
Risâla li-al-Mâhânî î al-mushkil min amr al-nisba(Épitre d’al-Mâhânî sur la difficulté relative à la question du rapport). Edition and French translation in Vahabzadeh 1997. Reprinted, with English translation, in Vahabzadeh 2002: 31–52; asîr al-maqâla al-‘âshira min kitâb Uqlîdis (Explication du Dixième Livre de l’ouvrage d’Euclide ). Edition and French translation in Ben Miled 2005: 286–92. Sharh al-mustaglaq min musâdarât al-maqâla al-ûlâ wa-l-hâmisa min Uqlîdis . Te text was translated into Hebrew by Moses ibn ibbon. See Freudenthal 1988: 104–219. Sharh musâdarât Uqlîdis. Partial edition, English translation and commentaries in Sude 1974. Maqâla harh s al-nisba (Commentaire sur le rapport). Facsimile o manuscript Algier 1466/3, os. 74r–82r and English translation in Plooij 1950. Edition and French translation in Vahabzadeh 1997. Risâla î sharh mâ ashkala min musâdarât Kitâb Uqlîdis(Épitre sur les problèmes posés par certaines prémisses problématiques du Livre d’Euclide ). French translation in Djebbar 1997 and 2002: 79–136. Edition o Arabic text with French translation in Rashed and Vahabzadeh 1999: 271–390.
47
48 49
50
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translations. Tis hypothetical archetype represented the state o the text prior to the re-edition o Teon, a re-edition rom which he believed any o the preserved Greek manuscripts stemmed. Te adoption o this point, one suspects, would overturn the entire ancient history o the text and have grave consequences or the establishment o the text, not only at the structural level, but also or the redaction o each proo as is shown in the example o .17 analysed in detail by Knorr. In order to present my results (and my doubts), I must rst give the reader some idea o the size and nature o the collection o textual divergences ound by the comparison o the direct Greek tradition with the indirect medieval tradition.
Extent and nature of the textual divergences between versions of the Elements ypology of deliberate structural alterations
It is obviously not possible either to give an exhaustive list o deliberate alterations which the text o the Elements has undergone or to detail the relatively complex methods o detection and identi cation o speci c divergences. I am not interested in the variants that the philologists use: variant spellings, small additions and/or microlacunae, saut du même au même, and dittographies (that is, reduplications o lines o text). Te errors shared between copies o the same text make it possible to establish the genealogy o manuscripts. Tey constitute textual markers, all the more interesting because they are reproduced by generations o copyists who did not notice them because they could not understand the text or did not try to understand it. I have tried to determine the variants which are connected with the deliberate modi cations made by those responsible or the re-edition o the Greek text or the possible revisers o the Arabic translations, such as Tâbit ibn Qurra, not those related to the ‘mechanical’ errors directly associated with the process o copying. Tis concern goes particularly or the global modi cations o proos.51 When such variations existed among the Greek manuscripts, they had a good chance o surviving the process o translation. Even the structure o the text o the Elements, composed
51
For the local variants o the Greek text, another phenomenon must be taken into account: the multiple uses o the margins o manuscripts afer the adoption o the codex. See Euclid/Vitrac 2001: 44–5.
Te Elements
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o rather easily identi able textual units, acilitates this work. In the same way, the ormulaic character o Greek geometrical language has been maintained in the translations and permits the identi cation o local variants which would probably be more diffi cult in a philosophical or medical text. My sample size is sufficiently large to propose a typology, although the qualitative considerations are provisional and clearly depend on the given range o the analysed corpus.52 In the absence o critical editions o the Arabic versions and in accounting or the multitude o recensions, epitomes and annotated versions inspired by Euclid’s work, we cannot pretend to determine with any degree o certainty the extent o the corpus to be taken into consideration. For the present purposes, I use the various components o the direct tradition, the so-called Greco-Latin version53 and the available inormation concerning the Arabic translation attributed to Ishâq ibn Hunayn and revised by Tâbit ibn Qurra, as well as the ragments attributed to al-Hajjâj in the manuscripts o the Ishâq–Tâbit version, the Arabo-Latin translations attributed respectively to Adelard o Bath and Gerard o Cremona. Tis group corresponds to what the specialists o the Arabic Euclid call the ‘primary transmission’, in order to distinguish it rom the secondary elaborations (recensions, epitomes, …).54 I currently work with a list o about 220 structural alterations o which the principal genres and species appear in Figure 1.2. Tey relate to wellde ned textual units: De nition, Postulate, Common Notion, Proposition, Case, Lemma, Porism, even a collection o such units, particularly when there is a change in the order o presentation. Te debate which divided Klamroth and Heiberg in the 1880s concerned a corpus o this genre, itsel strongly determined by the indications provided in the medieval recensions such as those o Nasîr at-Din at-ûsî and o the author known as pseudo-ûsî.55 Te ‘global/local’ distinction is necessary because o the question o the proos. It is easy to identiy the phenomenon o double proos. Generally the second proos are introduced by an indicator ‘ἄλλως’ (‘in another way’) 52
53
54
55
I add that the inormation which I have gleaned about the medieval Arabic (and Hebrew) tradition is second-hand and depends on the accessibility o the publication or the goodwill with which my riends and colleagues have responded to my requests. Particular thanks are due to S. Brentjes, . Lévy and A. Djebbar. A very literal version, directly translated rom Greek into Latin in southern Italy during the thirteenth century, discovered and studied by J. Murdoch in 1966 and edited by H. L. L. Busard in 1987. See Brentjes 2001: 39–41 and De Young 2004: 313–19. Other inormation is likewise accessible, thanks to the Greek or Arabic commentators, as well as through the scholia in Greek and Arabic manuscripts. See Rommevaux, Djebbar and Vitrac 2001: 235–8 and 284–5.
89
90 DELIBERATE ALTERATIONS Addition/Suppression of Material
Modification of Presentation
Change in Order
Change of Status
Fusion of 2 Propositions into 1 Division of 1 Proposition into 2
Different Formulations Alteration of Proofs
Global
Local Addition/Suppression of Cases
Stylistic Interventions
Substitution of Proof Double Proofs (Existence of Alternative Proof)
Logical Interventions
Abridged Construction or Shortened Proof
Figure 1.2 Euclid’sElements. ypology o deliberate structural alterations.
or ‘ἤ καὶ οὕτως . . . ’ (‘Or, also thus …’).56 In the same way, in the AraboLatin translation o Gerard o Cremona, the great majority o the second proos are explicitly presented as such, thanks to indications o the type ‘in alio libro … invenitur’ (‘in another book is ound …’). On the other hand the identi cation o proos as distinct is much more delicate when it is a question o comparing two solitary proos appearing in different versions – or example, when one compares a proo rom a Greek manuscript and its corresponding proo in the Arabic translation, or one rom Adelard o Bath and the other rom Gerard o Cremona. Te intricacies o the manuscript transmission prevent two proos which have only minimal variations rom being considered as truly different. I this were not so, there would be as many proos o a Proposition as there are versions or, even, manuscripts! Tis is why it has proven necessary to introduce the division between local and global. Ideally, it ought to be possible to identiy the ‘core argument’ which characterizes a proo and to distinguish it rom the type o ‘packaging’ which is stylistically or didactically relevant but which is neither mathematically nor logically essential. Te expression ‘substitution o (global o modi will be reserved orTe those cases where there is aproo’ replacement one cation) core argument by another. distinction between ‘core’ and ‘packaging’ is not always easy to establish, but it may be thought that the distinction will be better understood i the different methods o ‘packaging’ have been previously delineated. In other words, in order that 56
Nonetheless, there are conusions. Tus, the addition at alternative proo (ἄλλως). See EHS: 231.2.
.27 is introduced as i it were an
Te Elements
and uncertainties in Heiberg’s edition
the category o global differences – that is, substitution o proo – be well de ned, it is necessary also to propose a typology57 o changes or which I will reserve the quali er local (see the gure 1.1 above). Let us also give a ew explanations or examples or the variations or which the designation is perhaps not immediately apparent: • Tere is a doubling when a Proposition concerning two Cases is replaced by two distinct, consecutive Propositions. Tis expansion is observed in the indirect medieval tradition or .31 and 32, .31 and 34. Te inverse operation is usion. O course, these alterations are not the same as the substitution o a proo. Tus, the doubling might correspond to a logical or (in the case o very long proos) pedagogical concern. Even stylistic concerns might be represented, but they would not alter the mathematical content o the proos. • Te change o status may, or example, affect a Porism (corollary). Tis is the case o the Porism to Heib. .72, transormed into an independent Proposition in the indirect medieval tradition. According to another example, the (apocryphal) principle that ‘two lines do not contain an area’ is presented as Postulate No. 6 in some o the Greek manuscripts (PF), in the translation by al-Hajjâj58 and in the work o Adelard, but as Common Notion No. 9 in another part (BVb) o the direct tradition, in the translation o Ishâq–Tâbit, and in the work by Gerard o Cremona. • Tere is, or example, a different ormulation in Proposition .14. Te translations o al-Hajjâj and the Adelardian tradition propose to present the quadrature o a triangle, while the Greek manuscripts, the Ishâq– Tâbit and Gerard o Cremona translations undertake the quadrature o an unspeci ed rectilinear gure. Tis is related to another category o variations represented by the absence o Proposition .45 in the rst group o witnesses just mentioned. In the same way, the Porism to .19 is ormulated differently in the manuscriptP (or a gure) and in the manuscript T (or a triangle). Here, too, the variant is connected with the existence o the Porism to .20, No. 2 (or a gure), ound in only the so-called Teonine manuscripts. Te divergences may thus be correlated at long • As or range. the local variants with some possible logical and pedagogical purpose, we will see some examples in what ollows. Let us speciy only those which approach the category ‘abridged demonstrations’. Tis category concerns the use o proos described as analogical proos (AP) and 57 58
See this point introduced in Euclid/Vitrac 2001: See De Young 2002–3: 134.
41–69, in particular the chart on p. 55.
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potential proos (PP) introduced by the ormulae: ‘So also or the same reasons …’ (= ‘διὰ τὰ αὐτὰ δὴ καὶ …’) (AP), ‘Similarly we will prove (alternatively, it will be shown) that …’ (= ὁμοίως ‘ δὴ δείξομεν (alternatively, δειχθήσεται) ὅτι…’) (PP). Tese phrases reer to the desire to shorten the text. Te rst is the equivalent o our mutatis mutandis; it allows the omission o a completely similar argument with a particular gure or elements rom a different gure. Tesecond is a alse ‘prophecy’. It is invoked precisely not to have to prove in detail what it introduces. Te ‘abbreviated’ proos are not uncommon in the Elements (they number about 250), but in certain cases, it is easy to imagine that a later editor has used this Euclidean stylistic convention to abridge his text. It is rather striking that the Arabo-Latin versions are on the whole much more concise than the Greek text and sometimes have complete proos, where the latter uses one o the ormulae just cited. In Proposition .6, the version carried by manuscript P uses a potential proo (‘δειχθήσεται’), whereas that o the so-called Teonine manuscripts advances an analogical proo (‘διὰ τὰ αὐτὰ δὴ’). Te appearance o these ormulae is thereore not independent o the transmission o the text. 59
Quantitative aspe ct
Te 220 structural modi cations in my database include: more than 60 De nitions out o about 130, 8 o 11 Common Notions, 29 o 35 Porisms, 41 o 42 Lemmas and additions, 173 Propositions o 474 (actually, 465 in the Greek tradition) which is a little more than a third o the total. 60 Tese modi cations are very unequally distributed through the Books, depending on the type o textual units. aking a cue rom medieval scholars, I have grouped together the principal global variations according to three (not completely, but almost) independent criteria: (a) Te presence or absence o certain portions o the text (35 De nitions, 8 Common Notions, 27 Porisms, 41 Lemmas and additions, 25 Propositions). (b) A change in the order o presentation. Tere are roughly 30 which relate to about 30 De nitions and more than 60 Propositions. (c) Te (structural) alteration o proos. For now, I have listed about 80 which concern a little ewer than 100 Propositions.61 59 60 61
For other examples, see the reerences given in Euclid/Vitrac 2001: Some relate to a group o Propositions, or a total greater than 220. See Vitrac 2004: 40–2.
46–7, n. 51,53.
Te Elements
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In comparing Heiberg’s text with the text o the Arabo-Latin translations by Adelard o Bath and by Gerard o Cremona, I have noted (at least) three textual dichotomies (in decreasing order o importance): 62 Dichotomy 1: Edition Heiberg (υ P ) versus medieval tradition (existence o 18 De nitions, 12 Propositions, 19 Porisms, all the additional material (!), numerous changes in order, the majority o substitutions o proos) Dichotomy 2 (in Books – ): Adelardian tradition versus Gerard o Cremona translation (al-Hajjâj / Ishâq–Tâbit?)63 (existence o 16 De nitions, 10 Propositions, 2 Porisms, some changes in order, double proos in GC) Dichotomy 3: P versus T (existence o 3 Propositions, 2 Porisms, 3 additions, 2 inversions o De nitions, several modi cations) o return to certain elements rom our rst part, the Heiberg edition is ounded on Dichotomy 3. Te Danish editor reused to account or Dichotomy 1 demonstrated by Klamroth. Knorr nally proposed an interpretation somewhat similar to that o Heiberg. His interpretation was linear and consisted o two terms (pre-Teonine/Teonine), simply replacing P with the hypothetical Greek archetype which he believed possible to reconstruct or the medieval tradition. aking into account the inormation at his disposal, Heiberg was not able to identiy Dichotomy 2. Knorr appears to have ignored it, which is at the very least surprising, as he declared that the Arabo-Latin versions which he used (Adelard and Gerard) were neither divergent, nor contaminated. Tis break in the indirect tradition in Books – dashes hopes o reconstructing a common archetype or the indirect medieval tradition.64 As or the local variants, they number in the hundreds, probably amounting to 1000–1500 and concerning about 80 per cent o the Propositions in the Greek text. It might be thought that a single instance o an analogical proo or a simple stylistic intervention in a Proposition is hardly signi cant. I examples o this type are disregarded, 70 per cent o the Propositions rom the Euclidean treatise nonetheless 62 63
64
For details, see the three tables given in the Appendix. Accounting or the Arabo-Latin versions adds a supplementary difficulty rom my point o view (to return to the Greek) since it is a doubly indirect tradition. But the structural divergences which we observe between Adelard o Bath and Gerard o Cremona nearly always nd an explanation in their Arabic precursors, in particular in thedifferences between al-Hajjâj and Tâbit, as they are described – or right or wrong – by the copyists, commentators and authors o the recension (or example at-ûsî). It is particularly clear in Book ; see Romm evaux, Djebbar and Vitrac 2001: 252–70.
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remain, the difference being especially apparent in the arithmetical Books – , as a matter o act more ‘salvaged’ by these variants than the geometric portions, in particular Book and the stereometric Books. An example of a local variant
Te rather simple example which I propose is that o Proposition .1. It shows how accounting or the indirect medieval tradition allows us to go beyond the conrontation between P and T to which Heiberg was conned. Te codicological primacy which he accords to the Vatican manuscript is not inevitable because all Greek manuscripts, including P, have been subjected to various late enrichments. It also probably indicates the intention o these speci c additions. As with several other initial proos in the stereometric books, in .1 Euclid tries to demonstrate a property he probably would have been better off accepting (i.e. as a postulate) – namely, the act that a line which hasome s part in a plane is contained in the plane.65 Here, the philological aspect interests me, even though the changes in the text were probably the result o the perception o an insufficiency in the proo. Te text is as ollows: (a) Εὐθείας γραμμῆς μέρος μέν τι οὐκ ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ ἐν μετεωροτέρῳ .
C
D
B
A
Εἰ γὰρ δυνατόν , εὐθείας For, i possible, let some part AB γραμμῆς τῆς ΑΒΓ μέρος o the straight line ABC be in μέν τι τὸ ΑΒ ἒστω ἐν τῷ the subjacent plane, another part, BC, in a higher plane. ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ τι τὸ ΒΓ ἐν Tere will then exist in the μετεωροτέρῳ . subjacent plane some straight Ἔσται δέ τις τῇ ΑΒ συνεχὴς εὐθεῖαἐπ’ εὐθείας ἐν τῶν ὑποκειμένῳ ἐπιπέδῳ. ἒστω ἡ ΒΔ˙ δύο ἄρα εὐθειῶν τῶν ΑΒΓ, ΑΒΔ κοινὸντμῆμά ἐστιν ἡ ΑΒ˙ , ὅπερἐστὶν ἀδύνατόν
65
Some part o a straight line is not in a subjacent plane and another part is in a higher plane.
line continuous with AB in a straight line. Let it be BD; thereore, o the two straight lines ABC and ABD, the common part is AB; which is impossible,
On the weaknesses o the oundations o the Euclidean stereometry, see Euclid/Vitrac, 4, 2001: 31 and my commentary to Prop. .1, 2, 3, 7.
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(b) Ten the two textual amilies distinguished by Heiberg diverge: P
Because, i we describe a circle with the ἐπειδήπερἐὰν κέντρῳτῷ Β καὶ , centre B and distance AB, the διαστήματι τῷ ΑΒ κύκλονγράψωμεν αἱ διάμετροι ἀνίσους ἀπολήψονται diameters will cut unequal arcs o the circle τοῦ κύκλουπεριφερείας .
BFVb εὐθεῖαγαρ εὐθεῖᾳοὐ συμβάλλει κατὰ or a straight line does not meet a πλείομασημεῖαἢ καθ' ἕν· εἰ δὲ μή, straight line in more points than one; ἐφαρμόσουσιν αλλήλαις αἱ εὐθεῖαι. otherwise the lines will coincide.
(c) Te general conclusion ollows, then the closing o the theorem: Tereore, it is not the case that some part Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος o a straight line is in a subjacent plane and another part is in a higher plane. δέ ἐν μετεωροτέρῳ· .66 ὅπερἔδειδεῖξαι
Which is what was to be proved.
Conorming to the general rule which he ollows, Heiberg has retained the reading o P in his text, and he consigns the reading o the Teonine manuscripts in his apparatus criticus.67 From the stylistic p oint o view, one can see that: • Te what pI’,call post-actum havetwo the variants orm ‘q, are because rather than ‘i pexplanations , then q’. Tebecause ‘cause’ (pthey ) is stated afer the act (q) o which it is supposed to be the cause.68 • Te variant P is introduced by the conjunction ἐ‘ πειδήπερ ’, which is sucient to arouse suspicions about its authenticity.69 Moreover, I call what appears here an ‘active, personal, conjugated orm’ γράψωμεν (‘ ’) since the normal Euclidean orm o conjugation in the portion o the deductive argument is the middle voice,70 which reinorces the suspicion o inauthenticity. 66 67
68 69
70
See EHS: : 4.8–5.3. Tis same variant appears in the margin o P, but by a later hand, ollowed by the addition: ‘οὕτως ἐν ἄλλοις εὕρηται, ἔπειτα τὸ˙ εὐθείας ἄρα γραμμῆς’ (alternatively, this is ound in other [copies]: ‘O a straight line …’). See Euclid/Vitrac 2001: 50,56, 67–9. Tere exist, in the texto Book as edited by Heiberg, about feen passages introduced ’, all o which contain elementary explanations ound neither by the conjunction‘ἐπειδήπερ in manuscript b, nor in the Arabo-Latin translations by Adelard o Bath and by Gerard o Cremona. In the whole o theElements, 38 instances occur. As already indicated by Knorr 1996: 241–2, we know that there are relatively late interpolations in manuscripts used by Heiberg. A posteriori, we can see that Heiberg considered seven o these passages interpolations on the basis o criteria other than their absence in manuscriptb and the indirect tradition. See Euclid/Vitrac, 2001: 47.
95
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Let us now consult the indirect medieval tradition, or example the Arabo-Latin translation by Gerard o Cremona, 71 compared to parts (a) and (c) o the text edited by Heiberg: Parts (a) and (c) o Heiberg’s text Εὐθείας γραμμῆς μέρος μέν τι οὐκ ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ τι . ἐν μετεωροτέρῳ
Gerard o Cremona’s version Recte linee pars non est una in super cie et pars alia in alto.
G
D
B
A
Quoniam non est possibile ut ita sit, quod in exemplo declarabo. , εὐθείας γραμμῆς τῆς Εἰ γὰρ δυνατόν ΑΒΓ μέρος μέν τι τὸ ΑΒ ἒστω ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος δέ τι τὸ ΒΓ ἐν μετεωροτέρῳ .
Si ergo possibile uerit, sit pars linee ABG que est AB in super cie posita et sit alia pars que est BG in alto.
Ἔσται δέ τις τῇ ΑΒ συνεχὴς εὐθεῖαἐπ᾿ εὐθείας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ.
Protaham ergo a linea AB in data super cie lineam coniunctam linee AB
ἒστωἡ ΒΔ: δύο ἄρα εὐθειῶν τῶν ΑΒΓ, ΑΒΔ κοινὸντμῆμά ἐστιν ἡ ΑΒ˙
que sit BD. Linea ergo ABG est linea recta et linea ABD est linea recta, ergo linea AB duabus lineis BG et BD secundum rectitudinem coniungitur.
ὅπερἐστὶνἀδύνατόν .
Quod est omnino contrarium.
Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ ἒστιν ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ, μέρος ˙ δέ ἐν μετεωροτέρῳ
Non est ergo linee recte pars in super cie et pars in alto.
ὅπερἔδειδεῖξαι .
Et illus est quod demonstrare voluimus.
Despite the Arabic intermediary, the reader will easily recognize the aithulness o this Latin translation to the Greek, with two exceptions: • the Latin adds a clause intended to introduce an indirect reasoning (a systematic characteristic shared with several manuscripts o the Ishâq– Tâbit translation) • it has neither o the post-actum explanations o the Greek (part b). 71
Busard 1984: 338–9.
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It is possible to imagine (at least) two scenarios: either these post-actum explanations are inauthentic, or the translator (or the editor Tâbit), noting the divergence among the Greek manuscripts and the de ciency o the proposed explanations, rerained rom retaining one or the other. In other words, he has ‘cleaned up’ the text. Te mathematical de ciency o the explanation in P is obvious. It allows the points ABCD to be co-planar. In order to prove the co-planarity o lines ABC and ABD starting rom the act that they are secant (they even have a segment in common), one would have to use .2 – which in turn invokes .1! Tus, and this is Heiberg’s reading, an argument akin to lectio diffi cilior may be implemented and the text o the Teonine manuscripts may be declared an improvement. Hence, his editorial decision. Tis scenario is hardly likely. In act, in certain manuscripts o the T amily, particularlyV, there exists a scholium proposing a proo o the impossibility o two straight line s having a common segment, that is the concluding point o our indirect proo:72 For two straight lines, there is no common segment. Tus, or the two straight lines ABC and ABD, let AB be a common segment, and on the straight line ABC, let B be taken as the centre and let BA be the radius and let circle AEZ be drawn. Ten, since B is the centre o the circle AEZ and since a straight line ABC has been drawn through the point B, line ABC is thus a diameter o the circle AEZ. Now, the diameter cuts the circle in two. Tus AEC is a semi-circle. Ten, since point B is the centre o circle AEZ and since straight line ABD passes through point B, line ABD is thus a diameter o circle AEZ. However, ABC has also been demonstrated to be a diameter o the same AEZ. Now semi-circles o the same circle are equal to each other. Tereore, the semi-circle AEC is equal to semi-circle AED, the smallest to the largest. Tis is impossible. Tus, or the two straight lines, there is no common segment. Tereore, [they are completely] distinct. From that starting point, it is no longer possible to continuously prolong the lines by any given line, but [only] a [given] line and, that because, as has been shown, [namely] that or two straight lines, there is no common segment.
Tis scholium does not exist in P, but its absence may be explained i it is the srcin o the post-actum explanation, albeit in severely abbreviated orm, inserted in the text o the manuscript. Tus, there was no longer need to recopy the aorementioned scholium. It is likely that the explanations appearing in the Teonine manuscripts come rom the insertion o an abridgment o some (another) scholium into the text. Tere is even a chance that we know the source o these marginal annotations. In his commentary to Proposition .1, Proclus reports an objection by the Epicurean Zenon o 72
C. EHS: , 2, 243.27–244.22.
97
98
Sidon. Te Euclidean proo o .1 presupposes that there is not a common segment or two distinct straight lines,73 precisely what is here declared to be impossible. Te commentator denies the objection, using three arguments, the rst and last o which are close to the contents o the twopost-actum explanations (in T and P respectively), as well as to the scholia.74 In this example, there is every reason to believe that the rst scenario was the better one, that the ‘Euclidean’ proo o .1 was similar to that o the indirect tradition. Heiberg could not have known the Gerard o Cremona translation (discovered by A. A. Björnbo at the beginning o the twentieth century), but he could have consulted Campanus’s edition, which has neither o the post-actum explanations. It goes without saying that the difference, rom a mathematical point o view, is minuscule. However, rom the point o view o the history and use 75 – o the text, it is the number o alterations o this type – in the hundreds which is signi cant. Additions like those which we have just seen regarding .1 have been introduced on different occasions, undoubtedly independently o each other, since each version – including the Arabo-Latin translations which escape nearly uncorrupted by this phenomenon – has some which are proper to it.76 Tis work o improvement undoubtedly owes much to the marginal annotations eventually integrated into the text itsel. Yet it partially blurs the distinction between ‘text’ and ‘commentary’. For the majority o them, these additions ensure the ‘saturation’ o the text. Te interpretation o the Elements which the annotators presuppose is more logical than mathematical. Indeed, or them, Euclid’s text represents the very apprenticeship o deduction more than a means or the acquisition o the undamental results o geometry. Even i the role o the marginal annotations has probably been less effective in the case o structural divergences, we will see that the purpose which they pursue – when it can be determined – is requently the same. From the point o view o the history o the text, the abundance o these sometimes independent improvements implies that or theElements and or certain other mathematical texts the methods o transmission were much more exible than those postulated by philologists whose model rests on the tradition o poetic texts. It is not possible either to put the different examples o a text in a linearly ordered schema s(temma) or even to admit the simple primacy accorded to a manuscript, such as Heiberg accorded toP. Clearly, 73 74 75
76
See Friedlein 1873: 215.11–13, 215.15–16. See Friedlein 1873: 215. 17–216. 9. For example, about 600 sentences are intended to point out a hypothesis or what was the object o a previous proo. About twenty terminological explanations, mostly in Book, may be added. See Euclid/Vitrac 2001: 63.
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in the discussion o problematic places, variant readings o the indirect medieval tradition ought to be accounted or. Tis was exactly what Knorr recommended. He even thought that it was possible to reconstruct a Greek archetype or the whole o the medieval tradition. In other words, by comparing the different states o the text or each attested divergence, we ought to be able to identiy the least inauthentic version (or versions). aking into account the three principal types o structural variants that we have recognized, this amounts to: • solving the question o authenticity or each contested textual unit (the determination o the ‘materiel’ contained there) • selecting a method o presentation (in particular, an order) when several are known; and • knowing, or the cases o substitution or double proos, which o the two is older. o pronounce such judgements supposes criteria. Tere are essentially two o them: (i) the rst co ncerns the ‘quantity’ o material transmitted by various versions, and (ii) the second bears on the orm o this material (order o presentation, modi cation o proos). Tese criteria rest on the presuppositions that the historians accept regarding the nature o the text o the Elements and on the hypotheses that t hey imagine regarding its transmission. According to Klamroth (and Knorr), the textual history has essentially been an ampli cation. Tus, or example, except by accident, a Proposition missing rom a ‘thin’ version (containing less material than another or even several others) will be judged inauthentic. As or the transormations o orm, i it is not an accident o transmission but a deliberate alteration o the structure o the text (supposing that it is possible to discriminate between the two), the criterion, as stated explicitly by W. Knorr, will be improvement – that is, whether it met with success or ailure, whether it was really justi ed or invalid, the deliberate modi cation o the orm (order, proo) o the text sought to better the composition. Obviously, this is an optimistic vision o the history o mathematics. o see how to apply these principles and to understand the nature o the structural modi cations that we have called up, it is easiest to produce some examples. Te limitations o the aorementioned criteria will appear more clearly when we examine their application to the proos (see below, pp. 111–13).
99
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Questions of authenticity and the logical architecture of the Elements
I the different versions are considered rom the point o view o the ‘material contents’, the question o authenticity is perhaps the least complex o the three, at least as ar as the rst dichotomy is concerned. Tere exists in the Greek manuscripts material which I describe as ‘additional’. Tis additional material includes cases, some portions identi ed as additions, the double proos, and the Lemmas.77 Te critical edition o Heiberg, completed in 1888, our years afer the debate with Klamroth, condemns the lot o this material as inauthentic. In this regard, the (rather relative) thinness o P compared with the other Greek manuscripts is one o the criteria which justi es its greater antiquity.78 Now this additional material, to nearly a single exception,79 is absent rom the medieval Arabic and Arabo-Latin tradition. However, Heiberg did not alter his position and did not accept this conclusion about the ‘thinness’ o the indirect tradition as a gauge o its purity. According to Heiberg – and this too is a hypothesis about the nature o the treatise – the Elements could not be so thin that it suffered rom deductive lacunae, but such thinness is the case with the medieval versions. I do not believe that anyone (and certainly not Klamroth or Knorr) contested the global deductive structure o the Elements. I the Elements is compared with the geometric treatises o Archimedes or Apollonius, the local ‘texture’ may not be so different, but the principal variation resides in the act that the Elements was edited as i it supposed no previous geometric knowledge. Te identi cation o what would be a deductive lacuna in Euclid is thus a crucial point, but not always a simple one. Indeed, all the exegetical history o the Euclidean treatise, rom antiquity until David Hilbert, has shown that the logical progression o the Elements, probably like any geometric text composed in natural language, rests on implicit presuppositions.80 Te identi cation o the deductive lacunae supposes that consciously permitted ‘previous knowledge’ is always capable o clearly being distinguished rom ‘implicit presumption’. Let us take the example o Proposition .15. Here it is established that: Te bases o equal cones and cylinders are inversely [proportional] to the heights; and among the cones and cylinders, those in which the bases are inversely [proportional] to the heights are equal, 77 78 79 80
For details, see able 1 o the Appendix. See able 3 o the Appendix. Te addition o special cases in Prop. .35, 36 and 37. See the beautiul study by Mueller 1981.
and uncertainties in Heiberg’s edition
Te Elements
L Q
O
S
G
D M A K
P
N
H
F
C R
U
E
B
Figure 1.3
Euclid’s Elements, Proposition
.15.
a property likewise shown or the parallelepipeds ( .34) and pyramids ( .9). In the rst part o the proo, let us suppose the cones or cylinders on bases ABCD and EFGH, with heights KL and MN, are equal. I KL is not equal to MN, NP equal to KL is introduced and the cone (or cylinder) on base EFGH with height NP is considered (see Figure 1.3). Schematically, in abbreviated notation, we have (by .7) a trivial proportion: cylinder AQ = cylinder EO ⇒ cylinder AQ: cylinder ES:: cylinder EO: cylinder ES in which a substitution is made or each o the two ratios: cylinder AQ: cylinder ES:: base ABCD: base EFGH (which is justi ed by xii.11) cylinder EO: cylinder ES:: height MN: height PN (S). From which: base ABCD: base EFGH:: height MN: height PN (CQFD) However, the proportion (S) is an ‘implicit presumption’ in the AraboLatin versions. Admittedly, it may be easily deduced by those who understand Propositions .1 and 33, as well as .25, that is the way one employs the celebrated De nition .5. In(Sthe the situation is different. Proportion ) is Greek justi edmanuscripts, on the basis though, o previous knowledge: .13 in P and T, .14 in b.81 Tese Propositions .13–14 do not exist in the indirect medieval tradition and thus it may be inerred 81
Here, the indirect medieval tradition is not in accord with ms b which presents the most satisying textual state rom the deductive point o view! For details, see Euclid/Vitrac 2001: 334–44.
101
102
rom their absence, as Heiberg has done, that there is a deductive ‘lacuna’ in the proo o .15. However, rom the point o view o the history o the text, the question immediately arises about whether or not the insertion o Propositions .13–14 represents an addition aimed at lling a lacuna perceived in the srcinal proo o .15. Let us add that the assertions o Heiberg on this subject are ofen a little hasty because the status o authenticity cannot be judged independently o the status o the proos. For example, the indirect tradition does not contain Proposition .13 (‘I two magnitudes be commensurable and one o them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same’). Heiberg suggests that the absence o this Proposition introduces deductive lacunae in several Propositions which exist in the Arabic translations. In these Propositions, the Greek text explicitly uses .13. However, in act, when the proos in the aorementioned translations are examined, they are ormulated a little differently than in Greek and .12 (‘Magnitudes commensurable with the same magnitude are commensurable with one another too’) is employed in place o .13. Consequently, there is not a deductive lacuna!82 By consulting the indirect tradition o Greek citations in Pappus, the idea may be supported that .13 did not exist in his version o the Elements.83 Tus, the most natural conclusion is that .13 is effectively an inauthentic addition and its addition has allowed reconsideration o the proos o the other Propositions. Trough a simple comparison o the different versions, I have examined each o the Propositions whose authenticity has been called into question. My conclusion regarding this point – the details would exceed the scope o this essay – is that the real deductive lacunae, proper to the indirect tradition, are, so ar as can be judged, ar rom numerous: • wo in Book ,84 with the provision that in any event the stereometric Books constitute a particular case in the transmission o the Elements (see below). 82 83 84
See Vitrac 2004: 25–6. See Euclid/Vitrac 1998: 384–5. Te second is due to the absence, this time in b as well as in the indirect medieval tradition, o Proposition .6 and the Porisms to .7–8 which generalize the results established or pyramids on a triangular base to pyramids on an unspeci ed polygonal base, respectively in Propositions .5, 7 and 8. Tere also, Euclid may have considered this generalization as intuitively obvious given the decomposition o all polygons into triangles and the rule concerning proportions established in (Heib.) .12: ‘I any number o magnitudes be proportional, as one o the antecedents is to one o the consequents, so will all the antecedents be to all the consequents.’ Te non-thematization o pyramids on an unspeci ed polygonal base is comparable to what we have seen above regarding .14 (triangle unspeci ed rectilinear gure) in only the Adelardo-Hajjajian tradition. Te difference is that it introduces a deductive
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• One in Proposition .10 o the Adelardo-Hajjajian tradition, connected to the absence o .37, probably due to an accident in transmission, namely, the mutilation o the end o a Greek (or possibly Syriac?) scroll containing Book . Books – , perhaps the only ones to have been both translated by Ishâq and reworked by Tâbit,85 contain no supplementary deductive lacunae. In other words, the deductive lacunae which appear there already existed in the Greek text which served as their model. Te most striking case is that o the Lemmas designed to ll what can be regarded as a ‘deductive leap’, especially in Book .86 In act, there are in (some manuscripts o) the Ishâq–Tâbit and Gerard o Cremona translations a number o additions that ul l the same role o completion.87 When compared with the direct tradition, they are presented as additions, mathematically useul, but well distinguished rom the Euclidean text. Tose who composed our Greek manuscripts had no such scruples. Te addition o the so-called missing propositions and part o the additional material (Lemmas o deductive completion, some o the Porisms) serve with a certain uidity the obvious intention o improving the proos and reinorcing the deductive structure. Te second part o the Porism to .6 allows the resolution o the same problem as the lemma { .29/30}. Te Proposition .38 vulgo is clearly a lemma to .17. Te Proposition was probably inspired by a marginal scholium and then moved to the end o Book .88 Te textual variants o .6 suggest that perhaps it was initially introduced as a Porism to .5 and eventually transormed into a Proposition. For the other additional Porisms, it would certainly be excessive to speak about a deductive lacuna to be lled. However, .7 Por. and .19 Por explicitly justiy the use o inversion and conversion o ratios. Te Porisms to .20, .11, .35 serve to make explicit a deductive dependence on the Propositions .6 Por., .12 and .36, respectively. Our examples, ound in Books – , show that this work o enrichment began in the Greek tradition, but the Arabic and Arabo-Latin versions tell us that the
85 86
87 88
lacuna in the proos o Propositions .10–11. Here the properties established previously or pyramids and prisms are shown or cones and cylinders, by using the method o exhaustion. o do this, the pyramids are considered as having polygonal bases with an arbitrary number o sides, inscribed in the circular bases o the cones and cylinders. See below, pp. 116–19. I have called them the ‘lemmas o deductive completion’ in order to distinguish them rom lemmas with only a pedagogical use. See the list given in Euclid/Vitrac 1998: 391. o these might be added Lemma .4/5. See Euclid/Vitrac 1998: 392–4. See Euclid/Vitrac 2001: 229–30.
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enrichment was not con ned to the nal, more complicated portion o the text in question. It is even probable that the entire treatise has been subjected to such treatment. For example, the arithmetic books o the Ishâq–Tâbit and Gerard o Cremona versions possess our supplementary Propositions with respect to the Greek. Ishâq–Tâbit .30–31 are added to improve (Heib.) .30–31, and Ishâq–Tâbit .24–25 are the converses o (Heib.) .26–27. In act, the proo o (Ishâq–Tâbit) .24 (plane numbers) is nothing more than the second part o (Heib.) .2! Hence the idea, again suggested by Heron, to remove this portion in order to introduce it as a Proposition in its own right and to do the same or the converse o .27 (solid numbers) to simpliy the proo o .2.89 Insoar as the Euclidean approach is deductive, the work just described represents a real improvement o the text as much rom a logical perspective as rom a mathematical point o view. A number o implicit presumptions which might be described as harmless but real deductive lacunae have been identi ed and eliminated. However, the logical concerns have been sometimes pushed beyond what is reasonable. For example, in the desire to make the contrapositives appear in the text, Propositions .24–27 in the Ishâq–Tâbit version expect the reader to know that two numbers are similar plane numbers i and only i they have the ratio that a square number has to a square number to one another. Te Lemma .9/10 – an addition probably connected to Ishâq–Tâbit .24–25 – thence deduces that non-similar plane numbers do not have the ratio that a square number has to a square number to one another. Likewise, the (important) Propositions .5–6 establish that the ‘commensurable magnitudes have to one another the ratio which a number has to a number’ (5) and the inverse (6). In the Greek manuscripts, but not in the primary indirect tradition, two other Propositions (Heib.) .7–8 have been inserted: ‘Incommensurable magnitudes have not to one another the ratio which a number has to a number’ (7, contrapositive o 6) and its inverse (8, contrapositive o 5)! Propositions .14–15 show that ‘i a square (resp. cube) [number] measures a square (resp. cube) [number], the side will also measure the side; and, i the side measures the side, the square (resp. cube) will also measure the square (resp. cube)’. In the Greek manuscripts these Propositions are ollowed by their contrapositives (Heib. .16–17, or example): ‘I a square number does not measure a square number, neither 89
See Vitrac 2004: 25.
and uncertainties in Heiberg’s edition
Te Elements
will the side measure the side; and, i the side does not measure the side, neither will the square measure the square.’ I the indirect tradition is consulted, an interesting division is observed: • In the translation o Ishâq–Tâbit the contrapositives do not exist, but each o the Propositions .14–15 is ollowed by a Porism which expresses the same thing.90 • In the translation o al-Hajjâj91 and in the Adelardian tradition92 is ound a single Proposition combining the equivalent o Heib. .16–17. Te assertion about cube numbers is simply lef as a potential proo. • Gerard o Cremona transmits the two version successively.93 I think there is hardly any doubt in this case. Te Propositions .16–17 o the Greek manuscripts are inauthentic and all the versions, including those o the indirect tradition, contain augmentations or additions which proceed along different modalities and which are probably o Greek srcin. Logical concerns have certainly played a role in the transmission o the text.94 Te change in the order of
.9–13
Te examples that we have examined until now are rather simple in the sense that their to be athe improvement o a deective proomotivations (c. .1), orappear lling arather gap orclearly explaining deductive connection (supplementary material and Propositions). In a signi cant number o cases we have seen the advantages o taking into account the Arabic and Arabo-Latin indirect tradition. However, it ought not to be believed that this simplicity is always the case or that the indirect tradition systematically presents us with the state o the text least removed rom the srcinal. As we have already seen regarding the supplementary Propositions, the alteration o Books – is especially clear in the Greek, although among the 90 91 92 93
94
See De Young 1981: 151, 154–5, 431, 435. Tis we know thanks to Nâsir ad-Dîn at-ûsî. See Lévy 1997: 233. See Busard 1983 (Prop. . 15 Ad. I): 239.359–240.371. See Busard 1984, respectively, 201.11–16 (= . 14 Por. GC), 202. 11–16 (= .15 Por GC) and 202.19–40 (= .16 GC). One might add here the supplementary Porism to Prop. .5 ound in the Ishâq–Tâbit and Gerard o Cremona translations. .4 establishes that a cube, multiplied by a cube, yields a cube, and .5 states that i a cube, multiplied by a number, yields a cube, the multiplier was a cube. Te Porism to .5 affirms that a cube, multiplied by a non-cube, yields a non-cube and that i a cube, multiplied by a number, yields a non-cube, the multiplier was a non-cube. In a subamily o Ishâq–Tâbit manuscripts, this Porism has been moved afer .4. In Gerard o Cremona, there is a Porism afer .4 and one afer .5! See De Young 1981: 201, n. 7, 202–3, 480–1 and Busard 1984 213.29–31 and 213.51–6.
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106
Greek order
Medieval order
9: From a given straight line to cut off a prescribed part. 10: o cut a given uncut straight line similarly to a given cut straight line. 11: o two given straight lines to nd a third proportional. 12: o three given straight lines to nd a ourth proportional. 13: o two given straight lines to nd a mean proportional.
13: o two given straight lines to nd a mean proportional. 11: o two given straight lines to nd a third proportional. 12: o three given straight lines to nd a ourth proportional. 9: From a given straight line to cut off a prescribed part. 10: o cut a given uncut straight line similarly to a given cut straight line.
arithmetical books, the Ishâq–Tâbit version (itsel inspired by Heron) is the best evidence o this ‘betterment’. Te consideration o changes in order conrms the complexity o the phenomenon. In Book , Propositions .9–13 (according to the numbers o the Heiberg edition), resolve the ve problems listed in the table above. In the indirect tradition, the order o presentation runs 13–11–12–9–10. Te solutions o the problems are independent o each other. Tus the inversion has no in uence on the deductive structure, but .13 uses (part o)
.8 Por.:
From this it is clear that, i in a right-angled triangle a perpendicular be drawn rom the right angle to the base, the straight line so drawn is a mean proportional between segments o the base.95
Te Proposition has thus been moved in order to place it in contact with the used result. Since there are clearly two groups – one concerning proportionality, the other about sections – the coherence o the two themes has been maintained by also moving .11–12 (or, in the case o Adelard’s translation, only .11 because it lacks .12 as a result o a ‘Hajjajian’ lacuna).96 Tis order o the indirect tradition appears to be an improvement over the Greek. 95
96
Inalso the majority Greek manuscripts, second thatsegments each sideooita(which right angle is the meanoproportional between athe entireassertion base anddeclares one o the has a common extremity with the aorementioned side). It is absent in V, or example. Heiberg considered it inauthentic and bracketed it (see EHS : 57.1–3). Both parts exist in the Ishâq– Tâbit version and Adelard o Bath and Gerard o Cremona, but the complete Porism does not gure in the Leiden Codex (the an-Nayrîzî version). Moreover a scholium, attributed to Tâbit, explains that the Porism had not been ound among the Greek manuscripts. Without a doubt, this is in error. In (at least) two mss o the Ishâq–Tâbit version, a gloss indicates that Tâbit had not ound what corresponds to only the second part o the Porism (excised by Heiberg). See Engroff 1980: 28–9. Tis we know thanks to the recension o pseudo-ûsî. See Lévy 1997: 222–3.
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When the various versions are considered,97 the inversions are not the result o happenstance in binding or in later inexpert replacement o lost pages. As in our example, they leave practically all the deductive structure intact and they even improve it. O course, not all the examples are equally simple, and the same principle clearly cannot be applied to the inversions in the De nitions, or which it seems that a criterion, which I call ‘aesthetic’ or lack o anything better, has prevailed. Te evidence is divided but at this stage in my work, it seems to me that the preliminary conclusions about the orders con ict with what can be determined about the content.98 Namely, or problems regarding order, notably in Books – , the indirect tradition received the greatest number o improvements! Although changes in order may be limited, they are interesting because they have an advantage with respect to the authenticity or alteration o proos. Such changes are hardly conducive to contamination. Admittedly, we have several remarks by Tâbit ibn Qurra affirming that he had ound a different order o presentation in another manuscript,99 but no one saw t to reproduce the Propositions twice in each o the orders. In contrast, or the problems o authenticity, the contamination between textual amilies concerns the whole text, beginning particularly with the margins o the manuscripts. As or the substitutions o proos, we will see that they are the cause, at least in part, o the phenomenon o double proos. From the substitution of proof to the phenomenon of double proofs: the example of x.105
Te Propositions (Heib.) .66–70 and 103–107 establish that the twelve types o irrational lines obtained through addition and subtraction distinguished by Euclid are stable with respect to commensurability. In the Greek version, 97
98
99
Tings are a little different at the level o individual manuscripts which have not been preserved though the accidents o transmission. For example, in the Greek, the order othe Propositions (Heib.) .21–22 (each the converse o the other) runs opposite to the order in medieval indirect tradition. Te inversion has no in uence on the deductive structure, but the proo o (Heib.) .21 uses . 20. It is probable this time that the inversion wasconsecutive. made in the direct tradition, in order to make the two connected deductive theorems For example, in Book which was just discussed. In (at least) three mss o the Ishâq–Tâ bit version, the ollowing gloss appears afer (Ishâq–Tâbit) .9 = (Heib.) .13. ‘Tâbit says: we have ound, in certain Greek manuscripts, in the place o this Proposition, that which we have made the thirteenth.’ Undoubtedly, the existence otwo distinct orders ought to be understood as having been observed by the Editor among the Greek manuscripts which he consulted. (Tus, the change is Greek in srcin.) Te editor retained the better order (which was that already in al-Hajjâj). See Engroff 1980: 29, who mentions two mss. Te gloss also exists in ms ehran Malik 3586 (the oldest preserved copy o the Ishâq–Tâbit version), o.75a. I thank A. Djebbar or this inormation.
107
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two alternative proos or Propositions .105–106 are inserted at different places in the manuscripts.100 Called here ‘super cial’ as opposed to the srcinal ‘linear’ Greek proos, they apply to and argue about rectangular areas. Let us explain this difference by an example, Proposition (Heib.) .105: A [straight line] commensurable with a minor straight line is a minor. Aliter in Greek = rst proo in medieval tradition
First proo in Greek101 Let AB be a minor straight line and CD commensurable with AB; I say that CD is also minor.
Let A be a minor straight line and B [be] commensurable with A; I say that B is minor.
B A
C
F
H
D
E
G
E
C
F D
A
B
We will consider the two components (AE, EB) o AB and let DF be constructed so that (AB, BE, CD, DF) are in proportion. By .22, their squares
Let CD be a commensurate straight line. Let the rectangles be constructed: CE = square on A, width: CF, FG = square on B, width: FH.
will also be in proportion and, thence by .11, .23 Por. it will be shown (CD, DF) have the same properties as (AB, BE). Tus, by de nition, CD will be a minor.
CE is the square on minor A so CE is the ourth apotome ( .100). We have Comm. (A, B). Tus: Comm. (CE, FG) and Comm. (CF, FH). FH is the ourth apotome ( .103). Te square on B = Rect. (EF, FH), thus B is a minor ( .94)
• In each o the linear proos, the argument concerns the twoparts o an irrational straight line. Te same type o argument is repeated ten times. Tough repetitive, the approach has the advantage o not employing anything other than the De nitions o different types and the theory o 100
101
In the Greek manuscripts the proo aliter to .105–106 is inserted at the end o Book , afer the alternative proo to .115, which without a doubt implies that they had been compiled in this place, afer the transcription o Book , in a limited space. Tus, they are in the margins o manuscripts B and b. In one o the prototypes o the tradition, .107 aliter has been lost or omitted, probably or reasons o length, or because it was conused with .117 vulgo which ollows immediately (but which is mathematically unrelated). My diagrams are derived rom those ound in the edition o Heiberg (EHS: 191 and 229, respectively). Tose o the manuscripts are less general. Te segments AE, CF are very nearly equal (the same goes or A, B in aliter) and divided similarly.
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proportions. Deductively, the linear proos may be characterized as minimalist. • Te super cial proos introduce areas, which, rom the point o view o the linguistic style used, might seem more geometric than the proos using the theory o proportions, which is a second-order language. But, in act, they strengthen the deductive structure because they establish new connections by using Propositions (Heib.) .57–59 + 63–65 + 66 (resp. 94–96 + 100–102 + 103). In addition, these super cial proos – like the linear ones – present results expressed or commensurability in length but the ormer proos may be immediately generalized to commensurability in power. Te rst anomaly occurs in .107. No alternative proo exists, although this Proposition, along with two others, constitutes a triad o quite similar Propositions. Alternative proos are no longer known or the parallel triad o .68–70, which concerns the irrationals produced by addition, whereas the other triad .105–107 treats the corresponding irrationals produced by subtraction.102 However, in the indirect Arabic and Arabo-Latin tradition, there is a textual amily in which these two triads o Propositions have (only) super cial proos. Tis is the case in Arabic, with the recension o Avicenna, and in Latin, with the translation o Adelard I. Evidence rom the copyist o the manuscript Esc. 907 establishes a link between the super cial proos and the translation o al-Hajjâj.103 Te Ishâq–Tâbit version is less coherent. It contains the linear proos o the Greek tradition in the triad .68–70 and the super cial proos or the triad .105–107. In the manuscript rom the Escorial and the translation o Gerard o Cremona, which agree on this point, the situation is nearly the inverse to the Greek translation. Tere are only the super cial proos or .105–107 (like the indirect tradition), but they present proos o this type as aliter or the rst triad, whereas the Greek texts includes them only or (two Propositions o) the second triad! Let us add that the same type o substitution (and thus, generalization) is possible in Propositions (Heib.) .67 + 104 which concern the two corresponding types o bimedials and apotomes o a bimedial.104 Such substitution what is ound in the recensions o at-ûsî and pseudo-ûsî,isbutprecisely not in the Arabic or Arabo-Latin translations. 102 103 104
On the plan o Book , see Euclid/Vitrac 1998: 63–8. See De Young 1991: 659. However, this is not possible or Prop.Heib. .66 (binomials) and 103 (apotomes) because, in this case, it is required to show that the order (rom one to six) o the straight lines commensurable in length is the same. Tis crucial point is required or the super cial proos concerning the other ten types o irrationals.
109
110
I the principle o improvement advanced by Knorr is applied, we are led to think that the linear proos o the Greek are authentic, with the super cial proos clearly being ameliorations rom a mathematical point o view. Tis attempt at strengthening the deductive structure and generalizing was begun in Greek, as demonstrated by the proos aliter to .105–106. It is likely that there was also a proo aliter to .107 which has disappeared. Te opposite hardly makes any sense. Its disappearance is probably due to codicological reasons. However, the question o knowing who produced the alternative proos or the Propositions o the rst triad remains unanswered. A likely hypothesis is that the same editor is responsible or the parallel modi cation o the two triads and he happened to be a Greek. But it could also be imagined that it was a contribution rom the indirect tradition, occurring as the result o an initiative by al-Hajjâj. Tis latter explanation is the interpretation o Gregg De Young.105 Te examples o at-ûsî and pseudo-ûsî show that improvements continued into the medieval tradition, but it should not be orgotten that these were authors o recensions, not translators. As or the structure or the Ishâq–Tâbit version, it may be explained in different ways – either by the existence o a Greek model combining the two approaches or by an attempt at compromise on the part o the editor Tâbit. In the rst case, there would have been at least three different states o the text. In the second case, Tâbit would have combined the rst (linear) triad rom the translation o Ishâq (considered closer to the Greek) and the second (super cial) triad presented in the earlier translation! In neither o these scenarios does recourse to the indirect tradition simpliy the identi cation o the oldest proos. Whatever scenario is chosen, it must be admitted that there was a substitution o proos in one branch o the tradition. Te substitution occurred in the model(s) o al-Hajjâj, i the super cial proos are considered later improvements, but in the Greek, i the opposite explanation is adopted. Tis act is not surprising.106 In the situations in which the Greek tradition contains double proos, the medieval versions contain only one o them. (Tis is con rmed by the remarks o Tâbit and Gerard when they make such comments as ‘in another copy, we have ound …’ and thus, probably, in Greek models o which we have no evidence.) It is possible to take a lesson rom this example. Te existence o double proos in the Byzantine manuscripts could be explained, or the majority o 105 106
See De Young 1991: 660–1. It is noted or .44p; .14; .7p, 8p, 25, 31, 33p, 35, 36; .5; .5, 18; .9p, 20p, 31; .11p12p; 22–23; .1, 6, 14, 26p, 27–28, 29–30, 68–70, 105–107, 115; .30, .5. Te note ‘p’ signi es that the variant pertains only to a portion o the proo.
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cases, by the act that the aorementioned manuscripts have compiled the proos rom different versions which contained these proos in isolation. I what we have seen about Propositions .107 and .68–70 is recalled, the process o transliteration and the desire to saeguard a ourishing tradition seems to us to constitute a propitious occasion or compiling proos, however incomplete. Returning to discussions concerning the history o the text, we ought to rst note that the double proos do not all within what is called authorial variants. Euclid did not propose several proos with the same results. Tus the Greek manuscripts closest to the operation o transliteration (P and B) are most likely the results o a compilation o the tradition, rather than o simple reproduction – changing only the writing – o a venerably aged model.107 Te limits of Knorr’s criteria
It is ofen possible to perceive one or more reasons or the other types o structural changes that I described earlier (additions, modi cations o the order). Tus Knorr thought it possible to order the different states o the text, i not according to authenticity, then at least relative to the degree o alteration. We have already noted that this criterion o improvement applies locally, and the example o changes in the order suggested to us that it does not seem always to have been exercised or the bene t o the one and the same version. Te phenomenon o the substitution o proos evidences another difficulty. Te criterion o improvement works well enough as long as there is only a single parameter (or even more,108 but all acting in the same direction) which governs the replacement o a proo or the modi cation o a presentation. But, when there are at least two acting in opposite directions, the change which is more sophisticated rom a certain point o view may be less desirable rom another point o view. Let us reconsider our example o Proposition .105. Admittedly, rom a mathematical point o view, there is an improvement (generalization), but rom the logical, or metamathematical, point o view – and it is no doubt one o the points o view adopted by 107 108
See n. 33. Te most requent parameters governing the replacement o proos are the reinorcement o the deductive structure, the substitution o a direct proo with an indirect proo (a criterion notably explained by Heron – see Vitrac 2004: 17–18 (regarding .9 aliter) – and Menelaus), the addition o the case o a gure and the level o discourse used (geometric objects versus proportions; a criterion clearly noted by Pappus).
111
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those who deliberately changed the text o the Elements – different criteria could be used. From the logical, or metamathematical, point o view, the criteria are: • Render the deductive structure more dense, as the super cial proos have done, or conversely minimize the structure in order not to introduce what would eventually become accidental ‘causalities’, that is, links o dependence, as ound among the linear proos. • Preer either a type o object language over a second-order language – that is, a relational terminology, like the theory o proportions – or, on the contrary, privilege a concise but more general second-order language. A choice o this kind explains the aliter amily o proos conceived or Propositions .20, 22, 31, .9, .37.109 Te same choice exists also in our amilies o proos, but in these instances it acts in the opposite direction with respect to reinorcing the deductive structure. It would then be welcome to be able to organize these criteria hierarchically. Te deductively minimalist attitude seems well represented in the Elements. For example, deductive minimalism may saely be assumed to underpin the decision to postpone as long as possible the intervention o the parallel postulate in Book . It appears again in the decision to establish a number o results rom plane geometry beore the theory o proportions is introduced at the beginning o Book , even though this theory would have allowed considerable abbreviation. Te idea that geometry ought to restrict itsel to a minimal number o principles had already been explained by Aristotle.110 Deduction is not neglected, but emphasis is placed on the ‘ertility’ o the initial principles, rather than on the possible interaction o the resultants which are deduced rom them. Tere are thus different ways to put emphasis on the deductive structure. Te case o our proos rom Book is not unique. Te ten Propositions rom Book and the rst ve rom Book are successively set out in a quasi-independent manner based on the least number o principles, even i this means reproducing several times certain portions o the arguments. 111 Remarkably, we know that or the sequences .2–10 and .1–5 alternative proos had been elaborated, annulling this deductive mutual independence in order to construct a chain in the case o .2–10 or to deduce .1–5 rom certain results rom Books and . Even better, thanks to the testimony o 109 110 111
See Vitrac 2004: 18–20. De cælo, , 4, 302 b26–30. Similarly in the group El. .1, 3, 9, 10 (considering the rst proos o
.9–10).
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the Persian commentator an-Nayrîzî, we know that the author o the rst suggestion was Heron o Alexandria. It is thus tempting, as Heiberg did in his Paralipomena o 1903, to attribute to him the other alteration (in ) that shares the same spirit. 112 I, in order to strengthen the deductive structure, it is appropriate to argue about segments rather than the suraces described thereon as in the case o Books and ,113 it will be noted that the opposite is the case in the example rom Book which has just been discussed. Reinorcement o the aorementioned structure is realized through the introduction o suraces. For us to attribute it to Heron, it is necessary to be sure that the parameter most important to him was indeed the densi cation o the deductive structure. Without any external con rmation or other historical inormation, as in the case o Books and , this scenario remains a stimulating hypothesis, but only a hypothesis!114
Conclusions: contributions and limitations of the indirect tradition From the study o a better-known indirect tradition, several lessons may be drawn. Newly available inormation con rms certain results o the Klamroth–Heiberg debate. Consideration o a greater number o versions o the Elements than Heiberg or Klamroth could have used reinorces the existence o a dichotomy between the direct and indirect traditions. (1) Although they agree (albeit with opposite interpretations o the act), the ‘thinness’ o the indirect tradition is not so marked as Klamroth and Heiberg would have us believe, especially in Books – . Te most complete inventory o variants, probably Greek in srcin, which we have now (by induction or thanks to inormation transmitted by Arab scholars or copyists), has several consequences: • It puts into perspective the different textual dichotomies. For example, No. 3 (P / T), within the Greek direct tradition, is quite modest with See Heiberg 1903: 59. I have espoused the same hypothesis in Euclid/Vitrac 2001: 399–400. Te insertion o .10 aliter, explicitly attributed to Heron by an-Nayrîzî, has the same effect o strengthening the deductive structure. 114 A single thing seems likely. Te version o Euclid which Pappus had – i he is indeed the author o the second table o contents o Book contained in the rst Book o Commentary to the aorementioned book transmitted under his name – contained the linear proos. In effect, Propositions .60–65 and .66–70 were inverted (similarly or .97–102/103–107) and this act precludes the existence o super cial proos or .68–70. 112 113
113
114
respect to No. 1 (direct tradition/indirect medieval tradition) or even to No. 2, within the Arabic and Arabo-Latin translations.115 • It convinces us that some part o what exists in Greek, and preserved by Heiberg in his edition, is very probably inauthentic. • It gives a possible interpretation to some ‘isolated’ variants in Greek by integrating them into a broader picture which makes sense. For example, it makes sense o amilies o alternative proos created by the same editorial principles.116 (2) However, because o the number o variants, the homogeneity o the entire indirect tradition, which Klamroth believed existed, no longer exists in Books – . I have called this dichotomy 2, within the Arabic, Arabo-Latin and, it seems, the Hebraic traditions. For certain portions, notably Books , and , it seems that (at least) two rather structurally different editions existed and they contaminated each other signi cantly. Consequently, it will be impossible to reconstitute a unique Greek prototype or this portion o the whole o the medieval tradition as Knorr had wanted.117 I the study o the material contents, order, presentation, and proos o the preserved versions o the thirteen books is resumed, it is not to be expected to nd that among the preserved versions, one o them, or instance Adelard I or Ishâq–Tâbit, may be declared closer to the srcinal in all its dimensions than all the other versions. Te ‘local’ criteria used by Klamroth, Heiberg and Knorr, either ocusing on the material contents (according to the principle o expansion) or on the improvement o the orm, do not converge upon a global criterion which applies to the entirety o the collection o the thirteen books. Te result is thus that the indirect tradition appears more authentic in regard to the material contents but not or the order o presentation. For the problems o order and o presentation, conversely, the indirect 115 116
117
See ables 1–3 o the Appendix. We have seen an example o this with the super cial proos o .105–106. Another amily o double proos may be reconstituted or Propositions . 20p, 22, 31, .9, .37. See Vitrac 2004: 18–20. It should be emphasized that Knorr had not considered the problem at its ull scale: • He considered at most a group o 21 Propositions and proceeded by induction. • He did not take into account more than one single criterion o structural divergence – that o material contents – with one exception: the proo o .17, poorly handled in the indirect tradition and interpreted not as an accident o transmission but in terms o development. • He took into account neither changes in order nor the rich collection o double proos. • He did not ask himsel the question o whether the two Arabo-Latin translations that he used, Adelard and Gerard, were representative o the whole o the indirect tradition. Whether these translations are representative is not at all certain in the stereometric books (c. below, pp. 118–19).
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tradition has the bene t o many more improvements, and the Greek tradition seems to have been very conservative in this area. (3) Furthermore, the conclusions drawn rom the results o the comparison o versions change according to the book or group o books being studied. For example, interaction between Euclid and the Nicomachean tradition has had an impact on the text o the arithmetical Books. I .68–70 and 105–107 and .1–5 are judged by the criteria o improvement, the medieval versions (particular Adelard’s) are more sophisticated than the Greek text, at least as ar as the contents are concerned. At the end o Book (and perhaps also in response to an initiative by Heron), the medieval versions are also more sophisticated with regard to the material contents,118 although the opposite is much more requent. Along the same lines, the mathematically insufficient proos (according to the criteria o the ancients) in the Elements are our in number i the direct and indirect traditions are combined: .22–23, .19 and .17. I, as Knorr argues, we assume the errors are rom Euclid and not textual corruptions, we arrive then, by applying his criteria, at the ollowing conclusions: • For .22–23, the srcinal proos are those common to both the Greek and to the Hajjajian tradition; the proos presented by the Ishâq–Tâbit version are improvements.119 • For .19, the srcinal proo is that o manuscript P; those o T and o the indirect tradition are improvements. • For .17, the srcinal proo is that o the indirect tradition; those o b as well as o P and T are improvements.120 Te type o statements must also be taken into account. Te De nitions occupy a privileged place in philosophical exegesis. Te Porisms are particularly prone to the vagaries o transmission because they may easily be conused with additions. 121 118
119 120
121
Tere is the addition o the case o gures in thePropositions (Heib.) .25, 33, 35, 36; .5. Te copyists ascribe them to the version o al-Hajjâj, and even to his second version i al-Karâbîsî is to be believed. See Brentjes 2000: 48, 50. Other cases are also added in .37 without al-Hajjâj being mentioned. See De Young 1991: 657–9. For my part, contrary to Knorr, I believe that the criterion o improvement does not apply or .19 or .17. I also believe that the proos o P in one case and the proos o the indirect tradition in the other are corrupt. For .19, see Vitrac 2004: 10–12. For .17, see Euclid/ Vitrac 2001: 369–71. Heiberg 1884: 20 observed that with the De nitions and Corollaries (Porisms) ‘die Araber … sehr rei verahren haben’. In act, it is not even simple to say exactly how many Porisms there are in the Greek text. Heiberg identi es 30 o them as such but makes a second Porism
115
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(4) o explain this state o affairs, I see at least two explanations, that perhaps work in tandem: • Either our different witnesses o the text re ect a general contamination122 and a global criterion – at the scale o the complete treatise – cannot be reached. • Or the principles that underpin the local criteria are inadequate. I certain branches o the tradition have epitomized the Elements, then the principle used by Klamroth and Knorr that the text o the Elements grew increasingly ampli ed proves inadequate. Tese principles may also miss their goals i it is not possible to identiy the motivations o the ancient re-editors when they sought to improve the orm o a mathematical text. We have seen that the criterion o mathematical re nement is sometimes difficult to use. (5) Certain characteristics o the preserved versions and different external con rmations have convinced us that there has been both contamination and epitomization. Tus, not only is the text o the version by Ishâq, as revised by Tâbit, without any additional deductive lacuna in Books – , but the medieval evidence teaches us that the revision o Tâbit implied the consultation o other manuscripts and, con123
sequently, the collation o alternative proos. In so doing, various versions o the Greek or Arabic texts, i not contaminated by, were at rom what, in the manuscripts, is nothing more than an addition to the Porism to .20 and an insertion o a heading [Porism] beore the large recapitulation ollowing .111, although he did not do this or the summary ollowing .72! For feen Porisms, there is oneor more Greek manuscripts in which the heading
is missing. Fifeen Porisms are placed beore the standard clause (‘what ought to be proved’), particularly true or P. Eleven are inserted afer the clause. Normally, a Porism begins with the expression, ‘From this, it is clear that’ (‘ἐκ δὴ τούτου φανερὸν ὅτι …’), but in seven cases ( .5, .20, .11, .9, .111, .114 and .17), the ormulation is not canonical. Te possibility o conusion appears in the act that ten Porisms retained by Heiberg were ampli ed by inauthentic additions. I the indirect tradition is consulted, it ought not to be orgotten that two Porisms rom the Greek are related to substitutions o proo ( .31, .5) and to an addition ( .114) which do not exist in this tradition. Tus, it is not at all surprising that these Porisms did not exist in it. By holding to comparable cases, the indirect tradition counts eleven Porisms rom the Greek, but two exist
122
123
in a different orm. Te Porism to .111 exists as a Proposition and the one to .17 appears as part o a proo (as is also the case in certain Greek manuscripts). Tis ‘πόρισμα’ exists only in the margin o P and not in the other manuscripts! It may be remarked that neither has the standard ormulation and that the indirect tradition has none o the other ve Porisms ‘heterodox’ to the Greek text. For the others, their number decreases (to seven rom nine in – to which could be added three supplementary Porisms rom the Ishâq–Tâbit version (to .14, 15; .5), to one rom our in , to nil rom six in – ). Tis is the opinion o Brentjes, at least as concerns the Arabic and Arabo-Latin traditions. See Brentjes 1996: 205. See Engroff 1980: 20–39.
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least compared with each other in order to produce Tâbit’s revision o Ishâq’s translation. Tere is no reason or astonishment: these scholars were not working to provide guidance to modern philologists who want to establish the history o the text o the Elements. Tey sought to procure a complete and stimulating mathematical text. Knowing the hazards o manuscript transmission, they compared different copies, and I believe that Tâbit ibn Qurra used other Arabic translations, probably that o al-Hajjâj, and even some Greek commentaries, in particular that o Heron o Alexandria, which has some consequences or the structure o the revised text. At some points, it is more sophisticated than the Greek text o Heiberg.124 In the Arabo-Latin domain, the Gerard o Cremona version also proceeds by juxtaposition o dierent texts, some o which Tâbit had already combined, but also the alternate proos that the tradition attributes to al-Hajjâj and which ofen appears in the Latin o Adelard o Bath. (6) Te case o the translation (or translations) o al-Hajjâj is much more difficult to judge because we know it only very incompletely and indirectly through several citations by copyists o manuscripts o the Ishâq–Tâbit versions and through the evidence o ûsî and pseudoûsî.125 Virtually all the characteristics that distinguish it – primarily its thinness and the structure o several amilies o proos – appear in the Arabo-Latin version o Adelard o Bath. 126 Its antiquity and its thinness make it tempting to ascribe to it a privileged role. Nonetheless, the evidence rom the preace o the Leiden Codex introducing the commentary o an-Nayrîzî is troubling.127 Te principle o ampli cation, to which Klamroth (and Knorr) subscribe concerning the textual development, suppose that no deliberately abridged version has played a role in the transmission o the text. It is to precisely this phenomenon o abbreviation that the preace to the second translation (or revision) o al-Hajjâj makes reerence. Tus, I am not sure that this principle, 124
125
126
127
Tis is particularly clear in Books – , rst o allor the alternative proos proposed or .22–23, then the insertion o the converses to Prop. (Heib.) .24–25 and the simpli cation o the proo o .2, nallythe addition o the Propositions (Ishâq–Tâbit) .30–31 to simpliy the proos o .32–33 (= Heib. .30–31), without orgetting the addition o Porisms (c. n. 121). See Engroff 1980: 20–39. Recently Gregg de Young has discovered an anonymous commentary relatively rich in reerences to divergences between the versions o Ishâq–Tâbit and al-Hajjâj. See de Young, 2002/2003. wenty structural divergences are supposed to characterize the version o al-Hajjâj. O these, sixteen appear in Adelard. Te other our rom Book and the rst part o Book – thelost portion in Adelard’s translation – appear in the related Latin versions by Herman o Carinthia and Robert o Chester. See the text and French translation in Djebbar 1996: 97, 113, partially cited below as n. 142.
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which unctions rather well in the case o dichotomy 1, also applies to dichotomy 2.128 (7) Moreover, the case o the stereometric books, onwhich Knorr ounded his argument, seems problematic to me. Te Arabo-Latin translations are particularly close to each other in these books. Knorr relied on this point to deduce that the same thing would happen to their Arabic models and thus also the versions o al-Hajjâj and Ishâq–Tâbit.129 What I have called dichotomy 2 hardly occurs there at all.130 However, there are, in two manuscripts o this last version (Copenhagen, Mehrens 81; Istanbul, Fâtih 3439), glosses indicating that Book is the last which Ishâq has translated and that what ollows is ‘Hajjajian’. Te author o the gloss to the manuscript in Copenhagen speci es exactly that it 131 ‘comes rom the second translation o al-Hajjâj’, i.e., the abridgement. From this reerence, Klamroth deduced that Ishâq had translated only Books – and that Tâbit had taken – rom the translation o al-Hajjâj. Tis thesis has been challenged by Engroff and I obviously have no expertise on this point, but it seems to me that the stereometric books undeniably constitute a particular case.132 Even then, at-ûsî had remarked that there is no structural divergence between what he believed to be the two versions o the stereometric books. 133 I would add that there is not, to 134 my knowledge, any mention o the sort ‘Tâbit says …’ beyond Book . A nal element must be taken into account. In Proposition .11 it is established that the side o a pentagon inscribed in a circle with a rational diameter is irrational, o the ‘minor’ type. Tus, in Book , ‘ἄλογος’ is translated as ‘asamm’ (‘dea’) by al-Hajjâj and ‘ghayr muntaq’ (‘un-expressible’) by Ishâq–Tâbit. Te divergence appears, or example, between Avicenna and the manuscript Petersburg 2145 on the one hand and the other Ishâq–Tâbit manuscripts on the other 128
129 130 131 132
133 134
It seems to me that Brentjes equally admits the idea that the so-called al-Hajjâj version No. 2 represents an improved and abridged re-edition. See Brentjes 1996: 221–2. See Knorr 1996: 259–60. See able 2 o the Appendix. See Engroff 1980: 9. See Engroff 1980: 9–10, 12–13. Let us add that at the end o Book in themanuscript ehran Malik 3586, a gloss indicates that Tâbit ibn Qurra had revised only Books – and that Books , and are Hajjajian! See Brentjes 2000: 53. See Rommevaux, Djebbar and Vitrac 2001: 275, n. 184. In the anonymous commentary cited above at n. 125, the reerences relative to the divergences between the versions o Ishâq–Tâbit and al-Hajjâj stop afer the rstthird o Book . Tis observation is well explained in the line o the gloss inserted in the manuscript ehran Malik 3586 (c. above, n. 132).
Te Elements
and uncertainties in Heiberg’s edition
hand.135 It is interesting to note that in Proposition (Heib.) .11 (= IsT 14), the manuscript Petersb. 2145, as well as ehran Malik 3586 and Rabat 1101, record ‘asamm!’136 Tis does not necessarily mean that Ishâq did not translate Books – ,137 but it at least suggests that at some moment o transmission, the stereometric books existed only in a single version.138 Tis homogeneity, recorded by ûsî, might even be the cause o the glosses inserted in the three manuscripts o the Ishâq– Tâbit version that I just mentioned.139 (8) wo consequences may be drawn rom the preceding considerations. First, Knorr’s hypothesis that the indirect tradition derived rom a single Greek archetype, based only on the stereometric books – in act only on the portion .36– .17 – is challenged. Second, I have said that there are, in the versions o al-Hajjâj and Ishâq–Tâbit, three and two deductive lacunae respectively. Tose o Ishâq–Tâbit occur in Book . But, i the hypothesis o Klamroth or one o his variations is adopted, we know the translation o Ishâq–Tâbit only or Books – . Te translation here is without deductive lacunae, which, considering the work o the Reviser, is to be expected. As or the translation o alHajjâj, the evidence o the preace in the Leiden manuscript suggests that it could scarcely be other than an epitome! (9) Tese consequences being noted, it ought not to be orgotten that it is thanks to the indirect tradition itsel that we have been able to determine some o its limitations. Te medieval versions, notably those o Ishâq–Tâbit and Gerard o Cremona, are more attentive to problems o textual srcin than the Greek manuscripts and thereby more inormative about the divergences between versions observed by their authors. Te contamination is clearly not the doing o medieval scholars only. Te subject o double proos demonstrates this. Te abundance o additional material and local alterations o the 135 136 137
138
139
See Rommevaux, Djebbar and Vitrac 2001: 259, 288–9. I thank A. Djebbar or this inormation. It is possible to doubt such an abstention by Ishâq given that there are two series o de nitions or Book in ehran Malik 3586, the latter being attributed to Hunayn ibn Ishâq and, probably, there was some conusion here between the ather and the son (see Brentjes 2000: 54). However, Ishâq may well have brought his translation to an end with the De nitions or Book , which have been (piously) conserved, though he did not translate what ollowed. Tus, one again arrives at the thesis o Klamroth. Although she disagrees with the thesis o Klamroth, Brentjes pointed out that in regard to De nition , the rst version o ehran Malik 3586 (the Ishâq–Tâbit version) and the version given by al-Karâbîsî, who, (according to Brentjes), ollows Hajjajian version, have minuscule differences. See Brentjes 2000: 53. Tis seems to me to concur with the preceding remark. See above, nn. 131–132.
119
120
sort o post-actum explanations in Byzantine Greek manuscripts (c. above the example o .1) shows that the Greek text is itsel enriched through recourse to the relevant elements o the commentary, probably through the intermediacy o marginal annotations by simple readers or by scholars. (10) Te intervention o the epitomes in the indirect tradition is quite probable. Tere are, however, different ways o abridging a text like that o the Elements. An editor could eliminate portions considered inauthentic or some theorems dealing with a theme judged too particular. Regroupings could be made. Abbreviated proos could be substituted, using in particular the previously discussed ormulae or potential and analogical proos or by removing the uninstantiated general statements, which are ofen less comprehensible than the example (set out in ecthesis and diorism) accompanied by a diagram and labelled with letters. More radically, all the proos could be removed, and only what Bourbaki called a ‘ascicule de résultats’ might be retained, or some number o books no longer considered indispensable might be cut out. In this case, the very structure o the treatise and its plan, which have ofen been criticized, would be changed. Such recensions are not at all rare beginning rom the sixteenth century, but in the majority o ancient and medieval versions, even in a recension like that o Campanus which introduces numerous local changes, the Euclidean progression through thirteen books is maintained, even i at some stage supplementary books ( , , , …) were added. Alternatively, the other operations o abbreviations listed above are all mentioned in the medieval preaces, such as those o al-Maghribî,140 the recension now called pseudo-ûsî141 or the Leiden Codex, wherein the authors described recensions or epitomes. Moreover, as we have noted above, according to the preace o the Leiden Codex, al-Hajjâj, in order to win the avour o the new Khali al-Ma’mûn, improved his rst translation ‘by rendering it more concise and shortening it. He did not nd an addition without removing it, nor a lacuna without lling it, nor a ault without repairing and correcting it, until he had purged, improved, summarized and shortened it all.’142 It is possible 140
141 142
One can read a Latin translation in Heiberg 1884: 16–17, with several errors o identi cation about the cited Arabic authors (and even about the author o the preace! See Rommevaux, Djebbar and Vitrac 2001: 230, 239). It allows us however to have some idea o the liberties taken by the authors o recensions. Completed by Sabra 1969: 14–5 who corrects the identi cations and Murdoch 1971: 440 (col. b). It is taken up again by Murdoch in the article cited in the preceding note. ranslation in Djebbar 1996: 97.
Te Elements
and uncertainties in Heiberg’s edition
that this passage contains some rhetorical exaggerations or stock phrases about the improvement o a text. I the quest or conciseness seems hardly debatable, the preace indicates neither the motivations or the suppressions nor the criteria used to identiy the ‘additions’. It is conceivable that al-Hajjâj knew o other Greek versions, more concise than the text or texts initially translated, to which the phenomenon o the epitomization had itsel already been applied. (11) We know that at least one abridged version o the Elements had been produced in antiquity by Aigeias o Hierapolis. Mentioned by Proclus, he wrote thereore no later than the fh century o the modern era. Te difference with the second version o al-Hajjâj is that there is no evidence that it played a role in the transmission o the text. However, besides the obvious textual enrichment, it is not possible to completely exclude the intervention o one or several abridged Greek versions. Te relative ‘thinness’ o the al-Hajjâj version, as ar as can be known, can indeed be explained in different ways depending on the portion o text considered. Proposition .14, which treats the quadrature o the triangle (with the associated absence o .45), and Propositions .5, 7 and 8, which treat pyramids on a triangular base, proceed rom the same attitude, and, in these cases, there are good reasons to think that the srcin o this minimalist treatment has a Greek srcin. 143 For the absence o Proposition .37 I have noted that it was probably an accident o transmission. Te absence o the bulk o the additional material, o several De nitions in Books , , and and o the Porisms in the stereometric books may perhaps be explained because al-Hajjâj had identi ed them as additions. Similarly, several other Propositions missing rom his version ( .12, .11a– 12a, .16, .27–28), but present in the Ishâq–Tâbit translation, might be the result o additions lacking rom the Greek or Syriac manuscripts consulted by al-Hajjâj, or they might have possessed these assertions, but he judged them to be useless, as they very nearly are. (12) Te existence o abridged versions in Greek also made up part o the hypothesis o Heiberg, and he described the model o manuscript b in this way or its divergent part ( .36–
.17).144 Manuscript b is, however, very awed. It contains problems in the lettering o the 143
144
Let us recall that Proposition .6 and the Porisms to .7 and 8 are missing in manuscript b. For .14, Simplicius seemingly knew two versions o the theorem: the ‘rectangular’ version in his commentary to the Physics o Aristotle (CAG, 62. 8 Diels) and the ‘triangular’ version in his commentary to De cælo: (CAG, ed. 414.1 Heiberg). See above, pp. 81–2.
121
122
diagrams, saut du même au même, and even, as it seems to me, aults in reading the uncial script. Manuscriptb could thus be the result o a new transliteration, being more aulty since it was urther removed rom the ninth century, and produced (or reasons which elude us) at the same time as the copy, in the eleventh century, o the Bologna manuscript rom a model which was either truly ancient (the hypothesis o Knorr) or proceeding rom another archetype, such as an abridged version o the ‘Aigeias’ type. Here, I call upon the possibility o an ancient model, whereas Heiberg imagined a Byzantine recension. Whatever the case may have been, I do not believe that this really changes the attitude that the editor o the Greek text may have adopted toward it. Te appeal to b .36– .17 may prove useul or removing some cases o textual divergences between P and T, in the aorementioned portion. However, adopting these readings would probably create a philological monster which never existed. Perhaps it can yet be used to improve the edition o a similar Arabic version. Knorr wanted to adopt the text ob, rather than what he called ‘the wrong text’ o Heiberg, because he hoped that a comparison o the primary Arabic translations would permit the reconstitution o a Greek archetype o comparable antiquity or the remainder o the treatise. Tis reconstitution is impossible, at least or the present state o our knowledge. Tereore, the conception o a new critical edition o the Greek text seems useless to me or the moment. Te critical editions o the various identi ed Arabic, Arabo-Latin and Arabo-Hebrew versions would be preerable. It would be necessary to produce an ‘instruction manual’ or the reader to navigate these versions according to the problem, the time period, the language o culture, even the Euclid available to (another) interested author. Such a manual would be especially necessary in the cases o double proos or substitutions o proos, cases which the indirect tradition has considerably enriched. Tis necessity has long been perceived by the historians o the medieval and modern periods. Undoubtedly, the Hellenist would also admit the same necessity. Te movement to ‘return’ to the srcinal which inspired the work o the philologists o the nineteenth century seems to need a break. A less partial knowledge o the indirect tradition provides us not only with much richer inormation at a local level, but also with more uncertainty about its ancient components. Tus stripped o our (alse) certainties, we may eel a little rustrated, but the hope remains that new discoveries o ancient papyri, manuscripts o medieval translations o Euclid or o its commentators will allow us to move orward.
Te Elements
and uncertainties in Heiberg’s edition
Appendix Te appendix contains three tables (each describing one o the breaks observed in the textual tradition o Euclid’s Elements). I have used the ollowing abbreviations: D., De nition; Post., Postulates; CN, Common Notion; Prop. proposition; Por. Porism (= corollary); Te notationN/N + 1 designates the lemma between Propositions N and N + 1. Brackets indicate portions considered inauthentic by Heiberg, but which exist in Greek manuscripts. (+) or (−) signiy the presence or absence o a textual element, respectively; (÷2): usion o two elements into one; (× 2): subdivision o an element into two. aliter marks the existence o a second proo, possibly partial (indicated by ‘p’) or the existence o a second de nition. Ad., version called Adelard I (Busard 1983); GC, version attributed to Gerard o Cremona (Busard 1984); gr.-lat., Greco-Latin version (Busard 1987); Heib., Heiberg’s edition; IsT, Ishâq–Tâbit version;P, manuscript Vatic. Gr. 190; T, Greek manuscripts called Teonians (on P / T, see above, pp. 82–5); mg., marginalia.
123
a
; –)( 71 ,5 .1 );– ( 7– 5. . D ) (− 7– .3 . ;D ) + (l an itio dd a. . D
) n itoi da tr ni ta Lob ra A dn a ibc ra laA ve id e m
;
s u s er v
tns e m lee la ut xe
oin itd e s’ g erbi e H (
1 y m o t o h ic D
;) (+ 13 –0 .3X I itb â ;) – T -q (6 âh .1 s X ;I) ;)– (– (}o }o g
o pe y
ec ne gr e ivd
sn oi ti n e D
) (– 28– 5 .I2 X . ;)D (– 32 .I X . D
;) –( }o
lg u v
83 .I {;X gl lu )– u v }( v 3 o 22 1. gl . X{ vu ; { )– 17 –(); 13(. X.1{ } X ;) b ;) –( o lg – 4 u () 1 v 02 ii(i 1– 2 . II .9 1 v{ ;X .1 ;) ) X; – ) (– (8 (– 21 – 4 . 7 2 II . . I
sn itoi ops o Pr
X
;) (–. ro )–; P1 ( .1 r.o IX; P )+ )7 ( ro or ( P .4 5. V IX ;) ); – + .(r (r oP oP
X
–)( 1I.4 I X );– ( 13.I I X );– ( .6I I X
V I.5
,} .r oP 1 .I3 I {;I ) + ( } orP .4II {
s m s rio P
IX .2
;) (– .1r oP 0 .2 I V ,r. Po 91 . V
; –() .r oP 32 . ;)X (–. ro P .9X )–; ( r.o P 6. ;X –)( . orP .4 X
;) (– ro P 7.I I ;)X – .(r Po 5 .I3
)– (. orP 1I7. I
);–( r.o P 33 I. ;)X (–. ro P 41 .1
(.–); ro P 61 .II I X )–; ( ro P .8I
X
X
I X
I X
la rei ta m la oni ti dd A
)– / + (
;) –( 27, 42 ,0 2 I.I I in sea c ali ce pS ;) –( } .1I II ni e acs la ic ep -so d eus {P se as C la ic e Sp
)–; ( 8. V ni es ac ali ce pS ); + 37(, 6 ,53 3I. II ni es ac ali c peS
) (– }3 2.I X in es a lca ic ep ;{S –)( 7} .I2 V ni e acs la cie Sp{
;) (– 1. X ot no tii dd A );– ( }3 .3I V
ot no iti dd {A –);( r}o. P 61 .II I ot oni ti dd A {
s oni ti dd A
1,4 ,0 4, 93 83, , 37, 63 . X ot ;3 /23 3. X o ,r.t Po 32 . X o 8,t1 01, .X ot { }s no iitd d {A 21
; –)( }r
)– ( 81 .II
) (– }r
); itel etli + a ( } a p8 I X re .5 I 1 I t to lia {XI .III ?)( .1 ;) {X n X{ + ( ); itoi ); er –( ti }s – dd ( la is } A re 3 e }0; tli .–1 htn –95 3a1. XI; /sys .8 II ) isly X ;V – ( to; 03 r}e ana ii) ,p ilt yb se .02 ap r ri IV 7 eit se( ;p 1I.I la 5 X ,8 X . p7 ;22 –1I. D . . I ot I{II XI{ XI{ s oo rP el bu o D
er af a m m e ;4L 1/ 3 .I1I I X
3;/.2 I II X
;2 /12 2. X };1 /02 .X2 ;9{ /81 .1X ;7 /61 X .1; 41 /3 1. X ;0 1/ .X9 3; /22 2.I
V
c
as m m Le
}3; 3/ 2 .3 {;X }2 /13 3. X { }0; 3/ 9 .2 ;{X} {2, }1 { 9 /82 2. X ;} 8 /72 2. {X
;5 /4 . II X 3/; 2. II ;4X 2 32/ .I X ; 6/0 95 . ;4X 5/ 3 .5X ; 421/ 4. X };5 /43 .3X ;{} 4 /33 81 3. I.I X { XI
yl ni tar ec lla er a eyh tt ha gtni ni tan ia m el hi w hte ll a, sr eh to sp e k,e }){ ( e m o ess si m si d ly ict i plx e gr bei e H t.n er ffei d lyt ghil s si
n ios iv id usb ,n oi us e,c n
22 –9 .I X . ;0D 2– 71 .II V . ;D 41 – 1.I3 I
;2 3– 9 .2II V ;2 2– 12 .II V ; 32– 13 I.
D .V ;4 –3 .I V . ;3D 1– 1.2 .V D
;V02 –8 .I1 ;3V 1– 9.I V ; 1–3 21 . V
. . orp D P In In r dre o in s gne ah C
;9 –8 . II X ;4 3 33– .I X ro.; P1 1 –11 11 . 6;X2 –5 .X2 ;5 –41 1. ;X 11– 0 .1 X
51 –4 .1II I X
12;– .I8I I X ;5 4I.– II X
;2 –11 .1I I X
73 .I ;7X 01 –5 0 .1X );n oi ctu r nsto C ( 33 .II I ;7 ,3p 13 ,p 41 . II I
oo pr o oni utti bst Su sn ito ac dio M
r.o P 51 + .r oP 4 .I1I
) oni ist poo rP a o nit de orm nas r (t. ro )2 P 2 ÷ ( .7 2– ;.X 1.I or P .X 1.I D II
I V
32 . ;5V 2.I I I
C G = 16 C G = 51 d. A = 71 –6 .I1 II V
)2× ( 43 .I X ;) 2 (× 1 .3I X ; .)p o Pr ni ( n ito a m or nsa tr
seba e,c ne se rp y in g si )2 × ,)( 2 ÷ ( )–, ( ), + (: ec ne re are sa se rev s )9 8. ,ep v bao edn ed sa ( oni itd ar t al
.andr oP 11 1, 111 .X n iso r ven 93 I 1
≠ nso tai lu m r Fo
l tao
:s et o N
deiev m eh ,et lb at ish t In a
)}. 2 ,{} 1{ 92 /8 .2 o no i pet cx e e bli s osp e ol asm esh t m eL iht eh (w t d o eta sea ol c rp e et T in
.8 .n2 ee s, o
.n itio adtr si ht in yl ev tic pse er
lg u v
o gn in ae m eh t n O b
c
ot kos o B n I a
)a no m er C o d arr e G yb de v eic er n ditoa trt i âb T qâ– h Is s sru e v
nso tii ard t na di lare d A (
2 y m o t o h ic D
C G ni ) (– 9I. .X
d. A ni )– ( 22 .I X .
C G ni re itl a
.I X . D D D 2
) C G ( ni ) + ( anl o iitd da .6I V r,
o t kso o B In
.d .d A A n in i) )– – ( re ( 1,8 t 17 1 .V ,01 . .1V lD ; . an oi D ;2 ti e . d ilt IV d . si a 4a b , 9,;D .17V I.2V .6I II Is Is . . . D D D
st ne m el el a txu e
s on tii n e D
ec ne gr vie d o pe y
)– / + (
.d A ni la oni itd .d da A 61 in C –5 )– G 1 ( n .II 5 il V 3I–. an T I Is V o . itid . d D ;D }5 isba o 4,{, I.9V tion tui .I3 V Is ts . . b D D Su
;r
.d A ni ) (– 5}/ 4 N {C
no it no on m m o C
.d A in )– ( a)( 21 11– . II I V
;2 1.I
V 7; .I3I ;5I 4. I
sn oi ti so po rP
C G in ) + ( anl io idt ad 5 –2 .I24
d. A in ) (– 32; 8 II 2 V – 72 T Is X.
C G in ) + ( 5 –41 .I1
C G in 5 .I3 to no it id ad
II V
C G ni )2 ° r.(no P0 2.I V
s m si orP
ot s m isr oP noal iitd d A
r= e itl a
p
53 .I
la rie ta m anl o tii dd A
nso tii dd A
C G ni 01 –9 ,7 , .5V . D o nost iit dd A
tei l a ; 33 etr li p, 1 a
,53 2, 0 ,91 . C III G e;r in tli 45 a . 4 X II. at er; oni ital itd p d .44 A I
el ubo D
5. V r; et il a
51 8,, .5 IV r; et li a
6 ,53 .I3 II
re t li et a il 0 a .3 I
;r
3 X –22 t;re 2.I li a II 5 ;rV 11 , tei 1 l 1 a 1 13 ,1 ,2 ,9 ,92 07– I. 8 V 6 ,3 er;t 3 li ,0 a 3 8 , .V1 X6.
so C rop nG i
C G in )( + 33 /2 3. X
sa m m eL
) 42/ 14 gr eb ei H . (C 04/41 .X C G
3
; 20– 9 .I1 II V
3 2–1 2I. I V . D
. D In
s gne ah C
; 1–3 7.I ;VI6 2– 32 .I V
.p or P nI
re d or in
51 – 1–4 21 01– . ;7X 2– 6 –52 .X2 I 0;2 –9 –41 .1 IX 2;1 –1 1. X I
C G ni 32 –2 2.I
II V
,0 .2I V ;8 ,61 . V ni o o pr o
.d A in 07 –8 6. taX oo rP o
11 II. .I D ni no it lau m ro tionu tionu son t t o tsi its iiat bu bu ra S S V
ns ito uit st ubS
nso tai c dio M
oo pr o
n)o tir poo pr su ou ni tn oc .(d A ni .4V . D o t enm e acl pe R
9 N C ce rg = 01 N C C G = st.6o ,P. d A
9– 1 ro. II. P ro 51 . ≠ IV ts ro;. ne P m 5 et .I1 tSa
.d A ni 1I4. I ro oni t iar a l’veg n ira ‘
) 17– 61 I. II
II V
ro gn i ert t le in oi t ira a V
) C G in e blu od / d. nA i el p m is (
V bi. e H = ( 61 .I II V
r.o P 51 . II I V
.+ C ro G P 51 II14.I = V .d C G A =
.I II V
oni ti s poo rP leg n si a ot in 03 92– . Xo n iso uF
)n o tii da rt n iad r lea d A hte n (li ova m re dn a
C G in sn ioti osp or P ru o ot in 32– 1 .X3 o iionsv id b Su
no it id ar t ani rda le d A het ni e hret ot ni ro 83
≠ s oni t lau m orF
atl o
et: o N
a hit inr a C o nn a rem H o s sionr ev eh t, eiw v ot ino pl rau tc ur ts a m or F )1. 00 2
is. d A seu ac eb no tia re di sn coo tn i sn ios r ve see th ek at ot ray ss ec en ist .nI o iti
drsa u B ( J + )2 99 ,1s tr e lok F dn a dr sau (B C R + .d
rtad eh t ot gn ol be )5 00 2 dr as u (B su na p m a C dn a ) 79 –21 79 1– 76 19 dr as u B (
no:A i itd ar t ani rda l de A a
sti e T to .) no huc iitd m ar oo t t na m id he t arl g ed in A xi e m tho out se thi it w i ( arl sie uc il i rta m a p la alr ut ut xte cu o rt tw s s eh e t so bei pat .n rc x o i as jau sr e s stli no vit ai m b ce er â esp oC âq–T rd hs .T )x are eI h ko G t o o seh n c B o oisr oa dr e rp ih v p tt e ar sr .T eh j t eh âjja oe t h dn -H l t, a ae ino ik ti ko lg dar o hin t B t na o em id ss o ra lo els del het od A hg m hte uo an ot hrt oe rla ( c i edt ned ism itla ne tis u p s m de r
to s ok B In
.t a .rl g & T nP in i tss d xei an h o g y lv g u .b 31 m . X In
to sk oo B nI
P ni dn ah et la a .t .t y 9 al al .bt .2h r.g r.g al ry & P & P .r x n n g O T di T di & .p a ni n ni n sit hae ist hae nT inP xe alt xe alt sti its s o a o a i ex lg y lg y x e to u u v .b v .b r. n o s g g 0 2 .I2I m .I2I m .4P oe n n II D V I V I
2
T su re v
P
3 y m o t o h ic D
a
tn e m el el a txu e
sn itio so po rP
ec ne gr e ivd o ep y
)– / + (
s m si orP
P ni dn ha et al a inT by. st g ixs m e. in. orP orP 4. 4. V V
t.a -lr. g ni &
t. T a-l. P ni rg ni to n 1 i d tun & na b Th inP stin .byg st s m ixs ixe in e. r. r. orP oP oP 7. 1.9 1.9 V V V
P ly on stin si x .re Po 11 . X I
ali re ta m la no i itd d A
t. la. rg P n & di n inT etha 72 la I. a V y in .b + g sae m n C I
t.a .lr g & T in P tss din xi n 2e° eha t N .r la oP ay 61 .b . g III m nI nI
es ac l ica ep S
nso iit dd A
nda tn e m et at s e P ht ni o to sn tnu oita b, c .t id a .rl om g h in ti & w ’ T torc in e st ‘sr si o xe 3 4. 3.I V V nI nI
P in dn ah e atl a by. g m nI .t a .rl g dn a T n onii ss uc isd eh to
.t la. gr & P Tn ni i st d si an xe h r. y lite g a .b 1. m X In 1
.t la. rg & P Tn ni i st d si an xe h r. y lite g a .b 6. m X In 1
P ni ts xei ont eos .tD l.a rg & T ni st xie .r
t.a l. gr P & in T dn istn hae ixs alt .re ay ite et b . la li g 1 a m .3 .4II nI IIV
so or p el ubo D
.t la. rg & P Tn ni i st d si an xe h r. y itel g a .b 9. m X In 1
8 –72 2I X . D n rsioe vn I
P ni .c) ed o
s.;do ci (
.t al .r g & T ni .s) o i;c ecd. od (
o no i idt da ht i w 1I. X
ino o Pr
V. D o oni rs ven I
P ni de tp u orc 1.9 X I o oo rP
. D ni
≠ s oni t lau m ro F
P ni 7– 6
s gne ah C
re d or in
ns ito ac dio M
o T cea in pl dn in a d’ P e in pip ≠ el sn el oi lar t a anl idp px ol e ‘s
T itn ce rro c
er het ne h W .r het o e th n tio n nda sn oi rse v o w t het o e on ni st s xei hc ih w a m m e
1I7. I X
ni nig re T tte ni l 8 o .I3 no X it r a ’oe icd ubc o ‘ M
8
t.a .l gr ; & d n T ah in eta e)l la nga by ir , t P (= ni . ’n .n Por ono lipra 51 ig u . rt s V I :‘ n .,r ro ito oP P id 5. 91I. ad V I V &
1 d anh yb n iP tx et ni ’s od ie ‘
r.o P .7I I X
oL ;sn no isit op or P het ro re d or in eg na ch o ,n) !(
71
atl o
et: o N
rop o n ito uti ts usb o N a
a yb ro 1 dn ah = ts yip oc he tyb ( P o s oin ti dda anl ig ra m eh t in ylt os m sr uc co cen er e iffd e T . T. ni T sa yli P m ni a e eh m t sa o eh yp t o s c ay ha w ti al w is n re o rdo tail u hte ns ,o oc or re p f e a) blu dn od ha a et is la
130
Bibliography Editions and translations of versions of Euclid’sElements
Besthorn, R. O., J. L. Heiberg, then G. Junge, J. Raeder and W. Tomson (eds.) (1893–1932) Codex Leidensis 399/1: Euclidis Elementa ex interpretatione al’Hadschdschaschii cum Commentariis al’ Narizii. Arabic text and Latin trans., Hauniae, Lib. Gyldendaliana: , 1 and 2 (= L. I), 1893–7; , 1 and 2 (= L. H.and ), 1900–5; , 1,(eds.) 2 and(1992, 3 (= L.RC)–Robert ), 1910–32. Busard, L. L. and M. Folkerts o Chester’s (?) Redaction o Euclid’sElements, the so-called AdelardII Version. Basel. Busard, H. L. L. (ed.) (1967–1972–1977, HC) Te ranslation o the Elements o Euclid rom the Arabic into Latin by Hermann o Carinthia (?) . L. – in Janus 54, 1967, 1–140; L. – , Janus 59, 1972, 125–187; L. – , Mathematical Centre racts84, Amsterdam, Mathematisch Centrum, 1977. Busard, H. L. L. (ed.) (1983, Ad. I) Te First Latin ranslation o Euclid’s Elements Commonly Ascribed to Adelard o Bath. oronto. Busard, H. L. L. (ed.) (1984, GC) Te Latin ranslation o the Arabic Version o Euclid’sElements Commonly Ascribed to Gerard o Cremona. Leiden. Busard, H. L. L. (ed.) (1996) A Tirteenth-Century Adaptation o Robert o Chester’s Version o Euclid’sElements. Munich. Busard, H. L. L. (ed.) (2001, J) Johannes de inemue’s Redaction o Euclid’s Elements, Te so-called Adelard Version. Stuttgart. Busard, H. L. L. (ed.) (1987, gr.-lat.) Te Mediaeval Latin ranslation o Euclid’s Elements Made Directly rom the Greek. Stuttgart. Busard, H. L. L. (ed.) (2005) Campanus o Novara and Euclid’sElements. Stuttgart. De Young, G. (1981) ‘Te arithmetic books o Euclid’sElements’. Ph.D. thesis, Harvard University. Engroff, J. W. (1980) ‘Te Arabic tradition o Euclid’sElements: Book ’. Ph.D. thesis, Harvard University. Euclide d’Alexandrie, Les Eléments (Euclid/Vitrac). ranslation and commentary by Bernard Vitrac. 4 volumes: (general introduction by M. Caveing. L. – ): 1990; (L. – ): 1994; (L. ): 1998; (L. – ): 2001. Paris. Heiberg, J. L. and H. Menge (eds.) (1883–1916, EHM) Euclidis opera omnia. Elementa i–iv (1883); El. v–ix (1884); El. (1886); El. xi–xiii (1885); El. xiv–xv,Scholia, Prolegomena critica(1888); Data, Marini Commentarius in Eucl. Data(1896); Optica, Opticorum recensio Teonis, Catoptrica (1895); Phaenomena, Scripta musica, Fragmenta(1916); Supplementum. Leipzig. Heiberg, J. L. and E. S. Stamatis (eds.) (1969–77, EHS) Euclidis Elementa, post Heiberg ed. E. S. Stamatis. , El. i–iv (1969); , El. v–ix (1970); , El. (1972); , El. xi–xiii (1973); , 1. El. xiv–xv,Scholia in lib. i–v (1977); , 2. Scholia in lib. vi–xiii (1977). Leipzig. Peyrard, F. (1814–18) Les œuvres d’Euclide en grec, en latin et en rançais. Paris.
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and uncertainties in Heiberg’s edition
Editions and translations of commentators
Pappus o Alexandria Tomson, W. and G. Junge (eds.) (1930) Te Commentary o Pappus on Book X o Euclid’sElements, Edited and translated W. Tomson, comm. G. Junge and W. Tomson. Cambridge, Mass. (Reprinted. New York 1968.)
Teon o Alexandria Mogenet, J. and ihon, A. (eds.) (1985) Le ‘grand commentaire’ de Téon d’Alexandrie auxables acilesde Ptolémée, Livre , Studi e esti 315. Vatican City.
Proclus Lycaeus Friedlein, G. (ed.) (1873) Procli Diadochi in primum Euclidis Elementorum librum Commentarii. Leipzig. (Reprinted: Hildesheim 1967.)
al-Mâhânî Abû ‘Abdallâh Muhammad ibn ‘Isâ al-Mâhânî,Risâla li-al-Mâhânî î al-mushkil min amr al-nisba (Épitre d’al-Mâhânî sur la difficulté relative à la question du rapport). Edition and French translation in Vahabzadeh 1997 . (Reprinted, with English translation, in Vahabzadeh 2002: 31–52.) Abû ‘Abdallâh Muhammad ibn ‘Isâ al-Mâhânî,asîr al-maqâla al-‘âshira min kitâb Uqlîdis. Edition and French translation E ( xplication du Dixième Livre de l’ouvrage d’Euclide) in Ben Miled, M. (2005) Opérer sur le continu: raditions arabes du Livre X des Éléments d’Euclide. Carthage: 286–92 [completed with an anonymous commentary,ibid.: 296–333].
al-Fârâbî Abû Nasr Muhammad ibn Muhammad ibn arhân al-Fârâbî , Sharh al-mustaglaq min musâdarât al-maqâla al-ûlâ wa-l-hâmisa min Uqlîdis . Hebrew translation by Moses ibn ibbon in Freudenthal 1988.
an-Nayrîzî Arnzen, R. (ed.) (2002) Abû l-‘Abbâs an-Nayrîzî, Exzerpte aus (Ps.-?)Simplicius’ Kommentar zu den De nitionen, Postulaten und Axiomen in Euclids Elementa I, eingeleitet, ediert und mit arabischen und lateinischen Glossaren versehen von R. Arnzen. Cologne.
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132 ummers, P. M. J. E. (ed.) (1994), Anaritius’ Commentary on Euclid: Te Latin ranslation – . Artistarium Supplementa . Nijmegen.
al-Jayyânî Abû ‘Abdallâh Muhammad ibnMu‘âd al-Jayyânî al-Qâsî, Maqâla sha rh al-nisba (Commentary on Proportion). Facsimile o the Algier manuscript 1466/3, o. 74r–82r and English in Plooij 1950; edition and French translation in Vahabzadeh 1997.
Ibn al-Haytham Abû ‘Alî al-Hasan ibn al-Hasan Ibn al-Haytham,Sharh musâdarât Uqlîdis. Partial edition, English translation and commentary in Sude , B. H. (1974) ‘Ibn al-Haytham’s Commentary on the Premises o Euclid’sElements: Books – ’, Ph.D. thesis, Princeton University.
al-Khayyâm ‘Umar al-Khayyâm, Risâla î sharh mâ ashkala min musâdarât Kitâb Uqlîdis (Epître sur les problèmes posés par certaines prémisses problématiques du Livre d’Euclide). French translation (based on the Arabic edition o the text by A. I. Sabra) in Djebbar, A. (1997) ‘L’émergence du concept de nombre réel positi dans l’Épître d’al-Khayyâm (1048–1131) . Sur l’explication des prémisses problématiques du Livre d’Euclide’ Orsay, Université de Paris-Sud. Mathématiques. Prépublications 97–39. Re-edited, with corrections, in Fahrang, Quarterly Journal o Humanities and Cultural Studies14: 79–136 (2002). Arabic edition o the text with French translation by B. Vahabzadeh in Rashed, R. and Vahabzadeh, B. (1999) Al-Khayyâm mathématicien. Paris: 271–390. Studies
Blanchard, A. (ed.) (1989) Les débuts du codex. urnhout. Bourbaki, N. (1974) Éléments d’histoire des mathématiques, 2nd edn. Paris. Brentjes, S. (1996)transmission ‘Te relevance o non-primary sources or theinradition, recovery o the primary o Euclid’sElements into Arabic’, ransmission, ransormation, Proceedings o wo Conerences on PreModern Science held at the University o Oklahoma, ed. F. J. Ragep , S. P. Ragep and S. Livesey. Leiden: 201–25. (1997–98) ‘Additions to Book in the Arabic raditions o Euclid’sElements’, Studies in History o Medicine and ScienceXV: 55–117.
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and uncertainties in Heiberg’s edition
(2000) ‘Ahmad al-Karâbîsî’s Commentary on Euclid’s “Elements ”’, inSic Itur ad Astra: Studien zur Geschichte der Mathematik und Naturwissenschafen , Festschrif ür den Arabisten Paul Kunitzsch zum 70. Geburstag, ed. M. Folkerts and R. Lorch. Wiesbaden: 31–75. (2001a) ‘Observations on Hermann o Carinthia’s version o theElements and its relation to the Arabic ransmission’, Science in Context14(1/2): 39–84. (2001b) ‘wo comments on Euclid’s Elements? On the relation between the Arabic text attributed to al-Nayrîzî and the Latin text ascribed to Anaritius ’, Centaurus 43: 17–55. Dain, A. (1975) Les manuscrits, 3rd edn. Paris. De Young, G. (1991) ‘New traces o the lost al-Hajjâj Arabic ranslations o Euclid’sElements’, Physis 38: 647–66. (2002–3) ‘Te Arabic version o Euclid’sElements by al-Hajjâj ibn Yûsu ibn Matar’, Zeitschrif ür Geschichte der arabisch-islamischen Wissenchafen15: 125–64. (2004) ‘Te Latin translation o Euclid’sElements attributed to Gerard o Cremona in relation to the Arabic transmission’, Suhayl 4: 311–83. Djebbar, A. (1996) ‘Quelques commentaires sur les versions arabes desEléments d’Euclide et sur leur transmission à l’Occident musulman ’, inMathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich , ed. M. Folkerts. Wiesbaden: 91–114. Dorandi, . (1986) ‘Il Libro degli Elementi di Euclide’, Prometheus 12: 225. (2000) Le stylet et la tablette: Dans le secret des auteurs antiques . Paris. Folkerts, M. (1989) Euclid in Medieval Europe. Winnipeg. Follieri, E. (1977) ‘La minuscola libraria dei secoli e ’, inLa paléographie grecque et byzantine, ed. J. Glénisson, J. Bompaire and J. Irigoin. Paris: 139– 65. Fowler, D. H. (1987) Te Mathematics o Plato’s Academy . Oxord. Freudenthal, G. (1988) ‘La philosophie de la géométrie d’al-Fârâbî: son commentaire sur le début du er livre et le début du e livre des Eléments d’Euclide’, Jerusalem Studies in Arabic and Islam 11: 104–219. Heiberg, J. L. (1882) Litterargeschichtliche Studien über Euklid. Leipzig. (1884) ‘Die arabische radition der Elemente Euklids’, Zeitschrif ür Mathematik und Physik, hist.-litt. Abt. 29: 1–22. (1885) ‘Ein Palimpsest der Elemente Euklidis’, Philologus 44: 353–66. (1888) Om Scholierne til Euklids Elementer(with a French summary). Copenhagen. (1903) ‘Paralipomena zu Euklid’, Hermes, Zeitschrif ür classische Philologie XXXVIII: 46–74, 161–201, 321–56. Irigoin, J. (2003) La tradition des textes grecs: Pour une critique historique . Paris. Klamroth, M. (1881) ‘Über den arabischen Euklid’, Zeitschrif der Deutschen Morgenländischen Gesellschaf 35: 270–326, 788.
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134 Knorr, W. R. (1996) ‘Te wrong text o Euclid: on Heiberg’s text and its alternatives’, Centaurus 36: 208–76. Lévy, . (1997) ‘Une version hébraïque inédite desEléments d’Euclide’, inLes voies de la science grecque: Études sur la transmission des textes de l’Antiquité au dix-neuvième siècle, ed. D. Jacquart. Geneva: 181–239. Murdoch, J. (1971) ‘Euclid: transmission o theElements’, inDictionary o Scienti c Biography, ed. C. C. Gillispie, vol. . New York: 437–59. Pasquali, G. (1952) Storia della tradizione e critica del testo, 2nd edn. Florence. Plooij, E. B. (1950) Euclid’s Conception o Ratio and his De nition o Proportional Magnitudes as Criticised by Arabian Commentators, Rotterdam. Reynolds, L. G. and N. G. Wilson (1988) D’Homère à Érasme: La transmission des classiques grecs et latins. Paris. Rommevaux, S., A. Djebbar and B. Vitrac (2001) ‘Remarques sur l’histoire du texte des Éléments d’Euclide’, Archive or History o Exact Sciences 55: 221–95. Sabra, A. I. (1969) ‘Simplicius’s proo o Euclid’s parallels postulate’, Journal o the Warburg and Courtauld Institutes 32: 1–24. Vahabzadeh, B. (1997) ‘rois commentaires arabes sur les concepts de rapport et de proportionnalité’. Ph.D. thesis. Université de Paris . (2002) ‘Al-Mâhânî’s Commentary on the concept o ratio’,Arabic Sciences and Philosophy 12: 9–52. Van Haelst, J. (1989) ‘Les srcines du codex ’, inLes débuts du codex, ed. A. Blanchard. urnhout: 13–35. Vitrac, B. (2003) ‘Les scholies grecques aux Éléments d’Euclide’, Revue d’Histoire des Sciences 56: 275–92. (2004) ‘A propos des démonstrations alternatives et autres substitutions de preuve dans lesÉléments d’Euclide’, Archive or History o Exact Sciences 59: 1–44. Weissenborn, H. (1882) Die Übersetzungen des Euklid durch Campano und Zamberti. Halle.
2
Diagrams and arguments in ancient Greek mathematics: lessons drawn rom comparisons o the manuscript diagrams with those in modern critical editions
Introduction In some ways, the works o ancient Greek geometry can be regarded as arguments about diagrams. Anyone who has ever looked at a medieval manuscript containing a copy o an ancient geometrical text knows that the most conspicuous characteristic o these works is the constant presence o diagrams.1 Anyone who has ever read a Greek mathematical text, in any language, knows that the most prevalent eature o Greek mathematical prose is the constant use o letter names, which reer the reader’s attention to the accompanying diagrams. In recentin years, particularly entitled due to a‘Te chapter in diagram’ Netz’sTe Shaping o o Deduction Greek Mathematics lettered , historians Greek mathematics have had a renewed interest in the relationship between 2 Research prothe argument in the text and the gure that accompanies it. jects that were motivated by this interest, however, quickly had to come to grips with the act that the edited texts o canonical works o Greek geometry, although they contained a wealth o inormation about the manuscript evidence or the text itsel, ofen said nothing at all about the diagrams. For years, the classical works o Apollonius, Archimedes and, most importantly, the Elements o Euclid have been read in edited Greek texts and modern translations that contain diagrams having little or no relation to the diagrams in the manuscript sources. Because they are essentially mathematical reconstructions, the diagrams in modern editions are ofen mathematically more intelligible than those in the manuscripts, but they are ofen histori3 cally misleading and occasionally even mathematically misleading. 1
2 3
In some cases, the diagrams were never actually drawn, but even their absence is immediately evident rom the rectangular boxes that were lef or them. N1999: 12–67. In this chapter,we will see a number o examples o modern diagrams that are more mathematically consistent with our understanding o the argument and a ew that may have
135
136
In act, a ew scholars o the ancient mathematical sciences have or many years made critical studies o the manuscript gures,and Neugebauer ofen called or the critical and conceptual study o ancient and medieval diagrams.4 Tese scholars, however, were mostly working on the exact sciences, particularly astronomy and, perhaps due to the tendency o historians o science to divide their research along contemporary disciplinary lines that would have made little sense to ancient mathematicians, these works have generally ormed a minority interest or historians o ancient mathematics. Indeed, in his later editions, Heiberg paid more attention to the manuscript gures than he did in his earlier work, but by this time his editions o the canonical works were already complete. In act, or his edition o Euclid’s Elements, it appears that the diagrams were adopted rom the tradition o printed texts without consulting the manuscript sources. In this chapter, afer brie y sketching the rise o scholarly interest in producing critical diagrams, we investigate the characteristics o manuscript diagrams in contrast to modern reconstructions. o the extent that the evidence will allow, we distinguish between those eatures o the manuscript diagrams that can be attributed to ancient practice and those that are probably the result o the medieval manuscript tradition, through which we have received the ancient texts. We close with some speculations about what this implies or the conceptual relationship between the gure and the text in ancient Greek mathematical works.
Heiberg’s edition of Euclid’sElements Heiberg (1883–8), on the basis o a study o manuscripts held in European libraries, prepared his edition o the Elements rom seven manuscripts and the critical apparatus accompanying his text makes constant reerence to these sources.5 Nevertheless, there is usually no apparatus or the diagrams and hence no mention o their source. 6 An examination o the previous
4
5
6
led to historical misunderstandings or this reason. Mathematically misleading modern diagrams, on the other hand, are relatively rare; Neugebauer discusses one example rom the edition o Teodosius’On Days and Nights prepared by Fecht. Neugebauer 1975:752; Fecht 1927. For example see the section D,2, ‘Figures in exts’ in his A History o Ancient Mathematical Astronomy. Neugebauer 1975: 751–5. Heiberg 1903 later published a moredetailed account o the manuscript sources and the reasons or his editorial choices. For a more extended discussion o Heiberg’s work on the Elements and a discussion o the overall history o the text see Vitrac’s contribution in this volume. While this is largely the case there are someexceptions. For example, thediagrams orElem. .39 and .15 are accompanied with apparatus. Heiberg and Stamatis 1969–77: , 73 and 166.
Diagrams and arguments in Greek mathematics
A
B
Δ
Γ
Δ
12.
B
B Δ
Γ
Vatican 190
Γ
A
ugust A
Figure 2.1 Diagrams or Euclid’sElements, Book
A
Heiberg
, Proposition 12.
printed editions o the text, however, makes it clear that the diagrams accompanying Heiberg’s edition were drawn entirely, or or the most part, by copying those in the edition o August (1826–9).7 Te August edition would have been particularly convenient or copying the diagrams, since, as was typical or a German technical publication o its time, the diagrams were printed together in old-out pages at the end o the volumes. Although nearly all the diagrams appear to have been so copied, a single example may be used to demonstrate this point. ForElem. .12, concerning the construction o a perpendicular to a given plane, the diagrams in all the manuscripts consist simply o two equal lines, ΔA and BΓ, placed side by side and labelled such that points Δ and B mark the top o the two lines. In Figure 2.1, we compare the diagram or Elem. .12 in Vatican 190, as representative o all the manuscripts, with that in both the August and Heiberg editions.8 While Vatican 190 is typical o the manuscript diagrams, that in Heiberg’s text is clearly copied rom the August diagram. Although the given plane is not shown in the manuscript gures, it appears in both the printed editions and it is used with the techniques o linear perspective to make the two lines appear to be in different planes rom the plane o the drawing. Most signi cantly, however, there is a labelling error in the line BΓ. Point Γ is supposed to be in the given plane, and hence must be at the bottom o line BΓ, as inVatican 190. Tis error was transmitted when the diagram was
7
8
Te diagrams to the arithmetical books are a clear exception. Te August diagrams are dotted lines, whereas Heiberg’s edition returns to the lines we nd in themanuscripts. Tere also other, individual cases where the diagrams were redrawn, presumably because those in the August edition were considered to be mathematically unsatisactory. For example, the diagram to Elem . .38 has been redrawn or Heiberg’s edition, whereas all the surrounding diagrams are clearly copied. See also the diagram or Elem . .17. Compare Heiberg and Stamatis 1969–77: 75 and 128 with August 1826–9: ab. and ab. . In this chapter,we reer to manuscripts by an abbreviated name initalics. Full library shel marks are given in the reerences. For the Euclidian manuscripts see also Vitrac’s chapter in this volume.
137
138
A
E
E
E
A
A
Fig.B.
Γ
Vatican 190
B
Δ
Δ
B
Grynée
Gregory
Γ
Δ
B
Γ
August
Figure 2.2 Diagrams or Euclid’sElements, Book , Proposition 13.
copied, despite the act that it could have been easily corrected rom considerations o the orientation required by the text. Indeed, whereas through the course o the modern period, ollowing the general trends o classical scholarship, the editors o successive publications o the Elements tended to consult a wider and wider range o manuscripts and give their readers more and more inormation about these manuscripts, the diagrams that accompanied these editions were generally made on the basis o the diagrams in the previous editions. As an example o this practice, we may take Elem. .13, which concerns the sum o the angles on either side o a straight line that alls on another straight line. Te manuscripts all agree in depicting angle ABΓ as 9
opening to the lef, as shown in Figure 2.2 by the example o Vatican 190. Nevertheless, all printed editions, ollowing the editio princeps o Grynée (1533), print angle ABΓ opening to the right. In some sense, this may have been a result o the division o labour o the publishers themselves. Whereas the editions were prepared by classical scholars and typeset by printers who were knowledgeable in the classical languages and generally had some sensitivity to the historical issues involved in producing a printed text rom manuscript sources, the diagrams were almost certainly drafed by proessional illustrators, who would have been skilled in the techniques o visual reproduction but perhaps uninterested in the historical issues at hand. Nevertheless, ht e act that the scholars who prepared these editions and the editors who printed them were content to use the diagrams o the previous editions as their primary sources says a great deal about their views o the relative importance o the historical sanctity o the text and o the diagrams in Greek mathematical works. Already, during the course o Heiberg’s career, the attitudes o scholars towards the importance o the manuscript diagrams began to change. In the late 1890s, in the edition he prepared with Besthorn o al-Nayrīzī’s 9
See Saito 2006: 110 or urther images o the manuscript gur es.
Diagrams and arguments in Greek mathematics
commentaries to theElements, the diagrams were taken directly romLeiden 399, and hence ofen quite different rom those printed in his edition o the Greek.10 By the time he edited Teodosius’Spherics, he must have become convinced o the importance o giving the diagrams critical attention, because the nished work includes diagrams based on the manuscripts, 11 generally accompanied with a critical note beginning ‘In g.’
Editions of manuscript diagrams Because the manuscript diagrams or spherical geometry are so strikingly different rom what we have grown to expect since the advent o the consistent application o techniques o linear perspective in the early modern period, the editions o ancient Greek works in spherical astronomy were some o the rst in which the editors began to apply critical techniques to the gures. For example in the eighth, and last, volume o ht e complete works o Euclid, or his edition o thePhenomena, Menge (1916) provided diagrams based on the manuscript sources and in some cases included critical notes. One o the most in uential editions with regard to the critical treatment o diagrams was that made by Rome (1931–43) o the commentaries by Pappus and Teon to Ptolemy’sAlmagest. Te diagrams in this long work were taken rom the manuscript sources and their variants are discussed in critical notes placed directly below the gures themselves. 12 Rome’s practices in uenced other scholars working in French and the editions by Mogenet (1950), o Autolycus’ works in spherical astronomy, and Lejeune (1956), o the Latin translation o Ptolemy’sOptics, both contain manuscript gures with critical notes. More recently, the majority tendency has been to provide manuscript diagrams with critical assessment. For example, the editions by Jones (1986) and Czinczenheim (2000) o Book o Pappus’ Collection and 10 11
12
Besthorn et al. 1897–1932. Heiberg 1927. In act, these critical notes are difficult to notice, since they are ound among the notes or the Greek text. Te notes or the Greek text, however, are preaced by numbers reerring to the lines o the text, whereas the diagrams are always located in the Latin translation, which has no line numbers. Neugebauer 1975: 751–5 seems to have missed them, since he makes no mention o them in his criticism o the ailure o classical scholars to pay sufficient attention to the manuscript diagrams o the works o spherical astronomy. In connection with theearly interest that Rome and Neugebauer showed in manuscript gures , we should mention the papers they wrote on Heron’sDioptra, the interpretation o which depends in vital ways on understanding the diagram. Rome 1923; Neugebauer 1938–9; Sidoli 2005.
139
140
Teodosius’ Spherics, respectively, both contain critical diagrams, and a recent translation o Archimedes’Sphere and Cylinderalso includes a critical assessment o the manuscript gures.13 Nevertheless, although there are critically edited diagrams or many works, especially those o the exact sciences, the most canonical works – the works o Archimedes and Apollonius, the Elements o Euclid and the Almagest o Ptolemy – because they were edited by Heiberg early in his career, are accompanied by modern, redrawn diagrams. Hence, because a study o Greek mathematics almost always begins with the Elements, and because the manuscript diagrams o this work contain many distinctive and unexpected eatures, it is essential that we reassess the manuscript evidence.
Characteristics of manuscript diagrams In this section, ocusing largely on the Elements, we examine some o the characteristic eatures o the manuscript diagrams as material objects that distinguish them rom their modern counterparts. Manuscript diagrams are historically contingent objects which were read and copied and redrawn many times over the centuries. In some cases, they may tell us about ancient practice, in other cases, about medieval interpretations o ancient practice, and in some ew cases, they simply tell us about the idiosyncratic reading o a single scribe. In the ollowing sections, we begin with broad general tendencies that can almost certainly be ascribed to the whole history o the transmission, and then move into more individual cases where the tradition shows modi cation and interpretation. In this chapter, we present summary overviews, not systematic studies. Overspeci cation
One o the most pervasive eatures o the manuscript gures is the tendency to represent more regularity among the geometric objects than is demanded by the argument. For example, we nd rectangles representing parallelograms, isosceles triangles representing arbitrary triangles, 13
Netz 2004. In act, however, the gures printed by Czinczenheim contain some peculiar eatures. Although she claims to have based her diagrams on those o Vatican 204, they ofen contain curved lines o a sort almost never seen in Greek mathematical manuscripts and certainly not in Vatican 204. Tus, although her critical notes are useul, the visual representation o the gures is ofen misleading.
Diagrams and arguments in Greek mathematics Γ
Γ
Δ
B
A Vatican 190
Δ
B
A Heiberg
Figure 2.3 Diagrams or Euclid’s Elements, Book , Proposition 7.
squares representing rectangles, and symmetry in the gure where none is required by the text.14 Tis tendency towards greater regularity, which we call ‘overspeci cation’, is so prevalent in the Greek, Arabic and Latin transmissions o the Elements that it almost certainly re ects ancient practice. We begin with an example o a manuscript diagram portraying more symmetry than is required by the text. Elem. .7 demonstrates that two given straight lines constructed rom the extremities o a given line, on the same side o it, will meet in one and only one point. In Figure 2.3, where the given lines are AΓ and BΓ, the proo proceeds indirectly by assuming some lines equal to these, say AΔ and BΔ, meet at some other point, Δ, and then showing this to be impossible. As long as they are on the same side o line AB, points Γ and Δ may be assumed to be anywhere and the proo is still valid. Heiberg, ollowing the modern tradition, depicts this as shown in Figure 2.3. All o the manuscripts used by Heiberg agree, however, in placing points Γ and Δ on a line parallel to line AB and arranged such that triangle ABΔ and triangle ABΓ appear to be equal. 15 In this way, the gure becomes perectly symmetrical and, to our modern taste, ails to convey the arbitrariness that the text allows in the relative positions o points Γ and Δ. We turn now to a case o the tendency o arbitrary angles to be represented as orthogonal. Elem. .35 shows that parallelograms that stand on the same base between the same parallels are equal to each other. In Figure 2.4, the proo that parallelogram ABΓΔ equals parallelogram EBΓZ ollows rom the addition and subtraction o areas represented in the gure and would make no sense without an appeal to the gure inorder to understand these operations. In the modern gures that culminate in Heiberg’s edition, the parallelograms are both depicted with oblique angles, whereas 14
15
In this chapter,we give only a ew select examples. Many more examples, however, can be seen by consulting the manuscript diagrams themselves. For Book o the Elements, see Saito 2006. For Books – o theElements, as well as Euclid’sPhenomena and Optics, see the report o a three-year research project on manuscript diagrams, carried out by Saito, available online at www.hs.osakau-u.ac.jp/~ken.saito/. See Saito 2006: 103 or urther images o the manuscript gures.
141
142 Δ
E
Z
A
ΔE
Δ
A
Z
H
H B
E
Z
H B
Γ
[Γ ]
B
Γ
Bodleian 301
Vatican 190
Heiberg
Figure 2.4 Diagrams or Euclid’sElements, Book , Proposition 35.
A
B
E
M
Z H
Γ Vatican 190
Δ
N
Θ
Heiberg
Figure 2.5 Diagrams or Euclid’sElements, Book v , Proposition 20.
in the manuscripts the base parallelogram ABΓΔ is always depicted as a rectangle, as16seen in Bodleian 301, and ofen even as a square, as seen in Vatican 190. Once again, to our modern sensibility, the diagrams appear to convey more regularity than is required by the proo. Tat is, the angles need not be right and the sides need not be the same size, and yet they are so depicted in the manuscripts. We close with one rather extreme example o overspeci cation.Elem. .20 shows that similar polygons are divided into an equal number o triangles, o which corresponding triangles in each polygon are similar, and that the ratio o the polygons to one another is equal to the ratio o corresponding triangles to one another, and that the ratio o the polygons to one another is the duplicate o the ratio o a pair o corresponding sides. Although the enunciation is given in such general terms, ollowing the usual practice o Greek geometers, the enunciation and proo is made or a particular instantiation o these objects; in this case, a pair o pentagons. In Figure 2.5, the modern diagram printed by Heiberg depicts two similar, but unequal, irregular pentagons. In Bodleian 301, on the other hand, we nd two pentagons that are both regular and equal. Tis diagram strikes the modern eye as inappropriate or this situation because the proposition is not about equal, regular pentagons, but rather similar polygons o 16
See Saito 2006: 131 or urther images o the manuscript gur es.
Diagrams and arguments in Greek mathematics
any shape.17 In the modern gure, because the pentagons are irregular, we somehow imagine that they could represent any pair o polygons, although, in act a certain speci c pair o irregular pentagons are depicted. Te presence o overspeci cation is so prevalent in the diagrams o the medieval transmission o geometric texts that we believe it must be representative o ancient practice. Moreover, there is no mathematical reason why the use o overspeci ed diagrams should not have been part o the ancient tradition. For us, the lack o regularity in the modern gures is suggestive o greater generality. Te ancient and medieval scholars, however, apparently did not have this association between irregularity and greater generality, and, except perhaps rom a statistical standpoint, there is no reason why these concepts should be so linked. Te drawing printed by Heiberg is not a drawing o ‘any’ pair o polygons, it is a drawing o two particular irregular pentagons. Since the text states that the two polygons are similar, they could be represented by any two similar polygons, as say those in Bodleian 301 which also happen to be equal and regular. O course statistically, an arbitrarily chosen pair o similar polygons is more likely to be irregular and unequal, but statistical considerations, aside rom being anachronistic, are hardly relevant. Te diagram is simply a representation o the objects under discussion. For us, an irregular triangle is somehow a more satisying representation o ‘any’ triangle, whereas or the ancient and medieval mathematical scholars an arbitrary triangle might be just as well, i not better, depicted by a regular triangle. Indifference to visual accuracy
Another widespread tendency that we nd in the manuscripts is the use o diagrams that are not graphically accurate depictions o the mathematical objects discussed in the text. For example, unequal lines may be depicted as equal, equal angles may be depicted as unequal, the bisection o a line may look more like a quadrature, an arc o a parabola may be represented with the arc o a circle, or straight lines may be depicted as curved. Tese tendencies show a certain indifference to graphical accuracy and can be divided into two types, which we call ‘indifference to metrical accuracy’ and ‘indierence to geometric shape’. We begin with an example that exhibits both overspeci cation and indierence to metrical accuracy. Elem. .44 is a problem that shows how to 17
In act, the proo given in the proposition is also about more a speci c polygon in that it has ve sides and isdivided into three similar triangles, but it achieves generality by being generally applicable or any given pair o rectilinear gures. Tis proo is an example o the type o proo that Freudenthal 1953 calledquasi-general.
143
144
Θ
H
Δ
Γ
Z
Δ Z A
B
Γ Λ
M
K
E
E
K
H
Θ A
Vatican 190
M
B
Λ
Heiberg
Figure 2.6 Diagrams or Euclid’sElements, Book , Proposition 44.
construct, on a given line, a parallelogram that contains a given angle and is equal to a given triangle. As exempli ed by Vatican 190in Figure 2.6, in all the manuscripts, the parallelogram is represented by a rectangle, and in the majority o the manuscripts that Heiberg used or his edition there is no correlation between the magnitudes o the given angle and triangle and those o the constructed angle and parallelogram. 18 In the modern gure, printed by Heiberg and seen in Figure 2.6, however, not only is the constructed gure depicted asan oblique parallelogram, but the magnitudes o the given and constructed objects have been set out as equal. We turn now to an occurrence o metrical indifference that is, in a sense, the opposite o overspeci cation. In Elem. .7, Euclid demonstrates a proposition asserting the metrical relationship obtaining between squares and rectangles constructed on a given line cut at random. Te overall geometric object is stated to be a square and it contains two internal squares. Nevertheless, as seen in the examples o Vatican 190 and Bodleian 301 in Figure 2.7, the majority o Heiberg’s manuscripts show these squares as rectangles.19 We should note also the extreme overspeci cation oBodleian 301, in which all o the internal rectangles appear to be equal. In general, there seems to be a basic indifference as to whether or not the diagram should visually represent the most essential metrical properties o the geometric objects it depicts. 18
19
In this chapter,when we speak o the majority o the manuscripts, we mean the majority o the manuscripts selected by the text editor as independent witnesses or the establishment o the text. We should be wary o assuming, however, that the majority reading is the best, or most pristine. See Saito 2006: 140, or urther images o the manuscript gures. Vienna In 31, as is ofen the case with this manuscript, we nd the magnitudes have been drawn so as to accurately represent the stipulations o the text (see the discussion o this manuscript in ‘Correcting the diagram’, below). See Saito 2008 or urther images o the manuscript diagrams. In Vienna 31 and Bologna 18–19, the squares, indeed, look like squares.
145
Diagrams and arguments in Greek mathematics Γ
A
Λ K
H
B
Γ
A
B
Λ H
Z
Θ
M
Δ
E Vatican 190
Δ
M
E
K H M
Δ
Bodleian 301
N Heiberg
Figure 2.7 Diagrams or Euclid’sElements, Book , Proposition 7.
As well as metrical indifference, the manuscript diagrams ofen seem to reveal an indifference toward the geometric shape o the objects as specied by the text. Te most prevalent example o this is the use o circular arcs to portray all curved lines. As an example, we may take the diagram or Apollonius Con. .16. As seen in Figure 2.8, the diagram in Vatican 206 shows the two branches o an hyperbola as two semicircles. Indeed, all the diagrams in this manuscript portray conic sections with circular arcs. Heiberg’s diagram, on the other hand, depicts the hyperbolas with hyperbolas. Tis diagram, however, is also interesting because it includes a case o overspeci cation, despite the act that Eutocius, already in the sixth century, noticed this overspeci cation and suggested that it be avoided. 20 In Figure 2.8, the line AB appears to be drawn as the axis o the hyperbola, such that HK and ΘΛ are shown as orthogonal ordinates, whereas the theorem treats the properties o any diameter, such that HK and ΘΛ could also be oblique ordinates. Eutocius suggested that they be so drawn in order to make it clear that the proposition is about diameters, not the axis. Nevertheless, despite Eutocius’ remarks, the overspeci cation o this gure was preserved into the medieval period, and indeed was maintained by Heiberg in his edition o the text. 21 Tis episode indicates that overspeci cation was indeed in effect in the ancient period and that although Eutocius objected to this particular instance o it, he was not generally opposed, and even here his objection was ignored. As well as being used to represent the more complicated curves o the conics sections, circular arcs are also used to represent straight lines. As Netz has shown,22 this practice was consistently applied in the diagrams or 20 21
22
B
Λ
K
Z
Γ
A
Heiberg 1891–3: 224; Decorps-Foulquier 1999: 74–5. A more general gure, which would no doubt have pleased Eutocius, is given in aliaerro, Densmore and Donahue 1998: 34. Netz 2004.
Z
E
146
θ
H
Γ
A K
B
Λ
Δ E
Z N
M Vatican 206
Heiberg
Figure 2.8 Diagrams or Apollonius’ Conica, Book , Proposition 16.
Archimedes’ Sphere and Cylinderor a polygon with short sides that might be visually conused with the arcs o the circumscribed circle.23 In the manuscript diagrams o Elem. .16, however, we have good evidence that the curved lines are the result o later intervention by the scribes. Elem. .16 is a problem that shows how to construct a regular 15-gon in a circle (Figure 2.9). Te manuscript evidence or this gure is rather involved and, in act, none o the manuscripts that Heiberg used contain the same diagram in the place o the primary diagram, although there is some obvious cross-contamination in the secondary, marginal diagrams.24 Nevertheless, it is most likely that the archetype was a metrically inexact representation o the sides o the auxiliary equilateral triangle and regular pentagon depicted with straight lines, as ound in Bologna 18–19
23
24
In the present state o the evidence, it is difficult to determine with certainty whether or not the curved lines in the Archimedes tradition go back to antiquity, but there is no good reason to assert that they do not. All o our extant Greek manuscripts or the complete treatise o Sphere and Cylinder are based on a single Byzantine manuscript, which is now lost. Tis is supported by the ragmentary evidence o the oldest manuscript, the so-called Archimedes Palimpsest, whose gures also contain curved lines. Te diagrams in an autograph o William o Moerbeke’s Latin translation,Vatican Ottob. 1850, however, made on the basis o a different Greek codex, also now lost, have straight lines, but this does not prove anything. Te source manuscript may have had straight lines or Moerbeke may have changed them. Whatever the case, we now have three witnesses, two o which agree on curved lines and one o which contains straight lines. See Saito 2008: 171–3 ora ull discussion. Tis previous report, however, was written beore the manuscripts could be consulted in person. Since Saito has now examined most o the relevant manuscripts, it is clear rom the colour o the lines, the pattern o erasures and so on, that the curved lines are part o the later tradition. See www.hs.osakau-u.ac.jp/~ken.saito/ diagram/ or urther updates.
Diagrams and arguments in Greek mathematics
A
B
A B
A
B [E] E
E
Δ
Γ
Δ
B
Γ
Δ
E
Γ Vienna 31
Florence 28
Bologna 18–19
A
B E
A
A
Γ
B
B
Δ
E
E
Γ
Δ
Γ Δ Vatican 190
Paris 2466
Bodleian 301
Figure 2.9 Diagrams or Euclid’sElements, Book
, Proposition 16. Dashed lines were drawn in and later erased. Grey lines were drawn in a different ink or with a different instrument. 25
and in the erased part o Florence 28. In Bodleian 301 and Paris 2466 we see examples in which the scribe has made an effort to draw lines AB and AΓ so as to portray more accurately the sides o a regular pentagon and an equilateral triangle, respectively. InBodleian 301, the external sides o the gures are clearly curved, while in Paris 2466 this curvature is slight. In Vienna 31, the srcinal our lines were straight and metrically accurate, as is usual or this manuscript, and a later hand added urther curved lines. In Vatican 190, it appears that all the sides o the auxiliary triangle and pentagon were drawn in at some point and then later erased, presumably so as to bring the gureinto conormity with the evidence o some other source. Not only were circles used or straight lines, but we also have at least one example o straight lines being used to represent a curved line. Tis rather interesting example o indifference to visual accuracy comes rom one o the most ascinating manuscripts o Greek mathematics, the so-called Archimedes Palimpsest, a tenth-century manuscript containing various Hellenistic treatises including technical works by Archimedes that was 25
In Florence 28, the metrically inaccurate gure with straight lines was erasedand drawn over with a metrically accurate gure with curved lines. Te colour o the ink makes it clear that the rectilinear lines that remain rom the srcinal are AΓ and the short part o AB that coincides with the new curved line AB.
147
148
Γ
Μ
Δ
Ζ
Σ Λ Η
Α
[Ν]
Ε
Κ
Θ
Β
Palimpsest 159v–158r
Reversed X-ray fluorescence calcium image
Optical multispectral image
Figure 2.10 Diagrams or Archimedes’ Method, Proposition 12.
palimpsested as a prayer book some centuries later.26 In the section o the treatise that Heiberg called Method 14, Archimedes discusses the metrical relationships that obtain between a prism, a cylinder and a parabolic solid that are constructed within the same square base.27 In Figure 2.10, the base o the prism is rectangle EΔΓH, that o the cylinder is semicircle EZH, while that o the parabolic solid is triangle EZH. Tus, in this diagram, a parabola is represented by an isosceles triangle. Since the parabola is de ned in the text by the relationship between the ordinates and abscissa, and since the triangle intersects and meets the same lines as the parabola, this was apparently seen as a perectly acceptable representation. In this way, the triangle unctions as a purely schematic representation o the parabola. Indeed, without the text we would have no way to know that the diagram represents a parabola. Diagrams in solid geometry
Te schematic nature o ancient and medieval diagrams becomes most obvious when we consider the gures o solid geometry. Although there are some diagrams in the manuscripts o solid geometry that attempt to give a pictorial representation o the geometric objects, or the most part, they orego linear perspective avour o schematic representation. Tis means that they do not serve to in convey a sense o the overall spacial relationships 26 27
Te circuitous story o this manuscript is told yb Netz and Noel 2007. Tis section o the Method is discussed by Netz, Saito and chernetska 2001–2. Te diagram ound in the palimpsest is difficult to see in the srcinal. Here, we include two images developed by researchers in the Archimedes Palimpsest Project. Te diagram is in the lef-hand column o the text spanning pages 159v–158r. Tese images, licensed under the Creative Commons Attribution 3.0 Unported Access Rights, are available online at www.archimedespalimpsest.org.
Diagrams and arguments in Greek mathematics
Vatican 190
Figure 2.11 Diagrams or Euclid’sElements, Book
Vatican 206
Proposition , 33 and Apollonius’
Conica, Book , Proposition 13.
obtaining among the objects, but rather to convey speci c mathematical relationships that are essential to the argument. Some conspicuous exceptions to this general tendency should be mentioned. For example, the diagrams or the rectilinear solids treated in Elem. and and the early derivations o the conic sections in the cone, in Con. , appear to use techniques o linear perspective to convey a sense o the three-dimensionality o the objects. In Figure 2.11, we reproduce the diagram or Elem. .33 rom Vatican 190 and that or Con. .13 rom Vatican 206. In all o these cases, however, it is possible to represent the threedimensionality o the objects simply and without introducing any object not explicitly named in the proo merely or the sake o the diagram. For example, in Figure 2.1 above, the plane upon which the perpendicular is to be constructed does not appear in the manuscript gure. Hence, even in these three-dimensional diagrams, techniques o linear perspective are used only to the extent that they do not con ict with the schematic nature o the diagram. Auxiliary, purely graphical elements are not used, nor is there any attempt to convey the visual impression o the mathematical objects through graphical techniques. An example o this is the case o circles seen at an angle. Although it is not clear that there was a consistent theory o linear perspective in antiquity, ancient artists regularly drew circles as ovals and Ptolemy, in hisGeography, describes the depiction o circles seen rom an angle as represented by ovals,28 nevertheless, in the medieval manuscripts such oblique circles are always drawn with two 28
Knorr 1992: 280–91; Berggren and Jones 2000: 116.
149
150 Γ E
Δ A
B Z
H
Vatican 204
Figure 2.12 Diagrams or Teodosius’ Spherics, Book , Proposition 6.
circular arcs that meet at cusps, as seen in Figure 2.11.29 Tis con rms that the diagrams were not meant to be a visual depiction o the objects, but rather a representation o certain essential mathematical properties. Likewise, in the gures o spherical geometry, i the sphere itsel is not named or required by the proo, we will ofen see the objects themselves simply drawn ree- oating in the plane, to all appearances as though they were actually located in the plane o the gure. Teodosius’ Spher. .6 shows that i, in a sphere, a great circle is tangent to a lesser circle, then it is also tangent to another lesser circle that is equal and parallel to the rst.In Figure 2.12, we nd the great circle in the sphere, ABΓ, and the two equal and parallel lesser circles that are tangent to it, ΓΔ and BH, all lying at in the same plane, with no attempt to portray their spacial relationships to each other or the sphere in which they are located. Te diagram or Spher. .6 thus highlights the schematic nature o diagrams in the works o spherical geometry. Te theorem is about the type o tangency that obtains between a great circle and two equal lesser circles and this tangency is essentially the only thing conveyed by the gure. Te actual spacial arrangement o the circles on the sphere must either be imagined by the reader or drawn out on some real globe. 30 29
30
With respect to linear perspective, there is still a debate as to whether or not the concept o the vanishing point was consistently applied in antiquity. See Andersen 1987 and Knorr 1991. As Jones 2000: 55–6 has pointed out, Pappus’ commentary to Euclid’s Optics 35 includes a vanishing point, but it is not located in accordance with the modern principles o linear perspective. We argue elsewhere that Teodosius was, indeed, concerned with the practical aspects o drawing gures on solid globes, but that this practice was not explicitly discussed inthe Spherics; Sidoli and Saito 2009.
Diagrams and arguments in Greek mathematics
Z
A
Δ
A
Z
Δ
O B
M E
Ξ N
Γ K
Θ Vatican 204
Ξ
H
B M
N
Γ
O E
Λ
H
Λ
K
Θ
Perspective diagram
Figure 2.13 Diagrams or Teodosius’ Spherics, Book , Proposition 15.
Te schematic role o diagrams in spherical geometry becomes unmistakable when we compare the diagram o one o the more involved propositions as ound in the manuscripts with one intended to portray the same objects using principles o linear perspective. Spher. .15 is a problem that demonstrates the construction o a great circle passing through a given point and tangent to a given lesser circle. As can be seen in Figure 2.13, merely by looking at the manuscript diagram, without any discussion o the objects and their arrangement, it is rather difficult to get an overall sense o what the diagram is meant to represent. Nevertheless, certain essential eatures are conveyed, such as the conpolarity o parallel circles, the tangency and intersection o key circles, and so on. It is clear that the manuscript diagram is meant to be read in conjunction with the text as reerring to some other object, either an imagined sphere or more likely a real sphere on which the lines and circles were actually drawn. It tells the reader how to understand the labelling and arrangement o the objects under discussion, so that the text can then be read as reerring to these objects. Te modern gure, on the other hand, by selecting a particular vantage point as most opportune and then allowing the reader to see the objects rom this point, does a better job o conveying the overall spacial relationships that obtain among the objects.31
31
We should point out, however,that the modern diagram inFigure 2.13, as well as being in linear perspective, employes a number o graphical techniques that we do not ndin the manuscript sources, such as the use o non-circular curves, dotted lines, highlighted points, and so on.
151
152 B A
A
Γ
Z
B
E
Z
Δ
Γ E
Γ
B
Δ Heiberg
Bodleian 301
Figure 2.14 Diagrams or Euclid’sElements, Book
Z
Δ
, Proposition 36.
One diagram for multiple cases
In the oregoing three sections, we have discussed characteristics o the medieval diagrams that are so prevalent that they almost certainly re ect ancient practice. We turn now to characteristics that are more individual but which, nevertheless, orm an essential part o the material transmission through which we must understand the ancient texts. For a ew propositions that are divided into multiple cases, we nd, nevertheless, the use o a single diagram to represent the cases. Tere is some question about the srcinality o most o these, and in act it appears that, in general, Euclid did not include multiple cases and that those propositions that do have cases were altered in late antiquity.32 Nevertheless, even i the cases are all due to late ancient authors, they are historically interesting and the manuscript tradition shows considerable variety in the diagrams. Tis indicates that single diagrams or multiple cases were probably in the text at least by late antiquity and that the medieval scribes had difficulty understanding them and hence introduced the variety that we now nd. As an example, we consider Elem. .36. Te proposition shows that i, rom a point outside a circle, a line is drawn cutting the circle, it will be cut by the circle such that the rectangle contained by its parts will be equal to the square drawn on the tangent rom the point to the circle. Tat is, in Figure 2.14, the rectangle contained by AΔ and ΔΓ is equal to the square on ΔB. In the text, as we now have it, this is proved in two cases, rst where line AΔ passes through the centre o the circle and second where it 32
See Saito 2006: 85–90 or the case o a single gure conta ining two cases in Elem. .25, in which the division into cases was almost certainly not due to Euclid. Te Arabic transmission o the Elements gives urther evidence or the elaboration o a single gure into multiple gures. In the eastern Arabic tradition, we nd a single gure or both Elem . .31 and .5 (see or example, Uppsala 20: 42v and 38v), while in the Andalusian Arabic tradition, which was also transmitted into Latin, we ndmultiple gures or thesepropositions (compare Rabāt 53: 126–8 and 145–6 with Busard 1984: 83–5 and 102–5).
Diagrams and arguments in Greek mathematics
does not. In Heiberg’s edition, and Vienna 31 (which ofen has corrected diagrams), there is an individual gure or each case. In the majority o Heiberg’s manuscripts, however, there is only a single gure and it contains two different points that represent the centre, one or each case. In Figure 2.14, we reproduce the two diagrams rom Heiberg’s edition, which are mathematically the same as those in Vienna 31, and an example o the single gure taken rom Bodleian 301. In the single diagram, as ound in Bodleian 301, there are two centres, points E and Z, and neither o them lies at the centre o the circle. Nevertheless, i we suppose that they are indeed centres, the proo can be read and understood on the basis o this gure. Despite these peculiarities, there are a number o reasons or thinking that this gure is close to the srcinal on which the others were based. It appears in the majority o Heiberg’s manuscripts, and the other diagrams contain minor problems, such as missing or misplaced lines, or are obviously corrected.33 Moreover, the single gure appears to have caused widespread conusion in the manuscript tradition. In most o the manuscripts, there are also marginal gures which either correct the primary gure or provide a gure that is clearly meant or asingle case. Hence, although we cannot, at present, be certain o the history o this theorem and its gure, thecharacteristics and variety o the gures should be used in any analysis o the text that seeks to establish its authenticity or authorship. Tis holds true or nearly all o the propositions that were clearly subject to modi cation in the tradition. Correcting the diagrams
Medieval scribes also made what they, no doubt, considered to be corrections to the diagrams both by redrawing the gures according to their own interpretation o the mathematics involved and by checking the diagrams against those in other versions o the same treatise and, i they were dierent, correcting on this basis. We will call the rst practice ‘redrawing’ and the latter ‘cross-contamination’. We have already seen the example o Elem. .16, on the construction o the regular 15-gon (see Figure 2.9), in which the scribes corrected or metrical indifference and drew the lines o the polygon as curved lines to distinguish them better rom the arcs o the circumscribing circle. 33
See Saito 2008: 78–9 or a discussion o variants o this diagram in the man uscripts o the Elements.
153
154
A
E
E
A
E
A
Z Z
Z
Δ
B
Δ
B
Γ
Δ
B
Γ
Γ
Vatican 190
Bodleian 301
Figure 2.15 Diagrams or Euclid’sElements, Book
Vienna 31
, Proposition 21.
In a number o cases, the tendencies toward overspeci cation and graphical indifference resulted in a gurehat t was difficult to interpret as a graphical object. For example, we may reer again toFigure 2.14 in which two different centres o the circle are depicted, neither o which appears to lie at the centre o the circle. In such cases, the scribes ofen tried to correct the gure sothat it could be more readily interpreted without ambiguity. As an example o a redrawn diagram, we takeElem. .21, which proves that, in a circle, angles that subtend the same arc are equal to one another. As seen in Figure 2.15, Vatican 190 portrays the situation by showing the two angles BAΔ and BEΔ as clearly separated rom the angle at the centre, angle BZΔ, which is twice both o them. In the majority o Heiberg’s manuscripts, however, as seen in Bodleian 301 and Vienna 31, through overspeci cation the lines BA and EΔ have been drawn parallel to each other and at right angles to BΔ, so that the lines AΔ and BE appear to intersect at the centre o the circle. In the course o the proposition, however, centre Z is ound and lines BZ and ZΔ are joined. In order to depict centre Z as distinct rom the intersection o lines AΔ and BE, centre Z has been placed off centre, ofen by later hands, as seen in the examples oBodleian 301 and Vienna 31.34 Because o the variety o the manuscript gures, it does not seem possible to be certain o the archetype, but it probably either had point Z as the intersection o AΔ and BE, as in the example o Vienna 31, or it had 35 Later aexample second o centre called301 Z .but not located the centre circle, asconusin the Bodleian readers,atthen, ound o thisthe situation ing and corrected the diagrams accordingly. In this case, the redrawing was done directly on top o the srcinal gure.
34 35
See Saito 2008: 67 or urther discussion o this diagram. In Bodleian 301, a later hand appears to have crossed out this srcinal second centre, Z, and moved it closer to the centre o the circle.
Diagrams and arguments in Greek mathematics H
Θ
Z
Γ A
Λ
Δ
E
B
K
M
Vienna 31
Figure 2.16 Diagrams or Euclid’sElements, Book , Proposition 44.
K
Δ
K E ABΓ
Z
H
Λ Vatican 190
A
Θ
Δ
Z
H
B Γ
Θ
E
Λ Vienna 31
Figure 2.17 Diagrams or Euclid’sElements, Book , Proposition 22.
Te redrawing, however, might also be done at the time when the text was copied and the gures drafed. In this case, the source diagram is lost in this part o the tradition. O the manuscripts used by Heiberg, the diagrams in Vienna 31 are 36ofen redrawn or metrical accuracy, but less ofen or overspeci cation. For the diagram accompanyingElem. .44, the gure in Vienna 31 (see Figure 2.16) should be compared with that in Vatican 190 (see Figure 2.6). As can be seen, the given area Γ is indeed the size o the parallelogram constructed on line AB, but the parallelogram is depicted as a rectangle and this is re ected in the act that the given angle, Δ, is depicted as right. In this case, the diagram is metrically accurate but it still represents any parallelogram with a rectangle. For an example in which the diagram in Vienna 31 has been corrected both or metrical accuracy and overspeci cation, we consider Elem. .22, which demonstrates the construction o a triangle with three given sides. As seen in Figure 2.17, the older tradition, here exempli ed by Vatican 190, represents the constructed triangle with the isosceles triangle ZKH, and the given lines with the equal lines A, B and Γ. In some o the manuscripts, however, the constructed triangle ZKH is drawn as an irregular acute triangle.37 In Figure 2.17 we see the example o Vienna 31, in which the 36
37
As we saw in the oregoing example, in the case oElem. .21, however, the srcinal scribeo Vienna 31 did not correct the diagram, but a correction was added by a later hand. See Saito 2006: 118 or a larger selection o the manuscript gures. Te act that Vatican 190 belongs to the older tradition is con rmed by the Arabic transmission.
155
156
constructed triangle is depicted as an irregular, acute triangle and all o its sides are depicted as the same length as the sides that have been given or the construction. Indeed, here we have a gure that is ully in accord with modern tastes. For Elem. .22, o the manuscripts used by Heiberg in his edition, Bodleian 301 also depicts the constructed triangle as an irregular, acute triangle, similar to that in Vienna 31. Te act that Vienna 31 and Bodleian 301 have a similar irregular, acute triangle could either indicate that scribes in both traditions independently had the idea to draw an irregular, acute triangle and randomly drew one o the same shape or, more likely, a scribe in one tradition saw the gure in the other and copied it. Tere is considerable evidence that this kind o cross-contamination took place. As another example that we have already seen, we may mentionElem. .21 in which both Vienna 31 and Bodleian 301 show a second centre drawn in reehand at some time afer the srcinal drawing was complete. Moreover, in the case o Elem. .21, in Florence 28, which has the same primary diagram as Bodleian 301, we nd a marginal diagram like that in Vatican 190, while in Bologna 18–19, which has the same primary diagram as Vatican 190, we nd a marginal diagram like that in Florence 28. Hence, as well as being used as a cross-reerence or the primary diagram, the gures o a secondor third manuscript were ofen drawn into the margin as a secondary diagram. Although we are now only at the beginning stages o such studies, this process o cross-contamination suggests the possibility o analysing the transmission dependencies o the diagrams themselves without necessarily relying on those o the text. Indeed, there is now increasing evidence that the gures, like the scholia, were sometimes transmitted independently o the text.38 Te process o cross-contamination has lef important clues in the manuscript sources that should be exploited to help us understand how the manuscript diagrams were used and read.
Ancient and medieval manuscript diagrams Since the ancient and medieval diagrams are material objects that were transmitted along with the text, we should consider the ways they were copied, read and understood with respect to the transmission o the text. 38
For examples othe independent transmission othe scholia o Aristarchus’On the Sizes and Distances o the Sun and Moonand Teodosius’ Spherics see Noack 1992 and Czinczenheim 2000. Te independent transmission o the manuscript gures or Calcidius’ Latin translation o Plato’simaeus has been shown by ak 1972.
Diagrams and arguments in Greek mathematics
Although, or the most part, the text and diagrams appear to have been copied as aithully as possible, at various times in the Greek transmission, and perhaps more ofen in the Arabic tradition, mathematically minded individuals re-edited the texts and redrew the diagrams. For the most part, in Greek manuscripts the diagrams are drawn into boxes that were lef blank when the text was copied, whereas in the Arabic and Latin manuscripts the diagrams were ofen drawn by the same scribe as copied the text, as is evident rom the act that the text wraps around the diagram. Nevertheless, except during periods o cultural transmission and appropriation, the diagrams appear to have been generally transmitted by scribes who based their drawings on those in their source manuscripts, despite the act that the diagrams can largely be redrawn on the basis o a knowledge o the mathematics contained in the text. Hence, the diagrams in the medieval manuscripts give evidence or two, in some sense con icting, tendencies: (1) the scribal transmission o ancient treatises based on a concept o the sanctity o the text and (2) the use o the ancient works in the mathematical sciences or teaching and developing those sciences and the consequent criticism o the received text rom the perspective o a mathematical reading. For these reasons, when we use the medieval diagrams as evidence or ancient practices, when we base our understanding o the intended uses o the diagrams on these sources, we should look or general tendencies and not become overly distracted by the evidence o idiosyncratic sources.
Diagrams and generality Te two most prevalent characteristics o the manuscript diagrams are what we have called overspeci cation and indifference to visual accuracy. Te consistent use o overspeci cation implies that the diagram was not meant to convey an idea o the level o generality discussed in the text. Te diagram simply depicts some representative example o the objects under discussion and the act that this example is more regular than is required was apparently not considered to be a problem. In the case o research, discussion or presentation, a speaker could o course reer to the level o generality addressed by the text, or, in act, could simply redraw the diagram. Te indierence to visual accuracy implies that the diagram was not meant to be a visual depiction o the objects under discussion but rather to use visual cues to communicate the important mathematical relationships. In this sense, the diagrams are schematic representations. Tey help the reader navigate
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the thicket o letter names in the text, they relate the letter names to speci c objects and they convey the most relevant mathematical characteristics o those objects. Again, in the course o research, discussion or presentation, a speaker could draw attention to other aspects o the objects that are not depicted, or again could simply redraw the diagrams. We have reerred to the act that the diagrams could have been redrawn in the regular course o mathematical work, and, in act, the evidence o the medieval transmission o scienti c works shows that mathematically minded readers had a tendency to redraw the diagrams in the manuscripts they were transmitting.39 Tis brings us to another essential act o the manuscript diagrams. Tey were conceived, and hence designed, to be objects o transmission, that is, as a component o the literary transmission o the text. Nevertheless, the extent to which mathematics was a literary activity was changing throughout the ancient and medieval periods and indeed the extent to which individual practitioners would have used books in the course o their study or research is an open question. Tis much, however, is virtually certain: the total number o people studying the mathematical sciences at any time was much greater than the number o them who owned copies o the canonical texts. Hence, in the process o learning about and discussing mathematics the most usual practice would have been to draw some temporary gure and then to reason about it. In act, there is evidence that, contrary to the impression o the diagrams in the manuscript tradition, ancient mathematicians were indeed interested in making drawings that were accurate graphic images o the objects under discussion. We argue elsewhere that the diagrams in spherical geometry, as represented by Teodosius’ Spherics, were meant to be drawn on real globes and that the problems in the Spherics were structured so as to acilitate this process.40 As is clear rom Eutocius’ commentary to Archimedes’ Sphere and Cylinder, Greek mathematicians sometimes designed mechanical devices in order to solve geometric problems and to draw diagrams accurately.41 In contrast to the triangular parabola we saw in Method 14, Diocles, in On Burning Mirrors, discusses the use o a horn ruler to draw a graphically accurate parabola through a set o points.42 Hence, we must distinguish between the diagram as an object o transmission and the diagram as an instrument o mathematical learning and investigation. 39
40 41 42
See Sidoli 2007 or some examples o mathematically minded readers who redrew the gures in the treatises they were transmitting. Sidoli and Saito 2009. Netz 2004: 275–6and 294–306. oomer 1976: 63–7.
Diagrams and arguments in Greek mathematics
In act, we will probably never know much with certainty about the parabolas that were drawn by mathematicians investigating conic theory or the circles that were drawn on globes by teachers discussing spherical geometry. Nevertheless, insoar as mathematical teaching and research are human activities, we should not doubt that the real learning and research was done by drawing diagrams and reasoning about them, not simply by reading books or copying them out. Hence, the diagrams in the manuscripts were meant to serve as signposts indicating how to draw these gures and mediating the reader’s understanding o the propositions about them. We may think o the manuscript diagrams as schematic guides or drawing gures and or navigating their geometric properties. In some cases, and or individuals with a highly developed geometric imagination, these secondary diagrams might simply be imagined, but or the most part they would actually have been drawn out. Te diagrams achieve their generality in a similar way as the text, by presenting a particular instantiation o the geometric objects, which shows the readers how they are laid out and labelled so that the readers can themselves draw other gures in sucha way that the proposition still holds. Hence, just as the words o the text reer to any geometric objects which have the same conditions, so the diagrams o the text reer to any diagrams that have the same con gurations. We may think o the way we use the diagram o a difficult proposition, such as that o the manuscript diagram orSpher. .15 in Figure 2.13, in the same way that we think o the way we use the subway map o the okyo Metro.43 We may look at the manuscript diagram inFigure 2.13 beore we have worked through the proposition to get a sense o how things are laid out, just as we may look at the okyo subway map beore we set out or a new place, to see where we will transer and so orth. Although this may help orientate our thinking, in neither case does it ully prepare us or the actual experience. Te schematic representation o the sphere in Figure 2.13 tells us nothing o its orientation in space, an intuition o which we will need to develop in order to actually understand the proposition. Te okyo subway map tells us nothing about trains, platorms and tickets, all o which we will need to negotiate to actually go anywhere in okyo. In both cases, the image is a schematic that conveys only inormation essential to an activity that the reader is assumed to be undertaking. Tere is, however, also an important distinction. Te okyo subway map points towards a very speci c object – or rather a system o objects that are 43
Te okyo subway map, in a number o different languages, can be downloaded romwww. tokyometro.jp/e/.
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always in ux, and probably not nearly as determinate as we would like to believe – nevertheless, a system o objects with a very speci c locality and temporality. A okyo subway map is useless or Paris. I it was drawn this year, it will contain stations and lines that did not exist ten years ago and ten years rom now it will again be out o date. Te manuscript diagram in Figure 2.13, however, has no such speci city. It can reer to any sphere and does. Anyone who wants to draw a great circle on a sphere tangent to a given line and through a given point can use this diagram in conjunction with its proposition to do so. In the centuries since this proposition was written, a great many readers must have drawn gures o this construction – on the plane, on the sphere, in their mind’s eye – and this diagram, strange and awkward as it is, somehow reerred to all o them. It is in such a way that the overspeci ed, graphically inaccurate diagrams that we nd in the manuscript tradition achieve the generality or which they were intended.
Bibliography Manuscripts Bodleian 301: Bodleian Library, D’Orville 301. Ninth century. Bologna 18–19: Bologna, Biblioteca comunale, 18–19. Eleventh century. Florence 28: Florence Laurenziana 28.3 enth century. Leiden 399: Bibliotheek van de Universiteit Leiden, Or. 399.1. 1144–1145 (539 ). Paris 2466: Bibliothèque nationale de France, Gr. 2466. welfh century. Rabāt. 53: Rabāt., al-Maktaba al-Malikiyya, al-Khyzāna al-H. assaniyya, 53. 1607– 1608 (1016 ). Uppsala 20: Uppsala Universitetsbibliotek, O. Vet. 20. 1042–1043 (434 ). Vatican 190: Bibliotheca Apostolica Vaticana, Gr. 190. Ninth century. Vatican 206: Bibliotheca Apostolica Vaticana, Gr. 206. welfh–thirteenth century. Vatican 204: Bibliotheca Apostolica Vaticana, Gr. 204. Ninth–tenth century. Vatican Ottob. 1850: Bibliotheca Apostolica Vaticana, Lat. Ottob. 1850. Tirteenth century. Vienna 31: Vienna Gr. 31. Eleventh–twelfh century.
Modern scholarship Andersen, K. (1987) ‘Te central projection in one o Ptolemy’s map constructions ’, Centaurus 30: 106–13. August, E. F. (1826–9) ΕΥΚΛΕΙΔΟΥ ΣΤΟΙΧΕΙΑ, Euclidis Elementa. Berlin.
Diagrams and arguments in Greek mathematics Berggren, J. L., and Jones, A. (2000) Ptolemy’sGeography: An Annotated ranslation o the Teoretical Chapters. Princeton, N.J. Besthorn, R. O., Heiberg, J. L., Junge, G., Raeder, J. and Tomson, W. (1897– 1932) Codex Leidensis 399, vol. , Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii . Copenhagen. (Reprinted in 1997 by F. Sezgin (ed.), Frankurt.) Busard, H. L. L. (ed.) (1984) Te Latin ranslation o the Arabic Version o Euclid’s Elements, Commonly Ascribed to Gerard o Cremona. Leiden. Czinczenheim, C. (2000) ‘Edition, traduction et commentaire des Sphériques de Téodose’, PhD thesis, Université Paris . Decorps-Foulquier, M. (1999) ‘Sur les gures du traité des Coniques d’Apollonius de Pergé édité par Eutocius d’Ascalon’, Revue d’histoire des mathématiques5: 61–82. Fecht, R. (ed., and trans.) (1927) Teodosii, De habitationibus liber, De diebus et noctibus libri duo. Abhandlungen der Gesellschaf der Wissenschafen zu Göttingen, Philosophisch–Historische Klasse, n.s. 19(4). Berlin. Freudenthal, H. (1953) ‘Zur Geschichte der vollständigen Induktion’, Archives Internationales d’Histoire des Sciences6: 17–37. Gregory (Gregorius), D. (1703) ΕΥΚΛΕΙΔΟΥ ΤΑ ΣΩΖΟΜΕΝΑ, Euclidis quae supersunt omnia. Oxord. Grynée (Grynaeus), S. (1533) ΕΥΚΛΕΙΔΟγ ΣΤΟΙΧΕΙΩΝ ΒΙΒΛ▶ ΙΕ▶ ΕΚ ΤΩΝ ΘΕΩΝΟΣ ΣΥΝΟΥΣΙΩΝ. Basel. Heiberg, J. L. (ed. and trans.) (1883–1888) Euclidis Elementa, 4 vols. Leipzig. (Reprinted with minor changes in Heiberg and Stamatis 1969–77.) (1891–3) Apollonii Pergaei quae graece extant cum commentariis antiquis. Leipzig. (1903) ‘Paralipomena zu Euklid’, Hermes 38: 46–74, 161–201, 321–56. (1927) Teodosius ripolitesSphaerica. Abhandlungen der Gesellschaf der , n.s. 19(3). Wissenschafen zu Göttingen, Philosophisch–Historische Klasse Berlin. Heiberg, J. L., and Stamatis, E. (eds.) (1969–77) Euclidis Elementa, 5 vols. Leipzig. Jones, A. (ed., and trans.) (1986) Pappus o Alexandria: Book 7 o theCollection, 2 vols. New York. (2000) ‘Pappus’ notes to Euclid’sOptics’, in Ancient and Medieval raditions in the Exact Sciences: Essays in Memory o Wilbur Knorr, ed. P. Suppes, J. M. Moravcsik Mendell. Stanord, 49–58.in Euclid’s Optics’, Knorr, W. (1991)and ‘OnH.the principle o linearCali.: perspective Centaurus 34: 193–210. (1992) ‘When circles don’t look like circles: an optical theorem in Euclid and Pappus’, Archive or History o Exact Sciences 44: 287–329. Lejeune, A. (ed., and trans.) (1956) L’Optiquede Claude Ptolémée dans la version . Louvain. latine d’après l’arabe de l’èmir Eugène de Sicile
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162 (1989) L’Optiquede Claude Ptolémée dans la version latine d’après l’arabe de l’émir Eugène de Sicile. Leiden. (Reprint, with new French translation, o Lejeune 1956.) Menge, H. (1916) Euclidis phaenomena et scripta musica, Opera omniavol. . Leipzig. Mogenet, J. (1950) Autolycus de Pitane, histoire du texte suivie de l’édition critique . Louvain. des traités de la Sphère en mouvement et des levers et couchers Netz, R. (2004), Te Works o Archimedes,vol. , Te wo Books On the Sphere and the Cylinder. Cambridge. Netz, R. and Noel, W. (2007) Te Archimedes Codex. New York. Netz, R., Saito, K. and chernetska, N. (2001–2) ‘A new reading o Method proposition 14: preliminary evidence rom the Archimedes Palimpsest (Parts 1 and 2)’, SCIAMVS 2: 9–29 and 3: 109–25. Neugebauer, O. (1938–9) ‘Über eine Methode zur Distanzbestimmung Alexandria-Rom bei Heron I and II’, Det Kongelige Danske Videnskabernes Selskab 26 (2 and 7): 3–26 and 3–11. (1975) A History o Ancient Mathematical Astronomy. New York. Noack, B. (1992) Aristarch von Samos: Untersuchungen zur Überlieerungsgeschichte der Schrifπερὶ μεγεθῶν καὶ ἀποστρημάτῶν ἡλίου καὶ σελήνης. Wiesbaden. Rome, A. (1923) ‘Le problème de la distance entre deux villes dans la Dioptra de Héron’, Annales de la Société scienti que de Bruxelles 43: 234–58. (1931–43) Commentaires de Pappus et Téon d’Alexandrie sur l’Almageste, 3 vols. Rome. Saito, K. (2006) ‘A preliminary study in the critical assessment o diagrams in Greek mathematical works’, SCIAMVS 7: 81–144. (2008) Diagrams in Greek Mathematical exts, Report o research grant 17300287 o the Japan Society or the Promotion o Science. Sakai. (See www. hs.osakau-u.ac.jp/~ken.saito/diagram/ or urther updates.) Sidoli, N. (2005) ‘Heron’sDioptra 35 and analemma methods: an astronomical determination o the distance between two cities’, Centaurus 47: 236–58. (2007) ‘What we can learn rom a diagram: the case o Aristarchus’s On the Sizes and Distances o the Sun and the Moon’, Annals o Science 64: 525–47. Sidoli, N. and Saito, K. (2009) ‘Te role o geometrical construction in Teodosius’sSpherics’, Archive or History o Exact Sciences 63: 581–609. ak, J.Pontus’ G. van, Mnemosyne der (1972) ‘Calcidius’ illustration o the astronomy o Heraclides o 25 Ser. 4: 148–56. aliaerro, R. C. (with D. Densmore and W. H. Donahue) (1998) Apollonius o Perga: Conics Books – . Santa Fe, N.M. oomer, G. J. (1976) Diocles On Burning Mirrors. New York.
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Te texture o Archimedes’ writings: through Heiberg’s veil
Te reading o Archimedes will always be inextricably intertwined with the reading o Heiberg. Te great Danish philologer, involved with so many other projects in Greek science and elsewhere,1 had Archimedes become his lie project: the subject o his srcinal dissertation, Quaestiones Archimedeae (1879), which ormed the basis or his rst eubner edition o Archimedes’ Opera Omnia (1880) and then, ollowing upon the discovery o codices B and C, the second eubner edition o the Opera Omnia (1910–15). Te second edition appears to have settled the main questions o the relationship between the manuscripts, and has established the readings with great authority and clarity (it is this second and de nitive edition which I study here). Tis is especially impressive, given how ew technical resources Heiberg had or the reading o codex C – the amous Palimpsest. Even i today we can go urther than Heiberg did, this is to a large extent thanks to the ramework produced by Heiberg himsel: so that, even i his edition is superseded, his legacy shall remain. Let this article not be read as a criticism o Heiberg – the most acute reader Archimedes has ever had. Te historical signi cance o Heiberg’s publication is due not only to his scholarly stature, but also to his precise position in the modern reception o Archimedes. Classical scholarship is a tightly de ned network o texts and readers, organized by a strict topology. Te ‘standard edition’ has a special position. Its very pagination comes to de ne how quotations are to be made. Indeed, even more can be said or Archimedes speci cally. First, the rise o modern editions inspired by German philological methods, in the late nineteenth century, coincided with an early phase o an interest in the history o science. Tus Heath’s work o translating and popularizing Greek mathematics in the English-speaking world took place in the same decades that Heiberg was producing his edition o Archimedes. Te version o Archimedes still in use by most English readers – Heath 1897 – depends, paradoxically, on Heiberg’s rst (and ed cient) edition. Czwalina’s German translation ( 1922– 5) was based on the second edition, as was Ver Eecke’s French translation (1921). Perhaps the most useul version among those widely available today, 1
For Heiberg’s somewhat incredible bibliography,see Spang-Hanssen 1929.
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Mugler’s Budé’s text 1( 970–2) goes urther: it not merely translates the text o the second edition o Archimedes, but also provides a acing Greek text – which directly reproduces the srcinal edition by Heiberg! Mugler’s decision to avoid any attempt to revise Heiberg may well have been due to another curious twist o ate: by the 1970s, the Palimpsest had gone missing so that a new edition appeared impossible. Stamatis’ version 1970–4) ( repeats the same procedure, with modern Greek instead o French. An edition is ontologically distinct rom its sources. It is a synthesis o various manuscripts into a single printed text. Te editor, aiming to preserve a past legacy, inevitably transorms it. It is a truism that Heiberg’s version o Archimedes is not the same as the manuscript tradition – let alone the same as Archimedes’ srcinal ‘publication’ (whatever this term may mean). Once again: the point is not to criticize Heiberg. Te point is to try to understand the distinguishing eatures o his edition, which may even orm part o the image o Archimedes in the twenty- rst century. In this chapter I survey a number o transormations introduced by Heiberg into his text. Tese all into three parts, very different in character. First, Heiberg ignored the manuscript evidence or the diagrams, producing instead his own diagrams (this, indeed, may be the only point or which his philology may be aulted; I return to discuss Heiberg’s possible justi cations below). Second, at the local textual level, Heiberg marked passages he considered to be late glosses and thus not coming rom the pen o Archimedes. Tird, at the global textual level, through various choices o modern ormat as well as textual extrapolation, Heiberg introduced a certain homogeneity o presentation to the Archimedean text. Te net result o all those transormations was to produce an Archimedes who was textually explicit, consistent, rigorous and yet opaque. I move on to show this in detail.
Te texture of Archimedes’ diagrams Tis is not the place to discuss the complex philological question o the srcins o the diagrams as extant in our manuscripts. I sum up, instead, the main acts. O the three known early Byzantine manuscripts, one – the Palimpsest or codex C – is extant. Te two others are represented by copies: a plethora o independent copies o codex A, allowing a very con dent reconstruction o the srcinal; and Moerbeke’s Latin translation based in part on codex B (and in part based also on codex A). For most works we can reconstruct two early Byzantine traditions (codices A and C or SC , SC , SL, DC; codices A and B or PE ; codices B and C or FB , FB . For PE alone
Archimedes’ writings: through Heiberg’s veil
we have some evidence rom all three traditions).2 Te agreement between A and C is striking. We can also see that Moerbeke’s Latin translation involved a considerable transormation o the diagrams he had available to him rom codex A. Tis may serve to explain why, when we don’t have the separate evidence o A and just compare codices B and C, the two appear different: this is likely to be the in uence o Moerbeke’s transormation. In short, the evidence suggests that the various early Byzantine manuscripts wereprobably identical in their diagrams. Tis is certainly the case or the two independent early Byzantine manuscripts A and C, or the works SC, SC , SL and DC – representing the bulk o Archimedes’ extant work in pure geometry. In all likelihood, such resemblance stems rom a close dependence on a Late Ancient archetype. Whether or not this archetype can be pushed back to the srcinal publication by Archimedes – whatever that could mean – is an open question. o the extent that the manuscript evidence displays striking, srcinal practices, a kind o lectio difficilior makes it more likely that it is an srcinal practice. Te argument could never be very strong and it is probably or this cogent reason that Heiberg avoided offering an edition o the manuscripts’ diagrams. However, even i the ollowing need not represent the srcinal orm o Archimedes’ works, it certainly represents one important way in which Archimedes was read or at least some part o antiquity. In understanding Archimedes’ modern reception, it is helpul to compare this with the ancient reception to which the manuscripts testiy. In what ollows, then, I compare Heiberg’s diagrams with the Late Ancient archetype reconstructed or the two books on Sphere and Cylinder (concentrating on these two books or the reason that I have already completed their edition). I arrange my comments as three comparisons – three ways in which Heiberg transormed the srcinal ound in the manuscripts.
Heiberg goes metrical I put side by side the two diagrams or SC .16 (see Figure 3.1). Te differences as regards the triangle – in act, a ‘ at’ view o a cone – are immaterial. Neither do I emphasize at the moment the differences in overall layout (it is clear that Heiberg saves more on space, aiming at a more economic production; this may have been imposed by the press). Te major difference has to do with the nature o the circles Λ, Θ and K. Heiberg has them concentric, 2
Here and in what ollows I use a system o abbreviation o the titles o works by Archimedes, as ollows: SC (Sphere and Cylinder), DC (Measurement o the Circle), CS (Conoids and Spheroids), SL (Spiral Lines), PE (Planes in Equilibrium), Aren. (Arenarius), QP (Quadrature o Parabola), FB (Floating Bodies), Meth. (Method), Stom. (Stomachion), Bov. (Cattle Problem).
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Λ Θ K
B
Λ
K Δ
Z
E
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Δ
A
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Heiberg
Θ
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Archimedes (reconstruction)
Figure 3.1 Heiberg’s diagrams or Sphere and Cylinder .16 and the reconstruction o Archimedes’ diagrams.
in a descending order o size. Te manuscripts have them arranged side by side, all o equal size. Te proposition constructs the circles in a complex way which is then shown to determine that the circle Λ equals the surace o the cone BAΓ, circle K equals the surace o the cone BΔE, and Θ the difference between the suraces, that is the surace o the truncated cone at the lines AΔEΓ. It is thereore geometrically required that Λ > K, Λ > Θ (the relationship between K, Θ, though, is not determined by the proposition). It is clear that Heiberg’s diagram provides more metrical inormation than the manuscript diagrams do. In this particular case, indeed, Heiberg provides more metrical inormation than is determined by the proposition; while the manuscripts provide less than is determined by the proposition. Tis immediately suggests why the manuscripts’ practice is in act rational. Let us suppose that the manuscripts would set out to diagram the precise metrical relations determined by the proposition. It would make sense, then, to have both Θ and K smaller than Λ. However, how to represent the relationship between Θ and K? Once Λ appears bigger than both Θ and K, this is already taken to suggest that diagrams are metrically inormative; and so the reader would look or the diagram relationship between Θ and K so as to provide him or her with the intended metrical relation. Tus, a diagram where, say, Λ is greater than both Θ and K, the two, say, equal to each other, alsely suggests that the intended metrical properties are: Λ > Θ = K. Te difficulty o representing indeterminate metrical relations inside a metrical diagram is obvious.
Archimedes’ writings: through Heiberg’s veil
Te manuscripts’ diagram avoids this difficulty altogether. Te three equal circles – in agrant violation o the textual requirement that Λ > Θ, Λ > K – imply that the diagram carries no metrical consequences (at least so ar as these three circles are concerned) and thereore the diagram itsel leaves the metrical relationship between K and Θ indeterminate. Tis is a systematic eature o the manuscripts’ diagrams. Tere are twenty-our cases where a system o homogeneous, unequal magnitudes (typically all circles, or all lines) is represented by equal magnitudes set side by side, as well as ve cases where a system o homogeneous unequal magnitudes is represented by magnitudes some o which (in contradiction to the text) are represented equally. Tere are only our cases where a system o unequal magnitudes is allowed to be represented by a diagram where all traces are appropriately unequal. Te consequence o this convention is clear: the ancient diagrams are not read off as metrical. As a corollary, they are read more or their con gurational inormation. Tis is obvious rom the comparison with Heiberg: in the latter’s diagram o .16, the readers’ expectation clearly is not that the three circles should indeed all be concentric. Indeed, the reader must understand that such gures are pure magnitudes and do not stand to each other in any spatial, con gurational sense. While the conical surace ABΓ does indeed envelope the smaller suraces ΔBE, AΔEΓ, no such envelopment is understood between the three circles K, Λ and Θ that merely represent three magnitudes manipulated in the course o the proposition. Now, this does not make Heiberg’s diagramalse. It simply highlights what Heiberg’s reader – in direct opposition to the reader o the ancient diagrams – is supposed to edit away in his reading o the diagram. Heiberg’s reader is supposed to edit away a certain piece o con gurational inormation (the circlesmerely appearto envelop each other), whereas the ancient reader was supposed to edit away a certain piece o metrical inormation (the circlesmerely appear equal). One can say that both representational systems oreground one dimension o inormation, overruling the other dimension: Te metrical dimension o inormation is oregrounded in Heiberg and overrules the con gurational dimension; the con gurational dimension o inormation is oregrounded in the ancient diagram and overrules the metrical dimension. Tis may serve to elucidate the ollowing. Interestingly, the ve cases where the ancient diagrams represent unequals by unequals – propositions SC .15, 33, 34, 44 – all involve lines. Consider the typical case o .15 (see Figure 3.2). B is the radius o the circle A, Γ – the side o a cone set up on that circle, E – a mean proportional between the two. Te metrical relationship B < E < Γ is indeed determined. Further, the circle Δ is drawn
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Δ
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Γ
Figure 3.2 A reconstruction o Archimedes’ diagram or Sphere and Cylinder .15.
around the radius E. It thus ollows also that A < Δ. Te diagram displays the inequality between the lines B < E < Γ but not the equally determined inequality between the circles A < Δ. Tere are six other cases, however, where unequal lines are represented by equal diagram traces. Te rule then appears to be that the manuscripts’ diagrams have a very strong preerence to mark unequal plane gures as equal, but only a tendency to mark unequal line segments as unequal. Why should that be the case? Clearly, lines are less con gurationally charged than plane gures are. Te representation o a system o line traces does not suggest so powerully a con guration made o those lines in spatial arrangement, and it is easier to read as a purely quantitative representation (indeed, such lines orm the principle o representation used by Greek mathematicians when dealing with numbers or with general magnitudes, whose signi cance is purely quantitative, as in Euclid’s Elements , – ). Te principle is clear, then: the more the diagrams are taken to convey con gurational meaning, the less metrical they are made. Lines – whose non-con gurational character is easy to establish – may sometimes take metrical characteristics; but with plane gures, metrical characteristics are altogether avoided. Te upshot o this is obvious: diagrams which mostly carry con gurational inormation, to the exclusion o the metrical, can also be rigorous. As Poincaré pointed out long ago, diagrams may be geometrically correct, to the extent that they are taken to be purely topological.3 O course, Poincaré 3
Poincaré 1913 : 60. Needless to say, topology or ‘analysis situs’ (as Poincaré would say) meant something different a century ago: in particular, this to Poincaré had absolutely nothing to do with Set Teory and instead had everything to do with a study o spatial relations abstracted away rom any metrical conditions – which o course makes ‘topology’ even more obviously relevant to the study o schematic diagrams.
Archimedes’ writings: through Heiberg’s veil
himsel knew Greek mathematics only via editions such as Heiberg’s. Little could he guess that the ancient manuscripts or Archimedes had just the kind o diagrams he considered logically viable!
Heiberg goes three-dimensional A group o propositions early in Sphere and Cylinder involves the comparison o cones or cylinders with the pyramids or prisms they enclose: propositions 7–12. Proposition 12 selects a diagram ocused on the base alone, but the diagrams o propositions 7–11 require that we look at the entire solid construction. Te manuscripts’ diagrams (with a single exception, on which more below) produce a representation with a markedly ‘ at’ effect, whereas Heiberg produces several times a partly perspectival image with a three-dimensional effect. Te gure or .9 (see Figure 3.3) may be taken as an example. What is the view selected by the manuscripts’ diagram? Perhaps we may think o it as a view rom above, slightly slanted so as to make the vertex Δ appear to all not on the centre o the circle but somewhat below. Te view selected by Heiberg’s diagram is much ‘lower’, so that the point Δ appears higher above the plane o the base circle, allowing the pyramid to emerge out and produce an illusionistic three-dimensional effect. Te net result is that Heiberg’s gure impresses the eyewith the picture o an external object; the manuscripts’ diagram is reduced to a mere schema o interconnected lines. Tis de nitely should not be understood as a mark o poor draughtsmanship on the part o the manuscripts. Indeed, the one exception is telling: .11 has a clear three-dimensional representation o a cylinder, and here the motivation is clear: since the proposition reers in detail to both the top and bottom bases o the cylinder, a view rom ‘above’, where the bases coincide or nearly coincide, would have been useless. It turns out, thereore, that once the view rom above was excluded, the manuscripts were capable o producing a lower view, with its consequent threedimensional illusionistic effect. Strikingly and decisively, we note that the manuscripts’ diagrams or .11 represent the bases by almond-shapes (standardly used elsewhere or the representation o conic sections).4 Tis is a deliberate oreshortening effect – which Heiberg himsel eschews in his own diagram. Clearly, Heiberg has established a certain compromise between three-dimensional representation and geometric delity, to 4
Tis practice is commented upon, or the Arabic tradition, in oomer 1990 : lxxxv, and it is indeed widespread in the various manuscript traditions o Greek mathematics.
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Θ Δ
B
Z
E
B
Θ Z
E
Γ
A
Δ Γ
A
Heiberg
Archimedes(reconstruction)
Figure 3.3 Heiberg’s diagram orSphere and Cylinder .9 and the reconstruction o Archimedes’ diagram.
which he is consistent. Te manuscripts, on the other hand, insist on the preerence, where possible, o a more schematic representation, even while they mark their ability to produce a ull three-dimensional representation. Te manuscripts’ decision clearly is not motivated by simple considerations o space. As we have seen in the preceding section, the manuscripts tend to have much bigger gures. No one invests in an Archimedes’ manuscript or considerations o practical utility, so that these manuscripts should all be seen as luxury items,5 so that one is allowed more space. A printed book, o course, is not typically based on a patronage economy and its calculations are different. I do think that a certain consideration o layout is relevant, however: what we do see in the manuscripts’ diagrams is a certain preerence or the horizontal arrangement, perhaps re ecting the srcins o 6 such diagrams within the spaces o papyrus columns. Tis would in itsel make a three-dimensional representation less preerable. But note that this is a mere tendency in the manuscripts’ diagrams: as we will ese with .12 below, 5
6
Te main proo or the lack o practical purpose in Byzantine Archimedes manuscripts is in their plethora o uncorrected, trivial errors. Te extant Palimpsest shows not a single correction by a later hand (indeed, it was consigned to become a palimpsest!). We have a credible report rom one o the scribes copying codex A that this, too, was replete with uncorrected errors (a reported supported by the pattern o errors in the extant copies o A): see Heiberg 1915: x. On the tendency o papyrus illustrations to orient horizontally, see Weitzmann 1947.
Archimedes’ writings: through Heiberg’s veil
some diagrams in the manuscripts take a vertical arrangement (even though this arrangement is not determined by the geometrical situation). I do think the manuscripts avoid the three-dimensional representation, among other things, because o their preerence or the horizontal over the vertical; what I wish to stress is that this shows how little weight they allow the pictorial quality o the diagram – so that the minor consideration o a preerred orientation trumps over that o the three-dimensional representation. Note now that our discussion touches on a small stretch o text, but this is in act in itsel meaningul. Te Archimedean corpus is sometimes dedicated to purely plane gures (Spiral Lines, Planes in Equilibrium, Measurement o Circle, Stomachion, Quadrature o Parabola) but, even in the several cases where Archimedes studies solid objects, these are studied essentially via some plane section passing through them (Floating Bodies , Method, Conoids and Spheroids, Sphere and Cylinder ). Sphere and Cylinder orms an exception because o its mathematical theme o the comparison o curved, concave suraces – one which calls or a direct threedimensional treatment.7 Now consider .12, where Archimedes’ treatment o the three-dimensional cone is mediated via the plane base (where two lines orm tangents to the circle o the base). Such is the standard Archimedean diagram. In the manuscripts, the diagrams o .12 and o .9 are closely aligned together, displaying a similar con guration o crisscrossing lines; whereas Heiberg’s diagrams open up a chasm between the two situations, the solid picture o .9 marked against the planar view o .12 (see Figure 3.4). I would venture to say as much: that by making .9 appear more solid, Heiberg simultaneously makes .12 appear more planar. I .9 is designed to bring to mind a picture o what a pyramid looks like, then .12 should be seen to be designed so as to bring to mind a picture o what a circle looks like. But i .9 is a mere schematic representation o lines in con guration, then the same must be said o .12 as well: it is not a picture o a two-dimensional gure. It is, instead, a geometricallyvalid way o providing inormation, visually, about such a gure. Tis, o course, is an interpretation that goes beyond the evidence. Te acts on three-dimensional representation are simple: such representation is avoided as ar as possible by the manuscripts, but is produced, wherever 7
Among the lost works by Archimedes, the Centres o Weights o Solidsmay well have been based on planar sectional treatment – which Archimedes invariably pursues in the closely related Method (where various spheres, conoids and prisms are represented by planar cuts). One wonders how Archimedes’ treatment o semi-regular solids was handled: the account in Pappus (Hultsch 1876: 350–8) carries no diagrams and is based on a purely numerical characterization o the gures.
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H
H
K
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E
Δ
B
E
Z
Θ
Δ Γ
A
Z
A
Heiberg
K
Θ Γ
Archimedes (reconstruction)
Figure 3.4 Heiberg’s diagram or Sphere and Cylinder .12 and the reconstruction o Archimedes’ diagram.
applicable (which is rare), by Heiberg. My interpretation o this evidence is based on the acts shown above – the non-metrical character o the manuscripts’ diagrams – as well as those to which I now turn: their non-iconic character.
Heiberg goes iconic I have suggested that Heiberg goes beyond the manuscripts, in making the two-dimensional gures more o a ‘picture’ o the object they are designed to represent. So ar, my argument has been based purely on the contrast o such two-dimensional diagrams to their three-dimensional counterparts. What we require, then, is to see whether there are cases where Heiberg’s representation o two-dimensional gures inserts into them a visual ‘correctness’ absent in the manuscripts. We have to a certain extent seen this already with the quantitative, metrical character o Heiberg’s diagrams. Even more striking, however, is a certain systematic way by which Heiberg’s two-dimensional diagrams are qualitatively more ‘correct’ than those o the manuscripts. I turn now to SC .33 (see Figure 3.5). I note quickly the metrical acts. Te gure by Heiberg has A much bigger than the main circle, which is
Archimedes’ writings: through Heiberg’s veil
B
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Λ Γ
E
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Θ
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A
H
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Heiberg
Archimedes (reconstruction)
Figure 3.5 Heiberg’s diagram orSphere and Cylinder .33 and the reconstruction o Archimedes’ diagram. 8 indeed ‘correct’; the manuscripts’ smaller A is in a sense ‘alse’. Te manuscripts agree with Heiberg, however, in the arrangement o the line segments, all in keeping with the practice described above (pp. 167–8). Qualitatively, Heiberg represents the propositions’ requirement – o a 4n-sided regular polygon circumscribed and inscribed about a circle – by two octagons. Te manuscripts, instead, have a system made o two nested sequences o curved lines, 12 outside and 12 inside. Te visual effect could not have been more different and here we see the manuscripts’ diagrams becoming markedly non-iconic. A sequence o 12 curved lines, each nearly a semicircle, does not make the visual impression o a polygon. Te manuscripts, in this case, have a very good reason to choose their
non-iconic system o representation. As we can see rom Heiberg’s diagram, it is diffi cult to make the visual resolution between such a polygon and a 8
Incidentally, note that I did not count such alse planar inequalities in my treatment o the non-metrical character o the manuscripts’ diagrams. My survey ocused on the (very common) case where homogeneous objects are put side by side – typically unmarked circles or lines. I did not look into the case o heterogeneous objects, such as the simple circle A alongside the more complex main circle in .33.
B Δ Γ
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Figure 3.6 Te general case o a division o the sphere.
circle. A square perhaps could still do, but this offers a very special case o the 4n-sided regular polygon: considered as a division o the sphere, it reduces to a system o two cones, without any truncated cones. Te octagon already brings in a truncated cone, but this is the limiting truncated cone lying directly on the diameter o the sphere. Only with the dodecagon do we begin to see the general case o a division o the sphere based on 4 nsided regular polygons, with a limiting truncated cone lying directly on the diameter, another truncated cone next to it, and nally a non-truncated cone away rom the diameter (Figure 3.6). O course, a regular dodecagon is nearly impossible to distinguish, visually, rom a circle, but the entire point o avoiding a limiting case or the diagram is the desire to limit the extent to which the visual impression o the diagram creates alse expectations. Te same desire, then, accounts or the radical, non-iconic representation itsel: no one is going to base an argument concerning polygons on the visual impression made by the curved arcs. Indeed, the visual impression as such does not play into the argument. What matters, or the argument, is the similarity o the polygons and the purely topological structure they determine – a circle nested precisely between two polygons, triggering Archimedes’ results on concave suraces. Tis diagrammatic practice is not isolated: it de nes the character o Archimedes’ SC . As soon as the structure o a polygon inscribed inside the circle is introduced, in proposition 21, and right through the ensuing argument, the manuscripts systematically deploy such representations based on curved lines – in feen propositions altogether ( .21, 23–6, 28, 30,
Archimedes’ writings: through Heiberg’s veil
32–3, 37–42). I nd it hard to see how a scribe, asked to copy a manuscript where polygons are represented by polygons, would produce a manuscript where polygons are represented by a system o curved lines. Tis lectio dicilior argument is the best I have or showing that, i not introduced by a scribe, such diagrammatic practice is likely authorial. Perhaps our simplest hypothesis is that the diagrams as a whole derive, largely speaking, rom Archimedes himsel.
Te texture of Archimedes’ diagrams: summary Whether by Archimedes or not, the non-iconic character o the representation o polygons in SC is a striking example o how schematic the manuscripts’ diagrams are – and how Heiberg has turned such schematic representations into pictures. Tis is o course consistent with the manuscripts’ preerence or a ‘ at’ representation as against Heiberg’s pictorial pyramids, as well as with the much wider manuscript practice o metrical simpli cation, typically that o representing unequal magnitudes by equal gures. Heiberg has clearly transormed the manuscripts’ schematic diagrams into pictorially ‘correct’ ones. By so doing, however, he has also constructed diagrams o a different logical character. I diagrams are expected to be pictorially correct, then one is expected to read them or some metrical inormation; and i so, the inormation one gathers rom the diagrams is potentially alse (since no metrical drawing can answer the in nite precision demanded by mathematics) as well as potentially overdetermined (since a particular metrical con guration may introduce constraints that are less general than the case required by the proposition). Te schematic and more ‘topological’ character o the manuscripts’ diagrams, on the other hand, makes them logically useul. One can rely on the manuscripts’ diagrams as part o the argument, without thereby compromising the logical validity o the proo. A major claim o my book (N1999) was that diagrams play a role in Greek mathematical reasoning.9 I have suggested there – ollowing Poincaré – that the diagrams may have been used as i they were merely topological. My consequent study o the palaeography o Greek diagrams has revealed a striking and more powerul result: the diagrams, at least as preserved by early Byzantine manuscripts, simply were topological. Heiberg’s choice to obscure this character o the diagrams was not only philologically but also philosophically motivated. Clearly, he did not perceive diagrams to orm 9
N1999, especially chapters 1, 2, 5.
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part o the logic o the text and or that reason, on the one hand, did not value them enough to care or their proper edition and, on the other hand, preerred to produce them as mere ‘illustrations’ – as visual aids revealing to the mind a picture o the object under discussion. Te implication – alse or Archimedes as or Greek mathematics more generally – would be that the text is logically sel-enclosed, that all claims are textually explicit. Tis, then, was the rst transormation introduced by Heiberg into the texture o Archimedes’ reasoning.
Te texture of Archimedes’ text: the local level An overview of Heiberg’s practice of excision A characteristic eature o Heiberg’s edition is his use o square brackets in the sense o text present in the manuscripts, which however is to be excluded as non-authorial. Tis, incidentally, is not the current practice among classical philologers, where the ‘{}’ are used or the same purpose, whereas square brackets are used to signal text restored by the editor – or which Heiberg himsel used the ‘<>’ brackets.10 Tis practice should be compared with two other options Heiberg had available to him. (1) One was to omit excluded text rom his printed text altogether, relegating it into the critical apparatus alone. Such, indeed, is Heiberg’s practice whenever already any o the manuscripts exclude the passage . For instance, SL 68.15–16 has the printed textσυμπεσειται δε αυτατα Z, ‘Tis will meet Z’, which Heiberg has on the authority o codices BG. Heiberg’s apparatus has the comment:αυτα ‘ ] G, τα αυταA(C), ipsi B’ (G is the siglum used or one o the Renaissance copies o codex A), that is: the reconstructed manuscript A certainly readta auta ta (as this is the text read in all copies save the relatively mathematically sophisticated G), and so probably (Heiberg was unsure, but he was right) codex C; in codex B, Moerbeke translated the relevant words as i they wereauta ta alone – though once again, Moerbeke is relatively mathematically sophisticated. τα αυτα τα could ‘in ’, commenting inHeiberg the apparatus τα]principle del. prae.have BG’.printed Tis he‘[did] not do: his practice was to relegate such excluded words to the apparatus alone. On the other hand, in such cases where there was unanimous textual authority or a particular passage which Heiberg preerred to omit, his practice was to print that passage in the main text, surrounded by square brackets. 10
See e.g. http://odur.let.rug.nl/~vannij/epigraphy1.htm.
Archimedes’ writings: through Heiberg’s veil
(2) Another option was to avoid the square brackets altogether, leaving his doubts to ootnotes. He does so occasionally – particularly, it seems, when the exclusion involves both an excision as well as an addition to the text. So, or instance, ootnote 2 in PE , .149, where the text is printed simply as πεποιησθω: ‘πεποιησθω lin. 19 ortasse vestigium recensionis posterioris est. u. Quaest. Arch. p. 70. γεγονετω scripsit orellius cum Basil.’ , that is ‘let it be made in line 19 may be due to a late re-edition; see Quaest. Arch. p. 70 [Heiberg’s PhD]. orelli [Te Oxord 1792 edition] as well as Basil [the rst edition rom 1544] have let it come to be’.
Heiberg could have instead printed [πεποιησθω] γεγονετω, with a note in the apparatus ‘γεγονετω] πεποιησθω ABC, scripsi prae. or., Basil.’ By printing, simply, πεποιησθω, Heiberg shows in this case more respect to the manuscripts’ authority and allows a smoother reading o the main printed text. Heiberg’s strategy is well balanced. It is designed to help the reader navigate the main text as readable prose, without encumbering the apparatus (a necessary consequence o (1) above) or the ootnotes (a necessary consequence o (2) above). Te square brackets are a helpul eature o the text. Tey allow the reader to consider two possible ways o reading the text – with or without the excluded passage – and to see or hersel which she likes best. We should contrast Heiberg’s treatment o the text with his treatment o the diagrams. He made sure as much o the manuscript evidence as possible remained visible as regards the text, even taking pains to print text in whose inauthenticity he was certain – all o this, while removing the evidence or the manuscripts’ diagrams nearly in its entirety! However respectul Heiberg’s practice may have been towards the manuscripts’ textual evidence, its outcome was to de ne a certain set o expectations concerning the local texture o Archimedes’ writing. Heiberg effectively shares with us his view: ‘Archimedes could not write like this’, and readers would take notice o views with such authority. Let us consider, then, Heiberg’s judgements. I move on to describe the pattern o Heiberg’s square brackets. Te rst point to note is their unequal distribution among the treatises. I have gone through the corpus, counting all square brackets and classiying them as ‘single words’ (with the possible addition o the de nite article), ‘phrases’ (i.e. no more than a single claim or construction), ‘passages’ (consisting o several phrases) and ‘long passages’ (the border between these and ‘passages’ is difficult to de ne, but I mean an entire train o thought, going
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able 3.1 Heiberg’s use o square brackets Length (eubner Bracketed by pages o Greek) Heiberg (~BEPP)
reatise Floating Bodies 13 Arenarius
1 word (~0.05)
22
Method
3 words (~0.15)
41
Notes (discussed below) Doric, Palimpsest Doric, discursive
4 words, 2 phrases (~0.25)
Koine, Palimpsest
Spiral Lines
60
5 words, 2 phrases, 1 passage (~0.35)
Doric
Floating Bodies
~26
8 words (~0.35)
Doric, Palimpsest
Quadrature o Parabola
27
6 words, 3 phrases (~0.55)
Doric
Conoids and Spheroids
100
10 words, 10 phrases, 2 long passages (~0.95)
Doric
Planes in Equilibrium
25
3 words, 5 phrases, 2 passages (~1.4)
Doric, Eutocius extant
Planes in Equilibrium
20
7 words, 12 phrases, 2 passages (~2.6)
Doric, Eutocius extant
Measurement o the Circle
6
7 words, 1 phrase, 1 passage (~3.1)
Koine, Eutocius extant
Sphere and Cylinder
31
Sphere and Cylinder
83
12 words, 20 phrases, 12 Koine, Eutocius passages, 3 long passages extant (~8.7) 11 words, 48 phrases, 29 Koine, Eutocius extant passages, 12 long passages (~9)
Note: Te table is arranged by ascending BEPP (Stom. and Bov. are not included in this survey).
beyond a single argument or so). In able 3.1, I list or each treatise its length in eubner Greek pages, as well as its square-bracketed passages. I believe that a good way o quantiying the impact o such square brackets is not by mere word-count – excising ve times asingle-word passage is more signi cant than excising a single ve-word passage – and instead I develop an ad-hoc ‘logarithmic’ count, with each ‘single word’ counting or one unit, each ‘phrase’ counting or three units, each ‘passage’ or nine and each ‘long passage’ or twenty-seven. I then sum up this logarithmic value as the ‘Bracketing Equivalent’. I then calculate the ‘Bracketing Equivalent per Page’ or BEPP, which is the Bracketing Equivalent divided by the number o eubner pages. Tis entire exercise is o course somewhat absurd, but it does arrange the data in a useul way.
Archimedes’ writings: through Heiberg’s veil
Several actors emerge. Heiberg’s tendency was to introduce brackets much more into those texts or which we have an extant commentary by Eutocius (PE, DC, SC). Second, he introduced brackets into Koine treatises (DC, SC – , Meth.) more than to Doric treatises (thus, o the treatises or which we have a commentary by Eutocius, PE in Doric has ar ewer brackets than DC, let alone SC). On the other hand, he was reluctant to introduce brackets into texts or which he had textual authority rom the Palimpsest he introduced ew brackets into the text theintervene Method, even though(thus, it is extant in Koine). Finally, he practically didonot in the more discursive text o the Arenarius. I move on to comment on those actors. Eutocius A common source o square brackets (especially at the level o words) is the comparison o the manuscripts’ text to that o Eutocius’ quotation. Heiberg’s judgement here may be aulted on philological grounds: it is now widely understood that many ancient quotations did not aim at precision,11 and the transormations introduced by Eutocius (e.g. a different particle) can be explained by the new grammatical context into which the quotation is inset by Eutocius. Furthermore, the texts or which there is a commentary by Eutocius are the more elementary, and it appears that Heiberg suspected that such texts were more heavily retouched by their readers: a reasonable assumption, seeing that the more advanced works by necessity had much ewer readers. Te net result is to make the advanced works the benchmark against which all the treatises are judged. Dialect Archimedes the Syracusan may have written at least some o his works in Doric – even when addressing Koine readers in Alexandria. Te manuscripts present a variety o positions, between stretches o text written in what appears like pure Doric, through more mixed passages and all the way to texts in normal Hellenistic Koine. Heiberg’s edition turns this variety into just two options: treatises that Heiberg considered to have been transmitted in the Doric throughout antiquity (which we may call ‘Doric treatises’), and those he considered to have been turned into Koine at some point in antiquity (which we may call ‘Koine treatises’). Tus, the presence 11
A case studied in great detail is the quotations o Plato by his epitomizer Alcinous: Whittaker 1990: xvii–xxx.
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o Koine anywhere in the manuscript tradition o ‘Doric treatises’ – a presence which is ofen considerable, even preponderant – is taken by Heiberg to represent no more than the ailure o scribes whose Doric may not have not have been up to Archimedes’ text. I shall return to discuss all o this in considering the global texture o Archimedes. What is clear, however, is that Heiberg’s initial decision – whether or not to treat a treatise as ‘Doric’ – had consequences at the local level. Understandably enough, Heiberg elt less compelled to preserve the text o the ‘Koine treatises’, considering them the product o some late re-edition, as opposed to Archimedes’ pristine words preserved in the Doric. Tus the ‘Doric works’ come to serve as the benchmark against which the verbal texture o Archimedes as a whole is to be judged. Tis is comparable to the ‘Eutocius’ effect and indeed may be related to it. (Was the transition to Koine related to the presence o Eutocius’ commentaries?) Palimpsest Since the text o the Method is printed by Heiberg in its srcinal Koine, we would expect him to bracket its text more extensively. As I will point out in the next section, the Method provides enough textual difficulties to allow or such editorial intervention. In act, Heiberg leaves the text o the Method almost as it is. Te reason must be, I believe, what we may call a purely sociological or even psychological actor. Te text o the Method is recovered rom the Palimpsest, through Heiberg’s major palaeographic tour de orce. In sociological terms, Heiberg has already displayed his proessional skill by his very recovery o the text and is thereore less under pressure to scrutinize it so as to display his proessionalism. In psychological terms, I suspect Heiberg must have become attached to the words he did manage to read – it would be a pity to go through all the trouble just so as to discover some late gloss! (A reader o the Palimpsest mysel, I am all too amiliar with this urge.) For whatever reason, the act is that the texts recovered rom the Palimpsest are among those Heiberg trusts the most. Since these also happen to be among the more advanced works by Archimedes (in particular FB as well as the Method) this has the tendency o con rming the role o the advanced works as paradigmatic. Arenarius Te Arenarius is an outsider in the Archimedean corpus: written mostly in discursive prose rather than in the style o proos and diagrams, it presents
Archimedes’ writings: through Heiberg’s veil
12 many verbal and stylistic variations on the norm elsewhere. Te same goes or Heiberg’s interventions in this text. In .236.24, Heiberg brackets the particle men which is unanswered by the obligatoryde; in .258.11 he brackets the particleeti which seems to be a mere scribal error anticipating the ollowing prepositionepi. Te case o 222.31, with the words tou kulindrou bracketed, is more complex. Te text as it stands in the manuscript does not make any sense, as Greek grammar or as mathematics. Heiberg not only bracketstou kulindrou but also adds in a particleoun and changes the gender o a relative pronoun. In short, Heiberg’s interventions are philological rather than mathematical in character; that they are so ew is a mark o Heiberg’s tact as an editor. O course, Heiberg’s apparatus records many more variations that Heiberg introduced into the main text and indeed all three brackets could equally have been relegated to the apparatus alone. Needless to say, the Arenarius does not thereby obtain a canonical position or Heiberg’s reading o Archimedes: here, the lack o intervention signals, paradoxically, a marginal status. What theArenarius reminds us is that Heiberg’s exclusions are so closely ocused on the proos-and-diagrams style. Indeed, there are, I believe, no words bracketed inside the introductions to Archimedes’ works. o sum up: Heiberg intervened in Archimedes’ text mostly to exclude
words and passages that, in his view, do not square with what should have been Archimedes’ style o proo, as judged mostly by the advanced works extant in Doric.
Heiberg’s practice of excision: close-up onSphere and Cylinder Te mathematics o Archimedes, especially in the more advanced works, is very difficult. Generally speaking, Heiberg’s brackets tend to keep it that way. Many o the excluded passages take the orm o brie explanations to relatively simple arguments. Te excluded passages make the text o Archimedes locally transparent, and this is what Heiberg avoids – in this way also introducing a certain consistency which is absent rom the manuscripts’ evidence. Consider SC .4. Archimedes constructs a triangle ΘKΛ, with KΘ given and the angle at Θ right. It is also required that KΛ be equal to a certain line H. At this point the text comments ( 16.25): ‘For this is possible, since H is greater than ΘK.’ Tis comment is bracketed by Heiberg. Tere seem to be three reasons or Heiberg’s bracketing. 12
N1999: 199.
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First, this is an argument headed by the particle gar, usually translated ‘or’: having established a claim, the text moves on to offer urther grounds or it. Heiberg’s tendency, especially in the books on Sphere and Cylinder, was to excise a great proportion o gar statements. Tere are altogether 155 occurrences o the particle in the text o SC , outside o the introductions, but o these 58 occur not in the context o a backwards-looking argument but in the context o some meta-mathematical ormulaic expression using a gar, such as the heading o the reductio mode o reasoning: ‘or i possible’,ei gar dunaton. Remaining are 95 occurrences. O these 54 are inside Heiberg’s brackets; only 41 are considered genuine. Te 54 excised gars represent ewer than 50 excisions (a ew long passages excised by Heiberg include more than a single gar), all o them constituting at least a phrase (Heiberg never excises a gar alone – which o course would have produced an asyndeton). Heiberg excised altogether 124 phrases and passages rom the text o SC , , and so we see that about 40 per cent o these excisions are claimed by gars. Note however that many o the remaining excisions have a similar logical character, even while using a connector other than gar: e.g. a dēlon, ‘clearly’ phrase in SC .34 130.20–1, or even an ara, ‘thereore’ phrase in SC .32 120.8. In most cases, the excision is motivated by the elementary character o the claim made. Tis can be seen rom the distribution o excisions o gar between the two treatises. O the 68 gars in SC , Heiberg excises 45 or about two-thirds; o the 27 gars in SC , Heiberg excises 9, that is a third. Te major difference between the two treatises is that SC is usually much more complex than most o SC .13 Te rule then begins to emerge: Heiberg excises gars in the context o relatively simple mathematics. Going back nally to our example rom SC .4, we can now see one reason why Heiberg chose to bracket it: in this example, the text looks back to explain why a certain construction is possible. Tis condition, however, is relatively simple: in constructing a right-angled triangle, the hypotenuse must be greater than the side. Heiberg’s view was that Archimedes could well have just taken such a condition or granted. For this, Heiberg had something o a corroboration. Here I pass to the second ground or Heiberg’s excision: his search orconsistency. In the preceding proposition 3, Archimedes requires an analogous construction, and there the text does not provide an explicit backwards-looking argument, merely stating ( 14.8) ‘or this is possible’ (this is bracketed by 13
As a comparison: in the advanced treatise Spiral Lines, Heiberg brackets 2 out o 33gars – which orms, however, a large part o his overall editorial intervention in that treatise.
Archimedes’ writings: through Heiberg’s veil
Heiberg, or reasons that will be made clear immediately). Why should the text be uller here than in the preceding proposition? Consistency, thereore, requires an excision. I now move to the third reason or Heiberg’s bracketing. o understand it, let us note the ollowing: the received text or Archimedes’ propositions 3 and 4 seems to open a strange gap between propositions 3 and 4. Why would Archimedes offer no more than a brie ‘this is possible’ claim in proposition 3, expanding it in proposition 4? I anything, the opposite – going rom a more spelled-out expression to a brieer one – would be more natural. On the other hand, the entire picture makes perect sense i we pursue the ollowing hypothesis. Now, the text o Eutocius contains a commentary to proposition 3, starting with the ollowing words: ‘And let [the construction be made]. For this is possible, with KL being produced etc.’ ( 18.24–5). Let us assume that Archimedes’ text had none o the backwards-looking argument, and that some late reader has taken Eutocius’ commentary, rst inserting the words ‘or this is possible’ rom Eutocius’ commentary into the text o proposition 3, then using Eutocius as a kind o crib rom which to insert a very brie backwards-looking argument into proposition 4 (or which there is no commentary by Eutocius). We see how the various actors – the presence o Eutocius’ commentary, the elementary nature o the claims made, the use o a backwards-looking argument, textual inconsistency – all come together to inorm Heiberg’s considerations. Was Heiberg right? I tend to believe he was, at least in part. Tis, or the ollowing reason. Either we take the words ‘or this is possible’ in proposition 3 to represent Eutocius’ srcinal words, inserted into the text o Archimedes; or we take them as Archimedes’ srcinal words, quoted by Eutocius as part o his commentary. Now, the word order o those words isdunaton gar touto. Tis word order is natural as an anticipation o the genitive absolute used by Eutocius in his commentary; inside Archimedes’ ull phrase, the word order expected would more likely be touto gar dunaton. Te excision in proposition 3 thereore seems likely. And i so, it becomes somewhat more likely that the words in proposition 4, too, are due to some late reader. But then again, perhaps Archimedes’ text was strangely inconsistent, offering no argument in proposition 3 but some minimal argument in proposition 4? Obviously, such questions can be answered only based on some overarching argument concerning Archimedes’ style, an argument which would have to be derived – circularly – rom the established text o Archimedes. In some cases, and in particular in the longer passages, Heiberg’s excisions seem very reasonable. One o the clearest cases is SC .13 ( 56.10–24).
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Tis should be read in ull to get a sense o the manuscript evidence Heiberg had to contend with (I quote together with my numbering o claims in the argument. It should be clear that this is something o an extreme case, though not at all a unique one): (16) But that ratio which Δ has to H in square – Δ has this ratio to PZ in length [(17) or H is a mean proportional between Δ, PZ (18) through between Γ Δ, EZ, too; how is this? (19) For since Δ is equal to Γ, (20) while PE to EZ, (21) thereore Γ is twice Δ, (22) and PZ PE; (23) thereore it is: as ΔΓ to Δ, so PZ to ZE. (24) Tereore the by Γ Δ, EZ is equal to the by Δ PZ. (25) But the by Γ Δ, EZ is equal to the on H; (26) thereore the by Δ, PZ, too, is equal to the on H; (27) thereore it is: as Δ to H, so H to PZ; (28) thereore it is: as Δ to PZ, the on Δ to the on H; (29) or i three lines are proportional, it is: as the rst to the third, the gure on the rst to the gure on the second which is similar and similarly set up]
Te expression ‘how is this?’ inside claim 18 is without parallel in the corpus, and seems like a didactic order to a pupil (or, perhaps, an autodidact’s cri de coeur?). Te passage rom 19 to 21 is indeed extraordinarily simple (rom A = B to A + B being twice A). Te nalexplicit quotation rom Euclid’sElements is natural coming rom a didac tic context. And overall the argument is very simple, strikingly so given its length. It is thereore quite likely that the entire passage rom ‘how is this?’ in the end o claim 18 down to the end o claim 29 is a scholion inserted into the manuscript tradition. Heiberg’s choice, however, was to bracket starting rom step 17 itsel – this, apparently, merely because step 17 begins with a gar. It would be easy or us to condemn Heiberg’s use o square brackets as disrespectul to the manuscripts’ evidence, or as involving massive circular reasoning. But Heiberg’s practice is not unreasonable and is likely to be correct at least in part. I doubt any editor could have come up with a single system better than Heiberg – short, that is, o the conession o editorial ignorance which might have been best o all (and which Heiberg, in a sense, did nally ollow – by allowing the bracketed words to be printed inside the main text). I stand by my judgement o Heiberg as a superb, and superbly tactul, philologer. Having said that, however, the act remains that we cannot really say how correct he was. Tere are three texts at play here: (A) Heiberg’s text with the bracketed segments inserted, i.e. the manuscripts’ reading. (B) Heiberg’s text with the bracketed segments removed. (C) Archimedes’ srcinal text.
Archimedes’ writings: through Heiberg’s veil
Heiberg’s intention was o course to take A and, by transorming it into B, to make it come as close to C as possible. It is indeed certain that A and C are not identical. However, it is impossible to judge how close B is in act to C. Te only judgement we can make with con dence has to do with the relationship between A and B. Te transormation introduced by Heiberg into the manuscripts’ text is motivated by two main considerations: the avoidance o explicit argument in the context o relatively simple mathematics; and the avoidance o textual inconsistencies. Tis determines the image o Archimedes as projected by Heiberg’s method o excision: neither transparent nor inconsistent. I do not address right now the question whether this image is, or is not, correct. I merely point out the presence o this image, beore moving on to consider the in uence o this image in Heiberg’s treatment o the texture o Archimedes at the global level.
Te texture of Archimedes’ text: the global level As usual, my point is not to criticize Heiberg. In some ways, any edition involves a transormation at the global level. Te ‘eel’ o an Opera Omnia in its eubner print is very distinct rom that o codices A or C which, in turn, would have elt, possibly, even more different rom their antecedent o a basket o rolls in ancient Alexandria. Some o Heiberg’s decisions were o this inevitable character: so, or instance, an Opera Omnia must proceed in some order, and the act that this calls or editorial decision does not thereby make the editor unaithul to his author. On the other hand, in some other orms Heiberg made choices or presentation that went beyond the manuscripts’ evidence, mostly inormed by a sense o overall mathematical consistency.
Te order of Archimedes’ works Knorr was upset over that issue:14 Following the start made by orelli in 1792, Heiberg had in 1879 attempted to determine the relative chronology o the treatises then known to him. But in setting them out in his ensuing editions o Archimedes he chose to retain the traditional order in the principal manuscripts, based on the prototype A, and then tacked on the ew remaining works and ragments preserved in other sources. 14
Knorr 1978: 212–13.
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Heiberg’s ordering has been adopted in all subsequent editions and translations, notably those by . L. Heath, P. Ver Eecke, E. J. Dijksterhuis and C. Mugler. Indeed, Ver Eecke pronounced it to be o all possible orderings “le plus rationnel”. What began as merely a philological concern to keep strictly to the sequence o the manuscript sources has thus given rise to the astonishing view that this ordering has intrinsic rational merit, despite such patent incongruities as the placing o the Sand Reckoner and the Quadrature o the Parabolaand others to be discussed below.
Tis may, rst o all, serve as a nice reminder o the pre-eminent position o Heiberg in our contemporary reading o Archimedes. Further, I am not quite clear as to what ‘patent incongruities’ Knorr meant. Clearly his interest lay with the chronological sequence, and as such the order o the Opera Omnia makes no sense. It is not a random order, though, and its signi cance should be pondered. Here is the order o Heiberg’s second edition: SC – SC – DC – CS – SL – PE – PE – Aren. – QP – FB – FB Meth. – Book o Lemmas – Bov. – Fragments (in reality, estimonia).
– Stom. –
Up to QP, inclusive, this ollows (as explained by Knorr) the order o codex A (which was the only order available to Heiberg, on manuscript authority, or his rst edition). Te works extant on the Palimpsest ollow in the order FB – Stom. – Meth. (perhaps designed to keep the Method till later?), and then ollow several works rom diverse sources: the Book o Lemmas rom the Arabic, the Cattle Problem rom a different line o transmission altogether, and then o course the estimonia rom sources other than Archimedes himsel. One should note the outcome, that Heiberg oregrounded the works in which he detected most interpolations. Tis is not a paradox: the works oregrounded by Heiberg were the elementary works in pure geometry, and the detection o many interpolations could have meant to Heiberg an indication o the signi cance such works had or Archimedes’ ancient and medieval readers. While Heiberg’s principle was purely philological, he ollowed manuscripts that, themselves, made rational choices (so that Ver Eecke’s judgement is not necessarily alse). Te system underlying A is quite clear. A sequence o ve works in pure geometry (SC , SC , DC, CS, SL) is ollowed by a sequence o our works that reer in some way or another to the physical order (PE – PE – Aren. – QP; this is ollowed in codex A by Eutocius’ commentaries, and then by a treatise by Hero on Measures). Such an arrangement is suggestive o a previous ‘canonical’ selection o
Archimedes’ writings: through Heiberg’s veil
‘top ve Archimedean geometrical rolls’, ‘top our Archimedean physical rolls’, perhaps representing a previous arrangement o rolls by baskets, perhaps o some majuscule codices with only our to ve works each.15 In each sequence, the internal order is roughly rom the simpler to the more complex. It so happens that the works preserved via traditions other than codex A tend to be less ocused on pure geometry. Tree o the works preserved via C – FB , FB , Meth. – have a marked ‘physical’ character. Te Stomachion, also preserved via C, may be a unique study in geometrical combinatorics.16 And while the Book o Lemmas does touch on pure geometry, the Cattle Problem is an arithmetical work. Te ragments, nally, reer to such diverse topics as astronomy, optics or the arithmeticogeometrical study o semi-regular solids reported by Pappus .17 In short, the emphasis on pure geometry – very natural based on codex A alone – is less aithul to the corpus as a whole as recognized today. Or indeed as recognized by some other past traditions. For the order o codex C was distinct: PE (?)18 – PE
– FB – FB
– Meth. – SL – SC – SC
– DC – Stom.
Tis has ve works reerring to the physical world (PE – , FB – , Method) ollowed by ve works o a non-physical character (SL, SC – , DC, Stomachion). Once again, the srcin in some earlier arrangement is likely, and the main classi catory principle is the same – reerring, or ailing to reer, to an outside physical reality. Te striking difference is that codex C chose to position the physical works prior to the non-physical ones. At issue is a undamental question regarding Archimedes’ scienti c character. Was he primarily a pure geometer, who indulged in some exercises o a more physical or non-geometrical character? Or was he primarily an author o ‘mixed’ works, so that the more purely geometrical works – such as Sphere and Cylinder – should be seen as no more than one urther option in the spectrum o possible Archimedean variations? A very different Archimedes would emerge i we were to order his works, say, as ollows: 15
16 17 18
Tese two options, o course, do not rule each other out. See Blanchard 1989 or some suggestive comparisons. Netz et al. 2004. Hultsch 1876: 350–8. Te beginning o the Archimedes portion o the Palimpsest appears to be lost. Te text begins towards the end o PE . Tere could be works prior to PE , or the manuscript could start with PE only. Either option, however, is less likely than that the manuscript started with PE .
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Semi-Regular Solids19 Stomachion Book o Lemmas Measurement o the Circle Method Conoids and Spheroids Sphere and Cylinder Sphere and Cylinder Cattle Problem Planes in Equilibrium Planes in Equilibrium Spiral Lines Arenarius Quadrature o Parabola Floating Bodies Floating Bodies What would such a counteractual order suggest? Above all, a certain lack o order, and the sense o an author who reveled in variety. Tis, indeed, may not be too ar o the mark. But notice how different this is rom the impression made by Heiberg’s order chosen or theOpera Omnia! For his sober-minded eubner edition, based on the authority o the soberminded scribe o A, Heiberg has produced a sober-minded Archimedes – one who was above all a pure geometer. Tis, once again, may possibly be historically correct. But then again, perhaps it is not. Te one thing clear is that the order orms an editorial decision: a different ordering o the works would have given us perhaps a less sober, perhaps even a less geometrical Archimedes.
Te dialect of Archimedes’ works Te very language in which Archimedes’ works should be read orms a genuine philological puzzle. I do not think we are ready to solve this puzzle, yet, and sooI this merely outline here problem, somewhat the discussion problem rom pp.the 179–80 aboveexpanding and ocusing on the signi cance o Heiberg’s approach to it.
19
While not extant, Archimedes’ work on semi-regular solids is known through a report in Book o Pappus’Collection. I am envisaging how Archimedes’ works would have looked had a work such as this appeared rst.
Archimedes’ writings: through Heiberg’s veil
Some o the manuscripts that give evidence or Archimedes’ works contain a signi cant presence o Doric dialect orms, in particular ποτι or Koine προς, ειμεν or Koine ειναι, εσσειται or εσται as well as certain phonological variations, predominantly the use o long α or Koine η. Such dialect orms are very common in the manuscript evidence or PE , CS, QP, Arenarius (A alone), FB (C alone) and SL (both A and C). Te dialect orms are much less common, or totally missing, in SC , SC , DC, PE (both A and C), FB , Stomachion and Method (C alone). Heiberg’s comment on this last work ( .xviii) is telling: ‘And even though I do not doubt that this work, too, was written in Doric by Archimedes, I dare not reinstate the dialect that was so diligently removed by the interpolator.’20 In other words, Heiberg sees the Koine dialect as a kind o interpolation, inserted into the text o SC , SC and DC (works that Heiberg would anyway consider heavily mediated by their readers) as well as some other works. While SC , SC and DC are completely ree o Doric dialect, all the other works display a certain mixture o Doric and Koine, more Doric in such works as SL, much more Koine in works such as Method. Heiberg’s edition removes this sense o gradation, introducing instead a clear biurcation. SC , SC , DC and Method are printed mostly in pure Koine, no mention made in the critical apparatus or the (rather ew) cases where Doric orms are present. PE , PE , CS, QP, Arenarius, FB , FB and SL are printed in pure Doric, no mention made in the critical apparatus or the (rather many) cases where Koine orms are present. 21 Notice that Heiberg imposed Doric on PE and FB , against the manuscripts – which he avoided doing or Method – presumably because o a desire to preserve their continuity with PE and FB , respectively. Underlying this simple biurcation is an even simpler monolithic image o Archimedes’ language. As Heiberg said plainly, his position was that Archimedes wrote in Doric and in Doric alone. Heiberg, ever the philologer, did produce an explicit survey o the dialect variation. Tis however he did not in the critical apparatus itsel, but inside a dedicated index o manuscript variations, positioned as the major component o the introduction to the second volume. Tis doubly marginalizes the importance o the dialect variations. First, by taking them away rom the critical apparatus, and second, by positioning them in the second 20
21
‘et quamquam non dubito, quin hoc quoque opus Dorice scripserit Archimedes, dialectum de industria ab interpolatore remotam restituere ausus non sum.’ Te Stomachion – preserved in ragmentary orm and thereore more tactully handled – is the only work or which Heiberg simply prints, without comments, the orm o the manuscript (according to Heiberg’s readings), allowing a ‘mixed’ dialect.
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volume, rather than in the third and nal volume (which is where critical editions typically present their major philological observations). In this case as in the case o excisions (to which the question o dialect is afer all closely related, as Heiberg’s excisions, as we saw, centred on what he de ned as Koine-only treatises) Heiberg could well be right. We could never tell or sure whether Heiberg was indeed right on dialect, but his position is indeed plausible. What Heiberg did achieve however is to obscure the very question which, to my knowledge, has not been addressed at all to date. Which dialect(s) did Archimedes write in, and what was the signi cance o such choice? I do not have the expertise required to solve such questions, but I wish to emphasize that these questions have yet even to be posed. Would a choice to write in Doric, or in Koine, carry speci c cultural meanings? It is very intriguing that a late source tells us that Archytas is the model or Doric prose.22 Archytas o course was primarily a scienti c author, indeed known or his contribution to the exact sciences. Was there a cultural value attached to Doric as a marker o scienti c prose? (Eudoxus, rom the Doric-speaking island o Cnidus, could have written in Doric as well; or certain, he did not write in Koine which was not yet available in his time.)23 Clearly, dialectal choice was, in Archimedes’ time, a charged generic marker. Hellenistic authors were keenly aware o their position as heirs to a rich literary tradition, varied by genre and by dialect – the two ofen going hand in hand. Elegy would be written in (a speci c variety o ) Ionic, epic poetry in the Homeric dialect (which in itsel was a Kunstsprache, an ad-hoc amalgamation o several layers o Greek that never served together in any actually spoken Greek). 24 Heiberg’s implicit claim was that the question o dialect was minor, because it was unmarked: what would Archimedes write in, i not his native language? Even deeper lies the assumption that a mathematician’s language does not matter. Archimedes would write in Doric, the unmarked
22
23
24
Gregory o Corinth, On Dialects. (A6g in Huffman 2005: 279–80). Tis – Byzantine – source mentions Archytas and Teocritus as the models o Doric, Archytas clearly intended thereore as the model o Doric prose. While late, it is difficult to see how such a statement could emerge based on anything other than solid ancient testimony rom the time that Archytas’ works were still widespread. Nor should we think in terms o a monolithic ‘Doric’ opposed to a monolithic ‘Koine’ . It is completely unclear to me, or instance, whether the Doric prose o Archimedes’ usage could not have allowed των, instead o ταν, more ofen than Heiberg assumes (there are about twenty cases o such variation in each o SL and Arenarius, where Heiberg always prints ταν). Te locus classicus or an interpretation o this traditional observation is an essay by Parry rom 1932, ‘Studies in the epic technique o oral verse-making. . Te Homeric language as the language o an oral poetry’, most conveniently available as chapter 6 o Parry 1971 .
Archimedes’ writings: through Heiberg’s veil
orm he would speak anyway, since he would not even think about which language to use: the contents matter, and not their verbal orm. Such is the image projected by Heiberg’s editorial choice to minimize the question o dialect and to assume a purely Doric Archimedes. I am not sure this is true, and so I suspect that there is an open question as to the cultural signi cance o Archimedes’ choice o dialect. Tis question is elided by Heiberg’s editorial choices.25 Once again: I do not condemn Heiberg. I point, instead, to the signi cance o Heiberg’s move away rom the manuscripts, regardless o how close this may or may not have brought him to the ‘srcinal text’. Te main consequence o Heiberg’s move was to make the verbal texture o the text appear much more consistent than it was in the manuscript evidence. Te main implication o that would be to minimize the very signi cance o verbal texture: to make Archimedes, once again, into a pure geometer – one who cares about his mathematics and not at all about his style.
Te format of Archimedes’ works I Heiberg’s Archimedes ignores questions o verbal shape, this Archimedes certainly pays attention to mathematical shape or ormat. In the critical edition, the text is articulated throughout by a systematic arrangement based on two dualities: that o the introductory text as against the sequence o propositions; and, inside the propositions, that o the general statement as against the particular proo. Both are determined by the major eature o the ormat, namely the sequence o numbers o propositions inside each work. Te rst numeral, preceding the rst proposition, marks the transition rom introduction to the sequence o propositions; rom then onwards, each numeral is ollowed by a single paragraph written out without diagrammatic labels, which is the general statement preceding the main proo. Tis ormat has basis in the manuscripts’ authority and may to some extent re ect Archimedes himsel. In some ways, however, Heiberg tends to emphasize the regularity o this ormat and even to insert it against the manuscripts’ authority. Te layout itsel is signi cant. Heiberg has the proposition numerals written inside the block o printed text with clear spaces preceding and 25
All o this is closely parallel to the question o dialect in Teocritus – another third-century Syracusan extant, mostly, in some orm o Doric, poetic in this case – and even though the analogous problem has been researched or the case o Teocritus, scholars are ar rom consensus (see Abbenes 1996).
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ollowing them, serving in this way to articulate the writing in a highly marked orm. Following that, Heiberg writes out his text in accordance with the clear paragraph arrangement dictated by modern conventions, with the general statement always occupying a separate paragraph. Te Byzantine manuscripts ollowed a somewhat different layout. Numerals or propositions – where present – are marginal notes that do not break the sequence o the writing (this articulation is provided, however, by the diagrams, as a rule positioned at the end o their respective propositions). Division into paragraphs is less common in Byzantine manuscripts (where it is perormed by spacing inside the line o writing, where the break is to take place, together with an optional bigger initial in the ollowing line, positioned outside the main column o writing). ypically, general statements do not orm in this sense a paragraph apart, such division into paragraphs being reserved or more major divisions in the text – typically or the very beginning o a proposition or, occasionally, in such major transitions as the passage rom the ‘greater’ to the ‘smaller’ cases in the Method o Exhaustion (so, or instance, codex C in SL 25, 96.30). It is likely that Archimedes’ srcinal papyrus’ rolls were, i anything, less articulated than that.26 Not that this impugns Heiberg’s use o paragraphs: modern editions universally ignore such questions o layout, imposing modern conventions, and even though the layout o the manuscripts, as o Archimedes himsel, did not possess Heiberg’s visible articulation, it is air to say that the two divisions – o introduction rom main propositions, and o general statements rom proos – are genuinely part o Archimedes’ style. However, because Heiberg is committed to a visible layout, he is also orced to set clear-cut divisions where the srcinal may be less clearly de ned. First, even though the Archimedean text does operate between the polarities o discursive prose and mathematical proposition, it is not as i the transition between the two is typically handled as a break in the text. Rather, Archimedes negotiates the transition in varied ways that make it much smoother. o take a ew examples: ollowing the main introductory sequence in CS ( 246–258.18), Archimedes moves on to a passage ( 258.19–260.24) where several simple claims are either asserted without argument, or are accompanied by a minimal argument without diagrammatic labels (e.g. Archimedes explains that when a plane cuts both sides o a cone, it produces either a circle or an ellipse). Only ollowing that, at 260.25, Archimedes moves on to a longer and more complicated 26
On early papyrus practices o articulation o text, see Johnson 2000.
Archimedes’ writings: through Heiberg’s veil
proposition that also calls or a diagram. Codex A also marked this proposition with the marginal numeral A. Heiberg prints the entire sequence 246–260.24 preceding ‘proposition 1’ as a single paragraphed block o text, that is the ‘introduction’, ollowed by the sequence o ‘propositions’ starting at 260.25. But clearly Archimedes’ intention was to create a smooth transition mediated by the passage 258.19–260.24, which does not all easily under either ‘introduction’ or ‘propositions’. Very similar transitions are seen in SC , SC , QP and PE , with Heiberg making different choices: in SC and SC the transitional material is incorporated into the ‘introduction’; in QP and PE it is incorporated into the ‘propositions’. Further, while the rst proposition o theMethod has a complex argument that calls or a diagram, Archimedes rounds it off with a second-order comment that makes it appear rather like part o the ‘introduction’ ( 438.16–21). Heiberg, very misleadingly, prints this comment as i it ormed part o proposition 2: clearly Archimedes’ point was to smooth, once again, the transition rom introduction to propositions. I we bear in mind that the complex interplay o introduction and propositions is typical o the Arenarius, and that FB , PE and DC do not possess an introduction at all, we discover that Heiberg’s neat dichotomy o introduction divided rom text is ound in SL alone! Heiberg’s clear articulation o the text into ‘propositions’ alling into paragraphs tends to obscure, once again, the variety o ormats ound in the corpus. Quite ofen, the text relapses into brieer arguments set in a general language that does not call or a diagram. Heiberg marks such passages off and heads them as ‘corollaries’ or porisma, but this is done against the manuscripts’ evidence where, instead, such passages orm part o the unbroken ow o the text. Tis happens twenty times in the corpus. Heiberg systematically introduces the title porisma into the printed text, noting in the apparatus that the manuscripts ‘omit’ this title! For instance PE : Heiberg prints πορισμα α in 130.22 and πορισμα β in 132.4, with the ollowing apparatus: 130.22 om. AB Πο D, 132.4 om. AB. Tat is: one copy o A introduced, in the rst case, a marginal mark anticipating Heiberg’s own intrusion. But the srcinal text had no such headings. Te important consequence is that the srcinal text allowed stretches o text, inside the main ow o ‘propositions’, where no detailed, diagrammatic argument was required – and without segregating such passages by a title such as ‘corollary’. Te variety o the srcinal is wider than that. Tus, or instance, some propositions have a complex internal structure not neatly captured by the simple division into general statement and particular proo (such as the
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analysis-and-synthesis pairs typical o SC , as well as the extraordinarily complex internal structures – punctuated by several diagrams – o FB .8–10). Other propositions do not even display this simple division: or instance, several key propositions o SC , starting with 23, take the orm o a ‘thought experiment’ where a certain operation is carried out ollowed by an observation. Such propositions do not call or a general statement. Further, many o the propositions o QP do not have a general statement and start instead directly with diagrammatic labels. Now, Heiberg does report correctly the contents o such deviant propositions, but his overall system o articulating the text by explicit numerals tends to orce the readings o all propositions into a single mould. More, indeed, can be said or the case o QP. Te manuscripts do mark numerals or the rst our propositions (the rst three o which, however, dey easy counting, as they orm the transitional material rom introduction to propositions). Ten, rom proposition 5 onwards, no numerals are present. Heiberg dutiully notes this act but in a misleading ashion (analogous to his treatment o the title ‘corollary’): he goes on printing the numerals, noting in the critical apparatus to proposition 5 that rom this point onwards the numerals are ‘omitted’ by the manuscripts. Tis is not a unique case: the manuscripts or DC and Method never contain numerals or proposition numbering; Heiberg introduces the numbering and then makes the apparatus report their ‘omission’. A similar pattern can be seen inside the introductory material. Tere, Archimedes ofen includes material o substantive axiomatic import – certain assumptions, or de nitions, that he requires later on or his argument. ypically, Heiberg introduces titles to head such passages (that, in the srcinal, belong directly to the ow o theintroduction), and then numbers the individual claims made in such passages. Tus, Heiberg’s introduction o SC is divided (ollowing orelli) into three parts: a general discussion proper ( 2–4), αξιωματα or ‘de nitions’, so headed and numbered 1–6 ( 6), λαμβανομενα or ‘postulates’, so headed and numbered 1–5 ( 8). itles and numbers are not in the srcinal. Similar systematizations o the axiomatic material take place in Method, SL (inside the later axiomatic passage, 44.16–46.21) and PE . Heiberg’s position must have been that all such titles and numerals were required and so would have been lost only through some textual corruption. Otherwise, he could at the very least have marked off such editions by, say, pointed brackets, or, at the very least, commenting in the apparatus add. or ‘I added’ instead o om. or ‘the manuscripts omitted…’ Tis position blinded Heiberg to the serious textual question regarding the srcins
Archimedes’ writings: through Heiberg’s veil
o such numerals in general. While the manuscripts do usually possess numerals or proposition numbers, there seems to be some occasional disagreement between the manuscripts as to which numerals to attach. Tis disagreement is typically between the various copies o codex A, and so carries little signi cance (aside rom signalling to us that the scribes may have elt a certain reedom changing those numbers). In the ew cases (SL, SC , SC ) where Heiberg could compare the numbering reconstructed or codex A with that reconstructed or codex C, the numbers were indeed the same. However, it is interesting to observe that codex C has the number 11 or what Heiberg titles (based on codex A) PE .10.27 Heiberg almost certainly was unable to read this number but, once this evidence is considered, we nd a remarkable act: the two aerly Byzantine manuscripts or PE numbered their propositions differently. Tis o course raises the possibility that such numbers are indeed not part o the srcinal text but are rather (as their marginal position suggests) a late edition by Late Ancient or Byzantine readers. Here, remarkably, Heiberg may have ailed to be critical enough. Te possibility that the numbering was not authorial apparently did not even cross his mind. Tis phenomenon o systematization by titles and numerals is quite out o keeping with Heiberg’s overall character as an editor. Tere must have been a major reason or Heiberg to intervene in the text so radically, and so blindly. Tis act complements the evidence we have seen or Heiberg’s treatment o Archimedes’ verbal style. Just as Heiberg considered Archimedes indifferent to his verbal style, so we see Heiberg imputing to Archimedes meticulous attention to mathematical style. And this, even though such an imputation ies in the ace o the evidence. Whereas Archimedes’ text shows a great variety o orms o presentation, a gradation between more or less ormal, more or less general, and a merely discursive arrangement, Heiberg produces a text marked by the dichotomies o introductory and ormal, general and particular, throughout producing a neatly signposted text. Tis is a consistent Archimedes – and a consistently ormal one.
A close-up on the Method Archimedes’ Method orms a special case. First, Heiberg aced here a task somewhat different rom elsewhere: he needed not only to judge a text, but also, to a certain extent, to ormulate it himsel. Much o the text o the Palimpsest was illegible to him and so much had to be supplied. Second, 27
Te Archimedes Palimpsest 14r col. 1, margins o line 11.
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here we can test Heiberg’s judgement. Heiberg’s decisions elsewhere – that this or that was by Archimedes himsel, this or that was by an interpolator – will probably never be veri ed or reuted. But whenever we can now read passages o the Palimpsest that were illegible to Heiberg, we thereby test a conjecture. Te issue o course is not to see how good Heiberg was as a philologer. He was a superb one and, indeed, the new readings o the Palimpsest ofen corroborate Heiberg’s guesses to the letter. I shall now concentrate, however, on three alse guesses – which together orm a systematic whole, characteristic o Heiberg’s overall approach to the text o Archimedes. Tis is also a good example o the enormous sway Heiberg’s edition had over Archimedes’ destiny through the twentieth century. Heiberg’s edition was careul and prudent: pointed brackets surrounding passages that he ully guessed, dots to mark lacunae that he could not read at all (ofen with remarks in the apparatus asserting the length o such lacunae), dots underneath doubtul characters. It is true that today we nd that a number o characters Heiberg printed with con dence were wrong, but this is a natural phenomenon in a palimpsest where the overlaying text occasionally creates the illusion o alse characters. All o this was accompanied by a Latin translation – as was Heiberg’s practice elsewhere – where doubtul passages were careully marked by being printed in italics. In short, any careul reader could tell which part o the text was Heiberg’s, and which was Archimedes’. And yet, Heiberg’s in uence was such that all later editors, translators and readers operated, as it were, on the basis o Heiberg’s Latin translation, largely speaking ignoring the difference between the Latin printed in Roman characters (which Heiberg read con dently) and the Latin printed in italics (which Heiberg merely guessed or supplied). Here, more than anywhere else, Heiberg’s text supplanted that o Archimedes. Tis had real consequences, subtle but consistent – so as to change the overall texture o the treatise. (1) Te rst case is the most clear-cut. W e now recognize Method proposition 14 (to ollow Heiberg’s misleading numerals) as one o the most important proos ever written by Archimedes, but this is on the strength o a new reading, illegible to Heiberg. As read by Heiberg, this proposition is a mere variation on themes developed elsewhere in the Method, o little deep value. Te Method typically operates by the combination o two principles: a method o indivisibles (conceiving an n+1-dimensional object as constituted by a continuity o n-dimensional objects), and the application o results rom geometrical mechanics or the derivation o
Archimedes’ writings: through Heiberg’s veil
results in pure geometry. Tis is ofen done by obtaining a common centre o gravity to all pairs, suitably de ned, o the n-dimensional objects; assuming that the centre o gravity is then inherited by a pair o n+1-dimensional objects constituted by the n-dimensional objects; and nally applying the results that ollow rom the geometrical proportions inherent in the Law o the Balance. Tis is illustrated by Archimedes through a variety o results arranged by Heiberg as propositions 1–11. As Archimedes clari es in the introduction, his intent is to provide also ‘classical’ or purely geometrical proos or a couple o new results, measuring the volumes o (a) the intersection o a cylinder and a triangular prism, (b) the intersection o two orthogonally inclined cylinders. Nothing survives o the proos or (b), but we have considerable evidence or no ewer than three proos or (a). Te rst, arranged by Heiberg as the two propositions 12–13, is a proo based on both a method o indivisibles as well as geometrical mechanics. Te second is proposition 14, on which more below; the third – called by Heiberg ‘proposition 15’ – survives in ragmentary orm, but it is clear beyond reasonable doubt that this orms, indeed, a ‘classical’ proo based on standard geometrical principles applied elsewhere. Tis is in act a proo based on the method o exhaustion. Proposition 14 thereore occupies a middle ground between the special procedures o the Method, and the standard geometrical principles applied elsewhere. Indeed, it uses only one part o the procedures o the Method. It makes no use o geometrical mechanics, based instead on indivisibles alone. Archimedes considers a certain proportion obtained or any arbitrary slice in the solid gures – so that a certain triangle A is to another triangle B as a certain line segment C is to another line segment D. Te set o all triangles A constitutes the triangular prism; the set o all triangles B constitutes the intersection o cylinder and triangular prism that Archimedes sets out to measure; the set o all line segments C constitutes a certain rectangle; the set o all line segments D constitutes a parabolic segment enclosed by that rectangle. Heiberg’s readings reached this point, and then Heiberg hit what was, or him, a lacuna in his readings. He picked up the thread o the argument as ollows. It is assumed that, since the proportion holds between all n-dimensional gures, it will also hold between all n+1-dimensional gures. We thereore have the proportion: a triangular prism to the intersection o a cylinder and a triangular prism, as rectangle to parabolic segment. Since the ratio o a rectangle to the parabolic segment it contains is known, and since the triangular prism
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is measurable, the intersection o the triangular prism and the cylinder is measured as well. All this makes sense and we can thereore even understand why Heiberg was content: his reading, though lacunose, was mathematically sound. He did remark on the lacuna ‘Quid in tanta lacuna uerit dictum, non exputo’28 – ‘I do not guess what were to be written in such a long lacuna’. Tis comment may be prudent, but it accompanies a text that, otherwise, is meant to be read as mathematically meaningul. In other words, the implication would be that the missing lacuna was no more than ornament that does not impinge on the mathematical contents o proposition 14, and it was certainly in this way that proposition 14 was read through the twentieth century.29 Te upshot o this reading is indeed to make the proposition less important, because it contains nothing new. It applies the method o indivisibles – previously applied in the Method – by assuming that a certain property obtained or n-dimensional objects is inherited by the n+1-dimensional objects they constitute. It differs rom the previous propositions in a merely negative way – it does not apply geometrical mechanics – and thereore it makes no contribution to our understanding o Archimedes’ mathematical procedures. Tis understanding o proposition 14 was revolutionized by the readings o Netz et al. (2001–2), where the lacuna was nally read. It is clear that this lacuna adds much more than ornament. Indeed, it orms the mathematical heart o the proo. Archimedes applies certain results concerning the summation o sets o proportions developed elsewhere, results that call or counting the number o objects in the sets involved, with the number o objects in this set equal to the number o objects in that set. And this – even though the sets involved are in nite! Tus, Archimedes does no less than count (by the statement o numerical equality) in nite sets. Te proo is thereore not a mere negative variation on the previous proos; to the contrary, it opens up a unique avenue, completely unlike anything else extant rom Greek mathematics. Heiberg’s minimal interpretation o the text is thus reuted. Tough, o course, this is not to blame Heiberg: what else could he do? (2) Te next example comes rom the nal, ragmentary proposition 15. Te rst page o this proposition survives on os. 158–9 o the 28 29
Heiberg 1913: 499, n. 1. See in particular Sato 1986, Knorr 1996, texts rare or paying any attention to proposition 14, both assuming that the text extant in Heiberg can be taken to represent Archimedes’ own reasoning.
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Palimpsest, in a orm which was mostly illegible to Heiberg. Tere ollows a gap in the text extant in the Palimpsest, ollowed by our considerable ragments extant on o. 165 o the Palimpsest. wo o those ragments were nearly ully read by Heiberg, and they ormed a basis or an interpretation o the proo as a whole, one which the much uller reading we possess today corroborates on the whole. Its main eature is the ollowing. In this, purely geometrical proo, Heiberg makes Archimedes ollow a route comparable to that used in the measurement o conoids o revolution in CS. A sequence o prisms is inscribed inside the curvilinear object; the difference between the sequence o prisms and the curvilinear object is made smaller than any stated magnitude; and the assumption that the curvilinear gure is not o the volume stated then leads to contradiction. All o this is well known rom elsewhere in Archimedes and Heiberg had many patterns to ollow – especially rom CS itsel – in his reconstruction o the text o os. 158–9 beginning the proo. In contrast to proposition 14, wherethe lacuna unread by Heiberg – no more than about hal a column o writing – proved to be much richer in mathematical meaning than Heiberg imagined, here, os. 158–9 contain three and a hal columns o writing, mostly unread by Heiberg, and they contain practically no mathematical signi cance. Here the surprise is the opposite to that o proposition 14. Heiberg in his reconstruction rather quickly establishes the geometrical construction required or inscribing prisms inside the curvilinear object. Archimedes himsel, however, went through what may have been the most detailed construction in his entire corpus. Te construction is much slower than that o the analogous proos in CS. At the end o these three and a hal columns o writing, Archimedes had not yet reached the explicit conclusion that the difference between the curvilinear object and the inscribed prisms is smaller than any given magnitude. It appears that in making the transition rom the unorthodox procedures o propositions 1–14, to the ‘classical’ procedure o proposition 15, Archimedes made a deliberate effort to make proposition 15 as ‘classical’ as possible – as explicit and precise as possible. (One o course is reminded o how Heiberg tends, elsewhere, to doubt passages where Archimedes is especially explicit and transparent. Would he have excised a good deal o proposition 15, had he been able to read more o it?) Archimedes’ motives are difficult to judge but the effect most certainly was to emphasize the gap between the two parts o the treatise, the unorthodox and the orthodox. Tis gap was somewhat smoothed
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over in Heiberg’s reconstruction though, once again, let this not be construed as a criticism o Heiberg: or, once again, there was no way or him to guess how different Archimedes’ construction here was rom that o Heiberg’s models in CS. (3) A nal example is rom rpoposition 6. Here Archimedes determines the centre o gravity o a hemisphere – as it appears rom the beginning o the proposition, the relatively legible verso side o o. 163. Heiberg thus knew what this proposition was about. Te text then moves on to the recto side o o. 163, which was barely legible to Heiberg, o. 170 – mostly illegible in 1906 and one o the three leaves to have disappeared since – and the recto o 157, completely unread by Heiberg. As mentioned, we have meanwhile lost o. 170 but, at the same time, through modern technologies, we have recovered practically the entire text o os. 163 (recto) and 157 (recto) As a result, we now know that Heiberg’s reconstruction o the parts he could not read was wrong. Heiberg’s modus operandi here was straightorward. While proposition 6 was mostly illegible, proposition 9 was mostly easy to read, especially in the well preserved (then) os. 166–7 and 48–41. Tis proposition 9 dealt with nding the centre o gravity o any segment o the sphere, i.e. proposition 6 can be seen as a special case o proposition 9. What Heiberg did, then, was to reconstruct proposition 6 on the basis o proposition 9. In proposition 9, Archimedes constructs an auxiliary cylinder MN, whose various centres o gravity balance with certain cones related to the segments o the sphere. Tis cylinder is then imported by Heiberg into proposition 6 itsel. But there is no need o such an auxiliary construction in proposition 6. Indeed, the nding o the centre o gravity o a hemisphere is much simpler than that o nding the centre o gravity o a general segment (which is not all that surprising as this happens ofen: a special case may have properties that make it easier to accomplish). Te position o the centre o gravity along the axis is ound, in an elegant manner, by considering just the cone which is already contained by the hemisphere. Heiberg’s reconstruction o proposition 6 made it appear as i it were a precise copy o proposition 9, merely plugging in the special properties o the hemisphere. But it appears that Archimedes took two different routes, a more direct and elegant one or nding the centre o gravity o the hemisphere, and an indirect one or nding the centre o gravity o a general segment. Once again, we can hardly blame Heiberg. He played it sae, reconstructing a passage diffi cult to read on the basis o a closely
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related passage that was easier to read – just as he reconstructed proposition 15 on the basis o CS, and proposition 14 on the basis o propositions 1–11. How else would you reconstruct, i not on the basis o what you have available? But this immediately suggests that the act o reconstruction has, automatically, a signi cant consequence: i reconstruction is necessarily based on what one has available, reconstruction necessarily tends to homogenize the text. Hence 14 appears like 1–11; 15 appears like CS; 6 appears like 9. Te Method as a whole loses something o its internal variety and o its difference rom other parts o the corpus. In truth, o course, the Method is all about difference. It is different rom the rest o the corpus; it highlights internal variety, where the srcinal procedure contrasts with ‘classical’, geometrical approaches. Afer all, what is the point o supplying three separate proos o the same result (propositions 12–13, proposition 14, proposition 15) i not to highlight the difference between all o them? Tis can be seen at all levels. I have concentrated on the global orms o marking difference, but one can nd such orms at a more local level. We may return to proposition 14 to take a closer look at its unolding. Te proposition alls into three parts: (a) a geometrical passage showing that a certain proportion holds, (b) a proportion theory passage showing that this proportion may be summed up or sets o in nite multitude and (c) an arithmetical passage calculating the numerical value o the segment o the cylinder measured. Heiberg did not read (b) at all, and had to reconstruct large parts o (a). Te only part he could read in ull was (c), which is indeed surprisingly careul and detailed. Heiberg’s reconstruction ignored (b), and produced a careul and detailed development o (a). In Heiberg’s reading, thereore, the proposition unolded in an uninterrupted progression o careul geometrical argument, ollowed by a transition based directly on the method o indivisibles (and thus merely reduplicating propositions 1–11) leading to another careul, arithmetical argument. Following Netz et al. (2001–2), we now know that the structure o the proo is much more unwieldy. Remarkably, passage (a) hardly possesses any argument. Te difficult and remarkable geometrical conclusion required by Archimedes is thrust upon the reader as a given. Tis is then ollowed by the subtle and difficult argument o (b), leading nally to the much simpler passage (c) which now, in context, is truly startling in its slow development o such an obvious claim. Archimedes rst states a difficult result as obvious, then outlines the most difficult
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claim imaginable, and then nally develops in ull a sequence o mere arithmetical equivalences. Tis proposition 14 orms a microcosm o the Method as a whole: its undamental principle o composition is sharp difference. Heiberg could hardly have guessed this, staring as he did at the nearly illegible pages o the Palimpsest. Perhaps he should have been more orthcoming in revealing his ignorance. Perhaps it would have been best to avoid all those passages translated in Latin printed in italics, so as to broadcast in all clarity the lacunose nature o Heiberg’s own reading. But then again, the temptation to reproduce, in ull, the mathematical contents o the Method was irresistible and the remarkable act, afer all, is that Heiberg came so close to achieving this reproduction. Where he erred, that was in the spirit o the text more than in its mathematical contents. And so he did reconstruct, mostly, the mathematical contents o the Method – transorming along the way the texture o Archimedes’ writings.
Te texture of Archimedes’ writings: summary We have seen several ways in which Heiberg manipulated the evidence o the manuscripts, transorming it to produce his text o Archimedes and, through that transormation, projecting his image o Archimedes. Te manuscripts’ diagrams were ignored, producing an image o Archimedes whose arguments were textually explicit. Te bracketing o suspected interpolations produced an image o Archimedes whose arguments were less immediately accessible. As or Heiberg’s overall conventions o presentation, these would serve to make the argument appear more consistent than it really was – visible most clearly in Heiberg’s reconstruction o the Method. Tere, obviously, Archimedes used a wide variety o approaches – which Heiberg tended to narrow down. Tis drive towards consistency marked Heiberg’s project as a whole. All in all, then, Heiberg’s interventions make Archimedes to be textually explicit, non-accessible and consistent. Now, it is not as i Heiberg, throughout, adopted this editorial policy. Te practices adopted or the edition o Archimedes display Heiberg’s assumptions concerning Archimedes himsel. Tus, Vitrac shows, in his analysis o Heiberg’s edition o Euclid, that, with the latter, Heiberg’s policies were quite different, emphasizing transparency – nearly the opposite o those o Archimedes. Very likely, this editorial policy reveals, thereore, a certain image o mathematical genius. Heiberg could well make his Euclid transparent and
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accessible; Archimedes had to be diffi cult. While perectly explicit and consistent, the mathematical genius is also remote and diffi cult. Tis, o course, is no more than guesswork, ascribing to Heiberg motives he may never have ormulated explicitly or himsel. I shall not linger on such possibilities. And, indeed, let us not orget: Heiberg could well be right. Tere are probably grounds or saying that Euclid was easier to read than Archimedes, that on the whole Euclid took more pains to make his text accessible. Te one point I would like to stress, nally– and the one which Heiberg almost inevitably would tend to obscure – is the variety o Archimedes’ writings. Heiberg’s editorial policy is in itsel consistent, and it can’t help re ecting a single image Heiberg entertained o the texture o Archimedes’ writings. But in truth, the major eature o the corpus is that so many o its constituent works are unlike the others. Some are extant in Doric, some in Koine. Is this an arteact o the transmission alone? Perhaps. But the argument or that is yet to be made. Te Arenarius stands apart: it is written in discursive prose. Te Cattle Problem stands apart – it is written in poetic orm. Te Method stands apart – it deals with questions o procedure, putting side by side various approaches. Even Sphere and Cylinder stands apart – it is the only work dedicated to problems alone. Many works diverge rom the imaginary norm o pure geometry. Some works are heavily invested in numerical values – not only the Measurement o the Circle, but also the Arenarius and (in part) Spiral Lines, Planes in Equilibrium and Quadrature o Parabola (as well as the no longer extant treatise on semi-regular solids and, likely, the Stomachion). Some works are heavily invested in physical considerations, such as Planes in Equilibrium – , Floating Bodies – and Quadrature o Parabola. Even a book with the straightorward theme and methods o Sphere and Cylinder becomes marked by the very striking ormat o presentation, with the polygons represented by series o curved lines (surely one o the most striking eatures to arrest the attention o the srcinal treatise – i indeed this convention is due to Archimedes himsel ). Which work by Archimedes remains ‘typical’? Perhaps Conoids and Spheroids… Inside many works, again, Archimedes plays throughout with variety: with putting side by side the physical and the geometrical, twice, in Quadrature o Parabola as well as Method; with putting side by side the numerical and the geometrical, in Spiral Lines, Planes in Equilibrium, Quadrature o Parabola, Semi-Regular Solids and Stomachion. And so, is it so unlikely, nally, that Archimedes should, on occasion, be more explicit, on occasion, more opaque? I the answer is positive, then
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much o the argument or Heiberg’s excisions – his major editorial intervention in the text o Archimedes – disappears. Perhaps the answer should be negative; perhaps Heiberg was right in his recon guration o the Archimedean text. But this article serves as a note o caution: authors possess complex individual styles, and it is always hazardous to revise them on the basis o any single editorial policy. Which, once again, reminds us that we should not blame Heiberg: is it air to ask anyone to make himsel, deliberately, inconsistent? Such is the editor’s plight: orever limping upon his crutches o a single method – gasping, out o breath, as he tries to catch up with an author who ies upon thewings o a creative mind.
Bibliography Abbreviations used in this chapter SC Sphere and Cylinder DC Measurement o the Circle CS Conoids and Spheroids SL Spiral Lines PE Planes in Equilibrium Aren. Arenarius QP Quadrature o Parabola FB Floating Bodies Meth. Method Stom. Stomachion Bov. Cattle Problem Abbenes, J. G. J. (1996) ‘Te Doric o Teocritus: a literary language’, in Teocritus, ed. M. A. Harder, R. F. Regtuit and G. C. Wakker. Groningen: 1–19. Blanchard, A. (1989) ‘Choix antiques et codex’, in Les débuts du codex, ed. A. Blanchard. urnhout: 181–90. Czwallina-Allenstein, A. (1922–5) Archimedes / Abhandlungen. Leipzig. Heiberg, J. L. (1879) Quaestiones Archimedeae. Copenhagen. (1880) Archimedes, Opera Omnia, 1st edn. Leipzig. (1910–15) Archimedes, Opera Omnia, 2nd edn. Leipzig. Hultsch, F. (1876–8/1965) Pappus, Collectio, 3 vols. Berlin. Johnson, W. A. (2000) ‘owards a sociology o reading in Classical antiquity’, American Journal o Philology 121: 593–627. Knorr, W. R. (1978) ‘Archimedes and the Elements: proposal or a revised chronological ordering o the Archimedean corpus’, Archive or History o Exact Sciences 19: 211–90.
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(1996) ‘Te method o indivisibles in ancient geometry’, in Vita Mathematica: Historical Research and Integration with eaching, ed. R. Calinger. Washington, D.C.: 67–86. Mugler, C. (1970–2) Les oeuvres d’Archimède4 Vols. Paris. Netz, R., Acerbi, F. and Wilson, N. (2004) ‘owards a reconstruction o Archimedes’ Stomachion’, Sciamus 5: 67–100. Netz, R. chernetska, N. and Wilson, N. 2001–2. ‘A new reading o Method Proposition 14: preliminary evidence rom the Archimedes Palimpsest’,
Sciamus 2: 9–29 and 3: 109–25. Parry, M. (1971) Te Collected Papers o Milman Parry. Oxord. Poincaré, H. (1913) Dernières Pensées. Paris. Sato, . (1986) ‘A reconstruction o Te Method Proposition 17, and the development o Archimedes’ thought on quadrature’, Historia Scientiarum 31: 61–86 and 32: 61–90. Spang-Hanssen, E. (1929) Bibliogra over J. L. Heibergs skrifer. Copenhagen. Stamatis, E. S. (1970–4) Archimedous Hapanta. Athens. oomer, G. J. (1990) Apollonius, Conics – . New York. Ver Eecke, P. (1921) Archimède, Les oeuvres complètes. Paris. Weitzmann, K. (1947) Illustrations in Roll and Codex. Princeton, N.J. Whittaker, J. (1990) Alcinous, Enseignement des doctrines de Platon. Paris.
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John Philoponus and the conformity of mathematical proofs to Aristotelian demonstrations
One of the central issues in contemporary studies of Aristotle’sPosterior Analytics is the conformity of mathematical proofs to Aristotle’s theory of demonstration. Te question, it seems, immediately arises when one compares Aristotle’s demonstrative proofs with the proofs in Euclid’sElements. According to Aristotle, demonstrative proofs are syllogistic inferences of the form ‘All A is B, all B is C, therefore all A is C’, whereas Euclid’s mathematical proofs do not have this logical form. Although the discrepancy between mathematical proofs and Aristotelian demonstrations seems evident, it is only during the Renaissance that the conformity of mathematical proofs to Aristotelian demonstrations emerges as a controversial issue.1 Te absence of explicit discussions of the conformity of mathematical proofs to Aristotelian demonstrations in the earlier tradition seems puzzling from the perspective of contemporary studies of Aristotle’s theory of demonstration. Te formal discrepancies between Aristotelian demonstrations and mathematical proofs seem so obvious to us that it is difficult to understand how the conformity between mathematical proofs and Aristotelian demonstrations was ever taken for granted. In this chapter I attempt to bring to light the presuppositions that led ancient thinkers to regard the conformity of mathematical proofs to Aristotelian demonstrations as self-evident. Neither an outright rejection nor an explicit approval of the conformity of mathematical proofs to Aristotelian demonstrations is found in the extant sources from late antiquity; however, two approaches to this issue can be detected. According to one approach, found in Proclus’ commentary on the rst book of Euclid’s Elements, the conformity of 1
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Te rst Renaissancethinker to reject the conformity of mathematical proofs to Aristotelian demonstrations is Alessandro Piccolomini. His treatise Commentarium de certitudine mathematicarum disciplinarum, published in 1547, initiated the debate known as the Quaestio de certitudine mathematicarum, in which other Renaissance thinkers, such as Catena and Pereyra, sided with Piccolomini in stressing the incompatibility between mathematical proofs and Aristotelian demonstrations, whereas other thinkers, such as Barozzi, Biancani, and omitano, attempted to reinstate mathematics in the Aristotelian model. I discuss this debate and its ancient srcins in the conclusions.
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certain mathematical proofs to Aristotelian demonstrations is questioned.2 According to the other approach, found in Philoponus’ commentary on Aristotle’s Posterior Analytics, the conformity of mathematical proofs to Aristotelian demonstrations is taken for granted.3 Nevertheless, these thinkers did not address the same question that Aristotle’s contemporary interpreters discuss. Whereas contemporary studies focus on the discrepancy between the formal requirements of Aristotelian demonstrations and mathematical proofs, the ancient thinkers focused on the non-formal requirements of the theory of demonstration – namely, the requirements that demonstrations should establish essential relations and ground their conclusions in the cause. In view of this account, I attempt to explain why the question whether mathematical proofs meet these non-formal requirements does not arise within the context of Philoponus’ interpretation of Aristotle’s theory of demonstration. Regarding the requirement that demonstrative proofs should establish essential relations, I show that Philoponus considers it nonproblematic in the case of all immaterial entities including mathematical objects. I show further that Philoponus’ assumption that mathematical objects are immaterial renders the requirement that the middle term should serve as a cause irrelevant for mathematical demonstrations, since according to Philoponus causes are required only to explain the realization of form in matter. Accordingly, the dependence of mathematical proofs on de nitions is sufficient, in Philoponus’ view, to guarantee their conformity to Aristotelian demonstrations. In substantiating this conclusion, I then discuss Proclus’ argument to the effect that certain mathematical proofs do not conform to Aristotelian demonstrations. I show that within the context of Proclus’ philosophy of mathematics, in which geometrical objects are conceived of as realized in matter, consideration of the question whether mathematical proofs meet the two non-formal requirements – a question which Philoponus ignores with regard to mathematical demonstrations – led Proclus to argue for the non-conformity of certain mathematical proofs to 2
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Proclus’ commentary on the rst book of Euclid’s Elements was translated into Latin in 1560 by Barozzi and it played an instrumental role in the debate over the certainty of mathematics. For the reception of Proclus’ commentary on the Elements in the Renaissance, se e Helbing 2000: 177–93. Philoponus’ commentary on the Posterior Analyticshas been hardly studied; hence it is difficult to assess its direct or indirect in uence on the later tradition. Nevertheless, it seems that the several traits of Philoponus’ interpretation of the Posterior Analyticsare found in the medieval interpretations of Aristotle’s theory of demonstrations, such as the association of demonstrations of the fact with demonstrations from signs which is found in Averroes (see n. 38) and the identi cation of the middle term of demonstration with real causes (see n. 27).
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Aristotelian demonstrations. As a corollary to this discussion, I conclude my chapter with an attempt to trace the srcins of contemporary discussions of the conformity of mathematical proofs to Aristotelian demonstrations to the presuppositions underlying Philoponus’ and Proclus’ accounts of this issue. I thereby outline a possible explanation for how concerns regarding the ontological status of mathematical objects and the applicability of Aristotle’s non-formal requirements to mathematical proofs evolved into concerns regarding the logical form of mathematical and demonstrative proofs.
Philoponus on mathematical demonstrations In the Posterior Analytics .9, Aristotle states that if the conclusion of a demonstration A ‘ ll A is C’ is an essential predication, it is necessary that the middle term B from which the conclusion is derived will belong to the same family (sungeneia) as the extreme terms A and C (76a4–9). Tis requirement is tantamount to the requirement that the two propositions ‘All A is B’ and ‘All B is C’, from which the conclusion A ‘ ll A is C’ is derived, will also be essential predications. Te example that Aristotle presents in this pass age for an essential predication is ‘Te sum of the interior angles of a triangle is equal to two right angles’. In his comments on this discussion Philoponus tries to show that the attribute ‘having the sum of its interior angles equal to two right angles’ is indeed an essential attribute of triangles. He does so by arguing that Euclid’s proof meets the requirements of Aristotelian demonstrations: For having [its angles] equal to two right angles holds for a triangle in itself ( kath’ auto). And [Euclid] proves this [theorem] not from certain common principles, but from the proper principles of the knowable subject matter. For instance, he proves that the three angles of a triangle are equal to two right angles, by producing one of the sides and showing that the two right angles, the interior one and its adjacent exterior angle, are equal to the three interior angles,4 so that such a syllogism is produced: the three angles of a triangle, given that one of its sides is produced, are equal to the two adjacent angles. Te two adjacent angles are equal to two right angles. Terefore the angles of a triangle are equal to two right angles. And that the two adjacent angles are equal to two right angles is proved from the [theorem] that two adjacent angles are either equal to two right angles or are two right angles. Whence [do we know] that adjacent angles are either equal to two right angles or
4
Te proof that Philoponus describes is not identical to Euclid’s proof. Philoponus’ reference to ‘two right angles’ implies that he envisages a right-angled triangle, whose base is extended so as to create two adjacent right angles. Euclid’s proof refers to an arbitrary triangle. Tis discrepancy does not affect Philoponus’ reasoning, as he states in the sequel that two adjacent angles are either equal to two right angles or are two right angles.
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are two right angles? We know it from the de nition of right angles, [stating] that when a straight line set up on a straight line makes the adjacent angles equal to each other, the two equal angles are right. Well, having brought [the conclusion] back to the de nition and the principles of geometry, we no longer inquire further, but we have the triangle proved from geometrical principles.5
In showing that Euclid’s proof conforms to the Aristotelian model of demonstration, Philoponus focuses on two issues: (1) he presents Euclid’s proofs in a syllogistic form, and (2) he grounds the proved proposition in the de nition of right angle. Te notion of rst principles, on which Phil oponus’ account is based, includes only one of the characteristics of Aristotelian rst principles – namely, their being proper to the discipline. In Philoponus’ view, the dependence of Euclid’s geometrical proof on geometrical rst principles, rather than on principles common to or proper to other disciplines, is sufficient to establish that this proof conforms to the Aristotelian model. wo other characteristics of Aristotelian rst principles are not taken into account in this passage. First, Philoponus does not raise the question whether the middle term employed in this proof is related essentially to the subject of this proof; that is, he does not consider the question whether a proposition regarding an essential attribute of adjacent angles can by any means serve to establish the conclusion that this attribute holds essentially for triangles.6 Nor does he express any reservations concerning the auxiliary construction, in which the base is extended and two adjacent angles are produced. Second, Philoponus does not mention Aristotle’s requirement that the rst principles should be explanatory or causal; he does not raise the question whether the middle term in his syllogistic reformulation of Euclid’s proof has a causal or explanatory relation to the conclusion. Tus Philoponus’ account of the conformity of Euclid’s proofs to Aristotelian demonstrations raises two questions: (1) why Philoponus ignores the question whether mathematical propositions state essential relations; and (2) why the causal role of the principles of demonstration is not taken into account. Te following two sections answer these questions respectively.
Essential predications Philoponus addresses the question whether mathematical proofs establish essential predications in his comments on the Posterior Analytics .22. He 5 6
116. 7–22, Wallies. All translations are mine. For Philoponus’ syllogistic reformulation to be a genuine Aristotelian demonstration, one has to assume that adjacent angles and triangles are related to each other as genera and species. Tis assumption is patently false.
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formulates this question in response to Aristotle’s contention that sentences whose subject is an attribute, such as ‘the white (to leukon) is walking’ or ‘the white is a log’ cannot feature in demonstrations, because they are not predicative in the strict sense (Posterior Analytics 83a1–21). Tis contention jeopardizes, in Philoponus’ view, the status of geometrical proofs. Te subject matter of geometry, according to Philoponus, is shapes and their attributes. Hence, Aristotle’s narrow conception of predication may imply that proofs that establish that certain attributes belong to shapes are not demonstrative because they prove that certain attributes, such as having the sum of the interior angles equal to two right angles, belong to other attributes, such as triangles (239.11–14).7 Philoponus dismisses this implication saying: Even if these [attributes] belong to shapes accidentally, they are completive [attributes] of their being (symplērōtika tēs ousias) and like differentiae that make up the species they are [the attributes] by which [shapes] are distinguished from other things.8 … Just as ‘being capable of intellect and knowledge’ or ‘mortal’ or any of the [components] in its de nition do not belong to ‘man’ as one thing in another, but [man] is completed from them, so the circle is also contemplated (theōreitai) from all the attributes which are observed in it. Similarly, also the triangle would not be something for which ‘having three angles equal to two right angles’ or ‘having the sum of two sides greater than the third’ do not hold, but if one of these [attributes] should be separated, immediately the being of a triangle would be abolished too.9
Tis account does not answer Philoponus’ srcinal query; it does not tackle the question whether proofs that establish predicative relations between two attributes are demonstrative. Instead, Philoponus focuses here on the question whether the attributes that geometry proves to hold for shapes are essential, arguing that mathematical attributes like differentiae are parts of the de nitions of mathematical entities. However, the analogy between the differentiae of man and mathematical propositions is not as obvious as Philoponus formulates it. Te attributes ‘capable of knowledge’ and ‘mortal’ distinguish men from other living creatures; the former distinguishes human beings from other animals and the latter distinguishes 7
8
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Philoponus presupposes here Aristotle’s categorical scheme, in which terms belonging to the nine non-substance categories are attributes of terms belonging to the category of substance. According to Aristotle’sCategories the term ‘shape’ belongs to the category of quality. Hence, Philoponus claims that geometry studies attributes of attributes. Te term ‘completive attributes’ s(ymplērōtikos) refers in the neo-Platonic tradition to attributes without which a certain subject cannot exist. On these attributes and their relation to differentiae, see De Haas 1997: 201 and Lloyd 1990: 86–8. 239.14–25, Wallies.
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them from divine entities, which are also capable of knowledge but are not mortal. By contrast, the geometrical attributes that Philoponus mentions in this passage do not distinguish triangles or circles from other shapes. Admittedly, the attribute ‘having the sum of the interior angles equal to two right angles’ holds only for triangles, yet, unlike ‘having three sides’, it is not the feature that distinguishes triangles from other shapes. It seems, then, that in accounting for the essentiality of mathematical attributes, Philoponus expands the notion of differentia, so as to include all the attributes of mathematical entities. He does not distinguish between attributes that enter into the de nition of an entity and necessary attributes; he concludes from the statement that a triangle will cease to be a triangle if one of its attributes were separated from it that these attributes are essential. Tus, rather than explaining why mathematical attributes are essential in Philoponus’ view, this passage re ects his assumption that the essentiality of mathematical attributes is evident. Tis assumption, I surmise, can be understood in light of Philoponus’ interpretation of the principles of demonstration. In his comments on the Posterior Analytics .2,10 Philoponus accounts for the distinction between indemonstrable premises and demonstrable conclusions in terms of the distinction between composite and incomposite entities. Incomposite entities, according to this discussion, are simple or intelligible substances such as the intellect or the soul, which are considered (theōroumenon) without matter.11 In the case of such entities, Philoponus argues, the de ning attribute is not different from the de nable object and therefore propositions concerning such entities are indemonstrable or immediate. Another characterization of indemonstrable premises is found in Philoponus’ interpretation of Aristotle’s discussion of the relationship between de nitions and demonstrations in the Posterior Analytics .2–10. In addressing the question whether it is possible to demonstrate a de nition, Philoponus draws a distinction between two types of de nition: formal and material. Formal de nitions are the indemonstrable principles of demonstration that de ne incomposite entities; they include, according to Philoponus, the essential attributes (ousiodōs) of the de ned object. Material de nitions, by contrast, serve as demonstrative conclusions and 10
11
Te editor of Philoponus’ commentary on the Posterior Analytics, M. Wallies, doubted the attribution of the commentary on the second book of the Posterior Analytics to Philoponus (v–vi). Te authenticity of the commentary on the second book does not affect my argument, because all the references I make here to the commentary on the second book accord with views expressed in Philoponus’ other commentaries. 339. 6–7, Wallies.
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include the attributes that are present in matter.12 In this interpretation, then, the ontological distinction between incomposite and composite entities accounts for two characteristics of the principles of demonstration: their indemonstrability and their essentiality. Te question whether certain propositions meet Aristotle’s requirements is not answered by an examination of their logical characteristics, but by the ontological status of their subjects. It follows from this discussion that from Philoponus’ viewpoint the immateriality of the subject of predication is sufficient to guarantee the essential relation between a subject and its attributes.13 Tis assumption may explain Philoponus’ approach to the issue of the essentiality of mathematical propositions. Mathematical objects, according to Philoponus, are abstractions from matter14 – that is, they belong to the class of incomposite objects that serve as the subjects of formal de nitions. Tus, in light of Philoponus’ characterization of these de nitions, it plausible to regard all attributes of mathematical objects as essential, because the immateriality of these objects seems to entail, in Philoponus’ view, the essentiality of their attributes. In what follows, I show that the ontological distinction between incomposite and composite entities also explains why the causal role of the middle term is not taken into account in Philoponus’ discussion of the conformity of Euclid’s proofs to Aristotelian demonstrations.
Causal demonstrations In his commentary on Aristotle’sPhysics .2, Philoponus examines the tenability of Aristotle’s criticism of the theory of Forms, which involves, according to Aristotle, separation from matter of the objects of physics, although they are less separable than mathematical objects. In so doing, Philoponus draws a distinction between separability in thought and separability in existence, claiming that he agrees with Aristotle that the forms 12 13
14
364.16–18, Wallies. wo reasons may explain why Philoponus does not consider the possibility that immaterial entities have accidental attributes. First, it is commonly held in the ancient tradition that only individuals have accidental attributes, which belong to their matter. Second, Philoponus’ notion of essential predication is more formal than Aristotle’s. In characterizing essential predications Philoponus appeals to extensional, rather than intensional, considerations. In his view, attributes that belong to all members of a species and only to them are essential (e.g., In An. Post. 63.14–20, Wallies;In DA 29.13–30.1, Hayduck;In Cat. 64.9, Busse). For Philoponus’ conception of mathematical objects, see (e.g.) In Phys. 219.10; In DA, 3.7–11. For a discussion of this view, see Mueller 1990: 465–7.
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of natural things cannot be separated in existence from matter, but he disagrees with Aristotle’s view if it implies that these forms cannot be separated by reason and in thought.15 Although Ph iloponus’ account of the indemonstrability of the principles of demonstration presupposes the possibility of separating the de nitions of both mathematical and physical entities, the ontological difference between these classes of objects is nevertheless maintained. In his commentary on Aristotle’sDe anima, Philoponus draws a distinction between physical and mathematical de nitions, arguing that physical de nitions should refer to the matter of physical substance, their form and the cause by virtue of which the form is realized in matter. 16 Mathematical de nitions, by contrast, refer only to the form: Te mathematician gives the de nitions of abstracted forms in themselves, without taking matter into account, but he gives these [de nitions] in themselves. For this reason he does not mention the cause in the de nition; for if he de ned the cause, clearly he would also have taken the matter into account. Tus, since he does not discuss the matter he does not mention the cause. For example, what is a triangle? A shape contained by three lines; what is a circle? A shape contained by one line. In these [de nitions] the matter is not mentioned and hence neither is the cause through which this form is in this matter. Unless perhaps he gives the cause of those characteristics holding in themselves for shapes, for instance, why a triangle has its 17
angles equal to two right angles.
Philoponus’ distinction between physical and mathematical de nitions has two related consequences for the methods employed in physics and mathematics. First, although both physical and mathematical demonstrations are based on indemonstrable formal de nitions, these de nitions adequately capture the nature of mathematical objects but they fail to exhaust the nature of physical objects. In the case of physical demonstrations, the formal de nition captures only one aspect of the object: its form. Full- edged knowledge of physical objects requires reference also to the matter of this object and the cause of the realization of the form in matter. Indeed, in both the commentary on Aristotle’s De anima and the commentary on the Posterior Analytics, Philoponus considers formal de nitions of physical objects de cient. In the commentary on De anima, Philoponus argues that de nitions that do not include all the attributes
15
16 17
225.4–11, Vitelli. For the relationship between Philoponus’ discussion of separability in thought of physical de nitions and his analysis of demonstrations in the natural sciences, see De Groot 1991: 95–111. 55.31–56.2, Hayduck. 57.35–58.6, Hayduck.
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of an object are not physical de nitions, but are dialectical or empty. His example of such an empty de nition is the formal de nition of anger: ‘anger is a desire for revenge’. Te adequate de nition of anger, according to Philoponus, is ‘anger is boiling of the blood around the heart caused by a desire for revenge’.18 Tis de nition refers to the form, the matter and the cause. Similarly, in the commentary on the Posterior Analytics, Philoponus claims that neither the formal nor the material de nition is a de nition in the strict sense; only the combination of these two yields an adequate de nition.19 Tis conception of de nition is evidently inapplicable to mathematics. Mathematical objects are de ned without reference to matter or to their cause, hence formal de nitions provide an exhaustive account of these objects. Te second consequence of Philoponus’ distinction between physical and mathematical de nitions concerns the explanatory or causal relations in demonstrative proofs. Although in the above-quoted passage Philoponus contends that the cause is also studied in mathematics when a relation between a mathematical object and its attributes is proved, it seems that this cause is different from the one studied in physics. According to the above passage, physics studies the cause of the realization of form in matter, but since mathematics does not deal with the matter of its objects, its explanations do not seem to be based on this type of cause. Furthermore, Philoponus’ analysis of physical demonstrations in terms of the distinction between formal and material de nitions gives rise to a problem that has no relevance for mathematical demonstrations. Tis interpretation gives rise to the question of how the material aspect of a physical entity, which is a composite of form and matter, can be demonstratively derived from the formal de nition, given that this de nition does not exhaust the nature of the composite entity. Stating this question differently, how, in Philoponus’ view, can a proposition regarding a substance taken with matter be demonstratively derived from a proposition regarding its form, which is considered in separation from matter? Evidently this question does not arise in the mathematical context. Mathematical de nitions do not refer to matter; hence, they give an exhaustive account of mathematical objects. In what follows, I show that Philoponus answers this question by appealing to extra-logical considerations. More speci cally, I show that the causal role of the middle term in demonstrations provides Philoponus with the means of bridging the gap between formal de nitions and material de nitions. 18 19
43.28–44.8, Hayduck. 365.1–13, Wallies.
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In his comments on the Posterior Analytics .2, Philoponus presents the following explanation for Aristotle’s remark that the questions ‘what it is’ (ti esti) and ‘why it is’ (dia ti) are the same: For if the ‘what it is’ and the ‘why it is’ are different, it is insofar as the former is sought with regard to simple [entities] and the latter with regard to composite [entities]. Yet these [questions] are the same in substrate, but different in their mode of employment. Both the ‘what it is’ and the ‘why it is’ are studied in the case of the eclipse being an affection of the moon. And we use these, the ‘what it is’ and the ‘why it is’, differently. But if we take an eclipse itself by itself, we seek what is the cause of an eclipse, and we say that it is a privation of the moon’s light due to screening by the earth. But if we seek whether an eclipse exists ( hyparkhei) in the moon, namely why it exists, we take the ‘what it is’ as a middle term, namely privation of the moon’s light coming about as a result of screening by the earth.20
Although this passage is presented to account for the identity between the questions ‘what it is’ and ‘why it is’, Philoponus dissociates these two questions. Te distinction he draws here is based on the ontological distinction between simple and composite entities. Te question ‘what it is’ is asked with regard to simple entities, whereas the question ‘why it is’ is asked with regard to composite entities. In the case of composite entities, Philoponus argues, ‘what it is’ and ‘why it is’ are different questions. Te de nition of an eclipse and the cause of its occurrence are not identical. Te exact signi cance of Philoponus’ distinction between these questions is not clear from this passage. Te examples presented by Philoponus seem to blur his distinction between an eclipse considered in the moon and an eclipse considered in separation from the moon, as the accounts given for both cases are identical – ‘privation of the moon’s light due to screening by the earth’. Tis difficulty in understanding Philoponus’ distinction between ‘what it is’ and ‘why it is’ may stem from his attempt to accommodate his view, which dissociates these questions, with Aristotle’s claim that these questions are identical. As a result, Philoponus follows Aristotle in exemplifying the answers to these questions by one and the same account. However, according to Philoponus’ other discussions of the de nitions of entities, which are considered in separation from matter, the account for the eclipse taken in separation from the moon should be the formal de nition ‘screening by the earth’, whereas ‘privation of the moon’s light due to screening by the earth’ is the full de nition, resulting from a demonstration that relates the formal de nition to the material de nition.21 Despite the diffi culty in 20 21
339.20–9, Wallies. 371.19–25, Wallies.
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understanding the distinction made in this passage, Philoponus clearly does not follow Aristotle here in assimilating de nitions with explanations. Tis conclusion nds further support in Philoponus’ comments on the Posterior Analytics .4. In the Posterior Analytics .4, Aristotle presents four senses in which one thing is said to hold for another ‘in itself ’. Te rst two senses are predicative and they constitute Aristotle’s account for the predicative relations that the premises of demonstration should express. According to the rst sense, a predicate holds for a subject in itself if it is a part of the de nition of the subject. According to the second sense, a predicate holds for a subject in itself if the subject is a part of the de nition of the predicate. Te third sense distinguishes substances that exist in themselves from attributes, which depend on substances, by virtue of their being said of them. Te fourth sense distinguishes a causal relation between events from an incidental relation between events. In his comments on this fourfold distinction Philoponus argues that only the rst two senses of ‘in itself ’ contribute to the demonstrative method, 22 yet he also regards the fourth sense (i.e. the causal sense) as relevant to the t heory of demonstration. According to Philoponus, the causal sense of ‘in itself ’, though it does not contribute to the formation of the premises of demonstration, 23
contributes to the ‘production of the whole syllogism’. More precisely, Philoponus argues that the causal sense of ‘in itself’ expresses the relation between the cause, taken as the middle term of demonstration, and the conclusion. Te example Philoponus presents of this contention is the following syllogism: Te moon is screened by the earth. Te screened thing is eclipsed. Terefore, the moon is eclipsed. Commenting on this syllogism, Philoponus remarks that the fact that screening by the earth is the cause of the eclipse of the moon is not expressed in the premises of this demonstration, but its causal force becomes evident from its role as a middle term. 24 In this discussion, then, Philoponus employs two different senses of ‘in itself ’ in accounting for the relations expressed in the premises of demonstration and the relation between the middle term and the conclusion. Te premises of demonstration, according to Philoponus, are ‘in itself ’ in one of the two rst senses delineated by Aristotle. Tat is, their predicate is either a part of the de nition of the subject or their subject is a part of the de nition of the predicate. By contrast, the middle 22 23 24
65.10–11, Wallies. 65.15, Wallies. 65.16–19, 65.20–3, Wallies.
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term and the conclusion of a demonstration are related according to the fourth sense of ‘in itself ’ – that is, they are related as cause and effect. 25 So, according to Philoponus, the derivation of the demonstrative conclusion is not solely based on the transitivity of the predicative relation stated in the premises. In addition to the transitivity of the predicative relation, the demonstrative derivation is based on causal relations between the middle term and the conclusion. Such a distinction between logical relations and extra-logical or causal relations is explicitly drawn at the beginning of Philoponus’ introduction to his commentary on the second book of the Posterior Analytics: In the rst book of the Apodeiktike (i.e. the Posterior Analytics), he showed how there is a demonstration and what is a demonstration and through what premises it has come about, and he showed further how a demonstrative syllogism differs from other syllogisms and that in other syllogisms the middle term is the cause of the conclusion and not of the thing and in demonstrative syllogism the middle term is the cause both of the conclusion and of the thing.26
It follows from this discussion that Philoponus’ ontological distinction between physical and mathematical entities yields different accounts for physical and mathematical demonstrations. Te distinction between the three facets of physical entities – i.e. the form, the matter and the cause for the realization of form in matter – is re ected in Philoponus’ interpretation of the theory of demonstration. In this interpretation, demonstrations, like physical entities, have three components: indemonstrable premises, regarded as formal de nitions, demonstrative conclusions, which are material de nitions, and the middle term, which serves as the cause that relates the formal de nition to the material de nition. Philoponus’ distinction between the form of a physical entity and the cause of the realization of form in matter nds expression in the distinction he draws between the formal de nition considered in itself and that formal de nition in its role as the middle term in demonstration. Tis distinction implies that 25
26
Te analysis of demonstrative derivation in causal terms is widespread in Philoponus’ commentary on the Posterior Analytics(e.g., 24.22–4; 26.9–13; 119.19–21; 173.14–20; 371. 4–19). Te causal analysis of demonstrative derivation underlies Philoponus’ introduction of a second type of demonstration, called ‘tekmeriodic demonstration’, in which causes are deduced from effects (In An. Post. 33.11; 49.12; 169.8; 424.13, Wallies;In Phys. 9.9–10.21, Vitelli). On Philoponus’ notion of tekmeriodic proofs and its reception in the Renaissance, see Morrison 1997: 1–22. 334.1–8, Wallies. Te distinction between the middle term as the cause of the thing and the middle term as the cause of the conclusion is also found in the Latin medieval tradition of interpreting the Posterior Analytics. See De Rijk 1990.
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demonstrative derivation rests on two relations: the transitivity of the predicative relation that the premises state and the causal relation between the middle term and the conclusion. Tis distinction is applicable to physical demonstrations, for which the cause of the realization of form in matter is sought. Te demonstrative derivation in these demonstrations is based not only on logical relations but also on causal relations. Mathematical entities, by contrast, have only one facet: the form. Accordingly, Philoponus’ account of the conformity of mathematical demonstrations to Aristotelian demonstrations focuses only on the formal requirements of the theory of demonstration. Te conformity of mathematical demonstrations to Aristotelian demonstrations is guaranteed if the conclusions can be shown to depend on the de nitions of mathematical entities. Since mathematical objects have no matter, mathematical demonstrations can be based only on logical derivation; the question whether the middle term is the cause of the conclusion does not arise in this context, as the separation from matter renders super uous questions concerning causes.27 Te analysis of Philoponus’ interpretation of Aristotle’s theory of demonstration reveals the importance of the ontological distinction between simple and composite entities for his account of conformity of mathematical proofs to Aristotelian demonstrations. Te assumption that mathematical objects are analogous to simple entities by being separated in thought from matter does not give rise to two questions that may undermine the conformity of mathematical proofs to Aristotelian demonstrations. Te rst question is whether mathematical predications are essential; the second is whether the middle term in mathematical proofs is the cause of the conclusion. Te rst question does not arise because the separation from matter implies that only the essential attributes of entities are taken into consideration. Te second does not arise because causal considerations are relevant only with regard to composite entities, as it is only in their case that the cause of the realization of form in matter can be sought. Hence, given the assumption that mathematical entities are separated in thought from matter, the question whether mathematical proofs conform to the non-formal requirements of Aristotle’s theory of demonstration does not arise. Tis conclusion gains further support from Proclus’ discussion of the conformity of mathematical proofs to Aristotelian demonstrations.
27
Tis conclusion may explain Proclus’ otherwise curious remark that the view in which geometry does not investigate causes is srcinated in Aristotle ( In Eucl. 202.11, Friedlein). If this explanation is correct, Philoponus’ conception of mathematical demonstrations seems to re ect a widespread view in late antiquity.
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Proclus on the conformity between mathematical proofs and Aristotelian demonstrations Proclus’ philosophy of geometry is formulated as an alternative to a conception whereby mathematical objects are abstractions from material or sensible objects.28 According to Proclus, mathematical objects do not differ from sensible objects in their being immaterial, but in their matter. Sensible objects, in Proclus’ view, are realized in sensible matter, whereas mathematical objects are realized in imagined matter. In Proclus’ philosophy of geometry, then, mathematical objects are analogous to Philoponus’ physical objects; they are composites of form and matter. Proclus’ philosophy of mathematics is at variance not only with Philoponus’ views regarding the ontological status of geometrical objects but also with Philoponus’ views regarding the conformity of Euclid’s proofs to Aristotelian demonstrations.29 In his discussion of the rst proof of Euclid’sElements in the commentary on the rst book of Euclid’s Elements, Proclus questions the conformity of certain mathematical proofs to the Aristotelian model: We shall nd sometimes that what is called ‘proof’ has the properties of demonstration, in proving the sought through de nitions as middle terms – and this is a perfect demonstration – but sometimes it attempts to prove from signs. Tis should not be overlooked. For, although geometrical arguments always have their necessity through the underlying matter, they do not always draw their conclusions through demonstrative methods. For when it is proved that the interior angles of a triangle are equal to two right angles from the fact that the exterior angle of a triangle is equal to the two opposite interior angles, how can this demonstration be from the cause? How can the middle term be other than a sign? For the interior angles are equal to two right angles even if there are no exterior angles, for there is a triangle even if its side is not extended.30
In this passage, Proclus claims that Euclid’s proof that the sum of the interior angles of a triangle is equal to two right angles (Elements .32) does not conform to Aristotle’s model of demonstrative proofs. In so doing, he focuses on the causal role of the middle term in Aristotelian demonstrations. Proclus argues that Euclid’s proof does not conform to the Aristotelian model because it grounds the equality of the sum of the interior angles of a triangle to two right angles in a sign rather than in a cause.
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In Eucl. 50.16–56.22, Friedlein. A discussion of the relationship between Proclus’ philosophy of geometry and his analysis of mathematical proofs is beyond the scope of this paper. For this issue, see Harari 2006. 206.12–26, Friedlein.
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Proclus’ reason for regarding this Euclidean proof as based on signs rather than on causes concerns the relationship between the auxiliary construction employed in this proof and the triangle. According to Proclus, the extension of the triangle’s base is merely a sign and not a cause of the equality of the triangle’s angles to two right angles because ‘there is a triangle even if its side is not extended’. Te exact force of this statement is clari ed in Proclus’ discussion of the employment of this auxiliary construction in another Euclidean proof – the proof that the sum of any two interior angles of a triangle is less than two right angles (Elements .17). In this discussion, Proclus claims that the extension of the triangle’s base cannot be considered the cause of the conclusion since it is contingent: the base of a triangle may be extended or not, whereas the conclusion that the sum of any two interior angles of a triangle is less than two right angles is necessary.31 Hence, in questioning the conformity of certain Euclidean proofs to Aristotelian demonstrations, Proclus raises the two questions that Philoponus ignores in the case of mathematical demonstrations. Unlike Philoponus, Proclus asks whether the middle term in Euclid’s proofs is the cause of the conclusion and whether it is essentially related to the triangle. Furthermore, Proclus’ attempt to accommodate Euclid’s proofs of the equality of the sum of the interior angle of a triangle to two right angles with Aristotle’s requirement that demonstrations should establish essential relations indicates that he shares with Philoponus the assumption that demonstrations regarding material entities require an appeal to causal considerations. In concluding his lengthy discussion of Euclid’s proof that the sum of the interior angles of a triangle is equal to two right angles, Proclus says: We should also say with regard to this proof that the attribute of having its interior angles equal to two right angles holds for a triangle as such and in itself. For this reason, Aristotle in his treatise on demonstration uses it as an example in discussing essential attributes … For if we think of a straight line and of lines standing in right angles at its extremities, then if they incline so that they generate a triangle we would see that in proportion to their inclination, so they reduce the right angles, which they made with the straight line; the same amount that they subtracted from these [angles] is added through the inclination to the angle at the vertex, so of necessity they make the three angles equal to two right angles.32
Te procedure described in the passage, in which a triangle is generated from two perpendiculars to a straight line that rotate towards each other 31 32
311.15–21, Friedlein. 384.5–21, Friedlein.
Philoponus and Aristotelian demonstrations
up to their intersection point, is also presented by Proclus in his comments on propositions .16 and .17 of the Elements. In both cases, he regards this procedure – and not Euclid’s auxiliary construction in which the triangle’s base is extended – as the true cause of the conclusion. 33 Proclus’ appeal to this procedure in searching for the true cause of these conclusions indicates that in attempting to accommodate Euclid’s proofs with Aristotle’s requirement that demonstrations should establish essential relations, he grounds mathematical conclusions in causal relations rather than in logical relations. Proclus considers the proposition that the sum of the interior angles of a triangle is equal to two right angles essential not because it is derived from the de nition of a triangle, as Aristotle’s theory of demonstration requires, but because the proposition is derived from the triangle’s mode of generation. Viewed in light of Philoponus’ interpretation of Aristotle’s theory of demonstration, Proclus’ attempt to accommodate Euclid’s proof with Aristotelian demonstrations seems analogous to Philoponus’ account of physical demonstrations. In both cases, causal considerations are employed in rendering proofs concerning material objects compatible with Aristotelian demonstrations. Tis examination of the presupposition underlying Philoponus’ and Proclus’ views regarding the conformity of mathematical proofs to Aristotelian demonstrations has led to the following conclusions. (1) Te pre-modern formulation of the question of the conformity of mathematical proofs to Aristotelian demonstrations concerns the applicability of the non-formal requirements of the theory of demonstration to mathematical proofs. More speci cally, this formulation concerns the questions whether mathematical attributes are proved to belong essentially to their subjects and whether the middle term in mathematical proofs serves as the cause of the conclusion. (2) Te emergence or non-emergence of the question of the conformity of mathematical proofs to Aristotelian demonstration is related to assumptions concerning the ontological status of mathematical objects. Tis question does not arise in a philosophical context in which mathematical objects of as separated in thought from matter, whereas it does are ariseconceived when mathematical objects are conceived of as realized in matter. (3) Demonstrations concerning composites of form and matter were understood in late antiquity as based on causal relations, viewed as additional to the logical necessitation of conclusions by premises. 33
310.5–8, 315.15, Friedlein.
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Causal considerations are employed with regard to mathematical demonstrations, when mathematical objects are considered material; they are not employed when mathematical objects are considered separated in thought from matter.
Conclusions In concluding this chapter, I examine the relationship between the modern formulation of the question of the conformity of mathematical proofs to Aristotelian demonstrations and its formulation in late antiquity. Te modern discussions of the relationship between Aristotle’s theory of demonstration and mathematical proofs focus on Aristotle’s formal requirement that demonstrations should be syllogistic inferences from two universal predicative propositions, which relate the subject and predicate of the conclusion to a third term, called the ‘middle term’. Te disagreement among Aristotle’s modern commentators concerns whether mathematical proofs can be cast in this logical form. For instance, Ian Mueller, who says they cannot, argues that in a syllogistic reformulation of Euclidean proofs the requirement that the inference should have only three terms is not always met, because the mathematical proofs depend on the relations between mathematical entities and not on their properties taken in isolation from other entities.34 Te possibility of expressing mathematical relations in syllogistic inferences is also central in modern attempts to render Aristotle’s theory of demonstration compatible with mathematical proofs. Henry Mendell, for instance, shows that Aristotle’s theory of syllogism does have the formal means that make possible syllogistic formulations of mathematical proofs. In so doing, he argues that the relation of predication, which is formulated by Aristotle as ‘x belongs to y’, can be read exibly so that it also accommodates two-place predicates, such as ‘x equals y’, or ‘x is parallel to y’.35 Mendell’s argument, like Mueller’s, focuses on the possibility of expressing relations within the formal constraints of the theory of syllogism. Te extra-logical consequences of the expansion of the theory of syllogism to relational terms and their compatibility with Aristotle’s theory of demonstration are not at the centre of either Mendell’s or Mueller’s argument. More speci cally, they do not address the question of whether relational terms or mathematical properties can be proved to 34 35
Mueller 1975: 42. Mendell 1998.
Philoponus and Aristotelian demonstrations
be essential predicates of their subjects.36 Tis question, as I showed, was central in the discussions of the conformity of mathematical proofs to Aristotelian demonstrations in late antiquity. Te non-formal requirements of the theory of demonstration were also central in the Renaissance debate over the certainty of mathematics.37 Piccolomini’s objective in his Commentarium de certitudine mathematicarum disciplinarum was to refute what he presents as a long-standing conviction that mathematical proofs conform to the most perfect type of Aristotelian demonstration, called in the Renaissance demonstratio potissima. Te classi cation of types of demonstrations that underlies Piccolomini’s argument is based on Aristotle’s distinction between demonstrations of the fact (hoti) and explanatory demonstrations or demonstration of the reasoned fact (dioti). Tis distinction has been further elaborated by Aristotle’s medieval commentators and it appears in the Proemium of Averroes’ commentary on Aristotle’s Physics as a tripartite classi cation of demonstrations into demonstratio simpliciter, demonstratio propter quid and demonstratio quid est. It is in this context that Averroes claims that mathematical proofs conform to the perfect type of demonstration, in his terminology demonstratio simpliciter.38 According to this classi cation, the different types of demonstration differ in the epistemic characteristics of their premises, hence in the epistemic worth of the knowledge attained through them. Following this tradition, Piccolomini’s argument for the inconformity of mathematical proofs to Aristotelian demonstrations focuses on these characteristics. According to Piccolomini potissima demonstrations are demonstrations in which knowledge of the cause and of its effects is attained simultaneously; the premises of such demonstrations are prior and better known than the conclusion; their middle term is a de nition, it is unique and it serves as the proximate cause of the conclusion. Mathematical demonstrations, so Piccolomini and his followers argue, fail to meet these requirements. Te importance of the non-formal requirements of the theory of demonstration for the Renaissance debate over the certainty of mathematics comes to the fore in the following passage from Pereyra’s De communibus omnium rerum naturalium principiis et affectionibus:
36
37
38
Tis question is not utterly ignored in modern interpretations of the Posterior Analytics. See McKirahan 1992; Goldin 1996; Harari 2004. For a general discussion of the Quaestio de certitudine mathematicarum , see Jardine 1998. For the in uence of this debate on seventeenth-century mathematics, see Mancosu 1992 and 1996. Aristotelis opera cum Averrois commentariis, vol. , 4.
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Demonstration (I speak of the most perfect type of demonstration) must depend upon those things which are per se and proper to that which is demonstrated; indeed, those things which are accidental and in common are excluded from perfect demonstrations … Te geometer proves that the triangle has three angles equal to two right ones on account of the fact that the external angle which results from extending the side of that triangle is equal to two angles of the same triangle which are opposed to it. Who does not see that this medium is not the cause of the property which is demonstrated? . . . Besides, such a medium is related in an altogether accidental way to that property. Indeed, whether the side is produced and the external angle is formed or not, or rather even if we imagine that the production of the one side and the bringing about of the external angle is impossible, nonetheless that property will belong to the triangle; but what else is the de nition of an accident than what may belong or not belong to the thing without its corruption?39
Pereyra’s argument for the inconformity of mathematical proofs to Aristotelian demonstrations is similar to Proclus’ argument. Like Proclus, Pereyra focuses on the question whether mathematical proofs meet the non-formal requirements of the theory of demonstration. More speci cally, he raises the two questions that were at the centre of Proclus’ discussion of this issue: (1) Do the premises of mathematical proofs state essential or accidental relations? (2) Are Euclid’s proofs, which are based on auxiliary constructions, explanatory? Tese questions are viewed in this passage as interrelated; real explanations are provided when the relation between a mathematical entity and its property is proved to be essential. Tis requirement is met if the premises on which the mathematical proof is based state essential relations. Te only allusion to the syllogistic form of inference made in this passage is to the middle term in syllogistic demonstrations. However, like Proclus, Pereyra considers the middle term only in its role as the cause of the conclusion. Its formal characteristics, such as its position, are not discussed here. Tus, pre-modern and modern discussions of the conformity of mathematical proofs to Aristotelian demonstrations concern different facets of the theory of demonstration. Whereas the modern discussions focus on the formal structure of Aristotelian demonstrations, pre-modern discussions concern its non-formal requirements. Accordingly, the questions asked in these discussions are different. Te modern question is whether syllogistic inferences can accommodate relational terms whereas the pre-modern question is whether mathematical proofs establish essential relations.
39
Te translation is based on Mancosu 1996: 13. Te complete Latin text appears on p. 214, n. 12 of Mancosu’s book.
Philoponus and Aristotelian demonstrations
Nevertheless, when the pre-modern discussion of the conformity of mathematical proofs to Aristotelian demonstrations is viewed in light of its underlying ontological presuppositions, a conceptual development leading to the modern formulation of this question may be traced. Discussions of the conformity of mathematical proofs to Aristotelian demonstrations in late antiquity were associated with discussions of whether mathematical objects are immaterial or material;40 that is, whether they are conceptual or real entities. Tis ontological distinction is re ected in different accounts of the relation of derivation, on which demonstrations are based. Whereas demonstrations concerning immaterial objects are based on de nitions and rules of inference alone, demonstrations concerning material objects require the introduction of extra-logical considerations, such as the causal relations between form and matter. Tus, the question of the ontological status of mathematical objects re ects the epistemological question: whether extra-logical considerations have to be taken into account in mathematics. When discussions of the conformity of mathematical proofs to Aristotelian demonstrations in late antiquity are viewed in isolation from ontological commitments, they seem to be conceptually related to modern discussions of the nature of mathematical knowledge. Te need to take into account extra-logical considerations when mathematical objects are considered material is equivalent to Kant’s statement that mathematical propositions are synthetic a priori judgements. Developments in modern logic led to a reformulation of Kant’s statement in terms of logical forms. Kant’s contention that mathematical knowledge cannot be based on de nitions and rules of inference alone was regarded by Bertrand Russell as true for Kant’s time. According to Russell, had Kant known other forms of logical inference than the syllogistic form, he would not have claimed that mathematical propositions cannot be deduced from de nitions and rules of inference alone.41 In light of this account, the modern discussions of the conformity of mathematical proofs to Aristotelian demonstrations, which focus on whether syllogistic inferences can accommodate relational terms, may be understood as evolving from the pre-modern discussions of whether mathematical proofs establish essential relations, and to establish this conclusion, two conceptual developments have to be traced: the process by which the question whether mathematical propositions are 40
41
Tis assumption seems to underlie the Renaissance discussions of this issue as well. In the eleventh chapter of his treatise Piccolomini attempts to reinstate the status of mathematics as a science by claiming that mathematical objects are conceptual entities, existing in the human mind. Russell 1992: 4–5.
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essential has become dissociated from questions concerning the ontological status of mathematical objects, and the process leading to the development of modern logic.
Bibliography Editions Philoponus, In Aristotelis Categorias commentarium, ed. A. Busse, CAG 13/1, Berlin 1898. Philoponus, In Aristotelis De anima commentaria, ed. M. Hayduck, CAG 15, Berlin 1897. Philoponus, In Aristotelis Physicorum libros commentaria , ed. H. Vitelli, CAG 16, Berlin 1897. Philoponus, In Aristotelis Analytica posteriora commentaria , ed. M. Wallies, CAG 13/3, Berlin 1909. Proclus, In primum Euclidis elementorum librum commentarii , ed. G. Friedlein, Hildesheim 1969.
Studies De Groot, J. (1991) Aristotle and Philoponus on Light. New York. De Haas, F. A. J. (1997) John Philoponus’ New De nition of Prime Matter: Aspects . of Its Background in Neoplatonism and the Ancient Commentary radition Leiden. De Rijk, L. M. (1990) ‘Te Posterior Analyticsin the Latin West’, in Knowledge and the Sciences in Medieval Philosophy: Proceedings of the Eighth International Congress of Medieval Philosophy, ed. M. Asztalos, J. E. Murdoch and I. Niiniluoto. Acta Philosophica Fennica 48: 104–27. Goldin, O. (1996) Explaining an Eclipse: Aristotle’s Posterior Analytics 2.1–10. Ann Arbor, MI. Harari, O. (2004) Knowledge and Demonstration: Aristotle’s Posterior Analytics. Dordrecht. (2006) ‘Methexis and geometrical reasoning in Proclus’ commentary on 30: 361–89. Euclid’s ’, Oxford Studies in Ancient Philosophy Helbing, M. O.Elements (2000) ‘La fortune des Commentaires de Proclus sur le premier livre des Eléments d’Euclide à l’époque de Galilée’, in La Philosophie des , ed. G. Bechtle and D. J. O’Meara. mathématiques de l’antiquité tardive Fribourg: 177–93. Jardine, N. (1988) ‘Epistemology of the sciences’, in Te Cambridge History of Renaissance Philosophy, ed. C. Schmitt, Q. Skinner and E. Kessler. Cambridge: 685–711.
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Lloyd, A. C. (1990) Te Anatomy of Neoplatonism. Oxford. Mancosu, P. (1992) ‘Aristotelian logic and Euclidean mathematics: seventeenth century developments of the Quaestio de certitudine mathematicarum’, Studies in History and Philosophy of Science23: 241–65. (1996) Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford. McKirahan, R. (1992) Principles and Proofs: Aristotle’s Teory of Demonstrative Science. Princeton, NJ. Mendell, H. (1998) ‘Making sense of Aristotelian demonstration’, Oxford Studies in Ancient Philosophy 16: 169–78. Morrison, D. (1997) ‘Philoponus and Simplicius on tekmeriodic proofs’, in Method and Order in Renaissance Philosophy of Nature: Te Aristotle Commentary radition, ed. D. A. Di Liscia, E. Kessler and C. Methuen. Aldershot: 1–22. Mueller, I. (1975) ‘Greek mathematics and Greek logic’, in Ancient Logic and Its Modern Interpretation, ed. J. Corcoran. Dordrecht: 35–70. (1990) ‘Aristotle’s doctrine of Abstraction in the Commentators’, in Aristotle , ed. ransformed: Te Ancient Commentators and Teir In uence R. Sorabji. Ithaca, NY: 463–80. Russell, B. (1992) Principles of Mathematics( rst edn 1903). London.
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Contextualizing Playair and Colebrooke on proo and demonstration in the Indian mathematical tradition (1780–1820)
Te social shaping o representations o so called non-Western astronomy and mathematics in eighteenth- and nineteenth-century European scholarship has been o recent scholarly interest rom the perspective o the politics o knowledge.1 A principal concern has been the changing estimation o non-Western mathematical traditions by European mathematicians and historians o mathematics between the end o the last decades o the eighteenth century and the early decades o the nineteenth century; that is rom the heyday o the Enlightenment to the post-Enlightenment period. While these studies have been inormed by Said’s Orientalism,2 they have sought to examine the question whether the history o mathematics (the least likely case) is also inscribed within the rame o European colonial adventure and enterprise, as happened in the arts, literature and social sciences.3 It has been suggested that the European scholarship on the sciences o India reveals ractures along national lines, which in turn re ected the diversity o educational and institutional contexts o the world o learning.4 Tis chapter examines the relationship between the histories o Indian astronomy and mathematics produced by French astronomers and the translation rom the Sanskrit o works on Indian algebra undertaken by a colonial administrator and British Indologist, Henry Tomas Colebrooke. Te contrast revealed the divergent disciplinary orientations o the interpreters themselves. Second, in elaborating upon the canonization o a very important translation o Indian mathematical works by Colebrooke,5 I shall argue that the standard European depiction o the Indian mathematical 1 2 3 4 5
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Charette 1995; Raina 1999. Said 1978. Assayag et al. 1997. Raina 1999. Sir Henry Tomas Colebrooke was the son o the Chairman o the East India Company Directors, and arrived in India as an official o the Company in 1782–3. In India he acquired a pro ciency in Sanskrit literature and commenced writing on Hindu law, the srcins o caste, etc. As a result he was appointed Proessor o Hindu Law and Sanskrit at the College o Fort William, Calcutta (Buckland 1908:87–8). His translation o texts o Bhaskara and Brahmagupta became classics o nineteenth-century history o Indian mathematics.
Contextualizing Playfair and Colebrooke
tradition as devoid o proo went contrary to the spirit o Colebrooke’s translation and the large number o proos and demonstrations therein contained. In other words, this chapter elaborates upon how the Indian tradition o mathematics came to be constructed as one that was devoid o the idea o proo. While this characterization acquired stability in the nineteenth century, the construction itsel was pre gured in the eighteenth century. However, in the second hal o the nineteenth century there were historians o mathematics who held that speci c kinds o proo were encountered in Indian mathematical texts. It could be suggested that the concerns possibly giving the several contributions in the present volume a thematic unity is the ocus upon the empirical reality o mathematical practices, which perhaps suggests that mathematical traditions the world over, in the past as in the present, were and are characterized by several cultures o proo. Furthermore, studies on the culture(s) o proving among contemporary mathematicians, pure and applied, appear to indicate that rather than there being a unique criterion o what constitutes a proo there exist several mathematical subcultures.6 Tis view pushes in the direction o a sociological view o proo, amounting to a consensus theory o proo. Clearly this runs contrary to the ormal veri cationist idea that proos are pinioned on their ‘intrinsic epistemic quality’.7 Tis naturally raises the question as to how and when will these issues surace in the efforts o historians o mathematics. For i, as is suggested, it was not until the middle o the nineteenth century that proo became the sole criterion o validating mathematical statements,8 then its re ection is to be ound in the constructions o histories o mathematics as well. In order to look at the more technical mathematical writing it is rst necessary to brie y describe the optic through which Europeans turned their gaze on India during this period and the tropes that de ned their literary production on India during these decades. Te eighteenth century has been considered the ormative period or the emergence o the discourse on colonialism, but this discourse was not yet ‘monolithic or univocal’. European writing on India comprised a network o intersecting and contending representations.9 Te representations o India in this writing are naturally very ‘diverse, shifing, historically contingent, complex and competitive’. Te texts themselves are shaped ofen by ‘national and religious rivalries, domestic concerns’, and the cognitive or intellectual cultures o 6 7 8 9
Heinz 2000; MacKenzie 2001; Heinz 2003. Heinz 2003: 234–5. Heinz 2003: 938. eltscher 1995: 2.
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the respective interlocutors.10 Critical studies on oriental scholarship have sought to situate these texts in national and religious contexts and to identiy the elements they share.11 It has been argued that until the eighteenth century it was possible to speak o a European tradition o writing about India that differentiated into several national traditions by the middle o the eighteenth century. Te birth o a speci cally British tradition is put around 1765 when the East India Company was granted rights to collect land revenues and administer civil justice in Bengal.12 With the ounding o the Asiatic Society, British writing on India especially rom the 1780s onwards was marked by the impulse o British writers to ‘oreground the textual nature o their activity’, in other words to anchor their writings on India in the speci c study o classical texts produced in India.13 Te French missionaries who came to India in the late seventeenth century were the rst to have spoken o India’s scienti c past. French Indology, according to Jean Filliozat, emerged in the early decades o the eighteenth century, when the King’s librarian requested Etienne Fourmont, o the Collège Royal, to draw up a list o works o note rom India and Indo-China, to be purchased or the King’s library. By 1739, a catalogue o Sanskrit works had been prepared, and copies o Vedas, epics, philosophical and linguistic texts and dictionaries had been procured.14 Curiously enough there were very ew, i any, scienti c texts that were included in the cargo to the King’s library.15 he Jesuit astronomers were the rst tostudy the Indian astronomical systems that Filliozat considers ‘the rst scienti c or evencultural achievements o India studied by Europeans’.16 Kejariwal goes so ar as to suggest that the ‘history o French Orientalism is also the history o the 17 rediscovery o ancient Indian astronomy in the modern period’. A ruitul approach into this archive o scienti c texts and not just literary or religious texts is to pay attention to moments where the standard cultural descriptions characterizing the early European writing on India are challenged or unsettled through the textual analysis o similar and different orms o reasoning.18 In examining these mathematical texts, it is thereby essential or our purpose to be alert to those moments and descriptions o 10 11 12 13 14 15 16 17 18
eltscher 1995: 2; Raina 1999; Jami 1995. Inden 1990; Zupanov 1993. eltscher 1995: 3. eltscher 1995: 6. Filliozat 1955: 1–3. Raina 1999. Filliozat 1957. Kejariwal 1988: 17. eltscher 1995: 14.
Contextualizing Playfair and Colebrooke
mathematical results and procedures encountered within Sanskrit texts that were not accompanied by demonstrations or proo or exegesis. Te British mathematician and geologist John Playair (1748–1819) in introducing the Indian astronomy broadly speaking to an English speaking audience was to write: Te astronomy o India is con ned to one branch o the science. It gives no theory, nor even any description o the celestial phenomena, but satis es itsel with the calculation o certain changes in the heavens . . . Te Brahmin . . . obtains his result with wonderul certainty and expedition; but having little knowledge o the principles on which his rules are ounded, and no anxiety to be better inormed, he is perectly satis ed, i, as it usually happens, the commencement and duration o the eclipse answer, within a ew minutes, to his prediction.19
Tere are our ideas that are evident in this passage, and that run constantly throughout the construction o Indian astronomy and mathematics. Inasmuch as Indian astronomy is a science it differs rom modern astronomy in that (a) it lacks a theoretical basis, (b) it does not provide a description o celestial phenomena, and (c) it is not methodologically re ective (‘little knowledge o the principles on which his rules are ounded’), which in turn amounts to the idea that (d) the Indian astronomer computes but does so blindly. In other words these computations were perormed blindly by the Indian astronomers. On account o the predictive accuracy o the astronomy it merited the stature o a science, and the Indian astronomers were concerned no more with it than in this instrumental context.
Te srcins o British Indology : different starting points, different concerns British studies on Indian astronomy and mathematics may be said to lie at the conjuncture o two different historiographies: French and British. One o the earliest British Indologists to speak o the distinctive tradition o Indian algebra was Reuben Burrow (1747–92), a mathematician and a one-time assistant to Maskelyne, the Astronomer Royal in Greenwich. Te prior French tradition o the history o science had been preoccupied with the srcins o Indian astronomy. Burrow centred the question about the srcins o Indian mathematics. Tis will become evident urther ahead. Tat Burrow had a different optic rom the French is evident in his ‘Hints concerning the Observatory at Benaras’: 19
Playair 1790 (1971): 51.
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232 Notwithstanding the prejudices o the Europeans o the last century in avour o their own abilities, some o the rst members o the royal so ciety were sufficiently enlightened to consider the East Indies and China & c, as new worlds o science that remained undiscovered . . . had they not too hastily concluded that to be lost, which nothing but the prejudice o ignorance and obstinacy, had prevented being ound, we might at this time [be] in possession o the most nished productions o Asia as well as Europe; the sciences might, in consequence, have been carried to a much higher degree o perection with us than they are at present; and the elegance and superiority o the Asiatic models might have prevented the neglect and depravity o geometry, and that inundation o Algebraic barbarism which has ever since the time o Descartes, both vitiated taste, and overrun the publications, o most o the philosophical societies in Europe.20
Te encounter with other non-European scienti c traditions was encouraged by the ideological impulse to advance the rontiers o knowledge. In that sense Burrow’s philosophy o science resonated with that o the Enlightenment thinkers. Te most striking eature o the above passage is that the Indian tradition or Burrow is still not characterized as algebraic or geometric. In act, at this point the characterization is the very reverse o the late nineteenth century where Indian mathematics is constituted as one that is algebraic in spirit at the expense o geometry. Tis nineteenthcentury Indian mathematics traditions as algebraic or portraiture algorithmic,oand as one where the depicted geometricthe side o mathematics was underdeveloped. Modern European mathematics since Descartes, in Burrow’s words, had been overwhelmed by ‘algebraic barbarism’. An exposure to Asiatic models would then have prevented the neglect o geometry that marked contemporary sciences. I do not know i one could interpose the suggestion that there may have been some Anglo-French rivalry at stake. But then that is not immediately germane to the construction. Te relevant concern here is that until the end of the eighteenth century some British Indologists still entertained the hope that they would discover Indian geometrical texts that would unveil to them the foundations of an Indian geometrical tradition. Tus Playair would in 1792 pose six questions to
the researchers o the Asiatic Society, the rst o which was:‘Are any books 21
to be ound among the Hindus, which treat proessedly o Geometry?’ Playair was thus asking i it were possible to identiy elements o a corpus o knowledge albeit in a different disguise that could be considered geometry in the sense in which it was conceived in Europe. For one it could be 20 21
Burrow 1783 (1971): 94–5. Playair 1792: 151.
Contextualizing Playfair and Colebrooke
said that the question that the geometry o the Hindus could have a different basis rom the Greek ones is implied by the ‘proessedly’ in the question. Tat this is what Playair meant might be inerred rom his elaboration upon the question he posed: I am led to propose this question, by having observed, not only that the whole o the Indian Astronomy is a system constructed with great geometrical skill, but that the trigonometrical rules given in the translation rom the Surya Siddhanta, with which 22
Mr. Davis has obliged the world, point out some very curious theorems, which must have been known to the author o that ancient book.23
According to Playair, as he engages with Davis’ translation o the Surya Siddhanta the ‘trigonometrical canon’ o Indian astronomy is constructed on the basis o a theorem. Te theorem is stated as: I there be three arches24 o a circle in arithmetical progression, the sum o the sines o the two extremes arches is to twice the sine o the middle arch as the cosine o common difference o the arches to the radius o the circle.25
Tough the theorem was not known to Europe beore Viete, Playair continues, the method was employed by the Indian astronomers or constructing trigonometrical tables, and was based on the simpler procedure o calculating sines and arcs than through the use o methods that were based on extracting square roots.26 Te immediate task or Playair appears to have been to identiy those mathematical works where the theorem on which the trigonometrical rule employed in astronomy is rst laid out. Tis brings us back to Burrow’s concern with the srcins o Indian mathematics.
Contrasting approaches: sifing the mathematical rom the astronomical rexts In the late eighteenth century it would have been possible to differentiate between the efforts o the British Indologists and those o their French studying Indian astronomy andIndologists mathematics onbusy two counts. counterparts Methodologically speaking, while the British were 22
23 24 25 26
Samuel Davis (1760–1819) was a judge inBengal and produced one o the rst translations o the Surya Siddhanta. Playair 1792: 151. An ‘arc o a circle’ is what is meant here. I havekept the srcinal spelling. Playair 1792: 152. Playair 1792: 152.
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underlining the textual nature o their enterprise, the French astronomersavants relied a great deal on proto-ethnographic descriptions o the mathematical and astronomical practices o India. Secondly, the histories o Indian astronomy o Bailly and Le Gentil are preoccupied with the astronomy o India and the srcins o Indian astronomy.27 Even Montucla’s history o mathematics relies extensively upon the proto-ethnographic sources employed by Le Gentil and Bailly and draws inerences concerning Indian mathematics rom them.28 Te British Indological tradition, on the other hand, engaged with speci c texts and rom the astronomical rules presented there made a claim that these rules must be based on a mathematical system, and proceeded to discover mathematical texts. Teir ocus thus shifs rom the srcins o astronomy to the srcins o Indian mathematics, in particular Indian algebra and arithmetic. What were the rules encountered and what were the claims made? Te shif was precipitated by the desire to craf a history o mathematics independently o the history o astronomy. As scholars approached the corpus o Indian astronomical texts, they encountered a corpus o knowledge recognizable to them as algebra and arithmetic. Consequently, John Playair was later to insist upon the need to search or a geometrical tradition. Reuben Burrow was probably amongst the earliest o the British Indologists to engage with the textual tradition o Indian mathematics, although this search was prompted through his exposure to and study o astronomy, including Indian astronomy. Tis does not mean that these texts did not relate in any way to the histories o Le Gentil and Bailly. Actually, the texts o the ormer provided an initial rame or approaching the differences between the Indian and Modern traditions. For Burrow the study o the procedures employed by Indian astronomers in calculating eclipses would advance the progress o modern astronomy as well: ‘and the more so as our methods o calculation are excessively tedious and intricate’.29 Te sentiment echoes that o Le Gentil and Bailly; and it is certain that he was acquainted with the work o Le Gentil,30 though it is not possible to say the same o Bailly’s raité de l’astronomie indienne et orientale. Tis ascination with the computational procedures employed in astronomy led Burrow to iner in 1783 the existence o an advanced algebraic tradition:
27 28 29 30
Bailly 1775; Le Gentil 1781. Montucla 1799. Burrow 1783 (1971): 101. Burrow 1783 (1971): 116.
Contextualizing Playfair and Colebrooke
It is also generally reported that the Brahmins calculate their eclipses, not by astronomical tables as we do, but by rules . . . I they (the rules) be as exact as ours, . . . it is a proo that they must have carried algebraic computation to a very extraordinary pitch, and have well understood the doctrine o ‘continued ractions’, in order to have ound those periodical approximations . . .31
Te rules or computing eclipses employed by the Brahmins were not only different, but their complexity varied with the requisite degree o exactness: . . . which entirely agrees with the approximation deduced rom algebraic ormulae and implies an intimate acquaintance with the Newtonian doctrine o series . . . and thereore it is not impossible or the Brahmins to have understood Algebra better than we do.32
Tis was to become the central point rom which in subsequent papers Burrow would build his argument or the existence o an advanced algebra among the Indians. Te problem was taken up again by Colebrooke discussed below, and in a paper published slightly later by Edward Strachey, ‘On the early history o algebra’.33 Te paper emphasized the srcinality and importance o algebra among the Hindus and contained extracts that were translated rom the Bija-Ganita and Lilavati.34 Tese extracts were translations into English rom Persian translations o the srcinal Sanskrit 35
texts. But Burrow admits thattillthese were translated in 1784, but 36 But he deerred publishing them a ullextracts text was obtained. he prizes the moment: ‘when no European but mysel . . . even suspected that the Hindoos had any algebra’.37 Te rationale provided or the existence o treatises on algebra in India in Burrow’s 1790 paper on the knowledge o the binomial theorem among the Indians is the same as that suggested in the earlier one (1783). Many o the approximations used in astronomy were 38 Tese ‘deduced rom in nite series; or at least have the appearance o it’. included nding the sine rom the arc and determining the angles o a
31 32 33 34
35
36 37 38
Burrow 1783 (1971): 101. Burrow 1783 (1971): 101. Strachey 1818. Tese works were authored by the twelfh-century mathematician Bhaskara II, and while the rst o these deals with problems in algebra and the solution o equations, the latter ocuses more on arithmetic. Strachey’s paper will not be discussed here, since the ocus will be on the translation o versions o Sanskrit texts into English and not the manner in which these Sanskrit texts were reported in translations o Persian and Arab mathematical works. Burrow 1790. Burrow 1790: 115. Ibid.
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right-angled triangle given the hypotenuse and sides without recourse to a table o sines, etc. Te urgency o the moment was then to discover those texts beore they perished. Burrow thus emphasized the need or the collection o available astronomical and mathematical texts that till then had not been the ocus o attention o the French Académiciens. Te idea that the existing tradition was probably algebraic was being insinuated: ‘Tat many o their books are depraved and lost is evident, because there is now not a single book o geometrical elements to be met and yet that they had elements not long ago, and apparently more extensive than those o Euclid is obvious rom some o their works o no great antiquity.’39 At this liminal moment it appears as i the issue whether the geometric tradition prevailed over the algebraic or vice versa in India had not been settled. It cannot be decisively be said that Burrow had a xed view on the subject. But certainly the texts he encountered were not o a ‘geometric’ nature. But the trigonometrical calculations gave cause or belie that the semblance o such a system was in existence. And while Burrow promised to publish translations o Lilavati and the Bija-Ganita, the promise was not ul lled during his lie. Inspired by Burrow‘s research, Colebrooke embarked on a study o Sanskrit in order to probe some o the issues raised by Burrow more deeply. It was lef to Samuel Davis to publish the rst translation and analysis o an Indian scienti c work rom the Sanskrit into a European language, this being a translation o the Surya Siddhanta.40 Tis translation was based on the reading o an srcinal version o the text procured by Sir Robert Chambers in 1788. Davis encountered a number o obscure technical terms and had to rely upon a teeka or commentary procured by Jonathan Duncan.41 In act, i you examine the structure o Davis’ paper, it appears as a teeka on the Surya Siddhanta, with passages translated rom the text and Davis’ explanation intercalated between the translated passages. Davis begins by contesting the portrait o Indian astronomy and astronomers projected by Le Gentil and Bailly,42 without naming either o them. 39 40 41 42
Ibid.
Davis 1789. Ibid. More than Bailly and Le Gentil, Davis was reuting Sonnerat’ s constructions o Indian astronomy: . . . my present intention, which is to give a general account only o the method by which the Hindus compute eclipses, and thereby to show, that a late French author was too hasty in asserting generally that they determine by set orms couched in enigmatical verses &c. So ar are they rom deserving the reproach o ignorance, which Mons. Sonnerat has implied,
Contextualizing Playfair and Colebrooke
Te rst idea that he rejected was that this astronomical tradition was disgured over the years by idolatry and that the gems o Indian astronomy had been irretrievably lost over the centuries, in the absence o a textual tradition. Te second idea was that the Brahmins had shrouded their astronomy in mystery such that it was impossible to arrive at a cogent account o it. Further, they loathed sharing their ideas with others. Davis set out to show that: . . . numerous treatises in Sanskrit on astronomy are procurable, and that the Brahmins are willing to explain them . . . I can arther venture to declare, rom the experience I have had, that Sanskrit books in this science are more easily translated than almost any others, when once the technical terms are understood: the subject o them admitting neither of metaphysical reasoning nor of metaphor , but being delivered in plain terms and generally illustrated with examples in practice , . . .43
Te British Indologists were departing rom the reading o Académiciens grounded in Jesuit proto-ethnography, by textually locating their work. Tis textual grounding would revise the portrait o the French savants. A hundred years later in a review o the history o the history o Indian astronomy Burgess was to write: ‘Mr. Davis’ paper, however, was the rst analysis o an srcinal Hindu astronomical treatise, and was a model o what such an essay ought to be.’44 It appears then, as has been argued elsewhere, that the French savants in India were unable to establish trust with their Indian interlocutors, in total contrast to the rst generation o British Indologists such as William Jones,45 and i one takes Davis’ account literally then Davis himsel. wo papers o William Jones ollowed closely on the heels o Davis’ papers and a cursory glance at them reveals that they mutually respected and supported each other’s enterprise.46 And yet they both were in agreement with Bailly’s thesis o the independent srcins o the Indian zodiac, differing very strongly with Montucla on this count: that on inquiry, I believe the Hindu science o astronomy will be ound as well known now as it ever was among them, although perhaps, not so generally, by reason o the little encouragement men o science at present meet with . . . (Davis 1789: 177).
43 44 45 46
Evidently, Sonnerat unlike Davis could not enter the world o the Hindu astronomers on account o his inability to abandon a hermeneutic o suspicion. Pierre Sonnerat was a French naval official who travelled to India towards the last decades o the eighteenth century and published a book Voyages aux Indes Orientales et à la Chinein 1782 which discussed the history, religion, languages, manners, arts and science o the regions he visited. Davis 1790: 175 (emphasis added). Burgess 1893: 730–1. Raj 2001. An eighteenth-century Indian scholar who worked closely both with Jones and along with his associates with Colebrooke was Radhakanta arkavagisa (Rocher 1989 ).
237
238 I engage to support an opinion (which the learned and industrious M Montucla seems to treat with extreme contempt) that the Indian division o the zodiac was not borrowed rom the Greeks or Arabs, but having been known in this country rom time immemorial and being the same in part with other nations o the old Hindu race . . .47
But then they were also gradually transorming and re ning the portrait Bailly had lef behind. Tus Jones recognized that in Davis’ translation resided the hope that it would ‘convince Bailly that it is.’very possible or 48 an European to translate and explain theM. Surya Siddhanta
Playair’s programme and Colebrooke’s recovery o Indian algebraic texts In order to recapitulate a point made earlier, the French Jesuits o the seventeenth and eighteenth centuries were the inaugurators o a tradition, which 49 Bailly’s was to inspire the histories o Le Gentil and Jean-Sylvain Bailly. history inspired the work o the British mathematician John Playair and provided a stimulus to subsequent generations o British Indologists writing on Indian mathematics; though they were to disagree with the details o Bailly’s Histoire , adding some nuance here and digressing rom it in another context.50 Te antediluvian hypothesis proposed by Bailly was the source o both ascination and controversy, and was the outcome o his attempt to juxtapose observations o ancient Indian astronomy with astronomical theory o his day;51 rom which he went on to draw the inerence that 52 However, ancient Indian astronomy was the source o Greek astronomy. this reading was located within Jesuit historiography which sought to accommodate Indian history within the Christian conception o time.53 Bailly’s work was introduced to English-speaking readers through an article authored by John Playair entitled ‘Remarks on the Astronomy o the Brahmins’ published in the ransactions of the Royal Society of Edinburgh.54 47 48 49 50 51
52 53 54
Jones 1790a. Jones 1790b. Raina 1999. Raina 2001a. According to this hypothesis astronomy srcinated amongthe Indians, but the Indiansin turn had received it rom an even more ancient people. Te traces o this exchange had been lost in antiquity. Bailly 1775. Raina 2003. Playair 1790.
Contextualizing Playfair and Colebrooke
Te article draws extensively, need I say almost exclusively, upon the Mémoirs o Le Gentil published by the Académie des Sciences, Paris and Bailly’s Astronomie Indienne.55 Tis article o Playair’s was o prime importance or Indologists working on the history o Indian astronomy or the next our decades. Playair’s central contribution resided in re-appropriating Bailly’sraité in the light o the contributions o Davis and Burrow and proposing a set o tasks that could well be considered a research programme or the Asiatic Society. Tese included: (a) to search or and publish works on Hindu geometry, (b) to procure any books on arithmetic and to ascertain those arithmetical concerns whose trace is not to be ound among the Greeks, (c) to complete the translation o the Surya Siddhanta as initiated by Samuel Davis, (d) to compile a catalogue raisonné, with a scholarly account o books on Indian astronomy, (e) to examine the heavens with a Hindu astronomer in order to determine their stars and constellations, () to obtain descriptions and drawings o astronomical buildings and instruments ound in India.56 I Bailly had stirred a hornet’s nest in his time by suggesting that the srcins o astronomy were in India, albeit that this astronomy was inherited by the Indians rom an even more ancient people, Burrow’s paper did the same with the srcins o algebra. It is at this time diffi cult to separate the discussion on the history o astronomy rom the history o algebra; or both the Académiciens and the Indologists ofen turn to the history o astronomy to evoke computational procedures that were analysed mathematically. Tis programme o the recovery o the mathematical literature rom the astronomical literature was taken up by Colebrooke, who may be seen as providing translations rom the Sanskrit into English o the rst texts supposedly dedicated solely to algebra and arithmetic. I say supposedly because portions o some o the texts Colebrooke discovered or the English-speaking world were essentially the mathematical sections o larger astronomical canons o the Indian tradition. We come now to Colebrooke’s translation practices. In order to describe them we need to understand how Colebrooke identi ed an authenticated version o the texts that he set out to translate. It needs to be pointed out that at the very outset no nal version o the three texts, rom which only portions were translated, was readily available to him. Consequently, he worked with his Brahmin interlocutors and collected and collated 55 56
Le Gentil 1789; Bailly 1787. Playair 1792: 152–5.
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ragments o the works o Bhaskara and Brahmagupta beore proceeding to nalize versions o the three texts translated. But the enormous task was to nalize and authenticate a version as the version o these texts. Te central question then was: how were the ragments o the texts to be ordered into a sequence or other ragments spliced into appropriate sections o the sequence o ragments in order to complete the collation o the text. His native interlocutors were thus assigned the task o providing him with an exhaustive commentary(ies) on these texts and most certainly worked with him through the process o translation. Te larger the set o commentaries available on a given text, say the Lilavati, the greater the importance o the text within the canon. Te commentaries themselves served two exceedingly important unctions. In the rst instance the commentaries were employed to identiy the missing portions o the ragments available, and to x the sequence o chapters. In other words it is through the commentaries that the text was nalized. Second, the commentaries were employed to illustrate and explain semantically and technically obscure portions and procedures expounded in the main text. A typical page o Colebrooke’s translation thus comprises an upper hal or two-thirds that are translations rom the Sanskrit o nalized versions o the texts o Bhaskara and Brahmagupta, while the lower hal or third comprises: (1) Colebrooke’s explication o the text when need be, with reerences to other texts, which is done with ootnotes, (2) translations rom one or several commentaries that clariy the meaning o a term or terms or procedures mentioned in the portion o the text on the upper portion o the page, but at no point in Colebrooke’s text is the entire commentary translated. In act the text comprises translations rom portions o several commentaries, and it is Colebrooke who decided which part o one o several commentaries or portions o several commentaries best elaborates or clari es a portion o the master text being translated. But the commentaries are internally paired off against each other in order to arrange chronologically the commentaries and thus provide a diachronic relation between them. Colebrooke drew upon a rich commentarial tradition while working on his translation o the Lilavati. Te rst o these was a commentary by Gangādhara dated 1420. Te commentary was limited to the Lilavati, but as Colebrooke inorms us, it authenticated an important chapter rom the Bija-Ganita.57 Further, Suryadasa’s Ganitámrita dated 1538 was a commentary on the Lilavati and the Surya-pracāsa was a commentary 57
C1817: xxv.
Contextualizing Playfair and Colebrooke
on the Bija-Ganita that contained a clear interpretation o the text with a concise explication o the arithmetical rules.58 Te other important composition was Ganesa’s Buddhivilasini (c. 1545), comprising a copious exposition o the text with demonstration o the rules. However, Ganesa had not written a commentary on the Bija-Ganita and Colebrooke drew on the work o Krishna which explained the rules with a number o demonstrations. In addition to which two other commentaries were used, namely that o Ramakrishna Deva entitled Manoranjana, a text o uncertain date, and nally the Ganitakaumud, which was known through the works o Suryadasa and Ranganatha.59 A brie recapitulation is required beore we proceed to the translations o Colebrooke, or his work certainly marks a departure in the study o the history o Indian mathematics. wo main historiographic currents in the eighteenth century oriented the study o the history o the mathematics and astronomy o India. Te rst approach was that pursued by the Jesuit savants in India, who were observing the astronomical and computational procedures circulating among Indian astronomers. Teir audience did not merely comprise the devout back in France, but the Académiciens and astronomers, two o whom transcribed these proto-ethnographic accounts into a history o Indian astronomy. Administrator–scholars, who studied texts, collated ragments o texts and published translations with critical editions and commentaries, while indebted to the rst, pursued another approach. In the late eighteenth century, Sanskrit commentaries and canonized astronomical or mathematical works were considered the key to obscure technical terms and texts. What needs to be examined is whether by the late nineteenth century commentaries shared the same destiny as some o the Vedic texts. For it has been pointed out that by the second hal o the nineteenth century some Sanskritists belittled, marginalized and removed ‘explicit reerences to the intermediary process o transmission and exegesis o texts without which they would not have had access to them’.60 Te status o proos in the Indian tradition is related to how these commentaries on mathematical texts were read. 58
59 60
C1817: xxvi. Te term explication involves two different tasks when applied to literary texts and scienti c texts. In the case o literary texts explication means to unold; or to offer a detailed explanation o a story. In the case o a scienti c text or procedure, explication involves the transormation o the explicandum by the explicatum. However, explication in Colebrooke does not possibly conorm to the notion that the explicandum is pre-scienti c and inexact, while the explicatum is exact. Te explicandum and explicatum are related to each other in their difference and not in a hierarchy o exact/inexact. C1817: xxvii–xxviii. Vidal 1997: 25.
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Te point needs some reaffirmation since both Colebrooke and Davis, who worked with commentaries o canonized astronomical and mathematical texts respectively, do mention the existence o demonstrations, and rules in the texts they discuss. In Colebrooke’s introduction to his Algebra with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara, there are our terms o concern to us here, namely demon-
stration, rule, proo and analysis, that come up ofen, but it is only the last o these that Colebrooke clari es. Further, as will be noticed in the next section the terms demonstration and proo are used interchangeably by Colebrooke. Noted by its absence in the title is the term ‘geometry’, as a systematized science; on the contrary, the translation does allude to mensuration as discussed in the books he translates. Te crucial problematic or Colebrooke was, as with Burrow beore him, to determine the srcins o Indian algebra. Inspired, as it were, by the textual exemplars o Davis and Burrow, and guided by the research programme John Playair had drawn up or the researchers o the Asiatic Society, Colebrooke highlighted the pathway to his own work: In the history o mathematical science, it has long been a question to whom the invention o algebraic analysis is due, among what people, in what region was it devised, by whom was it cultivated and promoted, or by whose labours was it reduced to form and system.61
Te subsequent narrative ocuses upon establishing that ‘the imperect algebra o the Greeks’, that had through the efforts o Diophantus advanced no urther than solving equations with one unknown, was transmitted to India. Te Indian algebraists, through their ingenuity, advanced this 62 In his reading, ‘slender idea’ to the state o a ‘well arranged science’. Colebrooke shares a undamental historiographic principle, disputed by current scholarship, with Burrow, one that enjoyed currency among historians o mathematics into the twentieth century. In this historiographic rame: ‘. . . the Arabs themselves scarcely pretend to the discovery o Algebra. Tey were not in general inventors but scholars, during the short period o their successul culture o the sciences.’63 Te science o ‘algebraic analysis’, a term Colebrooke would later expand upon, existed in India beore the Arabs transmitted it to modern Europe.64 Te evidence or these claims resided in the translations o 61 62 63 64
C1817: ii (emphasis added). C1817: xxiv. C1817: ii (emphasis added). Ibid.
Contextualizing Playfair and Colebrooke
the Bija-Ganita and Lilavati o Bhaskara,65 as well as Brahmagupta’s (Colebrooke: ‘Brahmegupta’) Ganitadhyaya and Kuttakadhyaya (the chapter entitled ‘Te pulveriser’) (Colebrooke: Cuttacadhyaya), the last two as their name suggests being the mathematical sections o Brahmagupta’s Brahmasphutasiddhanta. Without ocusing too much on the antiquity o these texts, Colebrooke saw his oeuvre as disclosing that the: modes of analysis, and in particular, general methods for the solution of indeterminate problems both o the rst and second degrees, are taught in the Vija-Ganita,
and those or the rst degree repeated in the Lilavati, which were unknown to the mathematicians o the west until invented anew in the last two centuries by algebraists o France and England.66
Te terrain o historical studies on Indian mathematics was being transormed into a polemical one, with Colebrooke surreptitiously introducing categories that the French Indologists had denied the Indian tradition: typically or the rst time he speaks o ‘modes o analysis’, or the ‘general methods or the solution o indeterminate problems’. Te historians o astronomy had previously advanced the idea that the Indians had no idea o the generalizability o the methods they employed. In the absence o such generalizability, how could it have been possible to extend the idea o generalized methods dedicated to solving classes o problems in order to extract the different ‘modes o analysis’? Te intention here is not to paint Colebrooke’s construction as the diametrical opposite o that o the French historians o science that provided a context to his effort. On the contrary, Colebrooke’s project is naturally marked by a deep ambivalence. Te ambivalence arises rom the act that he attempted to draw the characterization o Indian mathematics away rom the binary typologies o the history o science that were already set in place. According to these typologies Indian mathematics was characterized as algebraic and pragmatic while European mathematics was geometric and theoretical (deductive). Since the British Indologists were not mathematicians by proession they lacked mathematical legitimacy amongst the network o historians o mathematics and deterred his ability to create a new vocabulary. Tis also explains why Playairowas so important to thetheIndological He was a mathematician repute who endowed Indologicalenterprise. accounts with authority. 65
66
I have given here the contemporary English spellings o the names oSanskrit books and Bija-Ganita as Vija-Ganita scholars and removed the diacritics. Colebrooke himsel spelled the and Bhaskaracharya as Bhascara Acharya. C1817: iv (emphasis added).
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Colebrooke begins by pointing out that Aryabhata was the rst o the Indian authors known to have treated o algebra. As he was possibly a contemporary o Diophantus, the issue was important or drawing an arrow o transmission rom Alexandria to India or vice versa. Colebrooke leaves the issue o the invention o algebra open by suggesting that it was Aryabhata who developed it to the high level that it attained in India; 67 this science he called an ‘analysis’.68 It is here or the rst time that a portion o the Indian mathematical tradition is reerred to as analysis, and it is important to get to the sense in which he employs the term. It is noticed that the use o a notation and algorithms is crucial to this algebraic practice; which Colebrooke then proceeds to elaborate upon, subsequently stating the procedures not merely or denoting positive or negative 69 quantities, or the unknowns but o manipulating the symbols employed. An important eature o this algebra is that all the terms o an equation do not have to be set up as positive quantities, there being no rule requiring that all the negative quantities be restored to the positive state. Te procedure is to operate an equal subtraction ( samasodhana) ‘or the difference o like terms’. Tis operation is compared with themuqabalah employed by the Arab algebraists.70 Te presence o this ‘analytic art’ among the Indians was apparent rom the mathematical procedures evident in the variety o mathematical texts that were becoming available to the Indologists. Te analytic art comprised procedures that included, according to Colebrooke, the arithmetic o surd roots, the cognizance that when a nite quantity was divided by zero the quotient was in nite, an acquaintance with the procedure or solving second degree equations and ‘touching upon’ higher orders, solving some o these equations by reducing them to the quadratic orm, o possessing a general solution o indeterminate equations in the rst degree. And nally, Colebrooke nds in the Brahmasphutasiddhanta (§18:29–49) and Bija-Ganita (§75–99) a method or obtaining a ‘multitude’ o integral solutions to indeterminate seconddegree equations starting rom a single solution that is plugged in. It was lef to Lagrange to show that problems o this class would have solutions that are whole numbers.71 Te analytic art o the Indians or algebraic 67
68 69 70 71
Te high level o attainment was ascribed to the ability o the Indian algebraists to solve equations involving several unknowns; and o possessing a general method o solving indeterminate equations o the rst degree (C1817: x). C1817: ix. C1817: x–xi. C1817: xiv. C1817: xiv–xv.
Contextualizing Playfair and Colebrooke
analysis is then or Colebrooke: ‘calculation attended with the manifestation of its principles’. Tis is maniest in the Indian mathematical texts being discussed since they intimate to the reader a ‘ method aided by devices, 72 In this sense among which symbols and literal signs are conspicuous’. Indian algebra bears an affinity with D’Alembert’s conception o analysis as the ‘method o resolving mathematical problems by reducing them to equations’.73 Delambre and Biot would subject these views o Colebrooke to trenchant criticism, but that is another subject.74 Te issue at stake here is that Colebrooke had insinuated the idea that Indian mathematics was not lacking in methodological re ection or generality, a eature that had hitherto been denied. Did Colebrooke’s view o algebraic analysis provide or demonstrations or proos o its rules or procedures? Citing speci c sutras rom the Brahmasphutasiddhanta, the Bija-Ganita and the Lilavati, Colebrooke moves to a characterization o Indian algebra, just as Diophantus is evoked to characterize early Greek algebra. Tus, we are inormed that these Indian algebraists applied algebraic methods both in astronomy and geometry, and in turn, geometric methods were applied to ‘the demonstration of algebraic rules’. Obviously, Colebrooke was construing the visual demonstrative procedures employed by Bhaskara to which we come as exempliying geometrical demonstration. Further, he goes on to state that: In short, they cultivated Algebra much more, and with greater success than geometry; as is evident in the comparatively low state o their knowledge in the one, and the high pitch o their attainments in the other.75
Tis passage came to be quoted ever so ofen in subsequent histories o science, and in the writings o mathematicians as evidence o the algebraic nature o Indian mathematics.76 Te power o its imagery resides in its ability to draw the boundary between different civilizational styles o mathematics. In this contrast between Western and Indian mathematics it could be suggested that Colebrooke’s quali cation concerning the ‘comparatively
72 73 74 75 76
C1817: xix–xx. Ibid. Raina 1999. C1817: xv. Te nineteenth-century British mathematician Augustus De Morgan, a sel-proclaimed a cionado o Indian mathematics, wrote a preace to the book o an Indian mathematician punctuated with aperçus rom Colebrooke’s introduction. Te introduction in act provides him the ground to legitimate the work o the Indian mathematician or a British readership (Raina and Habib 1990).
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low’ state o one and ‘high pitch’ o the other was lost sight o and the contrast between the two traditions came to be subsequently accentuated. Tis leads me to conjecture that Colebrooke’s translation is a watershed in the occidental understanding o the history o Indian mathematics on a second count as well, this being that it inadvertently certi ed the boundary line drawn between Indian algebra and Greek geometry. Tis was not Colebrooke’s intention at all, but a consequence o the comparative method he had adopted. Colebrooke’s particular comparative method consisted in displaying where India’s speci c contributions to mathematics resided, and he always contrasted these contributions with the Greek and Arab traditions o mathematics.77 Tis attempt to accentuate the contrast certainly revealed the differences, but with the loss o the context o the contrast, it was rst transormed into a caricature and then stabilized as a characterization. Te boundary lines had however been marked out beore Colebrooke’s time. Tis passage is crucial because it is ollowed by a discussion o some procedures o demonstration in Indian algebra that I shall brie y lay out. Tus the speci c areas in which ‘Hindu Algebra appears particularly distinguished rom the Greek’ are our.78 Some o these have been mentioned above. Te additional one that has not been mentioned concerns the application o algebra to ‘astronomical investigation and geometrical demonstration’, in other words algebra is applied to the resolution o geometrical questions. In the process the Indian algebraists, Colebrooke suggests, developed portions o mathematics that were reinvented recently. Tis last statement o his prompted a very severe reaction. He then takes up three instances, which he considers ‘anticipations o modern discoveries’ rom the texts he discusses and lays out their procedures o demonstration. Tere is nothing in the subsequent portion o the introduction to suggest that he did not consider these as demonstrations.
Proos and demonstrations in Colebrooke’s translations o Indian algebraic work Colebrooke’s Algebra with Arithmetic and Mensuration was completed shortly afer his departure rom India or England in 1814. Te volume comprises the translation o our Sanskrit mathematical texts, namely the Bija-Ganita and Lilavati o Bhaskara, and the Ganitadhyaya and Kuttakadhyayao Brahmagupta. Tese translations were undertaken during 77 78
Going by his text alone,he appears to have been totally oblivious o Chinese mathematics. C1817: xvi.
Contextualizing Playfair and Colebrooke
his homeward voyage – we are inormed o this through the biography o Colebrooke written by his son.79 Further, Colebrooke’s interest, as pointed out earlier, in the subject was aroused by Reuben Burrow’s paper that appeared in the second volume o the Asiatic Researches. Colebrooke’s son, Sir . E. Colebrooke, writes: It must be admitted that the utmost learning which may be employed on this abstruse subject leaves the question open to some doubt, and resembles in this respect, one o those indeterminate problems which admit a variety o solutions. Te treatises which have come down to us are variants o arithmetical and algebraical science, o whose antiquity ew would venture to suggest a doubt. Tey exhibit the science in a state o advance which European nations did not attain till a comparatively recent epoch. But they contain mere rules for practice, and not a work on the path by which they are arrived at. Tere is nothing o the rigour . . . 80
Tis biography o Colebrooke was published more than hal a century afer Colebrooke’s work had appeared, by which time the standard representation o Indian mathematics was more or less in place as evident rom the emphasis in the quotation.81 However, as I shall argue below, this understanding was quite at variance with the spirit and content o Colebrooke’s translation, which, not without ambivalence, made a strong case or the idea o analysis and demonstration in the Indian mathematical tradition. A point to be noted here is that when Colebrooke the son comments on the Indian mathematical tradition in the 1870s the historiographical context has totally changed and he writes about Indian mathematics and the absence o proo in a spirit quite at variance with his ather who wrote in the early decades o the nineteenth century. Te change in the historiographical context is evident in Haran Chandra Banerji’s publication o the rst edition o Colebrooke’s translation 82 o the Lilavati in 1892 and in the second edition that appeared in 1927.
79 80 81
82
Colebrooke, . E. 1873: 303. Colebrooke, . E. 1873: 309. Colebrooke’s son also raises the question o the reception o Colebrooke’s Algebra with Arithmetic and Mensuration by Delambre. In his work on the history o astronomy o the middle ages Delambre based his remarks on Colebrooke based on a review o the work by Playair (Colebrooke, . E. 1873:310). Delambre’s critique o Colebrooke’s work has been discussed in Raina 2001b. Re J. S. Mill who wrote the manual o imperial history o India, Colebrooke the son notes, ‘. . . in his laboured pleading against the claims o the Hindus to be regarded as a civilized race, devotes some space to an examination o Mr. Colebrooke’s work, and then does little more than repeat the doubts o Delambre whose criticisms on the weakness o the external proo he repeats almost verbatim’ (Colebrooke, . E. 1873 : 311). Evidently Colebrooke the son wishes to disabuse his readers o the prejudiced criticism o Colebrooke the ather’s work. Banerji 1927.
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Te really interesting eature is the convergence in the reading o Colebrooke the son and Banerji concerning the mathematical style o Bhaskara. In the introduction to this translation Banerji was to write about Bhaskara: ‘Te author does not state the reasons or the various rules given by him. I have tried to supply the reasons as simply and shortly as they occurred to me; but still some cases . . . and shorter demonstrations may possibly be given.’83 Banerji proceeded to edit Colebrooke’s translation o these mathematical works by keeping those demonstrations given chie y by Ganesa and Suryadasa ‘which are satisactory and instructive’ and omitting those which ‘are obscure and unsatisactory’.84 In other words Banerji exercises his editorial prerogative and omits some proos or demonstrations, insisting that the omitted geometrical proos or these ormulas were given in Euclid .5 and 9. Te reason he offers or omitting the ‘proos’ o Ganesa is because Banerji clari ed that he had introduced these proos to acilitate calculations required in §134 o the Lilavati.85 Whatever may be the reason, it is obvious that Banerji’s reading o these texts is located within the ‘historiography o the absence o proo ’.86 Colebrooke’s magnum opus was published in 1817 and the introduction to the work is hereafer reerred to as the ‘dissertation’, which is what it is titled in any case. Very brie y, I shall just mention the chapterization o this work. Te rst chapter consists o the de nitions o technical terms. Drawing upon these de nitions the second chapter deals with numeration and the eight operations o arithmetic, which included rules o addition and subtraction, multiplication, division, obtaining the square o a quantity and its square root, the cube and the cube root. Te discussion up to Chapter 6 comprises the statement and exempli cation o arithmetical rules or manipulating integers, and ractions. Te examples provided illustrate the different operations. It is in Chapter 6 that we come to the plane gures and it is here that §134 states the equivalent o the Pythagorean Teorem.87 Te discussion below will centre around rule §135 o the Lilavati in
Colebrooke’s translation, where Colebrooke suggests that Ganesa had
83 84 85 86
87
Banerji 1927: vi. Banerji 1927: xv. Banerji 1927: xvi. An equally insightul exercise would be to see how and where Banerji’s text differs rom that o Colebrooke; on which portions o the text does Banerji nd it necessary to comment upon Colebrooke’s translation and interpretation; and at what points does he insert his own commentary and replace that o Colebrooke. Tis would be a separate project, sufficient though it be to point out that Banerji is more o a practising mathematician than Colebrooke. C1817: 59.
Contextualizing Playfair and Colebrooke
offered both algebraic and geometrical proos. In a contemporary idiom these rules are stated as: 2ab + (a − b)2 = a2 + b2 (a + b) (a − b) = a2 − b2 §134 o the Lilavati is translated rom Sanskrit as: Te square root o the sum o the squares o those legs is the diagonal. Te square root, extracted rom the difference o the squares o the diagonal and side is the upright; and that extracted rom the difference o the squares o the diagonal and upright, is the side.88
§135 that ollows is translated as: wice the product o two quantities, added to the square o their difference, will be the sum o their squares. Te product o their sum and difference will be the difference o their squares: as must be everywhere understood by the intelligent calculator.89
And this theorem came in or much discussion rom the 1790s when Playair rst wrote about it in his discussion o Davis’ translation o the Surya-Siddhanta. Now §135 is marked with two ootnotes: the one indicates that §135 is a stanza o six verses in the anustubh metre and the next importantly indicates that Ganesa the commentator on Bhaskara’sLilavati provides both an ‘algebraic and geometrical proo’ o the latter rule, the one marked as above (my labelling), and an algebraic demonstration o the rstmarked as above (my labelling). Colebrooke is not just translating rom Bhaskara II’s Lilavati: in the ootnotes he intercalates a translation o Ganesa’s commentary. Te latter demonstration is taken rom the Bija-Ganita §148; and it is in §147 that the rst o the rules is given and demonstrated. 90 Colebrooke renders the term Cshetragatopapatti as geometrical demonstration and Upapatti avyucta-criyayaas proo by algebra.91 We come to one o the geometrical demonstrations o rule labelled as given in the Bija-Ganita §148 and §149 o Bhaskara to which Colebrooke reers as such. §148: Example: ell me riend, the side, upright and hypotenuse in a [triangular] plane gure, in which the square-root o three lessthan the side, being lessened by one, is the difference between the upright and the hypotenuse.92 88 89 90 91 92
Ibid. Ibid.
C1817: 222–3. C1817: 59. C1817: 223.
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250
In modern language this could be translated as a - 3 − 1 = c − b, where Bhaskara immediately suggests taking c − b as 2. In this demonstration the difference between one o the sides (upright) and the hypotenuse is assumed as 2. (a) Te square o that added to one to which 3 is added: (2 + 1)2 + 3 = 12 – this is the side. (b) 122 = 144 – this is the difference between the squares o the hypotenuse and side (upright). By the rule the difference o the squares is equal to the product o the sum and difference Which means a2 − b2 = (a + b)(a − b).
It is in this context that here Bhaskara includes a proo o the rule, to which Colebrooke reers. Tis proo as is evident is based on a orm o reasoning that draws upon gures with particular dimensions. Te text then gives the square o 7 as 49 represented as below (Figure 5.1):
Figure 5.1
Te square a2.
From this square o 7 × 7 subtract a square o 5, which is 25. Tis gives the ollowing (Figure 5.2). We are lef with a remainder o 24. a − b = 2 and a + b=12 and the product consists o 24 equal cells (Figure 5.3).
Contextualizing Playfair and Colebrooke
5× 5
Figure 5.2
Te square a2 minus the square b2.
Figure 5.3
Te rectangle o sides a + b and b − a.
Te text reads: ‘thus it is demonstrated that the difference o the squares is equal to the product o the sum and the difference’.93 Te text then proceeds on the basis o this example to construct other Pythagorean triples. Similarly, another visual demonstration ollows or §149. §149 Rule: Te difference between the sum o the squares o two quantities whatsoever, and the square o their sum, is equal to twice their product; as in the case o two unknown quantities.94
Te demonstration is worked out on the basis o a particular case, and provides a procedure thus or any two sets o numbers. Colebrooke’s translation o Bhaskara’s demonstration reads: ‘For instance, let the quantities be 3 and 5. Teir squares are 9 and 25. Te square o their sum is 64. From this taking away the sum o the squares the remainder is 30.’ 95 And then in the 93 94 95
C1817: 223. C1817: 224. C1817: 30.
251
252
3×3
Figure 5.4
Te square a2.
5 ×5
Figure 5.5
Te square b2.
translation Bhaskara exhorts his reader to ‘See’ the illustration that ollows (see Figures 5.4–5.6). Tus (3 + 5)2 = 64 . . ., (a + b)2 From this subtract 32 + 52 . . . a2 + b2 Which makes 64 – 34 = 30 . . . (a + b)2 – (a2 + b2) Te lef-over square cells are seen to be equal to twice the product (Figure 5.7). Afer which Bhaskara concludes: ‘Here square compartments, equal to twice the product are apparent, and (the proposition) is proved.’96 We have here two cases o visual demonstration (Colebrooke calls them geometrical demonstrations) though in his translations he vacillates between the terms proos and demonstrations. But clearly both are demonstrations rom particular cases ormulated within the ramework o particular cases treated in a general way. Furthermore, Colebrooke brie y discusses two different demonstrations o the Pythagorean theorem in Bhaskara’sBija-Ganita (§146). Te rst o 96
C1817: 224.
Contextualizing Playfair and Colebrooke
8×8
Figure 5.6
Te square (a + b)2.
In other words, rom Figure 5.6, delete the sum o the squares: which is 3 × 3 and 5 × 5. 3
5×3
3 ×5
5
Figure 5.7
Te area (a + b)2 minus the squares a2 and b2 equals twice the product ab.
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B
20
15
A
C D
Figure 5.8
A right-angled triangle ABC and its height BD.
these demonstrations, we are reminded, is similar to Wallis’ demonstration that appeared in the treatise on angular sections. Colebrooke sets Wallis’ and Bhaskara’s demonstrations side by side, such that Bhaskara’s method is apprehended in Wallis’ idiom(Figure 5.8).97 Wallis Bhaskara In a rectangular triangle, C and D Using the same symbols or the sides and designate the sides and B the hypotenuse. segments, Bhaskara’s demonstration Te segments are χ and δ. B : C :: C : χ B : C :: C : χ B:D:D:δ B:D:D: δ Tereore Tereore C2 = B χ χ = C2/B D2 = Bδ δ = D2/B Tereore Tereore C2 + D2=(Bχ + Bδ) = B(χ + δ) = B2 B = χ + δ = C2/B + D2/B B 2 = C2 + D 2
We shall now try to illustrate Bhaskara’s procedure above as it appears in Colebrooke’s translation, but I shall adopt a contemporary orm o the argument. Te problem that Bhaskara poses in §146 o the Bija-Ganita is: ‘Say what is the hypotenuse in a plane gure, in which the side and upright 98 So20? are equal to 15 and And show the demonstration o thewhose received mode o composition’. consider a right-angled triangle ABC sides are 15 and 20 and rotate the gure as above. Drop a perpendicular to the side AC and let AD = χ and DA = δ. Now AC is the hypotenuse o the triangle ABC and BC and AD o triangles BCD and DBA respectively. 97 98
C1817: xvi–xvii. C1817: 220.
Contextualizing Playfair and Colebrooke
Bhaskara then posits the ratios: AC BC and AC AB = = BC CD AB AD χ=
(BC)2 and δ (AB)2 = AC
AC
(BC)2 (AB)2 Now (χ + δ) = AC + AC Or (AC)2 = (BC)2 + (AB)2
And thus the value o AC is computed, and rom this the value o BD.99 Tus the procedure is reasoned again or a particular case with the sides o 15 and 20, but clearly the procedure is applicable or any set o numbers that constitute the sides o a right-angled triangle. It needs to be pointed out here that Colebrooke highlights the act that Bhaskara ‘gives both modes o proo’ when discussing the solution o indeterminate problems involving two unknown quantities. Te instances Colebrooke has selected in his dissertation are ‘conspicuous’ as he says, or as pointed out earlier his method is to accentuate the contrast to destabilize as it were the then received picture within the binary typologies o the history o mathematics mentioned earlier.100 But the task is undertaken with a great deal o caution. Te next example chosen is that o indeterminate equations o the second degree, wherein, according to Colebrooke, Brahmagupta provided a general method, in addition to which he proposes rules to resolve special cases. It is well known that Bhaskara solved the equation ax2 + 1 = y2 or speci c values o the variable a. But Colebrooke went on to suggest that Bhaskara proposed a method to solve all indeterminate equations o the second degree that were ‘exactly the same’ as the method developed by Brouncker. In effect, Colebrooke appeared to be suggesting that Bhaskara’s method was generalizable, that he was aware o the problem and its ‘general use’, a eature or whose discovery modern Europe had to await the arrival o Euler on the stage o European mathematics.101 99 100 101
C1817: 220–1. C1817: xviii. A contemporary mathematical review o the solution o Pell’ s equation indicates that the ‘Indian or English method o solving the Pell equation is ound in Euler’s Algebra’. However, it is subsequently clari ed that Euler, and his Indian or English predecessors, assumed that the method always produced a solution, whereas the contemporary understanding is that i a solution existed the method would nd one. Further, Fermat had probably proved that there was a solution or each value oa, and the rst published proo was that o Lagrange (Lenstra 2002: 182).
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On reading o the early responses rom a French savant to the work o Colebrooke, it is possible to discern that Delambre or one uses a very ne comb in rebutting several o the points taken up by Colebrooke. While Colebrooke himsel does not draw a very ne distinction between the use o the terms ‘proo’ and ‘demonstration’ in his reading, he does distinguish between algebra and analysis; and as mentioned earlier he speci es wherein the Indian tradition could be characterized as an algebraic analysis. A study o the reception o Colebrooke’s translations o the works on Indian arithmetic and algebra is a matter or a separate study. Te curious question to be examined by such a study is that despite its canonical status in Western scholarship on the history o Indian mathematics and algebra, neither Colebrooke nor Davis ever insinuated that it was a tradition devoid o proo or demonstration. And yet, as the nineteenth-century historiography o Oriental mathematics evolved, a theory o the absence o proo would become one o its salient elements. Te strong criticism o Colebrooke’s work at the time was possibly provoked by Colebrooke’s method o taking up those demonstrations rom Indian mathematics or which equivalents existed in eighteenth-century European mathematics. Tis would have vitiated both the claims o novelty and srcinality, both very important eatures o the new sciences. Second, up to the end o the eighteenth century British Indologists still believed that they could discover the srcins o an Indian geometry and the later work o the Indologist G. Tibaut may be seen to be in continuity with that tradition. But by the end o the nineteenth century the binary typologies o the history o mathematics, that portrayed the West as geometric and the East as algebraic, were well in place in the standard picture.
Acknowledgements I thank the participants at the Workshop on the History and Historiography o Proos in Ancient raditions, Paris, or their questions, comments and suggestions, and more recently Karine Chemla or a very close reading o the text. Te usual disclaimer applies.
Bibliography Assayag, J., Lardinois, R. and Vidal, D. (1997) Orientalism and Anthropology: From Max Mueller to Louis Dumont. Pondy Papers in Social Sciences. Pondicherry.
Contextualizing Playfair and Colebrooke
Bailly, J.-S. (1775) Histoire de l’astronomie ancienne depuis son srcine jusqu’à . Paris. l’établissement de l’école d’Alexandrie Bailly, J.-S. (1787) raité de l’astronomie indienne et orientale. Paris. Banerji, H. C. (1927) Colebrooke’s ranslation of the Lilavati with notes by Haran Chandra Banerji, 2nd edn. Calcutta. Buckland, C. E. (1908) Dictionary of Indian Biography. New York. Burgess, J. (1893) ‘Notes on Hindu astronomy and our knowledge o it’, Journal of the Royal Asiatic Society: 717–61. Burrow, R. (1783) ‘Hints concerning the Observatory at Benaras’, in Warren Hastings Papers, ff. 263–76, British Museum. (Reprinted in Dharampal 1971: 94–112.) Charette, F. (1995) ‘Orientalisme et histoire des sciences: L’historiographie des sciences islamiques et hindoues, 1784–1900’, MA thesis, Department o History, University o Montreal. Colebrooke, H. . (1873) Miscellaneous Essays by H. . Colebrooke with Notes by E. B. Coswell, 2 vols. London. Colebrooke, . E. (1873) Te Life of H. . Colebrooke. London. Davis, S. (1790) ‘On the early astronomical computations o the Hindus’, Asiatic Researches 2: 225–87. Dharampal (ed.) (1971) Indian Science and echnology in the Eighteenth Century: Some Contemporary Accounts. Delhi. Filliozat, J. (1951) ‘L’Orientalisme et les sciences humaines’, Extrait du Bulletin de la Société des études Indochinoises, : 4. (1955) ‘France and Indology’,Bulletin of the Ramakrishna Mission Institute of Culture, ransactions No. 12. (1957) ‘Ancient relations between India and oreign astronomical systems’, Te ( – ): 1–8. Journal of Oriental Research – Madras, Heinz, B. (2000) Die Innenwelt der Mathematik: Zur Kultur und Praxis einer beweisenden Disziplin. Vienna. (2003) ‘When is a proo a proo’, Social Studies of Science33(6): 929–43. Holwell, J. Z. (1767) ‘An account o the Manner o Inoculating or the smallpox in the East Indies’, addressed to the President and Members o the College o Physicians in London. Inden, R. (1990) Imagining India. Oxord. Jami, C. (1995) ‘From Louis XIV’s Court to Kangxi’s Court: an institutional analysis East Asian Science: o the French mission to China (1688–1712) ’, inL. , ed. K. Hashimoto, C. Jami and Skar . Osaka: 493–9. radition andJesuit Beyond
Jones, W. (1790a) ‘On the antiquity o the Indian Zodiac’, Asiatic Researches 2: 289–306. Jones, W. (1790b) ‘A supplement to the essay on Indian chronology’, Asiatic Researches ii: 389–403. Kejariwal, O. P. (1988) Te Asiatic Society of Bengal and the Discovery of India’s Past 1784–1838. Oxord.
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258 Le Gentil, G. H. J. B. (1779–81) Voyages dans les mers de l’Inde, fait par ordre du roi, à l’occasion du passage du Vénus, sur le disque du soleil, le 6 Juin 1761 & le 3 du même mois 1769, 2 vols. Paris. (1785) ‘Mémoires sur l’srcine du zodiaque’, in Mémoires de M. Le Gentil de l’Académie des sciences, Publié dans le volume de cette Académie pour les
années 1784 et 1785. Lenstra, H. W. (2002) ‘Solving the Pell equation’, Notices of the AMS 49(2): 182–92. MacKenzie, D. (2001) Mechanizing Proof: Computing, Risk, and rust. Cambridge, MA. Montucla, J. E. (1799–1802) Histoire des mathématiques4 vols. Paris. (Republished by Librairie Scienti que et echnique Albert Blanchard.) e siècle: Praxis, utopie, Murr, S. (1986) ‘Les Jésuites et l’Inde au préanthropologie’, Revue de l’Université d’Ottawa56: 9–27. Playair, J. (1790) ‘Remarks on the astronomy o the Brahmins’,ransactions of the Royal Society of Edinburgh, , part : 135–92. (Reprinted in Dharampal 1971: 48–93.) Playair, J. (1792) ‘Questions and remarks on the astronomy o the Hindus’, Asiatic Researches iv: 151–5. Raina, D. (1999) ‘Nationalism, institutional science and the politics o knowledge: ancient Indian astronomy and mathematics in the landscape o French Enlightenment historiography’, PhD thesis, University o Göteborg. (2001a) ‘Jean-Sylvain Bailly’s antediluvian hypothesis: the relationship between the lumières and the British indologists’, able ronde:L’Orientalisme: un héritage en débat, Paris: La Villette, 18 May 2001. (2001b) ‘Disciplinary boundaries and civilisational encounter: the mathematics and astronomy o India in Delambre’sHistoire (1800–1820)’, Studies in History 17: 175–209. (2003) ‘Betwixt Jesuit and Enlightenment historiography: the context o Jean-Sylvain Bailly’s History of Indian Astronomy’, Revue d’Histoire des mathématiques 9: 253–306. Raina, D., and Habib, S. I. (1990) ‘Ramchundra’s treatise through the haze o the golden sunset: the aborted pedagogy’, Social Studies of Science20: 455–72. Raj, K. (2001) ‘Reashioning civilities, engineering trust: William Jones, Indian intermediaries and the production o reliable legal knowledge’, Studies in 17
History : 175–209. Rocher, R. (1989) ‘Te career o Rādhākānta arkavāgīśa, an eighteenth-century pandit in British employ’, Journal of the American Oriental Society 109: 627–33. Said, E. W. (1978) Orientalism. London. Sonnerat, P. (1782) Voyage aux Indes Orientales et à la Chine, fait par ordre du Roi, depuis 1774 jusqu’en 1781. Paris. Strachey, E. (1818) ‘On the early history o algebra’, Asiatic Researches 12: 160–85.
Contextualizing Playfair and Colebrooke
eltscher, K. (1995) India Inscribed: European and British Writing on India 1600–1800. Oxord. Vidal, D. (1997) ‘Max Müller and the theosophist: the other hal o Victorian orientalism’, in Orientalism and Anthropology: From Max Mueller to Louis Dumont, ed. J. Assayag, R. Lardinois and D. Vidal. Pondicherry: 18–27. Županov, I. G. (1993) ‘Aristocratic analogies and demotic descriptions in the seventeenth-century Madurai Mission’, Representations 41: 123–47.
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Overlooking mathematical justi cations in the Sanskrit tradition: the nuanced case o G. F. W. Tibaut
Introduction Until the 1990s, the historiography o Indian mathematics largely held that Indians did not use ‘proos’ in their mathematical texts. 1 Dhruv Raina has shown that this interpretation arose partly rom the act that during the second hal o the nineteenth century, the French mathematicians who analysed Indian astronomical and mathematical texts considered geometry to be the measure o mathematical activity.2 Te French mathematicians relied on the work o the English philologers o the previous generation, who considered the computational reasonings and algorithmic veri cations merely ‘practical’ and devoid o the rigour and prestige o a real logical and geometrical demonstration. Against this historiographical backdrop, the German philologer Georg Friedrich Wilhelm Tibaut (1848–1914) published the oldest known mathematical texts in Sanskrit, which are devoted only to geometry. Tese texts, śulbasūtras (sometimes called the sulvasūtras) contain treatises by different authors (Baudhāyana, Āpastamba, Kātyāyana and Mānava) and consider the geometry o the Vedic altar.3 Tese texts were written in the style typical o aphoristic sūtras between 600 and 200 . Tey were sometimes accompanied by later commentaries, the earliest o which may be assigned to roughly the thirteenth century. In order to understand the methods that he openly employed or this corpus o texts, Tibaut must be situated as a scholar. Tis analysis will ocus on Tibaut’s historiography o mathematics, especially on his perception o mathematical justi cations.
1 2 3
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Srinivas 1990; H1995. See Raina 1999: chapter . I will adopt the usual transliteration o Sanskritwords, which will be marked in italics, except or the word Veda, which is ound in English dictionaries.
Te Sanskrit tradition: the case of G. F. W. Tibaut
Tibaut’s intellectual b ackground Tibaut’s approach to the śulbasūtras combines what hal a century beore him had been two con icting traditions. As described by Raina and by Charette, Tibaut was equal parts acute philologer and scientist investigating the history o mathematics. A philologer
Tibaut trained according to the German model o a Sanskritist.4 Born in 1848 in Heidelberg, he studied Indology in Germany. His European career culminated when he lef or England in 1870 to work as an assistant or Max Müller’s edition o the Vedas. In 1875, he became Proessor o Sanskrit at Benares Sanskrit College. At this time, he produced his edition and studies o the śulbasūtras, the ocus o the present article.5 Aferwards, Tibaut spent the ollowing twenty years in India, teaching Sanskrit, publishing translations and editing numerous texts. With P. Griffith, he was responsible or the Benares Sanskrit Series, rom 1880 onwards. As a specialist in the study o the ritualistic mimām . sa school o philosophy and Sanskrit scholarly grammar, Tibaut made regular incursions into the history o mathematics and astronomy. Tibaut’s interest in mathematics and astronomy in part derives rom his interest in mimām . sa. Te authors o this school commented upon the ancillary parts o the Vedas (vedā˙nga) devoted to ritual. Te śulbasūtras can be ound in this auxiliary literature on the Vedas. As a result o having studied these texts, between 1875 and 1878, 6 Tibaut published several articles on Vedic mathematics and astronomy. Tese studies sparked his curiosity about the later traditions o astronomy and mathematics in the Indian subcontinent and the rst volume o the Benares Sanskrit Series, o which Tibaut was the general scienti c editor, was the Siddhāntatattvavivekao Bhat.t.a Kamalākara. Tis astronomical treatise written in the seventeenth century in Benares attempts to synthesize the reworkings o theoretical astronomy made by the astronomers under the patronage o Ulug Begh with the traditional Hindu siddhāntas.7 Tibaut’s next direct contribution to the history o mathematics and astronomy in India was a study on the medieval astronomical treatise the 4 5 6 7
Te ollowing paragraph rests mainly on Stache-Rosen 1990 . See Tibaut 1874, Tibaut 1875, Tibaut 1877a, Tibaut 1877b. Te last being a study o thejyotisavedā˙nga, in Tibaut 1878. ˙ See Minkowski 2001 andCESS, vol. 2: 21.
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Pañcasiddhāntao Varāhamihira. In 1888, he also edited and translated this treatise with S. Dvivedi and consequently entered into a heated debate with H. Jacobi on the latter’s attempt to date the Veda on the basis o descriptions o heavenly bodies in ancient texts. At the end o his lie, Tibaut published several syntheses o ancient Indian mathematics and astronomy.8 His main oeuvre, was not in the eldo history o science but a three-volume translation o one o the main mimām . sa texts: Śa˙nkarācārya’s commentary on the Vedāntasūtras, published in the Sacred Books of the East, the series initiated by his teacher Max Müller.9 Tibaut died in Berlin at the beginning o the First World War, in October 1914. Among the śulbasūtras, Tibaut ocused on Baudhāyana (c. 600 )10 and Āpastamba’s texts, occasionally examining Kātyāyana’sśulbapariśis.t.a. Tibaut noted the existence o the Mānavasulbasūtra but seems not to have had access to it.11 For his discussion o the text, Tibaut used Dvārakānātha Yajvan’s commentary on the Baudhāyana sulbasūtra and Rāma’s (f l. 1447/9) commentary on Kātyāyana’s text.12 Tibaut also occasionally quotes Kapardisvāmin’s (f l. beore 1250) commentary o Āpastamba.13 Tibaut’s introductory study o these texts shows that he was amiliar with the extant philological and historical literature on the subject o Indian mathematics and astronomy. However, Tibaut does not reer directly to any other scholars. Te only14work he acknowledges directly is A. C. Burnell’s catalogue o manuscripts. For instance, Tibaut quotes Colebrooke’s translation o Līlāvatī but does not reer to the work explicitly.15 Tibaut also reveals some general reading on the history o mathematics. For example, he implicitly reers to a large history o attempts to square the circle, but his sources are unknown. His approach to the texts shows the importance he ascribed to acute philological studies.16 Tibaut ofen emphasizes how important commentaries are or reading the treatises: ‘the sūtra-s themselves are o an
8 9 10 11
12 13 14 15 16
Tibaut 1899, Tibaut 1907. Tibaut 1904. Unless stated otherwise, all datesreer to the CESS. When no date is given, theCESS likewise gives no date. For general comments onthese texts, see Bag and Sen 1983, inCESS, vol 1: 50; vol 2: 30; vol 4: 252. For the portions o Dvārakānātha’s and Venkateśvara’s commentaries on Baudhāyana’s treatise, see Delire 2002. Tibaut 1875: 3. Tibaut 1877: 75. Tibaut 1875: 3. Tibaut 1875: 61. See or instance Tibaut 1874: 75–6 and his long discussions on the translations ovr. ddha.
Te Sanskrit tradition: the case of G. F. W. Tibaut
enigmatical shortness . . . but the commentaries leave no doubt about the real meaning’.17 Te importance o the commentary is also underlined in his introduction o the Pañcasiddhānta: ‘Commentaries can be hardly done without in the case o any Sanskrit astronomical work . . .’18 However, Tibaut also remarks that because they were composed much later than the treatises, such commentaries should be taken with critical distance: rustworthy guides as they are in the greater number o cases, their tendency o sacri cing geometrical constructions to numerical calculation, their excessive ondness, as it might be styled, o doing sums renders them sometimes entirely misleading.19
Indeed, Tibaut illustrated some o the commentaries’ ‘mis-readings’ and devoted an entire paragraph o his 1875 article to this topic. Tibaut explained that he had ocused on commentaries to read the treatises but disregarded what was evidently their own input into the texts. Tibaut’s method o openly discarding the speci c mathematical contents o commentaries is crucial here. Indeed, according to the best evidence, the tradition o ‘discussions on the validity o procedures’ appears in only the 20
medieval and modern commentaries. rue,the the texts commentaries described mathematics o a period dierent than upon which they commented. However, Tibaut valued his own reconstructions o the śulbasūtras proos more than the ones given by commentaries. Te quote given above shows how Tibaut implicitly values geometrical reasoning over arithmetical arguments, a act to which we will return later. It is also possible that the omission o mathematical justi cations rom the narrative o the history o mathematics in India concerns not only the conception o what counts as proo but also concerns the conception o what counts as a mathematical text. For Tibaut, the only real mathematical text was the treatise, and consequently commentaries were read or clari cation but not considered or the mathematics they put orward. In contradiction to what has been underlined here, the same 1875 article sometimes included commentators’ procedures, precisely because the method they give is ‘purely geometrical and perectly satisactory’. 21 17 18 19 20
21
Tibaut 1874: 18. Tibaut 1888: v. Tibaut 1875: 61–2. Tese are discussed, in a speci c case, in the other chapter in this volume I have written; see Chapter 14. Tis concludes a description o how to transorm a square into a rectangle as described by . Dvārakantha in Tibaut 1875: 27–8.
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Tus there was a discrepancy between Tibaut’s statements concerning his methodology and his philological practice. Tibaut’s conception o the Sanskrit scholarly tradition and texts is also contradictory. He alternates between a vision o a homogeneous and a historical Indian society and culture and the subtleties demanded by the philological study o Sanskrit texts. In 1884, as Principal o Benares Sanskrit College (a position to which he had been appointed in 1879), Tibaut entered a heated debate with Bapu Pramadadas Mitra, one o the Sanskrit tutors o the college, on the question o the methodology o scholarly Sanskrit pandits. Always respectul to the pandits who helped him in his work, Tibaut always mentioned their contributions in his publications. Nonetheless, Tibaut openly advocated a ‘Europeanization’ o Sanskrit studies in Benares and sparked a controversy about the need or pandits to learn English and the history o linguistics and literature. Tibaut despaired o an absence o historical perspective in pandits’ reasonings – an absence which led them ofen to be too reverent towards the past.22 Indeed, he ofen criticized commentators or reading their own methods and practices into the text, regardless o the treatises’ srcinal intentions. His concern or history then ought to have led him to consider the dierent mathematical and astronomical texts as evidence o an evolution. However, although he was a promoter o history, this did not prevent him rom making his own sweeping generalizations on all the texts o the Hindu tradition in astronomy and mathematics. He writes in the introduction o the Pañcasiddhānta: these works [astronomical treatises by Brahmagupta and Bhāskarācarya] claim or themselves direct or derived inallibility, propound their doctrines in a calmly dogmatic tone, and either pay no attention whatever to views diverging rom their own or else reer to such only occasionally, and mostly in the tone o contemptuous depreciation.23
Trough his belie in a contemptuous arrogance on the part o the writers, Tibaut implicitly denies the treatises any claim or reasonable mathematical as we will see later. Tibaut attributed part o the clumsinessjusti whichcations, he criticized to their old age:
22 23
See Dalmia 1996: 328–30. Tibaut 1888: vii. I am setting aside here the act that he argues in this introduction or a Greek srcin o Indian astronomy. Te square brackets indicate the present author’s addenda or the sake o clarity.
Te Sanskrit tradition: the case of G. F. W. Tibaut Besides the quaint and clumsy terminology ofen employed or the expression o very simple operations (. . .) is another proo or the high antiquity o these rules o the cord, and separates them by a wide gul rom the products o later Indian science with their abstract and re ned terms.24
Afer claiming that the treatises had a dogmatic nature, Tibaut extends this to the whole o ‘Hindu literature’: Te astronomical writers . . . therein only exempliy a general mental tendency which displays itsel in almost every department o Hindu Literature; but mere dogmatic assertion appears more than ordinarily misplaced in an exact science like astronomy . . .25
Tibaut does not seem to struggle with de nitions o science, mathematics or astronomy, nor does he discuss his competency as a philologer in undertaking such a study. In act, Tibaut clearly states that subtle philology is not required or mathematical texts. He thus writes at the beginning o the Pañcasiddhānta: texts o purely mathematical or astronomical contents may, without great disadvantages, be submitted to a much rougher and bolder treatment than texts o other kinds. What interests us in these works, is almost exclusively their matter, not either their general style or the particular words employed, and the peculiar nature o the subject ofen enables us to restore with nearly absolute certainty the general meaning o passages the single words o which are past trustworthy emendation.26
Tis ‘rougher and bolder treatment’ is evident, or instance, in his philologically accurate but somewhat clumsy translation o technical vocabulary. He thus translates dīrghacaturaśra (literally ‘oblong quadrilateral’) vari27 ously; it is at some times a ‘rectangular oblong’, and at others an ‘oblong’. Te expression ‘rectangular oblong’ is quite strange. Indeed, i the purpose is to underline the act that it is elongated, then why repeat the idea? Te rst o Tibaut’s translations seems to aim at expressing the act that a dīrghacaturaśra has right angles, but the idea o orthogonality is never explicit in the Sanskrit works used here, or even in later literature. Tibaut’s translation, then, not literal but the coloured by verses his own ideatreatises, o whatthea dīrghacaturaśra is. is Similarly, he calls rules and o the Sanskrit sūtras, ‘proposition(s)’, which gives a clue to what he expects o a 24 25 26 27
Tibaut 1875: 60. Tibaut 1888: vii. Tibaut 1888: v. See or instance Tibaut 1875: 6.
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scienti c text, and thus also an inkling about what kind o scienti c text he suspected spawned the śulbasūtras. Tibaut’s historiography of science
For Tibaut, ‘true science’ did not have a practical bent. In this sense, the science embodied in the śulbas, which he considered motivated by a practical religious purpose, is ‘primitive’: Te way in which the sūtrakāra-s [those who compose treatises] ound the cases enumerated above, must o course be imagined as a very primitive one. Nothing in the sūtra-s [the aphorisms with which treatises are composed] would justiy the assumption that they were expert in long calculations.28
However, he considered the knowledge worthwhile especially because it was geometrical: It certainly is a matter o some interest to see the old ācārya-s [masters] attempting to solve this problem [squaring o the circle], which has since haunted so m[an]y unquiet minds. It is true the motives leading them to the investigation were vastly dierent rom those o their ollowers in this arduous task. Teirs was not the disinterested love of research which distinguishes true science , nor the inordinate craving o undisciplined theearnest solution o riddles which reason us all cannot be solved; theirs wasminds simplyorthe desire to render their sacritells ce in its particulars acceptable to the gods, and to deserve the boons which the gods coner in return upon the aithul and conscientious worshipper.29
Or again: . . . we must remember that they were interested in geometrical truths only as ar as they were o practical use, and that they accordingly gave to them the most practical expression.30
Conversely, the practical aspect o these primitive mathematics explains why the methods they used were geometrical: It is true that the exclusively practical purpose o the Śulvasūtra-s necessitated in some way the employment o practical, that means in this case, geometrical terms, . . .31
28 29 30 31
Tibaut 1875: 17. Tibaut 1875: 33. Te emphasis is mine. Tibaut 1875: 9. Tibaut 1875: 61.
Te Sanskrit tradition: the case of G. F. W. Tibaut
Tis geometrical basis distinguished the śulbasūtras rom medieval or classical Indian mathematical treatises. Once again, Tibaut took this occasion to show his preerence or geometry over arithmetic: Clumsy and ungainly as these old sūtra-s undoubtedly are, they have at least the advantage o dealing with geometrical operations in really geometrical terms, and are in this point superior to the treatment o geometrical questions which we nd in the Līlāvatī and similar works.32
As is made clear rom the above quotation, Tibaut was a presentist historian o science who possessed a set o criteria which enabled him to judge the contents and the orm o ancient texts. In another striking instance, Tibaut gives us a clue that Euclid is one o his reerences. Commenting on rules to make a new square o which the area is the sum or the dierence o two known squares, Tibaut states in the middle o his own translation o Baudhāyana’sśulbasūtras: Concerning the methods, which the Śulvasūtras teach or caturasrasamāsa (sum o squares) and caturasranirhāra(subtraction o squares), I will only remark that they are perectly legitimate; they are at the bottom the same which Euclid employs.33
Contemptuous as he may be o the state o Indian mathematics, Tibaut did not believe that the śulbasūtras were in uenced by Greek geometry.34 For Tibaut, history o mathematics ought to reconstruct the entire deductive process rom the srcin o an idea to the way it was justi ed. Although later commentaries may include some useul inormation, they do not give us the key to understanding how these ideas were developed at the time when the treatises were composed. Tis lack o inormation provoked Tibaut to complain about Indian astronomical and mathematical texts. Tibaut clearly considered the texts to have been arranged haphazardly because the order o the rules do not obey generative logic. He thus de ned his task: ‘I shall extract and ully explain the most important sūtra-s (. . .) and so try to exhibit in some systematic order the knowledge embodied in these ancient sacri cial tracts.’35 Here, Tibaut assumed that these works – not ‘tracts’ (presumably with derogatory connotations) – are not treatises clear andbutsystematic. Further, Tibaut elt the need to disentangle (‘extract’) the knowledge they contain. 32 33 34 35
Tibaut 1875: 60. Tibaut 1877: 76. ranslations within brackets are mine. Tibaut 1875: 4. Tis however was still being discussed as late as Staal 1999 . Tibaut 1875: 5.
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In his view, this knowledge might be quite remarkable but it was ill presented. Tus commenting a couple years later on the Vedān.gajyotis.a, he remarked: Te irst obstacle in our way is o course the style o the treatise itsel with its enigmatical shortness o expression, its strange archaic orms and its utter want of connection between the single verses.36
He thus sometimes remarked where the rules should have been placed according to his logic. All the various texts o the śulbasūtras start by describing how to construct a square, particularly how to make a square rom a rectangle. However, Tibaut objected: ‘their [the rules or making a square rom a rectangle] right place is here, afer the general propositions about the diago37 Consequently, nal o squares and oblongs, upon which they are ounded’. Tibaut considered the śulbasūtras as a single general body o text and selected the scattered pieces o the process he hoped to reconstruct rom among all the sūtras composed by various authors. At the same time, he distinguished the dierent authors o the śulbasūtras and repeatedly insisted that Āpastamba is more ‘practical’ than Baudhāyana, whom he preerred. For instance, an example o his method: Baudhāyana does not give the numbers expressing the length o the diagonals o his oblongs or the hypotenuses o the rectangular triangles, and I subjoin thereore some rules rom Āpastamba, which supply this want, while they show at the same time the practical use, to which the knowledge embodied in Baudhāyana’s sūtra could be turned.38
When alternating among several authors was insufficient or his purposes, Tibaut supplied his own presuppositions. Indeed, Tibaut peppered his text with such reconstructions: Te authors o the sūtra-s do not give us any hint as to the way in which they ound their proposition regarding the diagonal o a square; but we may suppose . . . Te question arises: how did Baudhāyana or Āpastamba or whoever may have the merit o the rst investigation, nd this value? . . . I suppose that they arrived at their result by.the ollowing method which or the exact reached . . Baudhāyana does not stateaccounts at the outset what thedegree shape o o accuracy his wheelthey will 39 be, but rom the result o his rules we may conclude his intention . . .
36 37 38 39
Tibaut 1877: 411; the emphasis is mine. Tibaut 1875: 28. Tibaut 1875: 12. Tibaut 1875: 11, 18, 49.
Te Sanskrit tradition: the case of G. F. W. Tibaut
Because he had an acute idea o what was logically necessary, Tibaut thus had a clear idea o what was su cient and insu cient or reconstructing the processes. As a result, Tibaut did not deem the arithmetical reasoning o Dvārakānātha adequate evidence o mathematical reasoning. Te misunderstandings on which Tibaut’s judgements rest are evident. For him, astronomical and mathematical texts should be constructed logically and clearly, with all propositions regularly demonstrated. Tis presumption compelled him to overlook what he surely must have known rom his amiliarity with Sanskrit scholarly texts: the elaborate character o a sūtra – marked by the diverse readings that one can extract rom it – enjoyed a long Sanskrit philological tradition. In other words, when a commentator extracts a new reading rom one or several sūtras, he demonstrates the ruitulness o the sūtras. Te commentator does not aim to retrieve a univocal singular meaning but on the contrary underline the multiple readings the sūtra can generate. Additionally, as Tibaut rightly underlined, geometrical reasoning represented no special landmark o correctness in reasoning to medieval Indian authors. Because o these expectations and misunderstandings Tibaut was unable to nd the mathematical justi cations that maybe were in these texts. Let us thus look more closely at the type o reconstruction that Tibaut employed, particularly in the case o proos.
Practices and readings in the history of science It is telling that the word ‘proo’ is used more ofen by Tibaut in relation to philological reasonings than in relation to mathematics. Tus, as we have seen above, the word is used to indicate that the clumsiness o the vocabulary establishes the śulbasūtras’ antiquity. No mathematical justi cations in the śulbasūtras
However, or Tibaut, Baudhāyana and probably other ‘abstractly bent’ treatise writers doubtlessly wanted to justiy their procedures. More ofen than not, these authors did not disclose their modes o justi cation. Tus, when the authors are silent, Tibaut developed ctional historical procedures. For instance: Te authors o the sūtra-s do not give us any hint as to the way in which they ound their proposition regarding the diagonal o a square [e.g. the Pythagorean proposition in a square]; but we may suppose that they, too, were observant o
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270 the act that the square on the diagonal is divided by its own diagonals into our triangles, one o which is equal to hal the rst square. Tis is at the same time an immediately convincing proo o the Pythagorean proposition as ar as squares or equilateral rectangular triangles are concerned . . . But how did the sūtrakāra-s [composers o treatises] satisy themselves o the general truth o their second proposition regarding the diagonal o rectangular oblongs? Here there was no such simple diagram as that which demonstrates the truth o the proposition regarding the diagonal o the square, and other means o proo had to be devised. 40
Tibaut thus implied that diagrams were used to ‘show’ the reasoning literally and thus ‘prove’ it. Tis method seems to hint that authors o the medieval period o Sanskrit mathematics could have had some sort o geometrical justi cation.41 Concerning Āpastamba’s methods o constructing re altars, which was based on known Pythagorean triplets, Tibaut stated: In this manner Āpastamba turns the Pythagorean triangles known to him to practical use . . . but afer all Baudhāyana’s way o mentioning these triangles as proving his proposition about the diagonal o an oblong is more judicious. It was no practical want which could have given the impulse to such a research [on how to measure and construct the sides and diagonals o rectangles] – or right angles could be drawn as soon as one o the vijñeya [determined] oblongs (or instance that o 3, 4, 5) was known – but the want o some mathematical justi cations which might 42
establish a rm conviction o the truth o theproposition.
So, in both cases, Tibaut represented the existence and knowledge o several Pythagorean triplets as the result o not having any mathematical justi cation or the Pythagorean Teorem. Tibaut proceeded to use this act as a criterion by which to judge both Āpastamba’s and Baudhāyana’s use o Pythagorean triplets. Tibaut’s search or an appropriate geometrical mathematical justi cation in the śulbasūtras may have made him overlook a striking phenomenon. wo different rules for the same result
Indeed, Tibaut underlined that several algorithms are occasionally given in order to obtain the same result. Tis redundancy puzzled him at times. 40 41
42
Tibaut 1875: 11–12. See Keller 2005. Bhāskara’s commentary on theĀryabhat. īya was not published during Tibaut’s lietime, but I sometimes suspect that either he or a pandit with whom he worked had read it. Te discussion onvis.amacaturaśra and samacaturaśra, in Tibaut 1875: 10, thus echoes Bhāskara I’s discussion on verse 3 o Chapter 2 o theĀryabhat. īya. Tibaut’s conception o geometrical proo is similar to Bhāskara’s as well. Tibaut 1875: 17.
Te Sanskrit tradition: the case of G. F. W. Tibaut
For instance, Tibaut examined the many various caturaśrakaran.a – methods to construct a square – given by different authors. 43 Āpastamba, Baudhāyana and Kātyāyana each gave two methods to accomplish this task. I will not expound these methods here; they have been explained amply and clearly elsewhere.44 Tibaut also remarked that in some cases, Baudhāyana gives a rule and its reverse, although the reverse cannot be grounded in geometry. Such is the case with the procedure to turn a circle into a square: Considering this rule closer, we nd that it is nothing but the reverse o the rule or turning a square into a circle. It is clear, however, that the steps taken according to this latter rule could not be traced back by means o a geometrical construction, or i we have a circle given to us, nothing indicates what part o the diameter is to be taken as the atiśayat. r. tīya (i.e. the segment o the diameter which is outside o the square).45
I am no specialist inśulba geometry and do not know i we should see the doubling o procedures and inverting o procedures as some sort o ‘proos’, but at the very least they can be considered eorts to convince the reader that the procedures were correct. Te necessity within theśulbasūtras to convince and to veriy has ofen been noted in the secondary literature, but has never ully or precisely studied.46 Tibaut, although puzzled by the act, never addressed this topic. Similarly, later historians o mathematics have noted that commentators on theśulbasūtras sought to veriy the procedures while setting aside the idea o a regular demonstration in these texts. Tus Delire notes that Dvārakānātha used arithmetical computations as an easy 47 Te use method o veri cation (in this case o the Pythagorean Teorem). o two separate procedures to arrive at the same result, as argued in another chapter in this volume,48 could have been a way o mathematically veriying the correctness o an algorithm – an interpretation that did not occur to Tibaut.
43 44
45 46 47 48
Tibaut 1875: 28–30. Tibaut 1875: 28–30; Bag and Sen 1983 inCESS, vol. 1; Datta 1993: 55–62; and nally Delire 2002: 75–7. Tibaut 1875: 35. See or instance Datta 1993: 50–1. Delire 2002: 129. See Keller, Chapter 14, this volume.
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Conclusion Tibaut, as we have thus seen, embodied contradictions. On the one hand, he swept aside the Sanskrit literary tradition and criticized its concise sūtras as obscure, dogmatic and ollowing no logic whatsoever. On the other hand, as an acute philologer, he produced nuanced studies on the dierences among the approaches o dierent authors. Trough his naive assumption o a practical mind o the ‘Hindu astronomers’, his ruitless search or proper visual demonstrations in an algorithmic tradition, and a disregard o commentaries in avour o the treatises, Tibaut envisioned a tradition o mathematics in India blind to the logic that could have been used to justiy the algorithms which he studied. Such arguments could have been perceived through the case o the ‘doubled’ procedures in the śulbasūtras, and maybe even through the arithmetical readings o these geometrical texts ound in later commentaries.
Acknowledgement I would like to thank K. Chemla and M. Ross or their close reading o this article. Tey have considerably helped in improving it.
Bibliography CESS (1970–94) Census of the Exact Sciences in Sanskrit, ed. D. Pingree, 5 vol. Philadelphia, PA. Dalmia (1996) ‘Sanskrit scholars and pandits o the old school: Te Benares Sanskrit College and the constitution o authority in the late nineteenth century’, Journal of Indian Philosophy 24: 321–37. Datta, B. (1993) Ancient Hindu Geometry: Te Science of the Sulba. New Delhi. Delire, J. M. (2002) ‘Vers une édition critique des śulbadīpikā et śulbamīmām.sā commentaires du Baudhāyana śulbasūtra: Contribution à l’histoire des mathématiques Sanskrites’, PhD thesis, Université Libre de Bruxelles. Keller, A. (2005a) ‘Making diagrams speak, in Bhāskara I’s commentary on the Āryabhat. īya’, Historia Mathematica32: 275–302. Minkowski, C. (2001) ‘Te pandit as public intellectual: the controversy over virodha or inconsistency in the astronomical sciences’, in Te Pandit: raditional Scholarship in India, ed. A. Michaels. New Delhi: 79–96. Raina, D. (1999) ‘Nationalism, institutional science and the politics o knowledge; ancient Indian astronomy and mathematics in the landscape o French Enlightenment historiography’, PhD thesis, University o Göteborg.
Te Sanskrit tradition: the case of G. F. W. Tibaut Srinivas, M. D. (1990) ‘Te methodology o Indian mathematics and its contemporary relevance’, in History of Science and echnology in India, ed. S. Prakashan, vol. II, Chapter 2. New Delhi. Staal, F. (1999) ‘Greek and Vedic geometry’, Journal of Indian Philosophy 27: 105–27. Stache-Rosen, V. (1990) German Indologists: Biographies of Scholars in Indian , 2nd edn. Studies Writing in German, with a Summary of Indology in German New Delhi. Tibaut, G. F. (1874). ‘Baudhayānaśulbasūtra’ , in Te Pan.d. it. Reprinted and re-edited in Chattopadhyaya, D. (1984) Mathematics in the Making in Ancient India. Delhi. (Page numbers reer to this edition.) (1875) ‘On the Sulvasutras’, inJournal of the Asiatic Society of Bengal. Reprinted and re-edited in Chattopadhyaya, D. (1984) Mathematics in the Making in Ancient India. Delhi. (Page numbers reer to this edition.) (1877a) ‘Baudhayānaśulbasūtra’, in Te Pan.d. it. Reprinted and re-edited in Chattopadhyaya, D. (1984) Mathematics in the Making in Ancient India. Delhi. (Page numbers reer to this edition.) (1877b) ‘Contributions to the explanation o the Jyotishavedān.ga’, Journal of the Asiatic Society of Bengal 46: 411–37. (1880) ‘On the Sūryaprajapti’, Journal of the Asiatic Society of Bengal 49: 107–27 and 181–206. (1882) ‘Katyayana śulbapariśis. t. a with the commentary o Ràma son o Sùryadasa’, Te Pan.d. it n.s. 4: 94–103, 328–39, 382–9 and 487–91. (1884) ‘Notes rom Varāha Mihira’s Panchasiddhāntikā’, Journal of the Asiatic Society of Bengal 53: 259–93. (1885) ‘Te number o stars constituting the several Naks.atras according to Brahmagupta and Vriddha-Garga’, Indian Antiquary 14: 43–5. (1894) ‘On the hypothesis o the Babylonian srcin o the so-called lunar zodiac’, Journal of the Asiatic Society of Bengal 63: 144–63. (1895) ‘On some recent attempts to determine the antiquity o Vedic civilization’, Indian Antiquary 24: 85–100. (1899) Astronomie, Astrologie und Mathematik. Strasburg. (1904) Vedānta sūtras with the commentary by Sha˙ nkarācarya, translated by George Tibaut. Delhi. (1907) ‘Indian astronomy’, Indian Tought 1: 81–96, 313–34 and 422–33. ͂
Tibaut, G. F.vol. and Dvivedi (1888) Pañcasiddhāntika Studies . Varanasi. (Reprinted 1968.). Chowkhamba Sanskrit
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Te logical Greek versus the imaginative Oriental: on the historiography o ‘non-Western’ mathematics during the period 1820–1920 ç
What makes Greek mathematics distinctive? In 1841, in an essay–review o Jean Jacques Sédillot’s (1777–1832) partial translation o a comprehensive thirteenth-century Arabic treatise on spherical astronomy and instrumentation written or the use o practical astronomers, and published posthumously by his son Louis Amélie (1808–75) in 1834–5 under the title raité des instruments astronomiques des Arabes, the French physicist Jean-Baptiste Biot (1774–1862) made the ollowing boldsounding statement: One nds [in this book] renewed veidence or this peculiar habit o mind, ollowing which the Arabs, as the Chinese and Hindus, limited their scienti c writings to the statement o without a series o rules, which, once given, ought only be veri edbetween by their applications, requiring any logical demonstration or to connections them: this gives those Oriental nations a remarkable character o dissimilarity, I would even add o intellectual ineriority, comparatively to the Greeks, with whom any proposition is established by reasoning, and generates logically deduced consequences.1
Apart rom the very ill-ounded nature o Biot’s judgement – which, incidentally, is contradicted on the very next page when he concedes that the book under review is not a representative work o Arabic astronomy, but rather a practical treatise or ‘vulgar’ use – this is nonetheless a clear ormulation o the idea that is at the core o the present investigation. For sure, such an opinion was not new. But the undeviating boldness and precision o Biot’s statement is really remarkable. I will thus take it as the starting point o my inquiry into the historiography o the mathematical demonstration 1
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‘. . . on y trouve une nouvelle preuve de cette singulière habitude deesprit, l’ en vertu de laquelle les Arabes, comme les Chinois et les Hindous, bornaient leurs compositions scienti ques à l’exposition d’une suite de règles, qui, une ois posées, devaient se véri er par leurs applications mêmes, sans besoin de démonstration logique, ni de connexion entre elles: ce qui donne à ces nations orientales un caractère remarquable de dissemblance, et j’ajouterai d’inériorité intellectuelle, comparativement aux Grecs, chez lesquels toute proposition s’établit par raisonnement, et engendre des conséquences logiquement déduites.’ Biot 1841 : 674–5.
Te logical Greek versus the imaginative Oriental
by labelling it the ‘orthright ormulation’ o the ideology under scrutiny. But a ew words on Biot are in order here to put his essay–review in context. Already in 1834, L. A. Sédillot had stridently claimed the srcinality o Arabic science, basing his argument on his alleged nding that a tenthcentury Arab astronomer had discovered the third inequality o the moon, 600 years beore ycho Brahe. Biot soon became a passionate opponent o Sédillot in an unending debate that occupied the Paris Academy o Science or more than 40 years. Biot, who in his polemic pieces against Sédillot revealed a prooundly anti-Arab ideology, was more candid with regard to Chinese and Indian science, about which he wrote numerous essays collected at the end o his lie in his Études sur l’astronomie indienne et sur l’astronomie chinoise (Paris, 1862). His son Edouard (1803–50), who had abandoned a liberal career or the study o sinology, was probably the rst European who, afer the Jesuits, made available new sources on Chinese mathematics; he published three papers on this topic between 1835 and 1841. ogether with K. L. Biernatzki’s amous paper on Chinese arithmetic and algebra printed in Crelle’s Journal ür reine und angewandte Mathematik in 1856 (and based entirely on various newspaper articles by the Protestant missionary in China Alexander Wylie), E. Biot’s contributions constituted the very ew ragments o Chinese mathematics available toEuropean histo2
rians until the beginning o the twentieth century.For Indian mathematics, Biot senior could rely on the widely available publications o British Sanskritists such as Henry Tomas Colebrooke (1765–1837), as well as on an increasing secondary literature based on them. Tis, o course, put J. B. Biot in a position o authority to judge Oriental science. Beore examining in more details the contexts and the evolution o the idea so precisely enunciated by Biot, let us contrast it with the view o a German historian o mathematics, Siegmund Günther (1848–1923), who, in 1908, nicely summarized the researches o the second hal o the nineteenth century on the matter. In a chapter devoted to Indian mathematics, Günther wrote the ollowing: But this [Indian] mathematics has such a peculiar character, that a study thereo is assured to guarantee the highest lure. In particular, one can only be ascinated by the undamental opposition between the Indian and Greek ways o thinking and o looking at things. Te Greek is – with a ew exceptions con rming the rule – a rigid synthetician, whose emphasis lies ully on rigorous demonstrations and who lives so much in spatial considerations that he will almost invariably attempt to cloth even arithmetical things into geometrical garments. Conversely, the Indian, being exceptionally gifed or everything computational, has very little appeal to 2
Martzloff 1997: 4–5.
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ç demonstrations: ‘look at the gure’ he says, allowing nothing but illustrative demonstrations [Anschauungsbeweise], whereas he could not have any eeling or the impressive but ofen awkward efforts o a Euclid or an Archimedes to really impose on reluctant [readers] the conviction o the validity o a theorem.3
Günther’s judgement is obviously more respectul and nuanced than that o Biot. I shall analyse the genealogy o the ideas expressed by Günther in the second hal o this chapter. First it is necessary to proceed towards the source o Europe’s knowledge o Indian mathematics. Te efforts o Strachey (1813) and aylor (1816) or making Bhāskara’s mathematical works available in English translation were very soon rendered obsolete by Colebrooke’s authoritative annotated translation o the mathematical parts o the works o Bhāskara and Brahmagupta in 1817. Te same year, the Scottish mathematician John Playair (1748–1819), who had been noted or his interest in the history o Indian astronomy, contributed an essay–review o Colebrooke to the Edinburgh Review. Playair noted the absence o demonstrations in the Lilavati and the Bīja-Gan.ita, but acknowledged that Bhāskara’s feenthand sixteenth-century commentators, such as Ganeśa, supplied demonstrations o the rules in several instances. He had to concede, however, that those occasional demonstrations were ‘ofen obscure, rom the want o reerence to a gure; or, though the gure be constructed on the margin, there is no reerence to it by letters’.4 Afer having presented a survey o the most important results achieved by Bhāskara, Playair made the ollowing observation: But in the midst o these curious results, there is a subject o regret that almost continually presents itsel. When such rules are laid down as the preceding, they are usually given without any analysis whatever, and even without any synthetic demonstration, so that the means by which the knowledge was obtained, remains quite unknown . . . In consequence o this, a mystery still hangs over the mathematical 3
4
‘Und doch ist diese Mathematik von so auszeichnender Eigenart, daß die Beschäfigung mit ihr den höchsten Reiz gewähren muß. Insonderheit esselt den Beschauer der grundsätzliche Gegensatz zwischen indischer und griechischer Denk- und Betrachtungsweise. Der Grieche ist – und die wenigen Ausnahmen bestätigen nur die Regel – strenger Synthetiker, der au rigorose Beweisührung das größte Gewicht legt und so durchaus in räumlichen Vorstellungen lebt, daß er selbst arithmetische Dinge ast ausschließlich in ein geometrisches Gewand zu kleiden bestrebt ist. Umgekehrt liegt dem ür alles Rechnerische ausnehmend beähigten Inder sehr wenig an der Demonstration; “siehe die Figur” sagt er und läßt keine anderen als Anschauungsbeweise zu, während er ür die imponierenden, aber of unbehil ichen Anstrengungen eines Euclides und Archimedes, die Überzeugung von der Richtigkeit eines Satzes örmlich einem Widerstrebenden auzuzwingen, gar keinen Sinn haben konnte.’ Günther 1908: 178. Playair 1817: 158.
Te logical Greek versus the imaginative Oriental knowledge o the East; and it is much to be eared that the means o removing it no longer exist.5
Playair regretted the absence o demonstrations, because he mainly expected them to illuminate the mechanisms o mathematical discovery among ancient authors. His interest in the innate heuristic patterns o mathematical creation thus stands in remarkable contrast to the usual strict concern orresults, which is characteristic o most nineteenth-century writings on history o mathematics (however naive it might be to hope that demonstrations would necessarily provide clues or understanding the underlying patterns o discovery). Concerning a particular geometrical theorem, he urther remarks that it ‘is demonstrated in a very ingenious and palpable manner, not altogether according to the rigour o the Greek geometry, but abundantly satisactory to those who are pleased with an argument when it is sound, though it be not dressed in the costume o science’.6 Another proo he sees as revealing ‘ingenious and simple’ reasoning that must stem rom ‘a system o geometrical demonstration that was not very re ned, or very scrupulous about 7 But even in those cases when demintroducing mechanical considerations’. onstration was wanting, Playair nevertheless believed that there existed an srcinal procedure o demonstration, no longer extant.8 In passing we must note the belie expressed by Playair that science, as or every other civilizational aspect o India, was ‘immoveable’ and deprived o progress.
Tree comparative views on Greek and Oriental mathematics: Hankel, Cantor and Zeuthen I now come to the major part o my inquiry, in which I offer a detailed analysis o the comparative views o three eminent historians o mathematics, Hankel, Cantor and Zeuthen, on Greek versus ‘Oriental’ (Indian, Chinese, Islamic) mathematics.
5 6 7 8
Playair 1817: 151. Playair 1817: 159–60. Playair 1817: 160. See on p. 159 the remarks on the theorem that in a circumscribed [this condition is not speci ed in the Sanskrit text] quadrilateral with sides a, b, c, d and diagonals g, h we have ac + bd = gh, a theorem ‘by no means very easy to be demonstrated’ and which ‘argues or a very extensive knowledge o elementary trigonometry, and such as is by no means easily acquired’.
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Contexts and predecessors
In France, the geometer Michel Chasles (1793–1880) and the exiled Italian mathematician Guglielmo Libri (1803–69) stirred interest in the history o mathematics among ellow mathematicians. Te rst published in 1837 a remarkable book entitled Aperçu historique sur l’srcine et le développement des méthodes en Géométrie, particulièrement de celles qui se rapportent à la Géométrie moderne, in which historical studies sought to inorm and inspire the renewal o modern geometry. Te second, whose scienti c contributions are today completely orgotten, united patriotic eelings with a liberal and enlightened historical erudition that ound its expression in his our-volume Histoire des sciences mathématiques en Italie. Tese two works certainly represent the nest pieces o scholarship in mathematical historiography rom the rst hal o the nineteenth century. In Germany, some men combined a command o science and o classical and orientalist philology which helped them produce remarkable works o historical erudition. We can mention Ludwig Ideler (1766–1846) in Berlin (astronomy and mathematical chronology) or, more importantly or us, Georg Heinrich Nesselmann in Königsberg (1811–81) who may have been the rst to offer lectures on the history o mathematics on a regular basis, which resulted in a much-praised history o Greek algebra (1842). Another important gure or our present concerns is the Heidelberg proessor o mathematics Arthur Arneth (1802–58), author o a now orgotten Geschichte der reinen Mathematik in ihrer Beziehung zur Entwicklung des menschlichen Geistes (Stuttgart, 1852), in which he clearly enunciated the undamental opposition between the Greek and Indian styles o practising mathematics. We shall return to his ideas below. Te works o Libri, Arneth, Nesselmann and Chasles impressed the three most important writers on history o ancient and medieval mathematics in the late nineteenth century mentioned above. Te distinguished Danish geometer and historian o mathematics Hieronymus Georg Zeuthen (1839–1920) was a pupil o Chasles in Paris and he himsel conceded how Chasles’s in uence had been decisive or his historical works. Moritz Cantor (1829–1920) also went to Paris where he met Chasles, whose historico-mathematical studies inspired an equally strong ascination in him. But another mathematician produced a very in uential book some years beore Cantor and Zeuthen would publish their major works. Hermann Hankel was born in 1839, the same year as Cantor. His essay on ancient and medieval mathematics srcinated rom the lectures he gave at theUniversity o übingen rom the year o his appointment in 1869 until his premature
Te logical Greek versus the imaginative Oriental
death in 1873. It was published posthumously the next year by his ather, who was Proessor o Physics in Leipzig. In spite o its being un nished, this book can be rightly quali ed as the most srcinal and rereshing view on the topic to have been offered in print up to its day. Hankel was notable or including an up-to-date summary – the only one available to date, especially or Arabic mathematics – o the ndi ngs o Colebrooke, Woepcke and other orientalists on the mathematics o the Hindus and Muslims. With his numerous thought-provoking interpretations, Hankel’s history represented a compelling source o inspiration or the orthcoming generation o ‘proessional’ historians o mathematics, among whom we mention the names o Cantor, Bretschneider, Zeuthen, annery, Heiberg, Eneström, Allman, von Braunmühl, Günther, Loria, Hultsch, Curtze, Suter, etc.9 Te éminence grise among them was undoubtedly Moritz Cantor, who enjoyed the privilege o studying mathematics in Göttingen with Carl Gauss and others. But another Göttingen proessor, Moritz Stern, instilled in him the taste or historical studies. Arneth’s lectures on the history o mathematics in Heidelberg, which Cantor heard in 1848, are also said to have exerted a strong in uence on him.10 Cantor’s ‘antiquarian’ style o scholarship – with its erudite, detailed and comprehensive narrative o every single episode o mathematical history within its own speci c context – is evident in his monumental Vorlesungen über Geschichte der Mathematik, whose rst volume appeared in 1880. Tis style is ofen contrasted with the ‘presentist’ and Platonic approach o H. G. Zeuthen, who insisted on the necessity to select the most signi cant episodes o the history o mathematics in order to illuminate our understanding o the development o mathematics rom a modern perspective, a vision embodied in his highly srcinal and in uential historical essay on the theory o the conics in antiquity (published in Copenhagen in 1885 and in German translation the next year). His introductory Geschichte der Mathematik im Altertum und Mittelalter(1896) o didactic intent (the intended readership were the uture teachers o mathematics in Denmark) had nevertheless a scope and depth similar to Hankel’s history, and remained or several decades the best work o its genre. India’s illogical lure
Playair’s essay on Indian mathematics provided the inspiration or Arthur Arneth’s ‘cultural’ history o mathematics, in which we nd the rst 9
10
On those historians omathematics, see the biographical notices inPart oDauben and Scriba 2002. See Homann 2008; Folkerts in Dauben and Scriba 2002:387–91, on 387.
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precise ormulation o the idea opposing the apodictic rationality o Greek mathematical practice to the more intuitive one o the Indians. He contended that whereas the Greeks were trying to recognize that which is given (das Gegebene) and has a orm (das Gestaltete), the Indians were creating orms (Gestaltungen) through active research, satisying themselves to know that something exists, without concern or knowing how it is so.11 Both styles were one-sided, but necessary. Te rapid development o modern mathematics, according to Arneth, owed much to the mingling o these two contrasting styles o mathematical practice. Let us turn our attention to Hankel, Cantor and Zeuthen’s writings. Hankel’s srcinal views on mathematical demonstration contrast with the coarse dogmatism o Biot, on the one side, and the more sophisticated conservatism o Cantor, on the other.12 Hankel devoted thirteen pages to the Greek concepts and practice o analysis and synthesis, presenting a competent and inspiring survey o the topic.13 For him, the painstaking care associated with analysis and synthesis and the ‘dry dogmatic syllogism’ so peculiar to Greek mathematicians was not a ‘useless burden’ to them; in act, he says, ‘or their mental strength, this orm, so annoying to us, was the appropriate one’.14 Hankel’s account o Indian mathematics is still permeated with the German romantic ascination or India and its philosophy. Like Playair, he noted the occasional and partial use o certain orms o demonstration in Indian mathematical texts: ‘Tere is also little to nd among the Indians o . . . a practice o proo. Only here and there does a commentator add some remarks to the rules and theorems, which can pave the way to their derivation.’15 Indian geometry, radically different rom that o the Greeks, was also characterized by the absence o demonstration in the traditional (Greek) sense; there is simply a reerence to a gure accompanied by the exclamation ‘Look!’ Tis kind o ‘illustrative demonstration’ Anschauungsbeweis ( ), as we have seen in our previous quotation o Günther, strongly ascinated historians o mathematics. Cantor saw this as a typically Indian mode o thought: ‘Tis orm o demonstration, which does not appear in Brahmagupta, must certainly be considered as (typically) Indian. Combined with the algebraic 11 12 13 14 15
Arneth 1852: 141. Cantor’s views are analysed urther below. Hankel 1874: 137–50. Hankel 1874: 208. ‘Von solcher Entwickelung undBeweisührung ist nun auch bei den Indern nicht eben so viel zu nden. Nur hie und da ügt ein Commentator zu den Regeln und Sätzen einige Bemerkungen, welche den Weg zu deren Ableitung geben können.’ Hankel 1874 : 182–3.
Te logical Greek versus the imaginative Oriental
orm o demonstration, it is tremendously characteristic or the mental capacity o those geometers. o compute with almost endless possibilities, they never go beyond that.’16 Compare Zeuthen: ‘In all cases they do not give such justi cations in words, but satisy themselves to make a drawing and, with the word “look!”, reer to the gure, which, or the Greeks, orms the starting point o the actual demonstration.’17 However, in a commentary on the geometry o Brahmagupta, Hankel ound a more promising kind o demonstration: ‘I we except the strange orm o the expression, we do nd in this passage the idea o the demonstration, brie y it is true, but hinted at with absolute clarity . . . But how different is this derivation rom a Euclidean one!’18 But why was Indian mathematical practice so different rom the Greek one? Hankel was convinced that the reasons lay in Indian philosophy: ‘Te Brahmans [have] an essentially different way o thinking than the Greeks; or them the reasons are less important than the results, the why less important than the how; they operate more with ideas and imaginations than with concepts. Te sharpness and certainty they thereby lose is compensated by increased depth and breadth.’19 And he then offered as an explanatory example the case o grammar: whereas the Greeks have a logical and syntactical grammatical system, Indian grammar – epitomized 20
by Pān. ini – is empirical and etymological. But these aspects o Pān . ini’s system he considered ruitul, or we are dealing with a ‘unique and absolutely scienti c grammar’. Te ormal constraints imposed on scienti c writings, however, namely the use o compact versi ed rules, he sees as an obvious obstacle to the ormulation o theorems and their logical proos. Tis negative remark notwithstanding, Hankel did not consider the Indian 16
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‘Diese Beweisorm, welche bei Brahmagupta nirgend aufritt, muss wohl als indisch betrachtet werden. Sie ist mit der algebraischen Beweisorm verbunden ungemein charakteristisch ür die Fassungskraf jener Geometer. Rechnen in nahezu unbegrenzter Möglichkeit, darüber kommen sie nicht hinaus.’ Cantor 1894:614. In the third edition (1914: 656), the word ‘Fassungskraf’ is replaced by ‘Darstellungsweise’ (mode o representation). ‘Jedenalls geben siesolche Begründungen nicht inWorten wieder, sondern sie begnügen sich damit zu zeichnen und durch das Wort ‘Siehe! ’ au die Figur hinzuweisen, die der wirklichen Beweisührung der Griechen zu Grunde lag.’ Zeuthen 1896 : 261. ‘Sehen wir von der remdartigen Form des Ausdruckes ab, so nden wir in dieser Stelle die Idee des Beweises zwar kurz, doch völlig klar angedeutet. . . . Wie verschieden aber ist diese Ableitung von einer nach Art des Euklid!’ Hankel 1874 : 208. ‘Die Brahmanen [haben] eine von den Griechen wesentlich verschiedene Art zu denken;sie legen weniger Werth au die Begründung, als au das Resultat, weniger au das Warum als das Wie; sie operieren mehr mit Ideen und Vorstellungen, als mit Begriffen. Was sie dadurch an Schäre und Bestimmtheit verlieren, gewinnen sie wieder durch größere iee und Weite.’ Hankel 1874: 173. Pān.ini (c. fh century ) is the author o the undamental grammar o classical Sanskrit.
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style as being intrinsically an impediment to mathematical progress, or he went as ar as declaring that a combination o Indian ideas (imagination!) and Greek principles (logic!) would improve the contemporary teaching o geometry, by giving students a sharper mathematical intuition. Such a combination, he maintained, could also have helped to advance mathematical progress, but logic, in the end, was wanting amongst the Indians. A very pervasive dogma among nineteenth-century historians o mathematics proclaimed the essentially geometrical character o the Greek mind, in contrast to that o the Orientals (Indian and Chinese), more akin to computational and algebraical operations. One consequence o this ideological assumption led Cantor to assume rigidly that all geometrical notions attested in India are necessarily in uenced by the Greeks, because ‘we should not and cannot expect a non-geometrical nation to have made essential progresses [in geometry]’.21 For the śulbasūtras, Cantor had rst postulated an in uence through Hero o Alexandria,22 but he had to retract his opinion in view o the evidence, put orward by Indologists, or the chronological impossibility o such a transmission.23 He later postulated a possible in uence rom Mesopotamia.24 Te Greek in uence on the geometry o Brahmagupta, or example, he considered certain, and again he had less a ‘rigorous Euclid’ in mind than a ‘calculator’ like Hero.25 Cantor’s mostly anti-Indian and Hellenocentrist attitude is evident in a letter to Paul annery dated 6 June 1880. Concerning an arithmetical method employed by Āryabhat . a, which, although similar, differs rom that o the Greek Tymaridas, he wrote: ‘the matter is that Āryabhat.a uses the method “epanthem o Tymaridas”, which proves indeed that those scienti c bandits o India did not content themselves with Greek geometry, but also appropriated Greek algebra, to which, 26 it is true, they have added much’. Zeuthen’s interpretation o Indian mathematical history stood closer to Hankel’s views than those o Cantor. He agreed with both o them that it was only through Greek in uence that the Indian computing skills (Rechenertigkeit) could lead to real mathematical progress. What they 21
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‘Wesentliche Fortschritte düren und könnenwir von einem nicht geometrisch angelegten Volksgeiste nicht erwarten.’ Cantor 1894 : 612. Cantor 1894: 603–4. See Chapter 6 by Agathe Keller in this volume on the work o Tibault. ‘Erinnern wir uns, wievieles an Babylon mahnt!’ Cantor 1907: 645 [3rd edn o Cantor 1894]. Cantor 1894: 615; 1907: 657. ‘C’est qu’ Āryabhat. a emploie la méthode dite «épanthème de Tymaridas», ce qui prouve bien que ces bandits scienti ques de l’Inde ne se contentaient guère de la géométrie grecque, mais qu’ils s’emparèrent encore de l’algèbre grecque, à laquelle, il est vrai, ils ajoutèrent beaucoup.’ annery 1950: 314; c. Cantor 1894: 583–4.
Te logical Greek versus the imaginative Oriental
inherited rom the Greeks they developed in this direction without being burdened by the logical circumspection that characterizes the Greeks. In this manner, they could appropriate new rules and methods without necessarily understanding the underlying reason or their validity. Te Indians could also go beyond Diophantus especially ‘because o their less sensitive (einühlig) logic’,27 which made the transer o existing rules rom rational to irrational numbers easier than it would have been to a Greek. Another example o Indian improvement over the Greeks is their use o negative numbers. In contrast to the limitations a ‘cautious Greek’ had to deal with, the ‘calculating Indian’ could take calculations ‘just as they present themselves’, as Zeuthen writes, without caring as to whether or not a quantity was positive or not; the Indians ‘arranged themselves’ with 28 One sees here such negative quantities, simply qualiying them as ‘debts’. a notable example o the Hellenocentrist tendency to systematically distort the interpretation o non-Greek mathematical thought by reducing the associated cognitive processes to irrational ortuities. Excursus: Hero and Diophantus – two ‘orientalized’ Greeks?
Another problematic aspect encountered by historians o mathematics was related to their interpretation o two ‘anomalous’ Greek mathematicians, Hero and Diophantus, whose styles, methods and preoccupations prooundly diverged rom those o classical Greek mathematics. Te tone is set very clearly by Hankel when he writes about Diophantus that ‘i his works were not written in Greek, it would occur to nobody to think that he sprang rom Greek culture; his mind and spirit is too ar away rom that which revealed itsel during the classical period o Greek mathematics.’29 Hankel sees Diophantus’Arithmetica, rom a historical point o view, as counting among the most signi cant mathematical works o Greek antiquity; he even added the surprising (over)statement that, in terms o srcinality and independence, his contributions stand perhaps higher than those o any other Greek mathematician!30 Hankel enthusiastically argued or the dependence o Diophantus on Indian sources which would have 27 28 29
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Zeuthen 1896: 279. Zeuthen 1896: 180. ‘Wären seine Schrifen nicht in griechischer Sprache geschrieben, niemand würde au den Gedanken kommen, dass sie aus griechischer Cultur entsprossen wären; so weit ist sein Sinn und Geist von dem enternt, der sich in der klassischen Zeit griechischer Mathematik geoffenbart hatte.’ Hankel 1874: 157. Hankel 1874: 170.
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been circulating in Alexandria beore his lietime.31 With the decline o Hellenism, the rigidly systematical spirit o classical geometry was supplanted by a de nitely orientalized orm o mathematics exempli ed by Diophantus’ Arithmetica. Siegmund Günther, whom we mentioned above, saw in Diophantus a ‘double nature’. In his earlier works, such as thePorismata, it was possible to detect purely Hellenistic demonstration practices which had obliged him to employ laborious roundabouts. But by the time o composing the Arithmetica, Diophantus had experienced a true emancipation rom his predecessors, notably in his use o symbolism and by his use o ‘clever 32 tricks’, his ‘boldness’ and his ‘skilulness’. Zeuthen implicitly ollowed Hankel by reusing to exclude an Indian in uence on Diophantus. Conversely, he wrote, later Indian authors may also have been in uenced by the Greek algebraist. For this statement Zeuthen harvested the criticism o his riend annery: Mr. Zeuthen shows a strong tendency to go back to Hankel’s thesis: the Greeks, wonderully gifed in geometry, had no talent whatsoever or arithmetic. Te composition o a work such as that o Diophantus can only be explained by supposing the in uence, in Hellenized Egypt, o a race particularly apt to numerical computations, such as that o the Hindus.33
annery, in this case, shared the opinion o Cantor, to whom he wrote in 1885: ‘As or the sources I assume or Diophantus, I would not want you to think that I am close to Hankel; I have always believed that Diophantus was exclusively Greek.’34 35 Compare Cantor: ‘He belonged to his own time and to his own nation.’ Note that Nesselmann in 1842 had expressed views similar to those o Cantor and annery, and rejected Bombelli’s (1579) earlier assumption o an Indian in uence, to which he was led by mistaking a scholion o Maximus 36 It was Planudes in a Vatican manuscript or a part o Diophantus’ work. 31 32 33
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Hankel 1874: 204–5. Günther 1908: 163–8. ‘M. Zeuthen accuse une propension assez marquéeà revenir à la thèse de Hankel: les Grecs, merveilleusement doués pour la Géométrie, ne l’étaient nullement pour l’Arithmétique; la rédaction d’un ouvrage comme celui de Diophante ne peut s’expliquer qu’en supposant l’in uence, dans l’Egypte hellénisée, d’une race particulièrement apte aux calculs numériques, comme celle des Hindous.’ annery 1950: 219. ‘Quant aux sources où je crois quepuisait Diophante, jene voudrais pas unseul instant que vous pensiez que je me rapproche de Hankel; j’ai toujours cru que Diophante était exclusivement grec.’ annery 1950: 328. ‘Er stand . . . innerhalb seiner Zeit, innerhalb seines Volkes.’ Cantor 1894: 450. Nesselmann 1842: 284–5.
Te logical Greek versus the imaginative Oriental
the Indologist Edward Strachey (1813, 1818) who rst positively ormulated the thesis that Hindu algebra had in uenced Diophantus, a conten37 tion repeated, albeit in a much more nuanced ashion, by Colebrooke. Te case o Hero o Alexandria – with his imaginative and practical problem-solving approach without emphasis on demonstrations – was less problematic.38 Cantor had argued or Old Egyptian in uence, a hypothesis – already insinuated by Hankel39 – that nobody took the trouble to challenge.40 In any case, there was a certain uneasiness in interpreting Hero and Diophantus. Te reason or our excursus is connected with the ollowing: i Greek mathematics can be essentially opposed to an Oriental style, how can it be that two important Greek authors are basically ‘oriental’ in style? Tis observation could not really undermine the main thesis: both authors were simply given a status o exceptions . . . Excursus 2: Cantor on inductive demonstrations in Ancient Eg ypt
A similar pattern is discernible in Cantor’s speculations about the geometrical knowledge supposedly acquired by Tales in Ancient Egypt, and the peculiar deductive shaping he, as a Greek, immediately conerred on the primitive rules and demonstrations o the Egyptians. Cantor had collaborated with Eisenlohr in his efforts to decipher the Rhind Papyrus, a translation o which was published in Leipzig in 1877. In the geometrical problems o Ahmes, ormulae are given as such, without derivation. But we are dealing with a book o exercises, Cantor says, so we should not ask or something which cannot be contained in it, namely derivations (Ableitungsverahren) o the ormulae. Ahmes must 41 have taken these derivations rom another, now lost, theoretical textbook. Tis hypothetical ‘theoretical’ textbook on geometry Cantor imagines to have contained primitive inductive demonstrations or even illustrative demonstrations (Beweisührung durch Anschauung), as with the Indians.42 But to assume strict geometrical demonstrations is not necessary in the context o Egyptian mathematics.43 37 38 39
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C1817. Günther 1908: 217. Hankel 1874 : 85. Hankel, who only had access to a summary description o the Rhind Papyrus by Birch (1868), recognized the similarities with Hero but was not sure whether the papyrus was older than the Alexandrian’s lietime or not. Cantor 1894: 365–7. Cantor 1894: 53; 1907: 91, 113. Cantor 1907: 113. Cantor 1907: 106.
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Tus Egyptian theory is inductive, and Greek theory is deductive. Tales, when he obtained his geometrical knowledge in Egypt, must have offered different kinds o demonstration than the Egyptians did (or example, the theorem stating that the diameter divides the circle in two equal parts), or the simple reason that he had a Greek mind! Cantor puts it as ollows: As a Greek he generalized, as a pupil o Egypt he grasped through the senses what he then made comprehensible to the Greeks. It was an ethnic characteristic [Stammeseigentümlichkeit] o the Greeks to get to the bottom o all things, and, starting rom practical needs, to reach speculative explanations. Nothing o the sort with the Egyptians.44
With the Egyptians, Cantor speculates, either the gu re sufficed or the proo, or it was done through computation o the areas o both semicircles according to the same, possibly uncomprehended, rule. Te problematic status of Islamic mathematics
Te conrontation with Arabic mathematical writings and the bibliographical inormation about it orced historians o mathematics to adopt a different approach than with Indian mathematics. First it became increasingly obvious that a large number o Greek mathematical works, including virtually all major ones, had been not only translated into Arabic but also studied, commented upon, adapted and transormed. Greek mathematics had been thoroughly assimilated within Islamic culture. On the other hand, Indian in uences were obvious in several works, such as the arithmetic o al-Khwārizmī, or (it was presumed) in the treatise on practical geometrical constructions by Abū al-Waā . How was it possible, then, to treat Islamic mathematics within the category ‘Oriental’? Which status did historians o mathematics grant to Arabic mathematics? Another, related problem was o course the question o its srcinality. In this respect late nineteenth-century historians o mathematics proved surprisingly severe, in spite o the excellent works o Franz Woepcke. ʾ
Te question o srcinality Although he presented an excellent summary o the available evidence (mostly thanks to Woepcke’s works), Hankel claimed at the outset that the Arabs had added little to what they had received rom the Greeks and the 44
‘Als Grieche hat er verallgemeinert, als Schüler Aegyptens sinnlich erasst, was er dann den Griechen wieder assbarer gemacht hat. Es war eine griechische Stammeseigentümlichkeit, den Dingen au den Grund zu gehen, vom praktischen Bedürnisse zu speculativen Erörterungen zu gelangen. Nicht so den Aegyptern.’ Cantor 1894 : 140.
Te logical Greek versus the imaginative Oriental
Indians. O course, it was no longer possible, at the end o the nineteenth century, to maintain the old myth o preservation. But this myth was replaced by another version that was as economical as possible: Cantor ormulated it this way: ‘[Te Arabs] have been capable not only to preserve, but also to expand the treasures entrusted to them.’45 Tese intellectual treasures, however, were regarded by Cantor as undamentally oreign elements, which could only live in the arti cial milieu o the princely courts.46 Symptomatically, Hankel, Cantor and Zeuthen ound only ew examples o srcinal and independent contributions by the Arabs. Zeuthen, indeed, introduced his chapter on Arabic mathematics by mentioning that he ‘would have liked to emphasize the ull extent and value o the mathematical works o the Arabs, in order to avoid negative conclusions rom the relatively ew positive results achieved beyond those known to the Greeks’.47 Tis act, he says, provides the very reason or restricting his presentation to a ew selected examples o the kinds o works the Arabs did. In this connection we should mention that, in 1888, Zeuthen had offered to the readers o Bibliotheca Mathematica a question that implicitly sought to undermine Woepcke’s view o the srcinality o Islamic contributions to algebra (especially the application o conic sections to the resolution o algebraic equations).48 Zeuthen suspected that the Greeks had already applied these techniques to the same algebraic problems, thereby raising serious doubts as to whether the Arabs had really been innovative in this regard.
Te obsessive search or in uences Otherwise, Hankel and especially Cantor were animated by a desire to identiy in Islamic mathematics as many oreign in uences they could, even on the basis o tenuous similarities. Tus Hankel did not hesitate to assign to the Indians a proo o a certain identity involving geometric series recorded by al-Karajī, even though Woepcke had been unable to detect any Indian in uence on al-Karajī in general.49 In the same manner, he saw nothing 50 in the indeterminate analysis o al-Karajī that went beyond Diophantus.
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‘[Die Araber] haben das Ihnenanvertraute Gutnicht nur zu bewahren, auch zu vermehren gewusst.’ Cantor 1894: 771. Cantor 1894: 741–2; 1907: 786–7. Zeuthen 1896: 297. Zeuthen 1888 . Woepcke 1853: 61–2; Hankel 1874: 42. Hankel 1874: 270.
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Identiying the srcin o al-Khwārizmī’s algebra presented more serious difficulties, since it differed rom both the Greek and Indian algebraic traditions. But or Cantor, Islamic algebra could under no circumstances be autochthonous: it could only eature Greek and Indian elements, so he assumed an amalgam o both traditions. In general, however, Cantor was convinced that there existed two separate schools in Islamic mathematics, bringing about a undamental opposition between the disciples o Indian methods and those, more numerous, who strictly adhered to the Greek tradition.51 Cantor’s entire section on Islamic mathematics shows precisely his constant concern or associating every single mathematician or result within one o the two groups.
How did the Arabs handle the Greek axiomatic–deductive methods? Now we come to the more crucial question: were the ‘Arabs’ up to dealing with Greek thought? How did ‘Oriental’ mathematicians come to terms with the Euclidean axiomatic–deductive method? Hankel described the nature o Euclid’s in uence on Islamic mathematical practice with the ollowing words: In same way, one his zealously occupied himseland withone theused logical analysis o his the (Euclid’s) method, de nitions and axioms, his demonstrations or the exempli cation o the rules o ormal logic in a similarly pedantic manner as what our German logicians still liked to do almost until our century.52
Tus the ‘Greek oversubtlety’, as Hankel says (griechische Spitz ndigkeit), was a level too high or the Arabs. Its perversion he exempli ed with 53 o [pseudo-]. ūsī’s vain attempt to prove Euclid’s postulate o parallels. this observation Hankel adds the ollowing statement concerning the status and use o demonstrations in Arabic treatises: Despite their even doctrinary acquaintance with the demonstrative method [o the Greeks], the Arabs, most o the time, have rerained rom providing the demonstrations, and have dogmatically strung the theorems and rules together, exactly as the 51 52 53
Cantor 1894: 718–19. Hankel 1874: 272. Te Arabic text o a recension o Euclid’sElements wrongly attributed to Nas . īr al-Dīn al-.ūsī had been printed in Rome in 1586 and was available through Wallis’ analysis thereo in his history o algebra, his interpretation being possible thanks to the collaboration o orientalist Pococke. See Molland 1994 and Stedall 2001.
Te logical Greek versus the imaginative Oriental Indians used to do in their siddhantas. Were they motivated by a concern or the shortness and corresponding cheapness o their books? 54
Cantor’s judgement was not as severe as Hankel’s, but he was nevertheless convinced that the practice o demonstration had almost a character o exception among Muslim mathematicians.55 Conronted with a highly srcinal treatise o practical geometry by Abū al-Waā (then known only through a mutilated Persian version which Woepcke had summarized), ʾ
Cantor had no recourse to contextualization to explain the absence o demonstrations – as he did in the case o the geometrical part o the Rhind Papyrus, where he accepted it precisely because o the practical nature o the work. For him Abū al-Waā ’s treatise recalled Indian geometry, so that ‘one would almost expect as a proo [o the validity o a particular construction] the request “look!”, with which Indian geometers are satis ed to conclude their construction procedures’.56 Tus or him there was no doubt that this work was an example oAnschauungsgeometrie, so thoroughly Indian in style that no deductive demonstrations, even on the part o one o the best Muslim mathematicians, could be possibly assumed. ʾ
Epilogue
For Zeuthen, the breakthrough in the history o mathematics occurred with the resolution o third-degree equations by means o roots (Cardano and artaglia), an achievement that closes the medieval periodization o mathematics and announces the rapid advances made thereafer, modelled on and inspired by a close study o Greek mathematics. Zeuthen’s periodization explains why nearly three-quarters o his book (245 pages out o 332 in the rst Germanedition) is devoted to Greek mathematics. In view o this, it is probably erroneous to assume that Zeuthen and his colleagues saw the practice o mathematical demonstration as the key to mathematical progress. Te Muslims had been competent and respectul 54
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‘rotz dieser selbst doktrinären Bekannschaf mit der demonstrativen Methode haben die Araber sich meistens aller Beweise enthalten und Lehrsätze wie Regeln dogmatisch aneinandergereiht, nicht anders als es die Inder in ihren Siddhanten zu thun p egen. Hat sie dazu die Rücksicht au Kürze und entsprechend grössere Wohleilheit ihrer Bücher bewogen?’ Hankel 1874: 273. Commenting on an srcinal geometrical problem solved by al-Kūhī in which the latter inserted a rigorous proo with diorismos, Cantor noted that ‘in general the imitators o the Greeks – Arabs not excluded – considered [this practice] by no means with the same regularity’ [‘. . . was die Nachahmner der Griechen im allgemeinen – die Araber nicht ausgeschlossen – keineswegs mit gleicher Regelmässigkeit zu beachten p egten’]. Cantor 1894 : 705. Cantor 1894: 700; c. 709–10.
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students o the Greeks, and, although relatively very little was known around 1900, their works revealed – even to the least interested historians such as annery – a sophisticated mathematical practice within which demonstration, closely ollowing the Greek model, played an important role. Yet, it was thought, this assimilation o Greek mathematical thinking and practice did not instil much progress among them; the Arabs’ contributions, however extensive and honest they may have been, did not bring about any major breakthrough, nothing comparable to what would happen in sixteenth- and seventeenth-century Europe. For the ‘Arabs’ were implicitly considered as immature custodians o a higher knowledge, who could not properly deal with it; being mere imitators o a oreign tradition, they were unable to reach the critical level beyond which real progress could have been initiated. For Zeuthen, the Eastern Arabs had been unable to emancipate themselves rom the rigid geometrical approach o the Greek; and the Western Arabs, who supposedly did liberate themselves rom this 57 approach, still remained ‘too reverential’ toward the Greeks. On the other hand, the Indians, in spite o the supposed laxness and lack o rigour o their mathematical practice and the primitiveness o the ew demonstrations revealed in their works, had indeed achieved results superior to those o the Arabs in arithmetic and algebra. Some authors even went as ar as comparing this ‘Indian’ style o mathematical practice with the intuitive works o certain modern mathematicians, probably because it was realized that absolute rigour had not played a undamental role in early modern Europe.58 Few believed that a stringent axiomatic–deductive system was a necessary condition or mathematical discovery. Nevertheless, historians o mathematics unanimously insisted that it was precisely the lack o a logical, rigorous system o mathematical thought similar to the Greek one that prohibited any urther progress in India. Imagination alone could at best generate haphazard discoveries. Te reusal o systematic rationality to the Orientals was saved whenever one encountered anything ingenious in their mathematics by explaining it through their having recourse to ‘tricks, dodges’ (Kunstgriffe), as in the case o al-Bīrūnī’s solution o the chessboard problem.59 For annery, theory provided with demonstration was the 57 58
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Zeuthen 1896: 314. Günther ( 1908: 127) says o Hero that he is a sort o ‘antique Euler’. Hankel1874: ( 202–3) compares Bhāskara’s numerical methods, especially in his solution o the so-called Pell equation, with those o modern mathematicians. On the relatively unimportant role o mathematical proos (compared to the concern or ‘exactness o contructions’) in early modern Europe, see Bos 2001: 8. Cantor 1894: 713–14.
Te logical Greek versus the imaginative Oriental
distinguishing eature between pre-scienti c and scienti c mathematics;60 but Indian mathematics, although categorized under the ‘scienti c’ genus by all historians o mathematics, eatured only primitive, indeed pre-scienti c, kinds o proos. Tis is an apparent paradox. Had ‘Oriental’ mathematics a special status? annery certainly perceived it this way. In act, he and his colleagues were convinced that no civilization other than the Greek ever attained the scienti c level autonomously. Tus, the prescienti c mathematics o India only became scienti c afer it had been nurtured by Greek in uence. Indian mathematics, however, remained stigmatized with a special and incomplete status o scienti city, because it had only imperectly assimilated the Greek model.
Concluding remarks Nineteenth-century historians o mathematics did not claim the practice o demonstration only or the Greeks, but they insisted on its character o exception in ‘Oriental’ mathematics. (Consequently, much o twentiethcentury historiography simply disregarded the evidence already available, and returned to the simpli ed view that a concern or proo and rigour never existed outside o ancient Greece and modern Europe – perhaps with the exception o medieval Islam.) Te criterion that really allowed a separation o ‘Western’ rom ‘non-Western’ science was one ostyle: systematic and axiomatic–deductive in one case, intuitive (at best inductive), illustrative and unre ected in the other.61 Te ideology associated with this undamental separation, even though its roots could be traced back to the Renaissance, did not crystallize until late in the nineteenth century. I will now conclude with a short sketch o the ideological landscape that avoured its dogmatic ormulation. With the accomplishment o the imperialist enterprise and the general con dence that space, people and nature could be successully dominated, Western Europeans acquired the ultimate certainty o their superiority over the rest o the world. It is no wonder, then, that the Romantic and Orientalist enthusiasm, omnipresent in the rst hal o the century, was quickly annihilated. Dismissing previous attempts to proclaim the srcinality o ‘Oriental’ science and consolidating the integrity o ‘Western’ 60 61
annery 1950: 25. Tis idea is still common today; a massive argumentation in avour o the distinctiveness o the Western scienti c style can be ound in Crombie 1994 . For a critical view, see Hart 1999.
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science was, indeed, a major characteristic o scholarship in the history o science during the last quarter o the nineteenth century. Te views became increasingly and consensually Helleno- and Eurocentrist, not in the ingenuous and instinctive manner o previous generations, but in systematic and dogmatic ways. In this context we can mention the in uence o Comtian positivism, and the rise, in some milieus, o racist and antisemitic theories. Te amous lecture ‘L’Islam et la science’ delivered by Ernest Renan at the Sorbonne in 1883 proclaimed the scienti c ineriority o Semitic races as opposed to Indo-Aryan ones, and emphasized the essentially antagonistic nature o the Islamic aith toward science.62 Te works o Pierre Duhem on ancient Greek and medieval Latin cosmology promulgated the relative insigni cance o extra-European science.63 Even the Arabist Bernard Carra de Vaux, a close collaborator o annery, ound recognition mainly or his work on Greek technological works preserved only in Arabic, and contributed an appendix to annery’sRecherches sur l’histoire de l’astronomie ancienne in which he misinterpreted one o the most interesting chapters o Islamic planetary theory by reducing it to an example o the way in which, when it attempted to be srcinal, Islamic science revealed only ‘weakness’ and ‘pettiness’.64 Such was indeed the dominant perspective in Europe around 1900 when the discipline o history o science was established as an international network o scholars under the aegis o Paul annery. History o science sought to and succeeded in promoting and deending the values and uniqueness o Western civilization.65 Bibliography Arneth, A. (1852) Geschichte der reinen Mathematik in ihrer Beziehung zur Entwicklung des menschlichen Geistes. Stuttgart. Biot, J. -B. (1841) [Review o] ‘ raité des instruments astronomiques des Arabes, traduit par J. J. Sédillot’, Journal des Savants 1841: 513–20, 602–10 and 659– 79. Bos, H. (2001) Rede ning Geometrical Exactness: Descartes’ ransormation o the Early Modern Concept o Construction. New York. Cantor, M. (1894) Vorlesungen über Geschichte der Mathematik, vol., Von den ältesten Zeiten bis zum Jahre 1200 n. Chr, 2nd edn. Leipzig. (1907) = 3rd edn o Cantor 1894.
62 63 64 65
Renan 1883. C. annery 1950: 391. On Duhem’s historiography oIslamic science, see Ragep 1990. Carra de Vaux 1893: 338. See Pyenson 1993.
Te logical Greek versus the imaginative Oriental Carra de Vaux B., (1893) ‘Les sphères célestes selon Nasîr-Eddîn Attûsî’, in Recherches sur l’histoire de l’astronomie ancienne, ed. P. annery. Paris: 337–61. Crombie, A. (1994) Styles o Scienti c Tinking in the European radition: Te History o Argument and Explanation Especially in the Mathematical and Biomedical Sciences and Arts, 3 vols. London. Dauben, J. W. and C. J. Scriba (eds.) 2( 002) Writing the History o Mathematics: Its Historical Development. Basel. Günther, S. (1908) Geschichte der Mathematik, vol. 1, Von den ältesten Zeiten bis Cartesius. Leipzig. Hankel, H. (1874) Zur Geschichte der Mathematik im Alterthum und Mittelalter . Leipzig. Hart, R. (1999) ‘Beyond science and civilization: a post-Needham critique’, East Asian Science, echnology, and Medicine 16: 88–114. Homann, J. E. (2008) ‘Cantor, Moritz Benedict’, inComplete Dictionary o Scienti c Biography, www.encyclopedia.com. Martzloff, J.-C. (1997) A History o Chinese Mathematics. Berlin. Michaels, A. (1978) Beweisverahren in der vedischen Sakralgeometrie: Ein Beitrag zur Entstehungsgeschichte von Wissenschaf. Wiesbaden. Molland, G. (1994) ‘Te limited lure o Arabic mathematics’, in Te ‘Arabick’ , ed. Interest o the Natural Philosophers in Seventeenth-Century England G. A. Russell. Leiden: 215–23. Nesselmann, G. H. F. (1842) Versuch einer kritischen Geschichte der Algebra der Griechen. Berlin. Playair, J. (1817) ‘On the algebra and arithmetic o the Hindus’ [Review o C1817], Edinburgh Review 39: 141–63. Pyenson, L. (1993) ‘Prerogatives o European intellect: historians o science and the promotion o Western civilization’, History o Science 31: 289–315. Ragep, J. (1990) ‘Duhem, the Arabs, and the history o cosmology’, Synthèse 83: 201–14. Renan, E. (1883) ‘L’islamisme et la science’, in Oeuvres complètes, ed. H. Psichari (1947–61), 10 vols., vol. (1947). Paris: 945–60. Stedall, J. (2001) ‘O our own nation: John Wallis’s account o mathematical learning in medieval England’, Historia Mathematica28: 73–122. annery, P. (1912–50) Mémoires scienti ques, 17 vols. oulouse. 2
Zeuthen, G. (1888)der ‘Question 20’, Bibliotheca : 63. Mathematica (1896) H. . Copenhagen. Geschichte Mathematik im Altertum und Mittelalter
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Te pluralism o Greek ‘mathematics’ . . .
Greek mathēmatikē, as has ofen been pointed out, is ar rom being an exact equivalent to our term ‘mathematics’. Te nounmathēma comes rom the verb manthanein that has the entirely general meaning o ‘to learn’. A mathēma can then b e any branch o learning, or anything learnt, as when in Herodotus (1 207) Croesus reers to the mathēmata – what he has learnt – rom his own bitter experiences. So the mathēmatikos is, strictly speaking, the person who is ond o learning in general, as indeed it is used in Plato’simaeus at 88c where the p oint at issue is the need to strike a balance between the cultivation o the intellect and that o the body, the principle that later became encapsulated in the dictum ‘ mens sana in corpore sano’. Yet Plato also recognizes certain special branches o the mathēmata, as when in the Laws at 817e the Athenian Stranger speaks o those that are appropriate or ree citizens as those that relate to numbers, to the measurement o lengths, breadths and depths, and to the study o the stars, in other words, very roughly, arithmetic, geometry and astronomy. In Hellenistic Greek mathēmatikos is used more ofen o the student o the heavens in particular (whether what we should call the astronomer or the astrologer) than o the mathematician in general in our sense. Whether we should think o either what we call mathematics or what we call philosophy as well-de ned disciplines beore Plato is doubtul. I have previously discussed the problems so ar as philosophy is concerned.1 Tose whom modern scholars conventionally group together as ‘the Presocratic philosophers’ are a highly heterogeneous set o individuals, most o whom would not have recognized most o the others as engaged in the same inquiry as themselves. Teir interests spanned in some, but not all, cases what we call natural philosophy (the inquiry into nature), cosmology, ontology, epistemology, philosophy o language and ethics, but the ways in which those interests were distributed among the different individuals concerned varied considerably. It is true that we have one good fh-century example o a thinker most o whose work (to judge rom the very limited inormation we have 294
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about that) related to, or used, one or other branch o mathematics, namely Hippocrates o Chios. He was responsible not just or important particular geometrical studies, on the quadrature o lunules, but also, maybe, or a rst attempt at systematizing geometrical knowledge, though whether he can be credited with a book entitled (like Euclid’s) Elements is more doubtul. Furthermore in his other investigations, such as his account o comets, reported by Aristotle in the Meteorology, he used geometrical arguments to explain the comet’s tail as a re ection. Yet most o those to whom both ancient and modern histories o preEuclidean Greek mathematics devote most attention were ar rom just ‘mathematicians’ in either the Greek or the English sense. Philolaus, Archytas, Democritus and Eudoxus all made notable contributions to one or other branches o mathēmatikē, but all also had developed interests in one or more o the studies we should call epistemology, physics, cosmology and ethics. A similar diversity o interests is also present in what we are told o the work o such more shadowy gures as Tales or Pythagoras. Te evidence or Tales’ geometrical theorems is doubtul, but Aristotle (who underlines the limitations o his own knowledge about Tales) treats him as interested in what he, Aristotle, termed the material cause o things, as well as in soul or lie. Pythagoras’ own involvement in geometry and in harmon2
ics has again been contested, and the more reliably attested o his interests relate to the organization o entities in opposite pairs, and, again, to soul. Tese remarks have a bearing on the controversy on the question o whether deductive argument, in Greece, srcinated in ‘philosophy’ and was then exported to ‘mathematics’,3 or whether within mathematics it was an srcinal development internal to that discipline. 4 Clearly when neither ‘philosophy’ nor ‘mathematics’ were well-de ned disciplines, it is hard to resolve that issue in the terms in which it was srcinally posed, although, to be sure, the question remains as to whether the Eleatic use o reductio arguments did or did not in uence the deployment o arguments o a similar type by such gures as Eudoxus. I we consider the evidence or the investigation o what Knorr, in other studies,5 called the three ‘traditional’ mathematical problems, o squaring the circle, the duplication o the cube and the trisection o an angle, those who gure in our sources exhibit veryvaried pro les. Among the ten or so individuals who are said to have tackled the problem o squaring the circle 2 3 4 5
Burkert 1972. Szabó 1978. Knorr 1981. Knorr 1986.
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it is clear that ideas about what counts as a good, or even a proper, method o doing so differed.6 At Physics 185a16–17 Aristotle distinguishes between allacious quadratures that are the business o the geometer to reute, and those where that is not the case. In the ormer category comes a quadrature ‘by way o segments’ which the commentators interpret as lunules and orthwith associate with the most amous investigator o lunules, whom I have already mentioned, namely Hippocrates o Chios. Yet even though there is another text in Aristotle that accuses Hippocrates o some mistake in quadratures (On Sophistical Refutations171b14–16), it may be doubted whether Hippocrates committed any allacy in this area.7 In the detailed account that Simplicius gives us o his successul quadrature o our speci c types o lunules, the reasoning is throughout impeccable. Quite what allacy Aristotle detected then remains somewhat o a mystery. But two other attempts are also reerred to by Aristotle and dismissed either as ‘sophistic’ or as not the job o the geometer to disprove. Bryson is named at On Sophistical Refutations 171b16–18 as having produced an argument that alls in the ormer category: according to the commentators, it appealed to a principle about what could be counted as equals that was quite general, and thus ar it would t Aristotle’s criticism that the reasoning was not proper to the subject-matter. Antiphon’s quadrature by contrast is said not to be or the geometer to reute (Physics 185a16–17) on the grounds that it breached the geometrical principle o in nite divisibility. It appears that Antiphon proceeded by inscribing increasingly many-sided regular polygons in a circle until – so he claimed – the polygon coincided with the circle (which had then been squared). Te particular interest o this procedure lies in its obvious similarity to the so-called but misnamed method o exhaustion introduced by Eudoxus in the ourth century. Tis too uses inscribed polygons and claims that the difference between the polygon and the circle can be made as small as one likes. It precisely does not exhaust the circle. I Antiphon did indeed claim that afer a nite number o steps the polygon coincided with the circle, then that indeed breached the continuum assumption. But o course later mathematicians were to claim that the circle could nevertheless be treated as identical with the in nitely-sided inscribed rectilinear gure. Other solutions were proposed by other gures, by a certain Hippias or instance and by Dinostratus. Whether the Hippias in question is the amous sophist o that name has been doubted, precisely on the grounds that the 6 7
Mueller 1982 gives a measured account. Lloyd 2006a reviews the question.
Te pluralism of Greek ‘mathematics’
device attributed to him, the so-called quadratrix, is too sophisticated or the fhcentury. Although much remains obscure about the precise claims made in different attempts at quadrature, it is abundantly clear rst that different investigators adopted different assumptions about the legitimacy o different methods, and second that those investigators were a heterogeneous group. Some were not otherwise engaged in mathematical studies at all, at least to judge rom the evidence available to us. An allusion in Aristophanes (Birds 1001–5) suggests that the topic o squaring the circle had by the end o the fh century become a matter o general interest, or at least the possible subject o anti-intellectual jokes in comedy. Among those I have mentioned in relation to quadratures several are generally labelled ‘sophists’, this too a notoriously indeterminate category and one that evidently cannot be seen as an alternative to ‘mathematician’. As is well known Plato does not always use the term pejoratively, even though he certainly has severe criticisms to offer, both intellectual and moral, o several o the principal gures he calls ‘sophists’. Yet Plato himsel provides plenty o evidence o the range o interests, both mathematical and non-mathematical, o some o those he names as such. As regards the Hippias he calls a sophist, those interests included astronomy, geometry, arithmetic, but also, or instance, linguistics: however, whether the music he also taught related to the mathematical analysis o harmonics or was a matter o the more general aesthetic evaluation o different modes is unclear. Again, the ragments that are extant rom Antiphon’s treatiseruth deal with questions in cosmology, meteorology, geology and biology.8 Protagoras, who is said by Plato to have been the rst to have taught or a ee, amously claimed, according to Aristotle Metaphysics 998a2–4, that the tangent does not touch the circle at a point, a meta-mathematical objection that he raised against the geometers. Tus ar I have suggested some o the variety within what the Greeks themselves thought o as encompassed by mathēmatikē together with some o the heterogeneity o those who were described as engaged in ‘mathematical’ inquiries. But in view o some persistent stereotypes o Greek mathematics it is important to underline the urther undamental disagreements (1) about the classi cation o the mathematical sciences and the hierarchy within them, (2) about the question o their useulness, and 8
Te identi cation o the author o this treatise with the Antiphon whose quadrature is criticized by Aristotle is less disputed than the question o whether the sophist is identical with the author called Antiphon whose etralogiesare extant.
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especially (3) on what counts as proper, valid, arguments and methods. Let me deal brie y with the rst two questions beore exempliying the third a little more ully. (1) Already in the late fh and early ourth centuries a divergence o opinion is reported as between Philolaus and Archytas. According to Plutarch (able alk 8 2 1, 718e) Philolaus insisted that geometry is the primary mathematical study (its ‘metropolis’). But Archytas privileged arithmetic under the rubric o logistikē (reckoning, calculation, Fr. 4). Te point is not trivial, since how precisely geometry and arithmetic could be considered to orm a unity was problematic. According to the normal Greek conception, ‘number’ is de ned as an integer greater than 1. In this view, arithmetic dealt with discrete entities. But geometry treated o an in nitely divisible continuum. Nevertheless both were regularly included as branches o ‘mathematics’, sister branches, indeed, as Archytas called them (Fr. 1). Te question o the status o other studies was more contested. For Aristotle, who had, as we shall see, a distinctive philosophy o mathematics, such disciplines as optics, harmonics and astronomy were ‘the more physical o the mathēmata’ ( Physics 194a7–8). Te issue o ‘mechanics’ was particularly controversial. According to the view o Hero, as reported by Pappus (Collection Book 8 1–2), mechanics had two parts, the theoretical which consisted o geometry, arithmetic, astronomy and physics, and the practical that dealt with such matters as the construction o pulleys, war machines and the like. However, a somewhat different view was propounded by Proclus (Commentary on Euclid’s Elements41.3 – 42.8) when he included what we should call statics, as well as pneumatics, under ‘mechanics’. (2) Tat takes me to my next topic, the issue o the useulness o mathematics, howsoever construed. Already in the classical period there was a clear division between those who sought to argue that mathematics should be studied or its practical utility, and those who saw it rather as an intellectual, theoretical discipline. In Xenophon’sMemorabilia 4 7 2–5 Socrates is made to insist that geometry is useul or land measurement, astronomy or calendar regulation and navigation, and so on, and he there dismissed the more theoretical or abstract aspects o those subjects. Similarly Isocrates too distinguished the practical and the theoretical sides o mathematical studies and in certain circumstances avoured the ormer (11 22–3, 12 26–8, 15 261–5). Yet Plato o course took precisely the opposite view. It is not or practical, mundane, reasons that mathematics is worth studying, but rather as a training or the soul in abstract thought. But even some who emphasized practical utility sometimes de ned that very broadly. It is striking that
Te pluralism of Greek ‘mathematics’
in the passage just quoted rom Pappus he included both the construction o models o planetary motion and that o the marvellous gadgets o the ‘wonder-workers’ among ‘the most necessary o the mechanical arts rom the point o view o the needs o lie’. Meanwhile the most ambitious claims or the all-encompassing importance o ‘mathematics’ were made by the neoPythagorean Iamblichus at the turn o the third and ourth centuries . He argued in On the Common Mathematical Science(ch. 32: 93.11–94.21) that mathematics was the source o understanding inevery mode o knowledge, including in the study o nature and o change. (3) From among the many examples that illustrate how the question o the proper method in mathematics was disputed let me select just ve. (3.1) In a amous and in uential passage in his Life of Marcellus (ch. 14, c. able alk 8 2 1, 718e) Plutarch interprets Plato as having banned mechanical methods rom geometry on the grounds that these corrupted and destroyed the pure excellence o that subject, and it is true that Plato had protested that to treat mathematical objects as subject to movement was absurd. Te rst to introduce such degenerate methods, according to Plutarch, were Eudoxus and Archytas. Indeed we know rom a report in Eutocius (Commentary on Archimedes Sphere and Cylinder2, 3 84.12–88.2) that Archytas solved the problem o nding two mean proportionals on which the duplication o a cube depended by means o a complex three-dimensional kinematic construction involving the intersection o three suraces o revolution, a right cone, a cylinder and a tore. Plutarch even goes on to suggest that Archimedes himsel agreed with the Platonic view (as Plutarch represents it) that the work o an ‘engineer’ was ignoble and vulgar. Most scholars are agreed rst that that mostprobably misrepresents Archimedes, and secondly that ew practising mathematicians would have shared Plutarch’s expressed opinion as to the illegitimacy o mechanical methods in geometry. (3.2) My second example comes rom Archimedes himsel and concerns precisely how he endorsed the useulness o mechanics, as a method o discovery at least. In his Method (2 428.18–430.18) he sets out what he describes as his ‘mechanical’ method which depends rst on an assumption o indivisibles and then on imagining geometrical gures as balanced against one another about a ulcrum. Te method is then applied to get the area o a segment o a parabola, but while Archimedes accepts the method as a method o discovery, he puts it that the results have thereafer to be demonstrated rigorously using the method o exhaustion standard throughout Greek geometry. At the same time the method is useul ‘even or the proos o the theorems themselves’ in a way he explains ( Method
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428.29–430.1): ‘it is o course easier, when we have previously acquired, by the method, some knowledge o the questions, to supply the proo, than it is to nd it without any previous knowledge’. We should note that what is at stake is not just the question o admissible methods, but that o what counts as a proper demonstration. (3.3) For my third example I turn to Hero o Alexandria. 9 Although he requently reers to Archimedes as i he provided a model or demonstration, his own procedures sharply diverge, on occasion, rom his. In the Metrica, or instance, he sometimes gives an arithmetized demonstration o geometrical propositions, that is he includes concrete numbers in his exposition. Moreover in the Pneumatica especially he allows exhibiting a result to count as a proo. Tus at 1 16.16–26 and at 26.25–28 he gives what we would call an empirical demonstration o propositions in pneumatics, expressing his own clear preerence or such by contrast with the merely plausible reasoning used by the more theoretically inclined investigators. In both respects his procedures breach the rules laid down by Aristotle in the Posterior Analytics, both in that he permits ‘perceptible’ proos and does not base his arguments on indemonstrable starting points and in that he moves rom one genus o mathematics to another. I we think o precedents or his procedures, then they have more in common with the suggestion that Socrates makes to the slave-boy in Plato’s Meno (84a), namely that i he cannot give an account o the solution to the problem o doubling the square, he can point to the relevant line. (3.4) Fourthly there is Ptolemy’s redeployment o the old dichotomy between demonstration and conjecture in two contexts in the opening books o the Syntaxis and o the etrabiblos. In the ormer ( Syntaxis 1 1, 1 6.11–7.4) he discusses the difference between mathēmatikē, ‘physics’ and ‘theology’. Te last two studies are conjectural, ‘physics’ because o the instability o what it deals with, ‘theology’ because o the obscurity o its subject. Mathēmatikē, by contrast, which here certainly includes the mathematical astronomy that he is about to expound in the Syntaxis, alone o these three is demonstrative, since it is based on the incontrovertible methods o geometry and arithmetic. Whatever we may think about the difficulties that Ptolemy himsel registers, in practice, in living up to this ideal when it comes, or instance, to his account o the movements o the planets in latitude, it is clear what his ideal is. Moreover when in the etrabiblos (1 1, 3.5–25, 1 2, 8.1–20) he speaks o the other branch o the study o the heavens, that which engages not in the prediction o the movements o the 9
C. ybjerg 2000: ch. 3.
Te pluralism of Greek ‘mathematics’
heavenly bodies, but in that o events on earth on their basis – astrology, in other words, on our terms – that study is downgraded precisely on the grounds that it cannot deliver demonstration. It is conjectural, though he would claim that it is based on tried and tested assumptions. (3.5) Fithly and inally there are Pappus’ critical remarks, in the opening chapters (1–23) o Book 3 o his Collection, on certain procedures based on approximations that had been used in tackling the problem o nding two mean proportionals in order to solve the Delian problem, o doubling the cube.10 Although certain stepwise approximations can yield a result that is correct, they all short, in Pappus’ view, in rigour. Pappus himsel distinguishes between planar, solid and linear problems in geometry and insists that each has its own procedures appropriate or the subject matter in question. What we nd in all o the cases I have taken is a sensitivity not just to the correctness o results or the truth o conclusions, but to the appropriateness or otherwise o the methods used to obtain them. It is not enough just to know the truth o a theorem: nor is it enough to have some means o justiying the claim to such knowledge. No: what is required is that the method o justi cation be the correct one or the eld o inqui ry concerned according to the particular standards o correctness o the author in question. Tat is the recurrent demand: yet it is clearly not the case that all Greek investigators who would have considered themselves mathēmatikoi agreed on what is appropriate in each type o case or had uniorm views on what counts as a demonstration. Similar second-order disputes recur in most other areas o inquiry that the Greeks engaged in, and this too is worth illustrating since it suggests that the phenomenon we have described in mathematics is symptomatic o more general tendencies in Greek thought. Sometimes we nd such disagreements within what is broadly the same discipline, sometimes across dierent disciplines. In medicine the Hippocratic treatise On Ancient Medicine provides examples o both kinds. Te author rst castigates other doctors who try to base medical practice on what he calls ‘hypotheses’, arbitrary postulates such as ‘the hot’, ‘the cold’, ‘the wet’, ‘the dry’ and anything else they ancy (CMG 1 1, 36.2–21). In this author’s view, that is wrong-headed since medicine is and has long been based on experience. Te investigation o what happens under the earth or in the sky may be orced to rely on such postulates, but they are a disaster in medicine, where they have the result o narrowing down the causal principles o diseases. While that drives a wedge between medicine and ‘meteorology’, he goes on in chapter 20 (51.6–18) 10
I may reer to the detailed analysis in Cuomo 2000: ch. 4.
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speci cally to attack the importation into medicine o methods and ideas that he associates with ‘philosophy’, by which he here means speculative theories about such topics as the constitution o the human body. For good measure he insists that i one were to engage in that study, the proper way o doing so would be to start rom medicine. Medicine provides particularly striking examples o second-order debates parallel to those in mathematics: indeed in the Hellenistic period the disagreements among the medical sects were as much about methods and epistemology as they were about medical practice. But other elds too exhibit similar undamental divisions between competing approaches. In music theory, Barker has explored the analogous disputes rst between practitioners on the one hand, and theoretical analysts on the other, and then, among the latter, between those who treated musical sound in geometrical terms, as an in nitely divisible continuum, and those who adopted an analysis based rather on arithmetic. 11 Further a eld I may simply remark that the methods and aims o historiography are the subject o explicit comment rom Herodotus onwards. His views were criticized, implicitly, by his immediate successor Tucydides, who contrasts history as entertainment with his own ambition to provide what he calls a ‘memorial or eternity’ (1 21). But to achieve that end depended, o course, on the critical evaluation o eyewitness accounts, as well as an assumption that certain patterns o behaviour repeat themselves thanks to the constancy o human nature. With the development o both the practice and the teaching o rhetoric – the art o public speaking – goes a new sense o what it takes to persuade an audience o the strength o your case – and o the weakness o your rivals’ position. Both the orators and the statesmen deployed a rich vocabulary o terms, such as apodeiknumi, epideiknumi and cognates, to express the claim that they have proved their point, as to the acts o the matter in question, as to the guilt or innocence o the parties concerned, or as to the bene ts that would accrue rom the policies they advocated. Yet that very same vocabulary was taken over rst by Plato and then by Aristotle to contrast what they claimed to be strict demonstrations on the one hand with the arguments that they now downgraded as merely plausible or persuasive, such as were used in the law courts and political assemblies – and this takes us back to mathematics, since it provides the essential background to the claims that some, but not all, mathematicians made about the strictest mode o demonstration that they could deliver. 11
Barker 1989, 2000.
Te pluralism of Greek ‘mathematics’
Aristotle was, o course, the rst to propose an explicit de nition o rigorous demonstration, which must proceed by way o valid deductive argument rom premisses that are not just true, but also necessary, primary, immediate, better known than, prior to and explanatory o the conclusions. Furthermore Aristotle draws up a more elaborate taxonomy o arguments than Plato had done, distinguishing demonstrative, dialectical, rhetorical, sophistic and eristic reasoning according rst to the aims o the reasoner (which might be the truth, or victory, or reputation) and secondly to the nature o the premisses used (necessary, probable, or indeed contentious). Yet while the ideal that Aristotle sets or philosophy and or mathematics is rigorous, axiomatic–deductive, demonstration, he not only allows that the rhetorician will rely on what he calls rhetorical demonstration, but concedes that in philosophy itsel there may be stricter and looser modes, appropriate to different subject matter.12 Te goal the philosophers set themselves was certainty – where the conclusions reached were, supposedly, immune to the types o challenges that always occurred in the law courts and assemblies. Yet rom some points o view the best area to exempliy this was not philosophy itsel (ontology, epistemology or ethics) but, o course, mathematics. However, the attitudes o both Plato and Aristotle themselves towards mathematics were distinctly ambivalent – not that they agreed on the status o that study. For Plato, the inquiries the mathematician engages in are inerior to dialectic itsel: they are part o the prior training or the philosopher, but do not belong to philosophy itsel. Te grounds or this that he puts orward in the Republic are twoold, that the mathematician uses diagrams and that he takes his ‘hypotheses’ or granted, as ‘clear to all’.13 So although mathematics studies intelligible objects and so is superior to any study devoted to perceptible ones, it is inerior to dialectic which is purportedly based ultimately on an ‘unhypothesised starting point’, the idea o the Good. Aristotle, by contrast, clearly accepts that mathematical arguments can meet the requirements o the strictest mode o demonstration, since he privileges mathematical examples to illustrate that mode in the Posterior 12 13
Lloyd 1996: ch. 1. Te interpretation o the expression ‘as clear to all’,hōs panti phanerōn, in the Republic 510d1, is disputed. My own view is that Plato is unlikely not to have been aware that many o the hypotheses adopted by the mathematicians were contested (including or example the de nitions o straight line and point). When Socrates says that the mathematicians give no account to themselves or anyone else about their starting-points, it would seem that this is their claim, rather than (as it has generally been taken) their warrant. Burnyeat (2000: 37), however, has argued that there is no criticism o mathematics in this text, but simply an observation o an inevitable eature o their methods.
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304 Analytics. But mathematics suffers rom a different shortcoming, in his
view, which relates to the ontological status o the subject matter it deals with. Unlike Plato, who suggested that mathematics studies separate intelligible objects that are intermediate between the Forms and sensible particulars, Aristotle argued that mathematics is concerned with the mathematical properties o physical objects.14 While physical objects meet the requirements o substance-hood, what mathematics studies belongs rather to the category o quantity than to that o substance. While Plato and Aristotle disagreed about the highest mode o philosophizing, ‘dialectic’ in Plato’s case, ‘ rst philosophy’ in Aristotle’s, they both considered philosophy to be supreme and mathematics to be subordinate to it. Yet mathematics obviously delivered demonstrations, and exempli ed the goal o the certainty and incontrovertibility o arguments, ar more effectively than metaphysics, let alone than ethics. Once Euclid’s Elements had shown how virtually the whole o mathematical knowledge could be represented as a single, comprehensive system, derived rom a limited number o indemonstrable starting points, that model exerted very considerable in uence as an ideal, not just within the mathematical disciplines, but well beyond them.15 Euclid’s own Optics, like many treatises in harmonics, statics and astronomy, proceeded on an axiomatic–deductive basis, even though the actual axioms Euclid invoked in that work are problematic.16 More remarkably Galen sought to turn parts o medicine into an axiomatic–deductive system just as Proclus did or theology in his Elements of Teology.17 Te prestige o proo ‘in the geometrical manner’,more geometrico, made it the ideal or many investigations despite the apparent difficulties o implementing it. Te chie problem lay not with deductive argument itsel, but with its premisses. Aristotle had shown that strict demonstration must proceed 14 15
16
17
Lear 1982. As noted, the question o whether Hippocrates o Chios had a clear notion o ultimate starting-points or axioms in his geometrical studies is disputed. In his quadratures o lunes he takes a starting-point that is itsel proved, and so not a primary premiss. Ancient historians o mathematics mention the contributions o Archytas, Eudoxus, Teodorus and Teaetetus leading up to Euclid’s ownElements, but while the commentators on that work identiy particular results as having been anticipated by those and other mathematicians, the issue o how systematic their overall presentation o mathematical knowledge was remains problematic. Tus one o Euclid’s de nitions in theOptics (de. 3, 2.7–9: c. Proposition 1, 2.21–4.8) states that those things are seen on which visual rays all, while those are not seen on which they do not. Tat seems to suggest that visual rays are not dense, a conception that con icts with the assumption o the in nite divisibility o the geometrical continuum. See Brownson 1981; Smith 1981; Jones 1994. Lloyd 2006c.
Te pluralism of Greek ‘mathematics’
rom premisses that are themselves indemonstrable – to avoid the twin aws o circular argument and an in nite regress. I the premisses could be proved, then they should be, and that in turn meant that they could not be considered ultimate, or primary, premisses. Te latter had to be sel-evident, autopista, or ex heautōn pista. Yet the actual premisses we nd used in different investigations are very varied. o start with, the kinds or categories o starting points needed were the subject o considerable terminological instability. Aristotle distinguished three types, de nitions, hypotheses and axioms, the latter being subdivided into those speci c to a particular study, such as the equality axiom, and general principles that had to be presupposed or intelligible communication, such as the laws o non-contradiction and excluded middle. Euclid’s triad consisted o de nitions, common opinions (including the equality axiom) and postulates. Archimedes in turn begins his inquiries into statics and hydrostatics by setting out, or example, the postulates, aitēmata, and the propositions that are to be granted, lambanomena, and elsewhere the primary premisses are just called starting points or principles, archai. As regards the actual principles that gure in different investigations, they were ar rom con ned to what Aristotle or Euclid would have accepted as axioms. In Aristarchus’ exploration o the heliocentric hypothesis, he set out among his premisses that the xed stars and the sun remain unmoved and that the earth is borne round the sun on a circle, where that circle bears the same proportion to the distance o the xed stars as the centre o a sphere to its surace. Archimedes, who reports those hypotheses in the Sand-Reckoner 2 218.7–31, remarks that strictly speaking that would place the xed stars at in nite distance. Te assumption involves, then, what we would call an idealization, where the error introduced can be discounted. But in his only extant treatise, On the Sizes and Distances of the Sun and Moon, Aristarchus’ assumptions include a value or the angular diameter o the moon as 2°, a gure that is ar more likely, in my view, to have been hypothetical in the sense o adopted purely or the sake o argument, than axiomatic in the sense o accepted as true. Meanwhile outside mathematics, we nd Galen, or example, taking the principles that nature does nothing in vain, and that nothing happens without a cause, as indemonstrable starting points or certain deductions in medicine. In Proclus, the physical principles that natural motion is rom, to, or around the centre, are similarly treated as indemonstrable truths on which natural philosophy can be based. Te disputable character o many o the principles adopted as axiomatic is clear. Euclid’s own parallel postulate was attacked on the grounds that it should be a theorem proved within the system, not a postulate at all,
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although attempts to provide a proo all turned out to be circular. Yet the controversial character o many primary premisses in no way deterred investigators rom claiming their soundness. Te demand or arguments that are unshakeable or immovable, unerring or inallible, in exible in the sense o not open to persuasion, indisputable, irreutable or incontrovertible is expressed by different authors with an extraordinary variety o terms. Among the most common are akinēton (immovable), used or example by Plato at imaeus 51e, ametapeiston or ametapiston (not subject to persuasion), in Aristotle’s Posterior Analytics 72b3 and Ptolemy’s Syntaxis 1 1 6.17–21, anamartēton (unerring), in Plato’s Republic 339c, ametaptōton (unchanging) and ametaptaiston (inallible), the rst in Plato’simaeus 29b and Aristotle’s opics 139b33, and the second in Galen, K 17(1) 863.3, and especially the terms anamphisbētēton, incontestable (already in Diogenes o Apollonia Fr. 1 and subsequently in prominent passages in Hero, Metrica 3 142.1, and in Ptolemy, Syntaxis 1 1 6.20 among many others) and anelegkton, irreutable (Plato, Apology 22a, imaeus 29b, all the way down to Proclus in his Commentary on Euclid’s Elements68.10).18 Te pluralism o Greek mathematics thus itsel has many acets. Te actual practices o those who in different disciplines laid claim to the title o mathēmatikos varied appreciably. Tey range rom the astrologer working out planetary positions or a horoscope, to the arithmetical proos and use o symbolism discussed by Mueller and Netz in their chapters, to the proo o the in nity o primes in Euclid or that o the area o a parabolic segment in Archimedes. Tere was as much disagreement on the nature o the claims that ‘mathematics’ could make as on their justi cation. One group asserted the pre-eminence o mathematics on the grounds that it achieved certainty, that its arguments were incontrovertible. Many philosophers and quite a ew mathematicians themselves joined together in seeing this as the great pride o mathematics and the source o its prestige. But the disputable nature o the claims to indisputability kept breaking surace, either in general or in relation to particular results. Moreover while there was much deadly serious searching afer certainty, there was also much playulness, the ‘ludic’ quality that Netz has associated with other aspects o the 18
It is striking that the term anamphisbētētonmay mean indisputable or undisputed, just as in Tucydides (1 21) the term anexelegkton means beyond reutation (and so also beyond veri cation). In neither case is there any doubt, in context, as to how the word is to be understood. Tat is less clear in the case o the chie term or ‘indemonstrable’, anapodeikton, which Galen has been seen as using o what has not been demonstrated (though capable o demonstration) although in Aristotle it applies purely to what is incapable o being demonstrated (see Hankinson 1991).
Te pluralism of Greek ‘mathematics’
aesthetics that began to be cultivated in the Hellenistic period.19 In the case o mathematics, there were occasions when its practitioners delighted in complexity and puzzlement or their own sakes. From a comparative perspective what are the important lessons to be learnt rom the material I have thus cursorily surveyed in this discussion? Te points made in my last paragraph provide the basis or an argument that tends to turn a common assumption about Greek mathematics on its head. While one image o mathematics that many ancients as well as quite a ew modern commentators promoted has it that mathematics is the realm o the indisputable, it is precisely the disputes about both rst-order practices and second-order analysis that mark out the ancient Greek experience in this eld. Divergent views were entertained not just about what ‘mathematics’ covered, but on what its proper aims and methods should be. Te very uidityand indeterminacy o the boundaries between different intellectual disciplines may be thought to have contributed to the construction o that image o mathematics as the realm o the incontrovertible – contested as that image was. But we may remark that that idea owed as much to the ruminations o the philosophers – who used it to propose an ideal o a ‘philosophy’ that could equal and indeed surpass mathematics – as it did to the actual practices o the mathematicians themselves. It may once have been assumed that the development o the axiomatic– deductive mode o demonstration was an essential eature o the development o mathematics itsel. But as other studies in this volume amply show, there are plenty o ancient traditions o mathematical inquiry that got on perectly well, grew and ourished, without any idea o the need to de ne their axiomatic oundations. In Greece itsel, as we have seen, it is ar rom being the case that all those who considered themselves, or were considered by others, to be mathematicians thought that axiomatics was obligatory. Tis raises, then, two key questions with important implications or comparativist studies. First how can we begin to account or the particular heterogeneity o the Greek mathematical experience and or the way in which the axiomatic–deductive model became dominant in some quarters? Second what were the consequences o the hierarchization we nd in some writers on the development and practice o mathematics itsel? In relation to the rst question, my argument is that there was a crucial input rom the side o philosophy, in that it was the philosopher Aristotle who rst explicitly de ned rigorous demonstration in terms o valid deductive argument rom indemonstrable primary premisses – an ideal 19
Netz 2009.
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that he promoted in part to create a gap between demonstrative reasoning and the merely plausible arguments o orators and others. Whether or how ar Aristotle was in uenced by already existing mathematical practice is a question we are in no position to answer de nitively. But certainly his was the rst explicit de nition o such a style o demonstration, and equally clearly soon aferwards Euclid’s Elements exempli ed that style in a more comprehensive manner than any previously attempted. From this it would appear that it was the particular combination o cross-disciplinary and interdisciplinary rivalries in Greece that provided an important stimulus to the developments we have been discussing. Elsewhere in other mathematical traditions there was certainly competition between rival practitioners. It is or the comparativist to explore how ar the rivalries that undoubtedly existed in those traditions conormed to or departed rom the patterns we have ound in Greece. Ten on the second question I posed o the consequences o the proposal by certain Greeks themselves o a hierarchy in which axiomatic–deductive demonstration provided the ideal, we must be even-handed. On the one hand we can say that with the development o axiomatics there was a gain in explicitness and clarity on the issue o what assumptions needed to be made or conclusions that could claim certainty. On the other there was evidently also a loss, in that the demand or incontrovertibility could detract attention rom heuristics, rom the business o expanding the subject and obtaining new knowledge. Tis is particularly evident when Archimedes remarks that conclusions obtained by the use o his Method had thereafer to be proved rigorously using the standard procedures o the method o exhaustion. I we can recognize – with one Greek point o view – that there was good sense in the search or axioms insoar as that identi ed and made explicit the oundations on which the deductive structure was based, we should also be conscious – with another Greek opinion indeed – o a possible con ict between that demand or incontrovertibility and the need to get on with the business o discovery.
Bibliography Barker, A. D. (1989) Greek Musical Writings, vol. . Cambridge. (2000) Scienti c Method in Ptolemy’s Harmonics. Cambridge. Brownson, C. D. (1981) ‘Euclid’s optics and its compatibility with linear perspective’, Archive for History of Exact Sciences 24: 165–94.
Te pluralism of Greek ‘mathematics’
Burkert, W. (1972) Lore and Science in Ancient Pythagoreanism. Cambridge, MA. Burnyeat, M. F. (2000) ‘Plato on why mathematics is good or the soul’, in Mathematics and Necessity, ed. . Smiley. Oxord: 1–81. Cuomo, S. (2000) Pappus of Alexandria and the Mathematics of Late Antiquity . Cambridge. (2001) Ancient Mathematics. London. Hankinson, R. J. (1991) ‘Galen on the oundations o science’, inGaleno: Obra, Pensamiento e In uencia, ed. J. A. López Férez. Madrid: 15–19. Jones, A, (1994) ‘Peripatetic and Euclidean theories o the visual ray ’, Physis 31: 47–76. Knorr, W. R. (1981) ‘On the early history o axiomatics: the interaction o mathematics and philosophy in Greek antiquity’, inTeory Change, Ancient Axiomatics and Galileo’s Methology, ed. J. Hintikka, D. Gruender and E. Agazzi. Dordrecht: 145–86. (1986) Te Ancient radition of Geometric Problems. Boston, MA. Lear, J. (1982) ‘Aristotle’s philosophy o mathematics’, Philosophical Review 91: 161–92. Lloyd, G. E. R. (1996) Aristotelian Explorations. Cambridge. (2006a) ‘Te alleged allacy o Hippocrates o Chios’, in G. E. R. Lloyd, Principles and Practices in Ancient Greek and Chinese Science . Aldershot: ch. . (Originally published in 1987 in Apeiron 20: 103–83.) (2006b) ‘Te pluralism o intellectual lie beore Plato’, in G. E. R. Lloyd, Principles and Practices in Ancient Greek and Chinese Science . Aldershot: ch. . (srcinally published as ‘Le pluralisme de la vie intellectuelle avant Platon’, inQu’est-ce que la philosophie présocratique?, ed. A. Laks and C. Louguet (2002). Lille: 39–53.) (2006c) ‘Mathematics as model o method in Galen’, in G. E. R. Lloyd, Principles and Practices in Ancient Greek and Chinese Science . Aldershot: ch. . (Originally published in Philosophy and the Sciences in Antiquity, ed. R. W. Sharples (2005). Aldershot: 110–30.) Mendell, H. (1998) ‘Making sense o Aristotelian demonstration’, Oxford Studies in Ancient Philosophy 16: 161–225. Mueller, I. (1982) ‘Aristotle and the quadrature o the circle ’, inIn nity and Continuity in Ancient and Medieval Tought , ed. N. Kretzmann. Ithaca, NY: 146–64. Netz, R. (1999) Te Shaping of Deduction in Greek Mathematics. Cambridge. (2004) Te ransformation of Mathematics inthe Early Mediterranean World. Cambridge. (2009) Ludic Proof. Cambridge. Smith, A. M. (1981) ‘Saving the appearances o the appearances: the oundations o classical geometrical optics’, Archive for History of Exact Sciences 24: 73–99.
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310 Szabó, Á. (1978) Te Beginnings of Greek Mathematics, trans. A. M. Ungar. Budapest. (Originally published as Anfänge der griechischen Mathematik. Munich (1969).) ybjerg, K. (2000) ‘Doing philosophy with machines: Hero o Alexandria’s rhetoric o mechanics in relation to the contemporary philosophy’, PhD thesis, University o Cambridge.
9
Generalizing about polygonal numbers in ancient Greek mathematics
Introduction Te main source or our inormation about the Greek handling o what are called polygonal numbers is the Introduction to Arithmetic o Nicomachus o Gerasa ( c. 100 ).1 Heath says o the Introduction that “Little or nothing in the book is srcinal, and, except or certain de nitions and re nements o classi cation, the essence o it evidently goes back to the early Pythagoreans.”2 I am not interested in this historical claim, the evidence or which is very slight; indeed I am not interested in chronology at all but only in certain eatures o Nicomachus’ treatment o polygonals, which I discuss in Section 1 , and in the general argumentative structure o a short treatise by Diophantus called On Polygonal Numbers,3 which I discuss in Section 2.
1. Nicomachus of Gerasa In the Introduction Nicomachus makes a contrast between the standard Greek way o writing numbers, in which, e.g., 222 is written σκβ, where σ represents 200, κ 20, and β 2, and what he says is a more natural way: .6.2 First one should recognize that each letter with which we reer to a number . . . signi es it by human convention and agreement and not in a natural way; the natural, direct (amethodos), and consequently simplest way to signiy numbers would be the setting out o the units in each number in a line side by side . . . : 1
2 3
Greek text: Hoche 1866; English translation: D’Ooge 1926;French translation: Bertier 1978. Tere is material parallel to Nicomachus’ presentation in Teon o Smyrna (Hiller 1878). For dates o individuals I use oomer’s articles inTe Oxord Classical Dictionary (Hornblower and Spaworth 1996). Heath 1921: 99. Greek text: 1893: 450,1–476,3; French translation: eVr Eecke 1926. I do not discuss the nal part o the treatise (476,4–480,2), a broken-off and inconclusive attempt to show how to nd how many kinds o polygonal a given number is. Te Oxord Classical Dictionary locates Diophantus in the interval between 150 and 280 . Heath 1921: 448 says that “he probably ourished A.D. 250 or not much later.”
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unit two three our ve
α, αα, ααα, αααα, ααααα,
and so on.
Nicomachus’ “natural” representation o numbers would seem to break down the customary Greek contrast between the numbers and the unit, but Nicomachus insists that it does not: .6.3 Since the unit has the place and character o a point, it will be a principle (arkhê) . . . o numbers . . . and not in itsel (oupô) . . . a number, just as the point is a principle o line or distance and not in itsel a line or distance.
We nd a close analog o Nicomachus’ “natural” representation o numbers in the account o nitary number theory in Hilbert and Bernays’ great work Grundlagen der Mathematik, except that in the Grundlagen the alphas are replaced by strokes. As that work makes clear, this representation provides a basis or developing all o elementary arithmetic, including everything known to the Greeks. Much the most important eature o the representation in this regard is the treatment o the numbers as ormed rom an initial object (the unit or one) by an inde nitely repeatable successor operation which always produces a new number. Tis treatment validates de nition and proo by mathematical induction, the core o modern number theory. Te nitary arithmetic o Hilbert and Bernays rests essentially on the intuitive manipulation o sequences o strokes (units) together with elementary inductive reasoning.4 It is difficult or me to see any substantial difference between the manipulation o sequences o strokes or alphas and the manipulation o lines and gures in what is requently called cut-and-paste geometry; the objects are different, but the reasoning seems to me to be in an important sense the same. I mention this modern orm o elementary arithmetic only to provide a contrast with its ancient orebears. Nicomachus relies heavily on the notion o numbers as multiplicities o units and the representation o them as collections o alphas, but, afer he has introduced his natural representation, it by and large vanishes in avor o a much more clearly geometric or con gurational representation in which three is a triangular number, our a square number, and ve a pentagonal number(Figure 9.1). 4
In this paper I use words like “inductive” and “induction” only in connection with mathematical induction.
Polygonal numbers in ancient Greek mathematics
α α
α
α
α
α
α
α α
α
α
α
Figure 9.1 Geometric representation o polygonal numbers.
Nicomachus also mentions hexagonal, heptagonal, and octagonal numbers, and there is no question that he has the idea o an n-agonal number, or any n, but he only expresses this with words like “and so on orever in the direction o increase” (aei kata parauxêsin houtôs; .11.4). It is clear that Nicomachus intends to make some kind o generalization, but it is not at all clear what, i any, theoretical or mathematical ideas underlie it. Any connection between what he says and the natural representation o numbers is at best indirect. Nicomachus is relying on the idea that the numbers go on orever, but much more central to his account o polygonal numbers is the geometric act that an n-agon is determined by the n points which are its vertices. I induction lies behind the reasoning, it is not made at all explicit. I turn now to some urther eatures o what Nicomachus says. Te rst sentence o his description o triangular numbers is quite opaque, but it is clearly intended to bring out their con gurational aspect. I quote it in the translation o d’Ooge: II.8.1 A triangular number is one which, when it is analyzed into units, shapes into triangular orm the equilateral placement o its parts in a plane. Examples are 3, 6, 10, 15, 21, 28, and so on in order. For their graphic representations (skhêmatographiai) will be well-ordered and equilateral triangles . . . .
Here again we have the thought o continuing inde nitely. Nicomachus now indicates the arithmetical procedure or generating these triangular numbers, again insisting on the distinction between the unit and a number even though leaving it aside would simpliy his description. And, proceeding as ar as you wish, you will nd triangularization o this kind, making the thing which consists o a unit rst o all most elementary, so that the unit may also appear as potentially a triangular number, with 3 being actually the rst. .8.2 Te sides will increase by consecutive number, the side o the potentially rst being one, that o the actually rst (i.e., 3) two, that o the actually second (i.e., 6) three, that o the third our, o the ourth ve, o the fh six, and so on orever.
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I we ignore the distinction between a unit and a number,5 we may express Nicomachus’ claim here as: Te side o the nth (actual or potential) triangular number is n.
Nicomachus now turns to deal more explicitly with the question o the relationship between the sequence o triangular numbers and the “natural” numbers: .8.3 riangular numbers are generated when natural number is set out in sequence (stoikhêdon) and successive ones are always added one at a time starting rom the beginning, since the well-ordered triangular numbers are brought to completion with each addition and combination. For example, rom this natural sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, i I take the very rst item I get the potentially rst triangular number, 1: α
then, i I add to it the next term, I get the actually rst triangular number, since 3 is 2 and 1, and in its graphic representation it is put together as ollows: two units are placed side by side under one unit and the number is made a triangle: α α
α
And then, ollowing this, i the next number, 3, is combined with this and spread out into units and added, it gives and also graphically represents 6, which is the actually second triangular number: α α α
α α
α
Nicomachus continues in this vein or the rst seven (potential and actual) triangular numbers, essentially showing that: Te nth triangular number is the sum o the rst n “natural” numbers.
5
As I shall sometimes do, without – I hope – introducing any conusion or uncertainty.
Polygonal numbers in ancient Greek mathematics
He proceeds to show in the same way that: Te nth square number is the sum o the rst n odd numbers and its side is n.
but in this case the odd numbers are added so as to preserve the square shape (Figure 9.2). α
α
α
α
α
α
αααα
1
α
α
α
α
α
αααα
α
α
α
αααα
1+3
1 + 3 + 5
αααα
1 + 3 + 5 + 7.
Figure 9.2 Te generation o square numbers.
Te ormulation corresponding to the presentation o the pentagonal numbers is: Te nth pentagonal number is the sum o the rst n numbers x1, x2, . . . , xn which are such that xi
1
+
=
xi + 3, and its side is n.
Te rst three are represented below (Figure 9.3). α
1
α
α
α
α
αα
α
α
α
1 +4
α
α
α
α
α
α
α
α
1+4+7
Figure 9.3 Te generation o the rst three pentagonal numbers.
We are not given a graphic representation o the the next pentagonal number 22, but its representation would certainly be the ollowing (Figure 9.4):
315
316
α
α
α
α
α
α
αααα
αααα
αααα
αααα
1 + 4 + 7 + 10
Figure 9.4 Te graphic representation o the ourth pentagonal number.
Nicomachus proceeds through the octagonal numbers without gures, making clear that: [Nic*]. Te sum o the rst n numbers x1, x2, . . . , xn which are such that xi is the nth j+2-agonal number and its side is n.
1
+
=
xi + j
He then turns to showing that his presentation o polygonal numbers is in harmony with geometry ( grammikê ), something which he says is clear both rom the graphic representation and rom the ollowing considerations: .12.1 Every square gure divided diagonally is resolved into two triangles and every square number is resolved into two consecutive triangulars and thereore is composed o two consecutive triangulars. For example, the triangulars are: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc., and the squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. I you add any two consecutive triangulars whatsoever you will always produce a square, so that in resolving any square you will be able to make two triangulars
Polygonal numbers in ancient Greek mathematics
rom them. And again i any triangle is joined to any square gure 6 it produces a pentagon, or example i the triangular 1 is joined to the square 4, it makes the pentagonal 5, and i the next , that is 3, is added to the next 9 it makes the pentagonal 12, and i the ollowing 6 is added onto the ollowing 16, it gives the ollowing 22, and 25 added to 10 gives 35, and so on orever.
Nicomachus states similar results or adding triangulars to pentagonals to get hexagonals, to hexagonals to get heptagonals, and to heptagonals to get octagonals, “and so on ad in nitum.” He introduces a table (able 9.1) as an aid to memory: able 9.1: riangles
1
3
6
10
15
21
28
36
45
54
Squares
1
4
9
16
25
36
49
64
81
100
Pentagons
1
5
12
22
35
51
70
92
117
145
Hexagons
1
6
15
28
45
66
91
120
153
190
Heptagons
1
7
18
34
55
81
112
148
189
2357
and describes some o the relevant sums, results which we might ormulate as: Te n+1th square number is the nth triangular number plus the n+1th triangular number; Te n+1th pentagonal number is the nth triangular number plus the n+1th square number,
or, generally, Te n+1th k+1-agonal number is thenth triangular number plus then+1th k-agonal number.
At this point I would like to introduce some o Heath’s remarks about Nicomachus’ Introduction: It is a very ar cry rom Euclid to Nicomachus. Numbers are represented in Euclid by straight lines with letters attached, a system which has the advantage that, as in algebraical notation, we can work with numbers in general without the necessity o giving them speci c values . . . . Further, there are no longer any proos in the proper sense o the word; when a general proposition has been enunciated, Nicomachus regards it as suffi cient to show that it is true in particular instances; sometimes we are lef to iner the proposition by induction rom particular cases which are alone given. . . . probably Nicomachus, who was not really a mathematician, intended his 6 7
Here some exaggeration, since the triangle and the square have to “ ttogether.” Apparently the octagons are missing.
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318
Introduction to be, not a scienti c treatise, but a popular treatment o the subject calculated to awaken in a beginner an interest in the theory o numbers . . . . Its success is difficult to explain except on the hypothesis that it was at rst read by philosophers rather than mathematicians . . . , and aferwards became generally popular at a time when there were no mathematicians lef, but only philosophers who incidentally took an interest in mathematics.8
Heath’s remarks here are aimed at the whole o theIntroduction, but I wish only to consider them in relation to Nicomachus’ treatment o polygonal numbers. Tere is no question that, as Heath also notes, Nicomachus’ owery and imprecise language is a “ar cry” rom Euclid’s sparse, ormal ormulations. But the representation o polygonal numbers by straight lines would obliterate their con gurational nature. Nicomachus shows how triangular con gurations o units can be generated as the series 1, 1+2, 1+2+3, etc. But I do not see what he could do to “prove” this act and, thereore, how he could “prove” any act about polygonal numbers as con gurations. O course, we know how to prove things about polygonal numbers, namely by eliminating all geometric content and transorming Nic*, which or Nicomachus expresses an arithmetical act about con gurations, into an arithmetical de nition in which the geometrical terminology is at most a convenience, perhaps as ollows: [Degeo/arith]. p is the nth j + 2-agonal number with side n i and only i p = x1 + x2 + ⋅⋅⋅ + x , where x = x + j and x = 1. n i 1 i 1 +
I assume that Fowler had something o this kind in mind when he advanced the hypothesis that lying behind Nicomachus’ presentation were ancient proos using mathematical induction.9 I doubt this very much, but the more important point or me is that, unless something like Degeo/arith is used to eliminate the con gurational aspect o polygonal numbers, anything like a Euclidean oundation or the theory o them lies well beyond the scope o Greek mathematics.
2. Te argument of Diophantus’ On Polygonal Numbers In annery’s edition oOn Polygonal Numbersthere are our propositions. Te propositions are purely arithmetical and in none o them is there a mention o polygonals.10 I quote them and give algebraic representations 8 9 10
Heath 1921: 97–9. Fowler 1994: 258. When I say that these propositions are purely arithmetical, I only mean to point out the absence o the notion o polygonality rom the ormulations and proos o the propositions.
Polygonal numbers in ancient Greek mathematics
o their content; in ootnotes I give simple algebraic proos o the results in order to show that the results are correct. Diophantus’ arguments are cumbersome and roundabout. [452,2] [Dioph 1] I three numbers exceed one another by an equal amount, then eight times the product o the greatest and the middle one plus the square on the least produces a square the side o which is equal to the sum o the greatest and twice the middle one. i x = y + k and y = z + k, then 8xy + z2 = (x + 2y)2.11 [454,6] [Dioph 2] I there are numbers in any multitude in equal excess, o the greatest over the least is their excess multiplied by one less than the multitude o numbers set out. i x1, x2, . . . xn 1 are such that xi +
1
+
=
xi + j then xn
+
1
−
x1 = nj.12
[456,2] [Dioph 3] I there are numbers in any multitude in equal excess, the sum o the greatest and least multiplied by their multitude makes a number which is double o the sum o the numbers set out. i x1, x2, . . . xn are such that xi 1 = xi + j then (xn + x1)n = 2(x1 + x2 + . . . + xn).13 +
[460,5] [Dioph 4] I there are numbers in any multitude in equal excess starting rom the unit, then the sum multiplied by eight times their excess plus the square o two less than their excess is a square o which the side minus 2 will be their excess
11
I do not mean to suggest, nor do I believe, that Diophantus’ reasoning does not include geometric elements o the kind we nd in the so-called geometric algebra o Book 2 o Euclid’s Elements. But discussion o that issue would require a detailed examination o Diophantus’ proos, a task which I cannot undertake here. Proo: Let x = z + 2k and y = z + k. Ten we should prove that: 8(z + 2k)(z + k) + z2 = ((z + 2k) + 2(z + k))2. But 8 ( z + 2k ) ( z + k ) + z 2 = 8 ( z2 + 3zk + 2k2 ) + z2 = 9z2 + 24zk + 16k2 = ( 3z + 4k) 2 = ( ( z + 2k ) + 2z + 2k ) 2 = ( ( z + 2k ) + 2 (z + k ) ) 2.
12
13
Dioph 2 is sufficiently obvious that there is really nothing to prove, the basic idea being that x2 = x1 + j, x3 = x2 + j = x1 + 2j, x4 = x3 + j = x1 + 3j, and so on. We give an inductive proo oDioph 3. Forn = 1 the theorem says that x − x = 1 . j. Suppose 2 1 (x1 + xn)n = 2(x1 + x2 + . . . + xn). We wish to show that: (x1 + xn 1)(n + 1) = 2(x1 + x2+ . . . + xn 1). +
+
But: (x1 + xn 1)(n + 1) = (x1 + xn + j)(n + 1) (x1 + xn)n + x1 + xn + (n + 1)j = 2(x1 + x2+ . . . + xn) + x1 + nj + xn + j = 2(x + x + . . . + x ) + x +x = 2(x + x + . . . + x ). 1 2 n n 1 n 1 1 2 n 1 +
=
+
(Tat xn
1
+
=
+
x1 + nj is a trivial reormulation o Dioph 2.)
+
319
320
multiplied by a certain number which, when a unit is added to it, is double o the multitude o all the numbers set out with the unit. i p = x1 + x2 + . . . + xn, where xi 1 = xi + j and x1 = 1, then p8j + (j − 2)2 = ((2n − 1)j + 2)2 [= ((n + n − 1)j + 2)2]. +
It is easy to prove Dioph 4 using Dioph 2 and 3, 14 as Diophantus does, although his argument is cumbersome. Here I wish only to present the very beginning o his argument and a diagram, provided by me, representing it. [460,13] For let AB, CD, EF be numbers in equal excess starting rom the unit. 15 I say that the proposition results. For let there be as many units in GH as the numbers set out with the unit. And since the excess by which EF exceeds a unit is the excess by which AB exceeds a unit multiplied by GH minus 1, 16 i we set out each unit, AK, EL, GM, we will have that LF is KB multiplied by MH. So LF is equal to the product o KB, MH. And i we set out KN as 2, we will investigate whether, i the sum is multiplied by 8 KB (which is their excess) and the square o NB (which is less than their excess by 2) is added, the result is a square o which the side minus 2 produces a number which is their excess (KB) multiplied by GH,HM together ( Figure 9.5). A
KN 1
B
K
E
B
L
F
M
H
n− 1
1
Figure 9.5 Diophantus’ diagram, Polygonal Numbers, Proposition 4. 14
Since:
=
((2n −1)j + 2)2 − ( j −2)2 4n2j2 − 4nj2 + j2 + 8nj − 4j + 4 − j2 + 4j − 4
= 4n2j2 − 4j2n + 8nj = 4j(n2j − jn + 2n), to prove Dioph 4 we need only prove:
2(x1 + x2 + . . . + xn) = n2j − jn + 2n, or, by Dioph 3: (xn + x1)n = n2j − jn + 2n, that is xn+ x1 = nj − j + 2. 15 16
j
xn
xn − 1
1 G
x2
j −2
2
Note that AB, CD, and EF are numbers, not the unit. C. Dioph 2.
n
Polygonal numbers in ancient Greek mathematics
As I have said, the material described thus ar in this section is purely arithmetical. However, i one accepts Degeo/arith, what Diophantus has shown is that: [Dioph 4geo/arith] i p is the nth j + 2-agonal number, then p8j + (j − 2)2 = ((2n − 1)j + 2)2.
It is clear rom Diophantus’ initial less speci c statement o what he will show that he does think that he can establish this : [450,11] Here it is established ( edokimasthê) that i any polygonal is multiplied by a certain number (which is a unction ( kata tên analogian) o the multitude o angles in the polygonal) and a certain square number (again a unction o the multitude o angles in it) is added, the result is a square. i p is a j + 2-agonal number, there are unctions and c such that ( j + 2)p + c( j + 2) is a square number.
Afer announcing this result Diophantus states the goal o the treatise: [450,16] We will establish this and indicate how one can nda prescribed polygonal with a given side and how the side o a given polygonal can be taken.
Tat is, how to nd (1) the j + 2-agonal p with side n and (2) the side o a j + 2-agonal p.
Tis last subject is the concern o the nal part o the treatise (472,21– 476,3). Nic* allows one to solve these in a slightly cumbersome mechanical way, but what Diophantus proves enables him to give what amounts to ormulae or the solutions: (1)
p
=
(2) n =
1 2
((2n −1) +j
2 2 2)− − ( j 2) 8j
2 p8j + ( j −2) – 2
j
+
1
,
.
Tis last material is quite mundane, and I shall not discuss it. My major concern will be with the material immediately ollowing the presentation o the arithmetical results Dioph 1 to 4. For those our propositions are purely arithmetical; they do not say anything about polygonal numbers and certainly do not establish anything about spatial con gurations o units. It is in the remainder o the treatise that Diophantus tries to establish a general truth corresponding to Nic*, but as I have indicated, I believe that it is impossible to prove this truth within the con nes o Greek mathematics.
321
322
What I will try to explain is the speci c reason why Diophantus’ attempt to do so ails, emphasizing that, although his reasoning is mathematically much more elaborate than Nicomachus’, his handling o generalization is essentially the same, namely the presentation o examples which make the general truth “obvious.” Beore turning to that material I want to signal the very rst statement in On Polygonal Numbers, which concerns the rst (actual) polygonal number o each kind: [450,1] Each o the numbers starting rom three which increase by one is a rst polygonal afer the unit. And it has as many angles as the multitude o units in it. Its side is the number afer the unit, i.e., 2. 3 is triangular, 4 square, 5 pentagonal, and so on.
Tis is, I think, the only application o Degeo/arith that Diophantus takes or granted, i.e., he takes or granted that: the rst j+2-agonal (afer 1) has side 2 and is j + 2.
Afer making a remark about the ordinary conception o square numbers,17 Diophantus gives (450,11 and 16) the announcement o what he is going to prove, which I have already quoted, and proves his our arithmetical propositions. It is at this point that he rst reintroduces the notion o a polygonal number in his announcement o what he intends to prove next, which is tantamount to Degeo/arith: [468,14] Tese things being the case, we say that i there are numbers starting rom the unit in any multitude and in any excess, the whole is polygonal. For it has as many angles as the number which is greater than the excess by 2, and the number o its sides is the multitude o the numbers set out with the unit.
He now invokes Dioph 4: [470,1] For we have shown that the sum o all the numbers set out multiplied by 8 KB plus the square o NB produces the square o PK.
Here Diophantus is working with a gure in which the line AKNB o Figure 6 or Dioph 4 is extended to the right so that PK is a representation o (2n − 1)j + 2.18 But to get a representation o j + 2 he also extends AKNB to the lef (Figure 9.6): [470,4] But also i we posit AO as another unit, we will have KO as two, and KN is similarly two. 17
18
[450,9] “It is immediately clear that squares have arisen because they come to be rom some number being multiplied by itsel.” Tis speci cation o PK occurs at 466,1–2.
Polygonal numbers in ancient Greek mathematics
O
A 1
O
K 1
A
N 2
j−2
B
K
N
B
K
N
B
P
Figure 9.6 Diophantus’ diagram,Polygonal Numbers.
But now Diophantus is only interested in OB (j + 2 = 1 + (1 + j)), KB (j), and BN (j − 2), and, in his only application o Dioph 1, he says: [470,6] Tereore OB, BK, BN will exceed one another by an equal amount. Tereore, 8 times the product o the greatest OB and the middle BK plus the square o the least BN makes a square the side o which is the sum o the greatest OB and 2 o the middle BK. Tereore OB multiplied by 8 KB plus the square o NB is equal to the square o OB and 2KB together. (j + 2)8j + (j − 2)2 = (j + 2 + 2j)2.
Tis is, o course, just the special case o Dioph 4 in which n = 2. o make this point clear Diophantus argues that j + 2 + 2j = (2·2 − 1)j + 2: [470,13] And the side minus two (OK) leaves 3 KB, which is KB multiplied by three. But three plus one is 2 multiplied by 2.
Diophantus underlines the analogy with Dioph 4 and then points out that OB (j + 2) is the rst j + 2-agonal number: [470,17] . . . the sum o the numbers set out with the unit produces (poiei) the same problem as OB, but OB is a chance number and is the rst polygon {o its kind} afer the unit (since AO is a unit and the second number is AB), and {OB} has two as side.
So, in addition to proving Dioph 4, Diophantus has proved a special case o it in which n =2, a case or which he has asserted that p is the rst j + 2-agonal number. Tese two propositions by themselves do not imply that whenever the conditions o Dioph 4 hold, p is a j + 2-agonal number with side n. But this is precisely what Diophantus asserts:19 [470,21] Tereore also the sum o all the numbers set out is a polygon with as many angles as OB and having as many angles as it is greater by 2 (i.e., by OK) than the excess, KB; and it has as side GH, which is the number o the numbers set out with the unit. 19
Commentators have standardly approved this “reasoning,” or at least not raised any doubts about it. See Poselger 1810: 34–5; Schulz 1822: 618; Nesselmann 1842: 475; Heath 1885: 252; Wertheim 1890: 309; Massoutié 1911: 26; and Ver Eecke 1926: 288.
323
(2n − 1)j + 2 + 2
j+2
j
324
Tere is no more basis or this generalization than or the generalizations we have seen in Nicomachus; indeed, in a sense there is even less since Diophantus has considered only rst polygonals and shown that they satisy Dioph 4. Like Nicomachus, he clearly could show the same thing or any particular example, but that hardly proves his claim or the next one: [470,27] And what is said by Hypsicles in a de nition 20 has been demonstrated, namely: I there are numbers in equal excess in any multitude starting rom the unit, then, when the excess is one the whole is triangular, when it is two, square, three, pentagonal. Te number o angles is said to be greater than the excess by two, and its sides are the multitude o numbers set out with the unit.
It is not clear exactly what the de nition o Hypsicles was.21 In Diophantus’ representation he said something about the rst three polygonals, but it seems reasonable to suppose that he at least intended a generalization and so can be credited with Degeo/arith.22 But we h ave no inormation about how he used it – i he did. In any case Diophantus would have been on rmer ooting had he made the de nition the basis o his treatise rather than purporting to do the impossible, namely demonstrate it. Had he done this he would not have had to worry about Dioph 1 and the special case o it which +
he invokes to deal with rst j 2-agonal numbers. Diophantus now applies Hypsicles’ de nition and his own results to triangular numbers. [472,5] Hence, since triangulars result when the excess is one and their sides are the greatest o the numbers set out, the product o the greatest o the numbers set out and the number which is greater by one than it is double the triangular indicated. I p = x1 + x2 + . . . + xn with xi 1 = xi + 1 and x1 = 1, then p is a triangular with side xn and xn(xn + 1) = 2p.23 +
Diophantus returns again to rst polygonal numbers. He recalls the application o Dioph 1 at 470,6. [472,9] And since OB has as many angles as there are units in it, i it is multiplied by 8 multiplied by what is less than it by two (that is by the excess; that will be 20
21 22 23
D’Ooge 1926: 246 endorses Gow’s (1884: 87) suggestion that en horôi might mean “in a book called De nition.” In itsel this suggestion seems to me unlikely, but the recurrences o the word horos in 472,14 and especially 472,20 seem to me to rule it out completely. Standard oruit: c. 150 . Contrast Nesselmann 1842: 463. Proo: It ollows rom Hypsicles’ de nition thatx1 + x2 + . . . + xn is a triangular number p with side n. But by Dioph 2xn = (n − 1) . 1 + 1 = n. And by Dioph 3 2(x1 + x2 + . . . + xn) = n(xn + x1) = xn(xn + 1).
Polygonal numbers in ancient Greek mathematics
8 × KB) the square o what is less than it by 4 is added (that is NB), it produces a square.
j + 2 is a j + 2-agonal and (j + 2)8j + (j − 2)2 is a square.
Tis, too, is immediately generalized with no justi cation. [472,14] And this will be a de nition ( horos) o polygonals: Every polygonal multiplied by 8 multiplied by what is less by two than the multitude o its angles plus the square o what is less than the multitude o angles by 4 makes a square. I p is j + 2-agonal, p8j + (j − 2)2 is a square. [472,20] In this way we have demonstrated simultaneously this de nition o polygonals and that o Hypsicles.
In this case the truth which Diophantus purports to establish as a de nition is not a de nition in the standard sense at all, since n8j + (j − 2)2 can be a square even when n is not a j + 2-agonal; 2·8·3 + (3 − 2)2 = 72, but 2 is not pentagonal.24 And his claim to have demonstrated it is just as weak as his claim to have established Degeo/arith.
Conclusion It is certainly not surprising that Diophantus’ treatise on polygonal numbers shows great mathematical skill. And it is perhaps also not surprising that its sense o logical rigor is at times not superior to that o Nicomachus. Within the limits o Greek mathematics there can be no mathematical demonstration o an arithmetical characterization o con gurationally conceived polygonal numbers. Within those limits Aristotle (Posterior Analytics 1.6 (Ross)) was correct to insist that the generic difference between arithmetic and geometry cannot be breached. 24
Tis shortcoming is already pointed out in the editio princeps o the Greek text o Diophantus (Bachet 1621: 21 o the edition oOn Polygonal Numbers). What Diophantus says at 472,14 could serve as a de nition or triangulars and squares. For, ignoring complications that would arise i one tried to avoid “numbers” less than 1, it is easy to prove that: p = 1 + 2 + . . . + n (i.e., p is a triangular) i and only i p8·1 + (3 – 4)2 is a s quare (i.e., i and only i 8p + 1 is a square); p = 1 +3 + . . . + 2n − 1 (i.e., p is a square) i and only i p8·2 + (4 – 4)2 is a s quare (i.e., i and only i 16p is a s quare, i.e., i and only i p is a square). It is tempting to think, although it cannot be proved, that Diophantus was misled by the truth o these biconditionals to the alse notion that Dioph 4 was the basis o a de nition o polygonality in general.
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Bibliography Bachet, C. (ed. and trans.) (1621) Diophanti Alexandrini Arithmeticorum Libri Sex et De Numeris Multangulis Liber Unus. Paris. Bertier, J. (trans.) (1978) Nicomaque de Gérase:Introduction Arithmétique. Paris. D’Ooge, M. L. (trans.) (1926) Nicomachus o Gerasa:Introduction to Arithmetic. University o Michigan Studies, Humanistic Series, vol. 16. New York. Fowler, D. (1994) ‘Could the Greeks have used mathematical induction? Did they use it?’, Physis 31: 253–65. Gow, J. (1884) A Short History o Greek Mathematics. Cambridge. Heath, . (1885) Diophantus o Alexandria: A Study in the History o Greek Algebra. Cambridge. (1921) A History o Greek Mathematics (2 vols.) Oxord. Hilbert, D., and Bernays, P. (1934) Grundlagen der Mathematik, vol. . Berlin. Hiller, E. (ed.) (1878) Teonis Smyrnaei Philosophi PlatoniciExpositio Rerum Mathematicarum ad Legendum Platonem Utilium. Leipzig. Hoche, R. (ed.) (1866) Nicomachi Geraseni PythagoreiIntroductionis Arithmeticae Libri . Leipzig. Hornblower, S., and Spaworth, A. (eds.) (1996) Te Oxord Classical Dictionary, 3rd edn. Oxord. Massoutié, G. (trans.) (1911) Le traité des nombres polygones de Diophante d’Alexandrie. Macon. Nesselmann, G. H. F. (1842) Die Algebra der Griechen. Berlin. Poselger, F. . (trans.) (1810) Diophantus von Alexandrien über die Polygonzahlen . Leipzig. Ross, W. D. (ed.) (1964) Aristotelis Analytica Priora et Posteriora. Oxord. Schulz, O. (trans.) (1822) Diophantus von Alexandria Arithmetische Augaben nebst dessen Schrif über die Polygonzahlen. Berlin. Ver Eecke, P. (trans.) (1926) Diophante d’Alexandrie, les six livres arithmétiques et le livre des nombres polygones. Bruges. Wertheim, P. (trans.) (1890) Die Arithmetik und die Schrif über Polygonalzahlen des Diophantus von Alexandria. Leipzig.
10
Reasoning and symbolism in Diophantus: preliminary observations
In memoriam D. H. F. Fowler
1. Introducing the problem Tis chapter raises two separate questions, one dealing with the role o reasoning in Diophantus, the other with the role o symbolism.1 Needless to say, this discussion o symbolism and reasoning in Diophantus is o philosophical interest, as the nature o symbolic reasoning is central to modern philosophy o mathematics. My main interest, or this philosophical question, is to underline our need to consider the demonstrative unction o symbolism cognitively and historically. Te promise o symbolic reasoning was ofen seen as a transition into a mode o reasoning where the subjective mind2 is excluded, and an impersonal machine-like calculation takes its place. But in reality, o course, the turn into symbolic proo must have involved the transition rom one kind o subjective operation to another, rom one set o cognitive tools to another. Te abstract question, concerning the role o ormalism as such in mathematics, may blind us to the actual cognitive unctions served by various ormal tools in different historical constellations. Tis chapter, then, may serve as an example or this kind o cognitive and historical investigation. Te speci c question concerning symbolism and reasoning in Diophantus is especially difficult and interesting. Ever since the work o Nesselmann 1
2
Te central idea o this article – that Diophantine symbolism should be primarily understood against the wider pattern o scribal practices – was rst suggested to me in a conversation with David Fowler. I will orever remember, orever miss, his voice. Te locus classicus or that is Wittgenstein’sractatus (Wittgenstein 1922) e.g. 6.126: ‘Whether a proposition belongs to logic can be calculated by calculating the logical properties o the symbol . . . ’ (italics in the srcinal); 6.1262: ‘Proo in logic is only a mechanical expedient to acilitate the recognition o tautology, where it is complicated.’ Probably, though, even the Wittgenstein o theractatus would not have denied the possibility o studying the cognitive and historical conditions under which a certain ‘mechanical expedient’ in act ‘acilitates the recognition o tautology’. But the thrust o the philosophy o mathematics suggested by Wittgenstein’sractatus was to turn attention away rom the proving mind and hand and on to the proo ’s symbols.
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(1842), it has been widely recognized that Diophantus’ symbols are not the same as those o modern algebra: his was a syncopated, not a symbolic algebra, the so-called symbols being essentially abbreviations (or a uller account o what that means, see Section 2 below). Building on this understanding, we need to avoid the Scylla and Charybdis o Diophantus studies. One, which may be called the great-divide-history-o-algebra, stresses that abbreviations are not symbols: Diophantus is not Vieta, and Diophantus’ symbols have no role in his reasoning.3 Te other, which may be called the algebra-is-algebra-history-o-algebra, stresses that symbols (even when abbreviations in character) are symbols: Diophantus is a symbolic author and his writings directly prepare the way or modern algebra (this is assumed with different degrees o sophistication in many general histories o mathematics).4 In this chapter, I shall try to show how Diophantus’ symbols derive rom his speci c historical context, and how they serve a speci c unction in his own type o reasoning: the symbols are neither purely ornamental, nor modern. So I do believe that Diophantus’ use o symbolism has a unctional role in his reasoning. But, even apart rom any such unction, it is interesting to consider the two together. Tis combination may serve to characterize Diophantus’ work. First, the work stands out rom its predecessors in the Greek mathematical tradition, indeed in the Greek literary tradition, by its oregrounding o a special set o symbols. Tis oregrounding is apparent not only in that the work in its entirety makes use o the symbols, but also in that the introduction to the work – uniquely in Greek mathematics – is almost entirely dedicated to the presentation o the symbolism.5 Second, the work stands out rom its predecessors in the Mediterranean tradition o numerical problems in its oregrounding o demonstration (in a sense that we shall try to clariy below). Te text takes the orm o a set o arguments leading to clearly demarcated conclusions, throughout organized 3
4
5
For this, see especially Klein 1934–6, a monograph that makes this claim to be the starting point o an entire philosophy o the history o mathematics. See e.g. Bourbaki 1991: 48; Boyer 1989: 204; besides o course being a theme o Bashmakova 1977). Bourbaki is laconic and straightorward (Bourbaki 1991: 48): ‘Diophantus uses, or the rst time, a literal symbol to represent an unknown in an equation.’ Boyer is balanced and careul. Noting Nesselmann’s classi cation, and stating that Diophantus was ‘syncopated’, he goes on to add that (Boyer 1989: 204) ‘with such a notation Diophantus was in a position to write polynomials in a single unknown almost as concisely as we do today’, however, ‘the chie difference between the Diophantine syncopation and the modern algebraic notation is in the lack o special symbols or operations and relations, as well as o the exponential notation’. Te introduction is in annery I.2–16, o which .4.6–12.21 is organized around the presentation o the symbolism.
Reasoning and symbolism in Diophantus
by such connectors as ‘since’, ‘thereore’, etc. Since the text is at the intersection o the Greek mathematical tradition with the Mediterranean tradition o numerical problems, it ollows that these two characteristics – oregrounding symbolism and oregrounding reasoning – may be taken to de ne it. Tis chapter ollows on some o my past work in the cognitive and semiotic practices o Greek mathematics. I bring to bear, in particular, three strands o research. I extend the theoretical concepts o deuteronomy (Netz 2004) and analysis as a tool o presentation (Netz 2000), arguing that Diophantus was primarily a deuteronomic author – intent on rearranging, homogenizing and extending past results – employing the ormat o analysis as a tool o presentation that highlights certain aspects o his practice. I urther contrast Diophantus’ use o symbolism with the geometrical practice o ormulaic expressions (N1999, ch. 4), arguing that Diophantus’ use o symbolism is designed to display the rationality o transitions inside the proo and that this display is better supported, in the case o Diophantus’ structures, by symbols as opposed to verbal ormulae. In short: because Diophantus is deuteronomic, he uses analysis; because he uses analysis, he needs to display the rationality o transitions; because he needs to display the rationality o transitions, he uses symbols.6 Further, Diophantus needs to display rationality in a precise way: both allowing quick calculation o the relationship between symbols, as well as allowing a synoptic – as well as semantic – grasp o the contents o the terms involved. o do this, he uses symbols in a precise way, which I call bimodal. Te symbols are simultaneously verbal and visual, and in this way they provide both quick calculation and a semantic grasp. What nally makes Diophantus’ symbols have this property? Tis, I argue, derives rom the nature o the symbolism as used in scribal practice in pre-print Greek civilizations. Tis involves the one main piece o empirical research underlying this chapter. I have studied systematically a group o Diophantine manuscripts, and consulted others, to show a result which is mainly negative: it must be assumed that, in the manuscript tradition, the decision whether to employ a ull word or its abbreviation was lef to the
6
By ‘Diophantus’ I mean – as we typically do – ‘the author o the Arithmetica’. I have no rm views on the authorship o On Polygonal Numbers, a work closer to the mainstream o Greek geometrical style. I indeed the two works had the same author (as the manuscripts suggest) we will nd that, or different purposes,Diophantus could deploy different genres – not a trivial result – but neither one to change our understanding o the genre o the Arithmetica. But we are not in a position to make even this modest statement so that it is best to concentrate on the Arithmetica alone.
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scribe’s discretion, and no pattern was assumed at the outset. Te two – ull word and its abbreviation – acted as allographs. Tis may be seen as a consequence o the scribal culture within which Diophantus operated. Te upshot o this chapter, then, is to situate Diophantus historically in terms o a precise deuteronomic, scribal culture, and within the context o practices available to him rom elite Greek mathematics.
2. Notes on symbolism in Diophantus We recall Nesselmann’s observation: Diophantus belongs to the category o ‘syncopated algebra’, where the text is primarily arranged as discursive, natural language (i o course in the rigid style typical o so much Greek mathematics), with certain expressions systematically abbreviated.7 In this, it is generally understood to constitute a stepping stone leading rom the rhetorical algebra o, say (i we allow ourselves such heresy), Elements Book , to the ully symbolic algebra o the moderns. As a rst approximation, let us take a couple o sentences printed in annery’s edition (prop. I.10, 1893, . 28.13–15): (1) Τετάχθω ὁ προστιθέμενος καὶ ἀϕαιρούμενος ἑκατέρῳ ἀριθμῷ ςΑ. κἂν μὲν τῷ Κ προστεθῇ, γίνεται ςΑ ΜοΚ. Let the which is added and taken away rom each number be set down, ςΑ <:Number 1>. And i it is added to 20, result: ςΑ ΜοΚ <:number 1 Monads 20>.
We see here the most important element in Diophantus’ symbolism: a special symbol or ‘number’, ς. We also see a transparent abbreviation or ‘monads’, Μο. o these should be added especially: two transparent abbreviations, or ‘dunamis’ (effectively, ‘square’),Δυ, and or ‘cube’, Κυ. Symbols or higher powers exist and are made by combining symbols or the low powers, e.g. ΔΚυ, dunamis–cube, or the fh power. An appended χ turns such a power into its related unit raction: a dunamis, Δυ, can become a dunamiston, Δυχ, or the unit raction correlated with a dunamis. (Te symbol itsel is reminiscent in orm especially o the standard scribal symbols or case endings.) Finally we should mention a special symbol or 7
For a previous, brie characterization o Diophantus’ symbolism inpractice, see Rashed 1984: lxxxi–lxxxii, whose position I ollow here. Heath 1885 : 57–82 may still be read with pro t. In general, many o the claims made in this section were made by past scholars already, and my apology or going through this section in detail is that the point is worth repeating – and should be seen in detail as an introduction to the ollowing and much more speculative discussion.
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‘lacking’, roughly an upside-downΨ (I shall indeed represent it in what ollows by Ψ, or lack o better onts. Note that this is to be understood as a ‘minus’ sign ollowed by the entirety o the remaining expression – as i it came equipped with a set o ollowing parentheses.) ogether with Greek alphabetic numerals (Α, Β, Ι, Κ, Ρ, Σ or 1, 2, 10, 20, 100, 200 . . .) one has the main system with which complex phrases can be ormed o the type, e.g. (2) ΚυΒΔυΑ ςΒ ΜοΓ Ψ Κυ Α ΔυΓ ςΔ ΜοΑ
Most o all, Diophantine reasoning has to do with manipulation o such phrases. Syntactically, note that such phrases have a xed order: one goes through the powers in a xed sequence (although in terms o Greek syntax, any order could be natural). Te numeral, also, always ollows the unit to which it reers (this, however, can be explained as natural Greek syntax). Finally, there is a xed order relative to the ‘lacking’ symbol: the subtrahend is always to the right o the symbol. Tis o course ollows rom the very meaning o ‘lacking’. Semantically, we may say that the ‘number’ unctions rather like an ‘unknown’, on which the ‘dunamis’ or the ‘cube’ depend as well (a single ‘number’ multiplied by itsel results in a single ‘dunamis’ which, once again multiplied by a ‘number’, yields a ‘cube’). Te monads, on the other hand, are independent o the ‘number’. Let us consider the wider context. When we discuss symbolism in Diophantus, we need to describe it at three levels. First, there is the symbolism which Diophantus had explicitly introduced in the preace to his treatise. Second, Diophantus has a number o airly specialized symbols which he did not explicitly set out. Tird, we should have a sense o the entire symbolic regime o the Diophantine page, bringing everything together – the markedly Diophantine, and the standard symbolism o Greek scribal practice. Te symbols explicitly introduced by Diophantus are those mentioned above (in the order in which Diophantus introduces them): Δυ, Κυ, ΔυΔ, ΔΚυ, Κυ Κ, ς, Μο, χ, Ψ. Tese then unmistakably belong to the phrases such as those o example (2), serving urther to underline the importance o this type o expression. Beyond that, the manuscripts display a variety o urther symbols. annery systematically represents symbolically in his edition such symbols as he eels, apparently, to be markedly Diophantine (on the other hand, he always resolves standard scribal abbreviations; more on this below). Te ollowing especially are noticeable among the markedly Diophantine:
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Te alphabetic numerals themselves. While Greek numbers are very ofen written out by alphabetic numerals, they are more requently spelled out in Greek writing as the appropriate number words – just as we have to decide between ‘5’ and ‘ ve’. Te avoidance o number words and the use o alphabetical numerals, instead, is thereore a decision involving a numerical code. , or ‘square’ (used here in the meaning o ‘a square number’). , or ‘the right sides’, in a right-angled triangle. Here they are studied as ul lling Pythagoras’ theorem and thereore offering an arena or equalities or square numbers. Strangely, annery does not print this symbol. Αʹ, Βʹ, Γʹ, etc. or ‘ rst’, ‘second’, ‘third’, etc. Tis is used in the important context where several numbers are involved in the problem, e.g. what we represent by ‘n1+n2=3n3’ which, or Diophantus, would be ‘the rst and the second are three times the third’, with ‘ rst’, ‘second’, etc. used later on systematically to reer to the same object. O course, such symbols are not to be conused with their respective numerals and they are differently written out. Βπλ, Γπλ, or ‘two times’, ‘three times’, etc. Tis symbolism is based on the alphabetic numerals, tucking on to them a transparent abbreviation o the Greek orm o ‘times’. ΕΙΓ: this is an especially dramatic notation whereby Diophantus rerains rom resolving the results o divisions into unit ractions, and instead writes out, like in the example above, ‘ ve thirteenths’ in a kind o superscript notation. annery urther transorms this notation into a sort o upside-down modern notation. As long as we do not mean anything technical by the word, we may reer to this as Diophantus’ ‘raction symbolism’. Te last ew mentioned symbols (with the possible exception o the raction symbolism) are not unique to Diophantus, but or obvious reasons the text has much more recourse to such symbols than ordinary Greek texts so that, indeed, they can be said to be markedly Diophantine. One ought to mention immediately that many words, typical to Diophantus, are not abbreviated. Tese all into two types. First, several central relations and concepts – ‘multiply’, ‘add’, ‘given’, etc. – are written in ully spelled out orms. In other words, Diophantus’ abbreviations are located within the level o the noun-phrase, and do not touch the structure o the sentence interrelating the noun-phrases. ‘Lacking’ is the exception to the rule that relations are not abbreviated, but it serves to con rm the rule that abbreviations are located at the level o the noun-phrase. Te ‘lacking’
Reasoning and symbolism in Diophantus
abbreviation is used inside the noun-phrase o the speci c orm o example (2) above, when a quantitative value is set out statically. Te relation o subtraction holding dynamically between such noun-phrases – when one engages in the act o subtracting a value rom a quantitative term – this operation is reerred to by a different verb, ‘take away’ (aphairein), which is not abbreviated. Further, the logical signposts marking the very rigid orm o the problem, such as ‘let it be set down’, ‘to the positions’, etc., are ully written out. In other words, just as symbolism does not reach the level o the sentence, so it does not reach the level o the paragraph. Te rule is con rmed: abbreviations are con ned to the level o the noun-phrase. I shall return to discuss the signi cance o this limitation in Section 4 below. For the time being, I note the conclusion, that Diophantus’ marked use o symbolism is not co-extensive with Diophantus’ marked use o language. Over and above Diophantus’ marked use o symbolism, it should be mentioned that Greek manuscripts, certainly rom late antiquity onwards, used many abbreviations or common words such as prepositions, connectors, etc.: our own ‘&’, or instance, ultimately derives rom such scribal practices. Tere are also many abbreviations o grammatical orms, especially case markings, so that the Greek nominal root is written, ollowed by the abbreviation or ‘ον’, ‘οις’, etc. as appropriate. Such abbreviations are o course in common use in the manuscripts o Diophantus. Most (but not all) o such symbols were transparent abbreviations and in general they could be considered as a mere aid to swif writing. Teir use is as could be predicted: the more expensive a manuscript was, the less such abbreviations would be used; they are more common in technical treatises than in literary works; humanists, proud o their mastery o Greek orms, would tend to resolve abbreviations, while Byzantine scribes – ofen scrambling to get as much into the page as possible – would also ofen tend to abbreviate. We should mention one scribal abbreviation, which is not at all speci c to Diophantus, but which is especially valuable to him: the one or the sound-sequence /is/. It so happens that this common sound-sequence is the lexical root or ‘equal’ in Greek. Since it is a very common sound-sequence, it naturally has a standard abbreviation, so that Diophantus has ‘or ree’ a symbol or this important relation. How are such symbols understood? Tat is, what is the relationship between Diophantus’ symbols, and the alphabetically written words that they replace? Te rst thing to notice, as already suggested above, is that the symbols are most ofen a transparent abbreviation o the alphabetical orm. Diophantus’ own strategy o choice in the symbols he had himsel coined was to clip the word into its rst syllable (especially when this is a simple,
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consonant–vowel syllable), which he then turned into a symbol by placing the vowel as a superscript on the consonant: Κυ, Δυ, Μο. Te symbols result rom two reductions – a word into its initial syllable, a syllable into its consonant. All o this makes sense in terms o natural language phonology so that, in such cases, Diophantus’ symbolism may be tied to the heard sound and not just to the visible trace. (It may be relevant that in all three words – monas, dunamis, kubos – the stress alls indeed on the rst syllable.) With arithm- and leipsei this simple strategy ails. Te symbols, in both cases, are more complex: perhaps some combination o the alpha and the rho o the arithmos (but this is a well-known palaeographic puzzle), certainly some reerence to the psi o the leipsei. Tis is in line with the standard symbolism, e.g. or prepositions: these are ofen rendered by a combination o their consonants (‘pros’, e.g., becoming a ligature o the pi and the rho). Note also that while alphabetical numerals do not directly represent the sounds o the number-words they stand or, the system as a whole is isomorphic to spoken numerals (two-number words, ‘two and thirty’ become two number-symbols, ΛΒ). In this, the alphabetical numeral system differs rom its main alternative in Greek antiquity, the acrophonic system where each symbol had, directly, a sound meaning ( Π or pente, ve, Δ or deka, ten, etc.: the only exception is the use o a stroke or the unit), but the acrophonic number symbolism as a whole was equivalent to the Roman system with which we are amiliar and was no longer isomorphic to spoken numerals: not ΛΒ, but ΔΔΔΙΙ. Te latter clearly is not meant to be pronounced as ‘deka-deka-deka-click-click’. In act, it is no longer a pronounced symbol: the trace has become ree o the sound. In the alphabetical system, everything can be understood as symbols standing or sounds in natural Greek: I believe this may be the reason why this system was nally preerred or most ordinary writing. With this in mind, we can see that Diophantus’ marked symbols are at least potentially spoken: the numbers, as explained above, as well as the symbols based upon them. A stroke turns a numeral into its dependent ordinal or unit-raction (identical in sound, as in symbol: compare English ‘third’, ‘ourth’, etc.). Further, ordinals are sometimes rendered in an even more direct phonological system, e.g. Δευ, abbreviating δευτερος, or ‘second’. (Tus the system or ordinals has three separate orms: the ully written-out word, the phonologically abbreviated orm and the alphabetic numeral-based orm. Tis is important, given the role o ordinals as a kind o unknown-mark in expressions such as ‘the rst number’.) Te ×-times symbolism, too, merely adds the onset consonants o the abbreviated words: Βπλ or ‘double’.
Reasoning and symbolism in Diophantus
Te symbols or square, and or sides in a right-angled triangle, are the exception, then. Tere the trace, and not the sound, becomes the vehicle o meaning. Te reason or this is clear, as the trace here has indeed such an obvious connotation. Te sign and the signi ed are isomorphic. Even so, note that the understanding is that stands not just or the concept ‘square’ but also and perhaps primarily or the sequence ‘tetragon’, as witnessed by the act that the symbol is ofen ollowed by case marking: οις or ‘tetragonois’, ‘by the squares’. Te most interesting exception is the orm , sometimes used to represent ‘squares’, the plural marked not by the sound o the case ending, but by the tracing o duplication (compare our use o ‘pp.’, or instance, or ‘pages’; notice also that the same also happens occasionally with the ‘number’ symbol). Speaking generally or Greek writing in manuscripts, the phonological nature o abbreviation symbolism becomes most apparent through the rebus principle. o provide an example: there is a standard scribal abbreviation or the Greek word ‘ara’, ‘thereore’. Tere is also an important preposition, ‘para’, meaning, roughly, ‘alongside’. Te letter pi, ollowed by the symbol or ‘ara’, may be used to represent the preposition ‘para’. Such rebus writing is common in Greek manuscripts and shows that the symbol or ‘ara’ stands not merely or the concept ‘thereore’ but, perhaps more undamentally, or the sound-sequence ‘ara’. Obviously, Diophantus’ symbolism does not lend itsel to such rebus combinations. One can mention, however, an important close analogue. We recall Diophantus’ symbol or ‘number’, meaning, effectively, the ‘unknown’. Tis may be said to be the cornerstone o Diophantus’ symbolism: on it ride the higher powers; it is the starting point or investigation in each problem. It is thus, perhaps, not inappropriate that this symbol is the least transparently phonological. It is, so to speak, Diophantus’ cipher. Crucially, it is also clearly de ned by Diophantus in his introduction: ‘Tat which possesses none o these properties [such as dunamis, cube, etc.] and has in it an indeterminate number o monads, is called a number and its symbol is ς’ (annery 6.3–5). Tus the symbol is, strictly speaking, only to be used or the indeterminate, or unknown, goal o the problem. It should be used in such contexts as ‘Let the which is added and taken away rom each number be set down, ςΑ <:Number 1>.’ Notice the two occurrences o ‘number’ in this phrase. Te rst is ‘number’ in its standard Greek meaning (which thereore, one would think, should not be abbreviable into the symbol ς). In the phrase ‘rom each number’, the word ‘number’ does not stand or an unknown number, but just or
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‘number’. It is only the second number – the one counted as ‘1’ - which serves as an unknown in this problem. Only this, then, by Diophantus’ explicit de nition, counts as a ς; appropriately, then, annery prints the rst ‘number’ as a ully spelled-out word and the s econd as a symbol. But as the reader may guess by now, there are many cases in the manuscripts where ‘number’ o the rst type is abbreviated, as well, using Diophantus ’ symbol ς.8 Tus the symbol is understoo d, at least by Diophantus ’ scribes, to range not across a s emantic range (the unknown number), but across a phonological or orthographic range (the representation o the sound, or trace, ‘arithm-’). It would indeed be su rprising i it were otherwise, given that scribal symbolism, as a system, was understood in such phonological or orthographic terms. Te text in example (1) above ollowed closely (with some variation o orthography) annery’s edition. It is clearly punctuated and spaced (as it is not in the manuscripts, not even the Renaissance ones). It has accents and aspiration marks (like the Renaissance manuscripts, but most probably unlike Diophantus’ text in late antiquity). It also sharply demarcates the two kinds o writing: explicit and markedly Diophantine symbols, which, in the proo itsel, annery systematically presents in abbreviated orm, on the one hand; and standard scribal abbreviations, which annery systematically resolves (as, indeed, philologers invariably do). As annery himsel recognized, his systematization o the symbolism was not based on manuscript evidence. I shall not say anything more on the unmarked symbolism, such as the case markings, whose usage indeed differs (as one expects) rom one manuscript to another. Tey should be mentioned, so that we keep in mind the ull context o Diophantus’ symbols. But even more important is that Diophantus’ own marked symbolism is not systematically used in the manuscripts. Te symbols described above are ofen interchanged with ully written words. Tis is as much as can be expected. Both Δυ and Δυναμις stand or exactly the same thing – the sound pattern or trace /dunamis/ – and so there is no essential reason to use one and not the other. Tus a ree interchangeability is predicted. Notice rst the orm o example (1) in all the Paris manuscripts, comparing the (translated) orm o annery’s text to that o the manuscripts: 8
Tis was pointed out already by Nesselmann 1842: 300–1. Indeed, my impression is that awareness o such quirks o Diophantus’ text was more widespread prior to annery: ollowing the acceptance o his edition, knowledge o the manuscripts (as well as o the early printed editions – whose practices, I note in passing, are comparable to those o the manuscripts) became less common among scholars o Diophantus’ mathematics.
Reasoning and symbolism in Diophantus
annery: Let the which is added and taken away rom each number be set down, ςΑ <:Number 1>. And i it is added to 20, result: ςΑ ΜοΚ <:number 1 Monads 20>. Manuscripts: Let the which is added and taken away rom each number be set down, One number. And i it is added to 20, result: One number, 20 Monads.
Here we see annery’s most typical treatment o the manuscripts: abbreviating expressions which, in the manuscripts, are resolved, within the problem itsel. Note the opposite, inside enunciations. For example, the enunciation to .10 which, in annery’s orm, may be translated: annery: o ndthree numbers so that the by any two, taken with a given number, makes a square.
Compare this with, e.g., Par. Gr. 2379: Manuscript: o ndthree numbers so that the by any two, taken with a given ς, makes a .
annery, we recall, ollowed a rational system: inside the proo, all markedly Diophantine symbols were presented in abbreviated orm, while in the enunciation no symbolism was used. We nd that the manuscripts sometimes have abbreviated orms where annery has ully written words, and sometimes have ully written words where annery has abbreviations. In other words, annery’s rational system does not work. I had systematically studied the marked Diophantine symbols through the propositions whose number divide by ten, in Books to , in all the Paris manuscripts. Tese are only eight propositions, but the labour, even so, was considerable: essentially, I was busy recording noise. As a consequence o this, I gave up on urther systematic studies, merely con rming the overall picture described here, with other manuscripts. One notices perhaps a gradual tendency to introduce more and more abbreviated orms as the treatise progresses (do the scribes become tired, in time?): Par. Gr. 2378, or instance, has no symbolism in my Book specimens at all, while they are requent in Book . Te ordinal numbers, with their three separate orms (ully spelled out, phonologically abbreviated, alphabetical numeral based), are especially bewildering. Consider once again .10, once again in Par. Gr. 2378. I plot the sequence o ordinals, using N or the alphabetic numeral, P or phonological abbreviation and F or the ull version: NNNNPPFFFFFFFNFFFFPFNNFFFPF.
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annery has all as alphabetical numerals. Te most we can say is that, in the manuscripts, there is an overall tendency to preer using the same orm within a single phrase, though exceptions to this are ound as well. Here we see annery homogenizing, turning numbers into numerals. But we may also nd the opposite, e.g. in .20, an expression we may translate as annery: Let the two be set down as ς3 τετάχθωσαν οἱ δύο ς3 Par. Gr. 2485: Let the 2 be set down as Numbers, Tree. τετάχθωσαν οἱ Β ἀριθμοὶ τρεῖς
annery has spelled out the word ‘two’, to signal that it unctions here in a syntactic, not an arithmetical way. But it is neither syntactic nor arithmetical, it is phonological/orthographic. In the manuscripts, we have the phonological/orthographic object /duo/ which may be represented, as ar as the scribes are concerned, by either B or δύο: both would do equally well. Signi cantly, it is difficult to discern a system even in the symbols introduced by Diophantus himsel. Consider Par. Gr. 2380, inside .10: ς ενος μοναδων Γ, that is ‘ς one, monads 3’ (I quote this as an elegant example where both Diophantus’ special symbols, as well as numerals, are interchanged with ully spelled out words). Very typical are expressions such υ
ο
υ
ο
as Par. Gr. 2378, .20: Δ Δ αριθμους Ε Μ , that is ‘Δ 4, numbers 5, Μ 1’. Te ‘numbers’ – alone in the phrase – are spelled out. In general, one can say that monads appear to be abbreviated more ofen than anything else in Diophantus’ symbolism: this may be because they are so common there. But the main act is not quantitative, but qualitative: one nds, in all manuscripts, the ull range rom Diophantine phrases ully spelled out in natural Greek, through all kinds o combinations o symbols and ull words, to ully abbreviated phrases. My conclusion is that symbols in Diophantus are allographs: ways o expressing precisely the same things as their ully spelled out equivalents. And once this allography is understood, the chaos o the manuscripts becomes natural. For why should you decide in advance when to use this or that, when the two are ully equivalent? One should now understand annery’s plight. Tat he systematized his printed edition is natural: what else should he have done? I am not even sure we should criticize him or ailing to provide a critical apparatus on the symbols. Te task is immense and its ruits dubious. In particular, given annery’s goal – o reconstructing, to the best o his ability, Diophantus’ srcinal text – a critical study o the abbreviations seems indeed hopeless. Te interrelationships between manuscripts, in terms o their choice o
Reasoning and symbolism in Diophantus
abbreviation as against a ully spelled out word, are tenuous. Sometimes one discerns affi nities: the same sequence o symbols is sometimes used in a group o manuscripts, suggesting a common srcin (and why shouldn’t a scribe be in uenced by what he has in his source?). But such cases are rare while, on the whole, patterns are more ofen ound inside a single manuscript: a tendency to avoid abbreviations or a stretch o writing, then a tendency to put them in . . . However, annery did not make appeal to this argument – which would have put his edition in the uncomortable position o being, in a central way, annery’s rather than Diophantus’. So he made appeal to another argument. When criticized by Hultsch (1894) or his ailure to note scribal variation or symbolism in his apparatus, annery replied that he had ound that tedious,9 because – so he had implied – Diophantus had purely abbreviated orms, that is in line with annery’s edition – which then were corrupted by the manuscript tradition. Tis question merits consideration. In the handul o thirteenth-century manuscripts we possess (the earliest), symbolism is more requent. Tus the tendency o scribes, during the historical stretch or which we have direct evidence , was to resolve abbreviations into words. Te simplest hypothesis, then, would be that o a simple extrapolation: throughout, scribes tend to resolve abbreviations –hence, Diophantus himsel must have produced a strict abbreviated text. Tis is alse, I think, or the ollowing reasons. First, the relevant consideration is not that o Diophantus’ manuscript tradition alone, but that o scribal practice in general. We may then witness a peak in the use o abbreviations in Byzantine technical manuscripts o the relevant period o the twelfh and thirteenth centuries – which are in general characterized by minute writing aiming to pack as much as possible into the page. Early minuscule manuscripts, and o course majuscule texts, ofen are more o luxury objects and have ewer abbreviations; humanist manuscripts, again, or similar reasons, tend to have ewer abbreviations. Tus the evidence o the process o resolution o abbreviations, rom the thirteenth to the sixteenth centuries, may not be extended into the past, as an hypothetical series o resolution stretching all the way rom as ar back as the ourth century . Second, I nd it striking that the Arabic tradition knows nothing o Diophantus’ symbols. Tere are o course good linguistic reasons why Arabic (as well as Syriac and Hebrew) would not rely as much on the kind o abbreviation typical to the Greek and Latin tradition. Indeed, to continue with the linguistic typology, symbolism is also independently 9
1893/5: xxxiv–xlii.
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used in Sanskrit mathematics.10 Indo-European words are a concatenation o pre xes, roots and suffixes. Each component is phonologically autonomous, so that it is always possible to substitute some by alternative symbols. A written word can thus naturally become a sequence concatenating symbols, or alphabetic representations, or pre xes, roots and suffixes. Semitic words, on the other hand, are consonantal roots inside which are inserted patterns o vocalic in xes. Te components cannot be taken apart in the stream o speech, so that iteach is no longer easible to substitute a word by a concatenation o symbols, standing or a root or a grammatical element. Quite simply, the language does not unction in terms o such concatenations. Arab translators, then, had naturally resolved standard Greek abbreviations into their ully spelled out orms. But they did respect some symbols: or instance, magical symbols, similar in character to those known rom Greek-era Papyri (though not derived rom the Greek), are attested in the Arabic tradition;11 most amously, the Arabs had gradually appropriated Indian numeral symbols. In such cases, the symbols were understood primarily not as phonological units, but as written traces. I suggest that, had Diophantus’ use o symbolism been as consistent as annery makes it, an astute mathematical reader would recognize in it the use o symbolism which goes beyond scribal expediency, and which is based on the written trace – especially, given Diophantus’ own, explicit introduction o the symbols. Te Arab suppression o the symbolism in Diophantus suggests, then, that they saw in it no more than the standard scribal abbreviation they were amiliar with rom elsewhere in Greek writing. I conclude with two comments, one historical, and the other cognitive. Historically, we see that Diophantus’ symbols are rooted in a certain scribal practice. Tis should be seen in the context o the long duration o Greek writing. In antiquity, Greek writing was among the simplest systems in use anywhere in human history: a single set o characters (roughly speaking, our upper case), used with ew abbreviations. Trough late antiquity to the early Middle Ages, the system becomes much more complex: the use o abbreviations becomes much more common, and a new set o characters (roughly speaking, our lower case) is introduced while the old set remains in use in many contexts. In other words, the period is characterized by an explosion in allography.12 Tis may be related to the introduction o the 10 11 12
See the lucid discussion in H1995: 87–90. Canaan 1937–8/2004, especially 2004: 167–75. It is difficult to nd precis e reerences or such claims that are rather the common stock o knowledge acquired by palaeographers in their practice. Te best introduction to the practices o Greek manuscripts probably remains Groningen 1955. For abbreviations in early Greek script, see McNamee 1982.
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codex, and with the overall tenor o the culture with which it is associated: a culture where writing as such becomes the centre o cultural lie, with much greater attention to its material setting. It is in this context that Diophantus introduces his symbols: they are the product o the same culture that gave rise to the codex. Cognitively, we see that those symbols introduced by Diophantus are indeed allographs. Tat is: they do not suppress the verbal reading o the sign, but reer to it in a different, visual way. It was impossible or a Greek reader to come across the symbol Μο and not to have suggested to his mind the verbal sound-shape ‘monad’. But at the same time, the symbol itsel would be striking: it would be a very common shape seen over and over again in the text o Diophantus and nowhere else. It would also be a very simple shape, immediately read off the page as a single visual object. Tus, alongside the verbal reading o the object, there would also be a visual recognition o it, both obligatory and instantaneous. I thus suggest that what is involved here is a systematic bimodality. One systematically reads the sign both verbally and visually. One reads out the word; but is also aware o the sign. o sum up, then, Diophantus’ symbolism gives rise to a bimodal (verbal and visual) parsing o the text (at the level o the noun-phrase). I shall return to analyse the signi cance o this in Section 4 below, where I shall argue that this bimodality explains the unction o Diophantus’ symbols within his reasoning. Beore that, then, let us acquaint ourselves with this mode o reasoning.
3. Notes on reasoning in Diophantus A sample of Diophantus Te ollowing is a literal translation o Diophantus’ .10. I ollow annery’s text, with the difference that, or each case where a symbol is available (including alphabetical numerals which, when symbolic, I render by our own Arabic toss a (25% couple o coins to ull decide whether it as symbol ornumerals), as resolvedIword. I make to be words, whichI print is what I postulate, or the sake o the exercise, might have been the srcinal ratio.) Te translation ollows my conventions rom the translation o Greek geometry,13 including the introduction o Latin numerals to count steps o construction and Arabic numerals to count steps o reasoning. 13
See Netz 2004.
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o two given ςς: to add to the smaller o them, and to take away rom the greater, and to make the resulting have a given ratio to the remainder. Let it be set orth to add to 20, and to take away rom 100 the same ς, and to make the greater 4-times the smaller. (a) Let the which is added and taken away rom each ς be set down, number, one. (1) And i it is added to twenty, results:ς1 Μο20. (2) And i it is taken away rom 100, results: Μο100 lacking number 1. (3) And it shall be required that the greater be 4- tms the smaller. (4) Tereore our-tms the smaller is equal to the greater; (5) but our-tms the smaller results: Μο400 Ψ ς4; (6) these equal ς1 Μο20 (7) Let the subtraction be added common, (8) and let similar be taken away rom similar . (9) Remaining: numbers, 5, equal Μο380. (10) And the ς results: monads, 76. o the positions. I put the added and the taken away on each ς, ς 1; it shall be Μο76. And i Μο76 is added to 20, result: monads, 96; and i it is taken away rom 100, remaining: monads, 24. And the greater shall stand being 4-tms the sm aller.
Diophantus the deuteronomist: systematization and the general o understand the unction o the text above, I move on to compare it with three other, hypothetical texts. I argue that all were possible in the late ancient Mediterranean. However, only the rst two had existed, while the third remained as a mere logical possibility, never actualized in writing. ext 1: A: I have a hundred and a twenty. I take away a number rom the greater and add it to the smaller. Now the smaller has become our times that which was greater. How much did I take away and add? B: ? A: Seventy six! Check or yoursel. ext 2: Hundred and twenty. I took away rom the greater and added the same to the smaller, and the smaller became our times that which had been greater. ake the greater, a hundred. Its our times is our hundred. ake away the smaller, twenty. Lef is three hundred eighty. Four plus one is ve. Divide three hundred eighty by ve: seventy six. Seventy six is the number taken away and added.
Reasoning and symbolism in Diophantus
ext 3: Given two numbers, the rst greater than the second, and given the ratio o a third number to unity, to nd a ourth number so that, added tothe second and removed rom the rst, it makes the ratio othe second to the rst equal tothe given ratio o the third number to unity. Let the ourth have been ound. Since the second number together with the ourth has to the rst lacking the ourth the ratio o the third to unity, make a fh number which is the third multiplied by the rst lacking the third multiplied by the ourth. Tis fh number is equal to the second together with the ourth. So the third multiplied by the rst lacking the third multiplied by the ourth is equal to the second with the ourth.So the third multiplied by the rst is equal to the second with the ourth with the third multiplied by the ourth, or to the second with the ourth taken the third and one times. Tat is, the third multiplied by the rst, lacking the second, is equal to the ourth taken the third and one times. Multiply all by the third and one raction. Tus the third multiplied by the rst, multiplied by the third and one raction, lacking the second multiplied by the third and one raction, is equal to the ourth taken the third and one times, multiplied by the third and one raction, which is the ourth. So the third multiplied by the rst, multiplied by the third and one raction, lacking the second multiplied by the third and one raction, is equal to the ourth. So it shall be constructed as ollows. Let one be added to the third to make the sixth. Let the seventh be made to be the raction o the sixth. Let the third be multiplied by the rst and by the eventh s to make the eighth. Again, let the second be multiplied by the seventh to make the ninth. Now let the ninth be taken away rom the eighth, to make the ourth. I say that the ourth produces the task. [Here it is straightorward to add an explicit synthesis, showing that the ratio obtains; or brevity’s sake, I omit this part.]
I suggest that we see Diophantus’ text with reerence to texts 1 and 2 – o which it must have been aware – and with reerence to text 3 – which it deliberately avoided.14 Based on Høyrup’s work,15 I assume that texts such as text 1 were widespread in Mediterranean cultures rom as ar back as 14
15
ext is an myidiom invention; perhaps not the one possible. Allanalysis I did was to try to write,3 in as close as possible to most that oelegant Diophantus, a general o the problem, ollowing a line o reasoning hewing closely to the steps o the solution in Diophantus’ own solution. (Tis is not a mechanical translation: obviously, a particular solution such as Diophantus’ underdetermines the general analysis rom which it may be derived, since any particular term may be understood as the result o more than one kind o general con guration.) See, or instance, H2002: 362–7. It is air to say that my summary is based not so much on this reerence rom the book, as on numerous discussions, conerence papers and preprints rom
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the third millennium (i not earlier), surviving, arguably, into our own time. Tey persisted almost exclusively as an oral tradition (sometimes, perhaps, taking a ride or a couple o centuries on the back o written traditions o the type o text 2, and then proceeding along in the oral mode). Such texts are called by Høyrup ‘lay algebra’. Occasionally, lay algebra gets written and systematized (to a certain extent) in an educational context. It then typically gets transormed into texts such as text 2: the mere question-and-answer ormat o text 1 is transormed into a set o indicative and imperative sentences put orward in the rigid, authoritarian style typical o most written education prior to the twentieth century. Tis is school algebra which has appeared several times in Mediterranean cultures. One can mention especially its Babylonian (early second millennium ), Greek (around the year zero) and Italian (early second millennium ) orms. Te Babylonian layer is important as the rst schoolalgebra o which we are aware; the Greek layer is important, or our purposes, as providing, possibly, a context or Diophantus’ work; the Italian layer is important, or our purposes, as providing a context or the interest in Diophantus in the Renaissance. Te historical relationship between various school algebras is not clear and it may be that they depend on the persistence o lay algebra no less than on previous school algebras. It should be said that, while essentially based on the written mode, this is a use o writing undamentally different rom that o elite literary culture. Writing is understood as a local, ad-hoc affair. Te difference between the literacy o school algebra and the oralcy o lay algebra is huge, in terms o theirarchaeology: clay tablets, papyri andlibri d’abbaco ofen survive, spoken words never do. But the clay tablets, papyri andlibri d’abbacoo school algebra do not belong to the world o Gilgamesh, Homer or Dante. Tey are not aithully copied and maintained, and the assumptions we have or the stability o written culture need not hold or them. What would happen when such materials become part o elite literate culture itsel? One hypothetical example is text 3: a reworking o the same material, keeping as closely as possible to the eatures o elite literate Greek mathematics (which was developed especially or the treatment o geometry). Tis may be called, then – just so that we have a term – Euclidean algebra.16 When transorming the materials o lay and school algebra into
16
the author, and that as such summaries go it is likely to deviate in some ways rom the way in which Høyrup himsel would have summed up his own position. I use the term ‘Euclidean’ to reer to elite, literate mathematical practices. It is true that Euclid – especially Books and – could have been occasionally part o ancient education (the three papyrus as ragments P. Mich. 3. 143, P. Berol. Inv 17469 and P. Oxy. 1.29, with de nitions
Reasoning and symbolism in Diophantus
elite-educated, literate orm, Diophantus chose to produce not Euclidean algebra, but Diophantine algebra. I note in passing that the character o Diophantus – as intended or elite literate culture – is in my view not in serious doubt. Te material does not conorm to elementary school procedures; it is ultimately o great complexity, suitable only or a specialized readership. It had survived only inside elite literate tradition; and, as is well known, it quickly obtained the primary mark o elite literate work – having a commentary dedicated to it (that o Hypatia).17 In other words, I suggest that Diophantus is engaged primarily in the rearrangement o previously available material into a certain given ormat, o course then massively extending it to cover new grounds that were not surveyed by school algebra itsel. Tis is very much the standard view o Diophantus, and I merely wish to point out here what seem to me to be its consequences. Let us agree that Diophantus is engaged in the re tting o previous traditions into the ormats o elite writing sanctioned by tradition. Ten it becomes open to suggest that he belongs to the overall practice o late antiquity and the Middle Ages which I have elsewhere called deuteronomic: the production o texts which are primarily dependent upon some previous texts.18 ypically, deuteronomic texts emphasize consistency, systematicity and completion. Tere is an attention to the manner o writing o the text. Tis means that they bring together various elements that might have been srcinally disparate. Te act o trying to bring disparate
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o Book I, Propositions .8–10 and .5, respectively – most likely derive rom a classroom context). However, the bulk o papyri nds with mathematical educational contents are different in character, involving basic numeracy and measuring skills or, in more sophisticated examples, coming closer to Hero’s version o geometry. Te impression is that, in antiquity itsel, Euclid was undamentally a cultural icon, which occasionally got inducted into the educational process. Te evidence is the imsiest imaginable –a mere statement in the Suidas (Adler IV:644.1–4: Yπατια . . . εγραψεν υπομνημα εις Διοφαντον) which, however, i not proving beyond doubt that Hypatia wrote a commentary on Diophantus, makes it at least very likely that someone did. Virtually everyone, rom annery to Neugebauer onwards, has agreed that Diophantus was acquainted with many arithmetical problems deriving rom earlier Mediterranean traditions and was thereore at least to some extent a systematizer. Some, such as Heath, had thought that Diophantus’ systematization o earlier problems may not have been the rst in the Greek world, making comparison with Euclid as the culmination o a tradition o writing Elements (I doubt this or Euclid and nd it very unlikely or Diophantus). Te dates are xed,based on internal evidence, as –150 to +350. What else is argued concerning Diophantus’ dates is based on scattered, late Byzantine comments which are best ignored. Te e silentio, together with Diophantus’ very survival, suggest – no more – a late date. (Te silence is not meaningless, as it encompasses authors rom Hero to the neo-Platonist authors writing on number.) A late date was always the avourite among scholars (not surprisingly, then, the thesis o an early
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components into some kind o coherent unity then would lead to a certain transormation. Te way this applies to Diophantus is obvious. He brings together previously available problems. He arranges them in a relatively clear order, ranging rom the simple to the complex. He classi es, creating clear units o text, or instance the Greek Book , all dedicated to right-angled triangle problems. In the introduction he discusses his way o writing down the problems, and introduces a special manner o writing or the purpose. Te structuring involves large-scale and small-scale transormations. Te large-scale transormation is a product o the arrangement o the disparate problems in a rational structure. Te problems ofen become combinatorial variations on each other, e.g. .11–13: 11. o add the same number to two given numbers, and to make each a square. 12. o take away the same number rom two given numbers, and to make each o the remainders a square. 13. o take away rom the same number two given numbers, and to make each o the remainders a square.
In such cases, it seems clear that Diophantus had used the rational structure as a guide, actively searching or more problems, bringing completion to his much more ragmentary sources. Te huge structure – thirteen books, o which, in some orm or another, ten survive, with perhaps our hundred problems solved – was built on the basis o such rational, combinatorial completion. Te small-scale transormation involves each and every problem, which is presented, always, in the orm above. It is immediately obvious that, in this respect, Diophantus consciously strove to imitate elite literate Greek mathematics though (as suggested by the examples above) this in itsel would not determine the orm o his text. Quite simply, there was more than a single way o producing numerical problems in elite literate Greek date was deended by Knorr 1993). I shall assume such a late date, while realizing o course the hypothetical nature o the argument: the dating o Diophantus is the rst brick o speculation in the ollowing, speculative edi ce. I would like to question, though, the very habit o treating the post quem and the ante quem as de ning a homogeneous chronological segment. One’s attitude ought to be much more probabilistic – and should appreciate the act that not all centuries are alike. Here are two probabilistic claims: (1) the rst century , and the rst century , both saw less in activity in the exact sciences; the second century , as well as the rst hal o the ourth century , saw more. (2) Te e silentio is more and more powerul, the urther back in time we go. I think it is thereore correct to say that Diophantus most likely was active either in the second century or the rst hal o the ourth century .
Reasoning and symbolism in Diophantus
ormat - not a single monolith to start with. In tting his text into the established elite Greek mathematical ormat, Diophantus had a certain reedom. Te rst decision made by Diophantus was to keep the basic dichotomy o presentation rom standard Greek mathematics, with an arrangement o a general statement ollowed by a particular proo. Tis indeed would appear as one o the most striking eatures o the Greek mathematical style. But most important, this arrangement is essential to the large-scale transormation introduced by Diophantus. o produce a structure based on rational completion, Diophantus needed to have something to complete rationally: a set o general statements reerring to each problem in terms transcending the particular parameters o the problem at hand. I thereore argue that Diophantus’ general statements can be understood, at two levels, as a unction o his deuteronomic project. He needs the general statements so as to conorm to the elite orm o presentation he sets out to emulate. Even more important, he needs them to provide building blocks or his main project o systematization. Te upshot o this is that Diophantus does not need the general statements or the logical ow o the individual problem. Tis is indeed obvious rom an inspection o the problems, where the general statements play no role at all. Tis observation may shed some light on the major mathematical question regarding Diophantus, that is, did he see his project in terms o providing general solutions? In some ways he clearly did. Te clearest evidence is in the course o the propositions (extant in Arabic only) .13–14. We are given a square number N which is to be divided into any three numbers (i.e. N=a+b+c) so that either N+a, N+b, N+c are all squares ( .13), or N−a, N−b, N−c are all squares ( .14). It is not surprising that, in both cases, we reach a point in the argument where we are asked to take a given square number and divide it into two square numbers19 – the amous Fermatian problem .8. Now, Diophantus (or his Arabic text) explicitly says that this is possible or ‘It has been seen earlier in this treatise o ours how to divide any square number into square parts.’20 Tere, o course, the divided square is a particular number, 16. (Te particular number chosen as example in .13–14 is 25.) Tis reerence is hardly a late gloss, as the very approach taken to the problem is predicated upon the reduction into .8. Indeed, the natural assumption on the part o any reader amiliar with elite Greek 19
20
By iteration, this allows us to divide a square number into any number o square numbers; Diophantus, in act, requires a division into three parts. Note however that even the basic operation o iteration itsel calls or a generalization o the operation o .8. Sesiano 1982: 166.
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geometry would be that results should be transerable rom one set o numerical values to any other soluble set, on the analogue o the transerability o geometrical results rom one diagram to another: this would be the implication o picking a mode o presentation which is so suggestive o that o elite geometry. It is also likely that the very exposure to certain quasi-algebraic practices (basically those o additions or subtractions o terms until one gets a simple equation o species) as well as the choice o simple parameters would instil the skills required or the nding o solutions with different numerical values rom those ound by Diophantus himsel, so that the text o Diophantus, taken as a whole, does teach one how to nd solutions in terms more general than those o the particular numerical terms chosen or an individual Diophantine solution.21 Having said that, however, the undamental point remains that Diophantus allows his generality, such as it is, to emerge implicitly and rom the totality o his practice. Tere is no effort made to make the generality o an individual claim explicit and visible locally. He does not solve the problem o dividing a square number into two square numbers in terms that are in and o themselves general – which he could have done by pursuing such problems in general terms. Why doesn’t he do that? Tere are three ways o approaching this. First, readers’ expectations on how generality is to be sustained would have been inormed with their experience in elite Greek geometry. Tere, generality is not so much explicitly asserted, as it is implicitly suggested. 22 It is true that the nature o Greek geometrical practice – based on the survey o a nite range o diagrammatic con gurations – does not map precisely into Diophantus’ practice. Greek geometry allows a rigorous, even i an implicit, orm o generality, which Diophantus’ technique does not support. Tis mismatch, in act, may serve as partial explanation or the emerging gap in Diophantus’ generality. Second, i indeed I am right and Diophantus’ goals were primarily completion and homogeneity, and that the general statement may have been introduced in the service o such goals, than our problem is to a large extent diffused. Diophantus did not provide explicit grounds or his generality, but this is because he was not exactly looking or them. He did not introduce general statements or the reason that he was looking or general solutions. Rather, he introduced general statements because he perceived such statements to be an obligatory eature o a systematic arrangement 21 22
Tis, i I understand him correctly, is the claim o Tomaidis 2005. As argued in N1999: ch. 6 (a comparison also made by Tomaidis 2005).
Reasoning and symbolism in Diophantus
o mathematical contents. O course, I imagine that he would still preer a general proo to a particular one – but only as long as other, no less important characteristics o the proo were respected as well. But this, I suggest, was not the case. I will try to show why in the next section. Even beore that, let us mention the third and most obvious account or why Diophantus did not present a more general approach. An argument that comes to mind immediately is that Diophantus did not produce more general arguments because he did not possess the required symbolism. Fundamentally, what we then do is to put side by side our symbolism and that o Diophantus so as to observe the differences and then to pronounce those differences as essential or a ull- edged argument producing a general algebraical conclusion. O course, the differences are there. In particular, Diophantus has explicit symbols or a single value in each power: a single ‘number’ (a single x), a single ‘dunamis’ (a single x2), a single ‘cube’ (a single x3), etc. Tere is thus no obvious way o reerring even to, say, two unknowns such as x and y. Tis is a major limitation, and o course it does curtail Diophantus’ expressive power. Some scholars come close to suggesting that this, nally, is why Diophantus does not produce explicit general arguments.23 But by now we can see how weak this argument is, and this or two reasons. First, it is perectly possible to express a general argument without the typographic symbolism expressing several unknowns, by the simple method o using natural language (over whose expressive power, afer all, typographic symbols have no advantage). Tis is the upshot o text 3 above. O course, even though a text such as text 3 does prove a general claim, it does so in an opaque orm that does not display the rationality o the argument. But this helps to locate the problem more precisely: it is not that, with Diophantus’ symbolism, it was impossible to prove general claims; rather, it was impossible to prove general claims in a manner that makes the rationality o the argument transparent. Second, and crucially, note that it was perectly possible or Diophantus to make the rather minimal extensions to his system so as to encompass multiple variables. Indeed, since the most natural way or him o speaking o several unknowns was to speak o ‘the rst number’, ‘the second number’, etc., he effectively had the symbolism required – all he needed was to make the choice to put together the less common symbol or ‘number’ together with the standard abbreviation or numerals: αʹ ʹς would be ‘the rstnumber’, β´ ς would be ‘the second number’, etc. A bit more cumbersome than 23
See e.g. Heath 1885: 80–2.
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x and y, or sure, a conusing symbolism, as well (one would need to develop procedures to differentiate ‘two numbers’ rom ‘the second number’) – but an effective symbolism nonetheless. Why did Diophantus not use it? Because he had no use or it. Te task Diophantus set himsel did not call or multiple symbols or multiple unknowns. He did not set out to produce general proos but rather to solve problems, where (with ew exceptions) a single unknown was to be ound. Diophantus’ project aimed not to obtain the generality o Euclidean theorems, but rather to solve problems, in a manner expressing the rationality o the solution. Tis task de ned, or Diophantus, his choice o symbolism. So let us then rerame accordingly our interpretation o Diophantus’ symbolism: not as a second-rate tool or the task o modern algebra, but, instead, as the perect tool or the task Diophantus set himsel. I proceed to discuss this task.
Diophantus the analyst: choosing a mode of persuasion Over and above the rigid structure o general enunciation ollowed by particular problem, Diophantus ollows a rigid orm or each o the problems. We should now explain Diophantus’ motivations in choosing this particular orm (that he chose some rigid orm – instead o allowing reely varying orms or setting out problems – is o course natural given his deuteronomic project). Te basic structure o the Diophantine proposition, as is well known, is that o analysis: that is, Diophantus assumes, or each proposition, that it has already been solved. ypically, he then terms the hypothetically ound element ‘number’ (the ς with which we are amiliar) and notes the consequences o the assumption that the conditions o the problem are met (in the case quoted above: 20, together with the number, is our times 100, lacking the number). Tis is then manipulated by various ‘algebraic’ operations (roughly, indeed, those later used by al-Khwarizmi, in his algebra) until the number comes to be de ned as monads. Tis then is quickly veri ed in a nal statement where the terms are put ‘in the positions’. In the Arabic Diophantus, besides the quick veri cation one also has a ormal synthesis, repeating the argumentation o the analysis backwards so that one sees that, given the solution, the terms o the problem cannot ail to hold. Sesiano believes this may be due to Hypatia; alternatively, this could be due to some Arabic commentator. In any case, the systematic addition o the synthesis may serve as another example o how deuteronomic texts seek the goal o completion.
Reasoning and symbolism in Diophantus
It is natural that, among the models available to him rom elite literate Greek mathematics, Diophantus would choose that o analysis. While not the most common orm o presenting propositions, it is very markedly associated with problems rather than with theorems – i.e. with those situations where one is aced not with a statement, whose truth is to be corroborated, but with a task which is to be ul lled.24 Tis is o course the nature o the material Diophantus had available to him. And, since he set out to produce a systematic, monolithic work, it is natural that he would use the same orm o presentation throughout – resulting in a unique text among the extant Greek works, consisting o analysis and nothing else. Te choice o the analytic orm has important consequence or the nature o the reasoning. Now, it is ofen suggested that analysis is a method o discovery: that is, it is a way by which Greek mathematicians came to know how to solve problems. I have written on this question beore, in an article called ‘Why did Greek mathematicians publish their analyses?’ I shall not repeat in detail what I had to say there, but the title itsel suggests the main argument.25 Whatever heuristic contribution the analytic move – o assuming the task ul lled – may have had, this cannot account or writing the analysis down. Te written-down analysis most certainly is not a protocol o the discovery o the solution. It must serve some other purpose in the context o presentation, which is what I was trying to explain in my article. Like most authors on Greek geometry, I had completely ignored Diophantus in that previous article o mine, but in act here is a clear case or my claim: no doubt, Diophantus in general knew the values solving his tasks, as part o his tradition. Te analysis, or him, was not a way o nding those values, but o presenting them. What is the contribution o analysis in the context o presentation? I have suggested the ollowing: when producing solutions to problems (unlike the case where one sets out proos o theorems) one aces a special burden o showing the preerability o the offered solution to other, alternative solutions. Tis, indeed, was a standard arena o polemic in Greek mathematics: 24
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Tis is the main theme o Knorr 1986. In general, or the nature o ancient analysis, the best starting-point today is the Stanord Encyclopedia o Philosophyentry, with its rich but wellchosen bibliography: http://plato.stanord.edu/entries/analysis/, by M. Beaney. Otte and Panza 1997 are the best starting point in print. I will state immediately my position, that much o the discussion o ancient analysis is vitiated by paying too much attention to Pappus’ pronouncements on the topic C ( ollectio .1–2): while Pappus was not an unintelligent reader o his sources, it is most likely that he presents not so much any earlier theory but rather his own interpretation, so that his authority on the subject is that o a secondary source. Te selective discussion in that article may be supplemented by my comments on a ew analyses by Archimedes, in Netz 2004: 207, 217–18.
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are problems solved in the most appropriate way? Te task, then, is to show how the offered solution to the problem comes out naturally, given the terms themselves. Tis is what the analysis does: it reaches the solution to the problem, as a demonstrative consequence o the terms that the problem had set out. Tus analysis need not discover a solution, nor prove its truth (though this is a by-product o a successul analysis). Its aim may simply be to display how the solution emerges naturally out o the conditions set out by the problem. Te aim o the proo in an analysis is not in its conclusion, but in the process itsel: it lays down a rational bridge leading rom the terms o the problem, to the solution offered. I this is true, then Diophantus should have similar expectations rom his own analyses. But in act this goal o the analyses emerges rom his choice o the orm itsel. He avoided schoolroom algebraical presentations with their take-it-or-leave-it approach: probably, within the overall expectations o elite literate Greek mathematics, this could not do. Such texts were driven by a culture whose central mode was persuasion, and the text thereore had to display a rational, persuasive structure. But neither did Diophantus aim primarily to show the reason why. He could easily have chosen to adopt a strictly theoretical approach to numerical problems, as, one may perhaps say, certain Arabic mathematicians did much later; his uency in extending numerical problems and solving quite complex ones suggest that, in sheer terms o mathematical intelligence, he was quite capable o such a theoretical approach. But he did not aim at such. He understood his task in a more limited way – not so much to open up a new eld o theoretical inquiry, but rather to arrange a eld inherited rom the past. Te only constraint was that this eld should display a rational, persuasive structure: Diophantus’ analyses served just that. Instead o the take-it-or-leave-it o lay and school algebra, Diophantus would have rational bridges leading rom the terms o the problems to their solutions. Tus he would show that the solutions are not arbitrary, but arise naturally given the terms set out by the problems. 26 26
It is interesting to notice in this context the cases where Diophantus departs rom the strict analytic presentation. Tis happens, in particular, where he has to make some arbitrary choices o numerical values. Ten he sometimes takes us into his con dence, explaining the rational basis or his next move. For example in .2: ‘but 16 monads are not some arbitrary number, but are a square which, added to 20 monads, makes a square as well. So I am brought to investigate: which square has a ourth bigger than 20 monads, and taken together with twenty monads makes a square. So the square results to be bigger than 80. But 81 is a square bigger than 80. . . ’ – this entire discussion is there to explain why, in an arbitrary move, Diophantus picks the numerical value 9 and none else. Te choice is arbitrary; but Diophantus shows that it is not irrational, and is somehow suggested by the values at hand.
Reasoning and symbolism in Diophantus
My interpretation o Diophantus thus relies on two theoretical contexts I developed elsewhere: deuteronomy, and analysis as a tool o presentation. What is Diophantus’ project? I interpret this within the theoretical context o what I call deuteronomy: it is to systematize and complete previously given materials, making them all conorm with some ideal standard. Tis systematic structure is two-dimensional. Horizontally, all units should conorm to each other. Vertically, all units should conorm to the ideals o Greek elite mathematics. How does Diophantus then ul l his project? I interpret this within the theoretical context o analysis as a tool o presentation. I all the units are to be the same, then the most natural ormat to take is that o a problem. And to make those problems conorm to the ideals o Greek elite mathematics, the method o analysis is deployed, so as to display the rationality o each o the moves made through the text. Tis, nally, I suggest, is the unction o reasoning in Diophantus: to build a rational bridge leading rom the terms o the problem, to the solution. I now need to show how Diophantus’ symbols may serve this unction.
Diophantus’ symbolism and the display of rationality My basic thesis is that the reasoning in Diophantus is designed, primarily, to display a rational bridge leading rom the terms o the problem to the solution. wo questions arise: (1) How does symbolism such as that used by Diophantus help with this goal? (2) Why would it help with such a goal here, and not elsewhere in Greek mathematics? Let me rst discuss the appropriateness o Diophantus’ symbolism or his goal. Diophantus’ goal, as I reconstruct, is in one sense limited, in another sense ambitious. Te goal is limited, because he does not aim at powerul, general theoretical insight into numerical problems. He merely aims at classiying and completing them as a system. Te goal is ambitious, because each solution, at each step, has to clear a high cognitive hurdle. It has to display, step by step, its rationality. Both the limit and the ambition explain why a general, theoretical approach such as text 3 above would not be appropriate. It is not called or, because o the limited ambition; and it is undesirable because, with the prolix phrases and the difficulty o ing x the identity o the entities involved, it becomes impossible to survey, step by step, the rationality o the argument as it unolds. Note that in a text with theoretical goals, local obscurity can be tolerated: the reader is then expected to work his or her way through the text. It is quite easible to have valid arguments expressed
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in roundabout, extremely subtle, or even paradoxical ashion, so that it is only by reading them several times over – effectively, producing a commentary – that one comes to see their validity. Indeed, such writing is very typical o the Western philosophical tradition. Diophantus’ world had also people reading, say, Stoic metaphysics, which is as opaque (and as precise) as text 3 above. But Stoic metaphysics is the product o a proessional community o specialists who pride themselves in their uency in a complex language. Its subject matter is perceived to have enormous inner signi cance. Tus readers preer the theoretical power o an argument to its apparent rationality: it is more important to derive a truth than to show that that truth arises naturally (indeed, there is a premium in a diffi cult-toparse argument, in whose production and parsing both author and reader may take pride). On the other hand, because the author offers solutions he is under a special obligation, as argued above, to display the rationality o the solution as it unolds, to show that it is not a contrived solution but instead derives naturally rom the terms o the problem.27 Tis immediately suggests a unction or Diophantus’ symbolism. Obviously, it makes the parsing easier: it abbreviates overall, and it brings about clear visual signposts with which the text is structured and its entities identi ed. But let us be more precise: just what is being more easily parsed, and how? o repeat the conclusion o Section 2 above: we see that Diophantus’ symbolism gives rise to a systematic bimodal reading, visual and verbal, at the level o the noun-phrase. Tis, I argue, directly serves the goal o constructing a rational bridge leading rom the terms o the problem to its solution. For what is a rational bridge like? It is a structure where everything is meaningully present to the mind, and is also under the mind’s control. Te relationships are all calculated and veri ed, but they are perceived as meaningul relationships and not as mere symbolic structures lacking in meaning. In modern terms, we may say that Diophantus needs to have a semantic derivation; it also ought to be cognitively computable. Since the derivation must be semantic, a bimodal reading is preerable to a strictly visual one. For Diophantus, it appears important that the derivation reers directly to numbers and monads, and does not make use o some opaque symbols. Te derivation should be conducted throughout at the level o the meanings: the signi ed – and not only the signs – should never 27
I ollow an explanatory mode comparable to that o Chemla 2003. Considering the closely analogous case o the use o particular examples in Chinese mathematics, Chemla argues that these were used because the authors were seeking generality above abstraction. My analogous argument is that Diophantus sought transparency above generality.
Reasoning and symbolism in Diophantus
be lost out o sight, or otherwise the derivation would appear as a conjurer’s trick out o which the solution happened to have emerged – precisely the opposite effect o the rational bridge Diophantus aims to construct. At the same time, the visual component o the bimodal reading serves in the computation o the expression. Te eye glances quickly to the correct spot in the phrase, nding the correct value. Even more important, perhaps: the mind is trained to look or the expressions, so that a visual–spatial arrangement or the phrase comes to aid the purely verbal computation. Tis is a speculative statement: I believe it to be true. Let me explain. First o all, independently o how a particular phrase may be spelled out, through abbreviations or through ully written-out words, it is certainly read by a mind that is already acquainted with the xed structure o the phrase on the page, and with its limited arsenal o symbols. Tus the reader would have triggered in him or her not only the verbal response, but also the visual response. In other words, it appears to me that, just as the mind involuntarily creates a verbal representation o a Diophantine abbreviation, so it involuntarily creates a visual representation o a Diophantine spelled-out word. Tus the reader has three resources available: (1) the actual trace o the page, (2) the verbal representation o the contents, kept by the mind’s working memory o phonological representations, (3) the visual representation o the contents, kept by the mind’s working memory o visual representations. Resource (1) would then serve to stabilize and keep in place both resources (2) and (3). It is obvious that the presence o a visual resource, over and above the verbal resource, helps in the computation o the expression: I shall return to explain this in more detail below. What is involved in the computation? Te reader, above all, veri es that a certain relation holds, in the rational bridge, leading rom one statement to the next. In other words, what we need is to have a tool or operating upon phrases expressing arithmetical values. We need to veriy that the product o an operation on the expression X is indeed the expression Y. So we can see why the operations themselves do not call or symbolism: they may be ully spelled out, instead. What we need is symbolism or the arithmetical values on which the operations operate. We can thus see why Diophantine symbolism stops at the level o the noun-phrase and does not reach the level o the sentence. Te computation is thus local to the level o the noun-phrase. Indeed, it is clear that the resources (2) and (3) – the verbal and visual representation o expressions in the reader’s working memory – are limited in capacity and duration. In act, all that they allow is the veri cation o the relation in a single stage o the argument – the rational bridge is built one link at a time. We can now return to .10 and consider the veri cation in action:
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(1) And i it is added to twenty, results: ς1 Μο20. (2) And i it is taken away rom 100, results: Μο100 lacking number 1. (3) And it shall be required that the greater be 4-tms the smaller. (4) Tereore our-tms the smaller is equal to the greater; (5) but our- tms the smaller results: Μο400 Ψ ς 4; (6) these equal ς1 Μο20 (7) Let the subtraction be added common, (8) and let similar be taken away rom similar . (9) Remaining: numbers, 5, equal Μο380. (10) And the ς results: monads, 76.
Tis – the entirety o the argumentative part o the proposition – all revolves around a single veri cation, the one connecting the statement o steps 5–6 taken as a whole, and the statement o step 9. Te operation to be veri ed is contained in steps 7–8; steps 1–5 (which are very simple, but somewhat convoluted) make sense as soon as their purpose becomes clear: to bring the two expressions o steps 5–6 into close proximity, in preparation or the veri cation o the operation. Finally, step 10 is a very simple consequence o step 9 and calls or no cognitive effort. Note, then, that steps 7–8 are ully spelled out: they do not include any o Diophantus’ symbolic terms. Te operation itsel is ully verbal and semantic: the meaning o the operation is directly told to the reader. On the other hand, the substratum or the operation – the phrases o steps 5, 6 and 9 – is presented in the bimodal orm o abbreviations. One knows throughout what one talks about: these are not abstract symbols, but ‘numbers’ and ‘monads’. On the other hand, the computations can relatively easily be carried out: a ‘lacking’ in the one can be translated into an addition to the other, which easily leads to 5; 400 with 20 taken away easily leads to 380; each result is attached to the correct rubric, ‘number’ in the rst case, ‘monads’ in the second. In all o this, the simpli cation introduced by a xed visual structure to which objects can be added or removed is o obvious help. Tis, then, is my suggestion or the role o symbolism in Diophantus’ reasoning. As Diophantus transormed the lay and school algebra material at his disposal, into the argumentative orm o Greek mathematical analysis, he added in a tool which served in this analytic orm – making the argument display the rationality o the passage rom the terms o the problem to the terms o its solution.28 We can see why the transition rom lay and school algebras, to elite literate algebra, would encourage Diophantus to introduce the type o symbolism he uses. But we should also consider the second transition leading to Diophantus’ text. His text differs not only rom 28
An analogous account can perhaps be provided or Diophantus’ raction symbolism. With ractions, as well, Diophantus does not develop a symbolic operation that allows him to calculate directly on ractions (e.g. rom a/b*c/d to get a*b/c*d). Tus the validity o the operations is lef or the reader to veriy explicitly. However, the symbolism – whose essence
Reasoning and symbolism in Diophantus
previous lay and school algebras, but also rom the established elite literate Greek mathematics Diophantus was amiliar with. Tis mathematics had included no such symbolism as Diophantus’. Why would Diophantus introduce such a symbolism, then? In other words, what is the unction served by symbolism, in the case o the problems studied by Diophantus – but which is not required in the case o the problems studied by previous elite literate Greek mathematicians? Tecomputable question canwithout be put symbolism, precisely: why areDiophantus’ Greek geometrical relations easily while numerical relations are not? Te question is cognitive, and so we should look or a cognitive divide between the character o geometrical and numerical relations. o begin with, then, let us remind ourselves o how Greek geometrical relations are expressed. As described in Chapter 4 o N1999, Greek geometrical texts are written in a system o ormulaic expressions, the most important o which is the amily o ratio-expressions, e.g. ‘the ratio o A to B is the same as the ratio o C to D’ (typically, the slots A, B, C and D are lled by spelled-out ormulae or geometrical objects, e.g. ‘the [two letters]’, the standard ormulaic representation o a line). One may then bring in urther inormation, always expressed within the same system o ormulaic expressions, e.g. that ‘C is equal to E’, or that ‘the ratio o C to D is the same as the ratio o G to H’. Extra inormation o the rst kind would license a conclusion such as ‘the ratio o A to B is the same as the ratio o E to D’, while extra inormation o the second kind would license a conclusion such as ‘the ratio o A to B is the same as the ratio o G to H’. o repeat, the system is based on ormulaic expressions – all within natural Greek grammar. No special symbolism is involved and the text is spelled out in ordinary alphabetical writing, so that the mind doubtless rst translates the written traces into verbal representation and then computes the validity o the argument on the basis o such verbal representations. Note now that the ormulaic expressions o Greek geometry are characterized by a hierarchical, generative structure. ypically, a ormulaic expression has, as constituent elements subordinate to its own structure, several smaller ormulaic expressions, all ultimately governing the characters o the alphabet indicating diagrammatic objects. Tus in ‘the ratio o A is that divisions are not represented as unit ractions but are lef ‘in the raw’ – makes such a veri cation possible. It is one thing to be told that 6 divided by 8 is 43 (where you directly veriy that 86 is the same as 43); another, to be told that 6 divided by 8 is 24 (where the veri cation depends on a relatively complex, separate calculation – usually, much more complex than in this simple example). ′
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to B is the same as the ratio o C to D’ one can detect three levels: the level o the proportion statement, which is in turn a structure o two ratio statements, each o which in turn is a structure o two object descriptions (which, in the Greek srcinal orm, would reer through characters o the alphabet indicating diagrammatic objects). Te structure is hierarchical in that its constituents are related to each other in relations o syntactic subordination; it is generative in that such constituents can be added and substituted at will. Tis substitution is in act one o the two bases o the computation o the validity o the geometrical argument in Greek mathematics – the other being the diagram, which we may ignore here. It is easible precisely because the ormulaic expression is hierarchical and generative. Mathematical computation here is parasitic upon syntactic computation. Te mind is equipped with a tool or computing substitutions on hierarchic, generative syntactic structures. It is thus a matter o immediate inspection that, rom the two expressions ‘the ratio o A to B is the same as the ratio o C to D’ and ‘C is equal to E’, the expression ‘the ratio o A to B is the same as the ratio o E to D’: one unailingly knows where to affix the correct substitution, based on one’s structural grasp o the expression ‘the ratio o A to B is the same as the ratio o C to D’. Since natural language syntax is the mental tool brought to bear when computing the validity o such arguments, it is only natural that they are represented verbally and not visually. We see then that, to the extent that expressions possess a hierarchic structure, they may be effectively computed through natural language tools. And it is important to notice that Greek geometrical ormulae are indeed characterized by such hierarchic structures, with proportion as the central operation in this type o mathematics. Not all expressions in natural language, however, have this hierarchic structure based on subordination. Alongside subordinate structure, natural language uses another structural principle, that o paratactic arrangement, i.e. the concatenation o phrases to create larger phrases without introducing an internal structure o dependency. Tis is the difference between expressions o the type ‘Te A o the B o the C’ and expressions such as ‘A and B and C’. Expressions o the rst kind contain, in their syntactic representation, internal structure, which the mind can use in manipulating them. Expressions o the second kind are syntactically represented as mere concatenation lacking internal structure, so that there is nothing syntactic computation can latch onto. My suggestion, then, is obvious: the central Diophantine expression – the phrase representing the sums o, e.g., dunamis, number and monads – is paratactic and not subordinate in structure. It thus essentially differs
Reasoning and symbolism in Diophantus
rom expressions such as ‘the ratio o A to B is the same as the ratio o C to D’. For this reason, purely verbal representations o the Diophantine phrase are o limited value, and Diophantus naturally was led to look or urther tools or easing computation, in the principle o allography present in his scribal culture. I nd it striking that the same seems to be true o numerical expressions in natural language as a whole. It seems that numerical expressions tend to be paratactic, rather than subordinate: this may be because they are essentially open-ended in character, ‘A and B and C and D’. Tus they always offer incentives or non-verbal representation in which their computation is aided by more than natural language syntax. Number symbolism itsel is the primary example. For afer all the earliest and most central case o symbolic argument is precisely that – the algorithm, manipulating number-symbolism via a translation o numbers rom natural language into a visual code.29
4. Summary Te suggestion o this article can now be put orward as ollows. Involved in the deuteronomic project o tting in previously available texts within established orms, Diophantus set himsel the task o presenting lay and school algebra within the ormat – and expectations – o Greek geometrical analysis. Tis entailed the task o constructing a rational bridge leading rom the setting o the problems to their solutions. Since the expressions involved were numerical in character (rather than standing or qualitative relations), their structure was not subordinate, but paratactic. As a consequence, the syntax o natural language no longer helped in their computation and could not support the task o constructing a rational bridge. Instead, Diophantus reached or the tool available to him in his culture – allography – to construct expressions whose visual structure could support the same task. Tese two eatures o Diophantus’ context – deuteronomy and allography – both may have to do, ultimately, with the material history o writing in late antiquity. And so, the relationship between reasoning and symbolism in Diophantus is ound to be dependent upon the very speci c historical conditions o late antiquity. Te complex, many-dimensional nature o the account sketched here is in itsel signi cant. Why does Diophantus use his particular symbolism? 29
Allard 1992.
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Because he has a particular task, and particular tools, all re ecting a complex historical setting. Everything argued here is tentative but o one thing I am certain: the history o mathematical symbolism is not linear. Let us discard the notion o a single linear trajectory rom ‘natural language’ to ‘symbolic algebra’, a gradual transition rom the concrete to the abstract, rom the less expressive to the more expressive, a simple teleological route leading to an ever more perect science. In truth, mathematics never did rely on natural language: rom its very inception it expressed itsel, in its various cultural traditions, through different complicated ormulaic languages, using various specialized traces or numerical values or or diagrams. History then takes off in a non-linear ashion. Symbolism is invented and discarded, employing this or that set o cognitive tools, inventing this or that orm o writing, in the service o changing goals: nothing is predetermined. Symbolism – just as mathematics itsel – is contingent. Te same, nally, must be true o our own (various uses o) symbolism: they should be seen not as the ‘natural’ achievement o precise abstraction but as a historical arteact. We should thereore study the precise cognitive tools our symbolism employs, the precise tasks that such symbols are made to achieve, and the precise historical route that brought us to the use o such symbols. Te modern equation is not the ‘natural’ outcome o a mathematical history destined to reach its culmination in the nineteenth century; it is a culturally speci c orm. Tis article, sketching a speculative account o Diophantus’ symbolism, offered one chapter rom the historical route leading to that equation.
Bibliography Allard, A. (1992) Al-Khwarizmi / Le Calcul Indien (Algorismus). Namur. Bashmakova, I .G. (1997) Diophantus and Diophantine Equations. Washington, DC. (rans. o Bashmakova 1972: Moscow.) Bourbaki, N. (1991) Elements o the History o Mathematics(tr. J. Meldrum). New York. Boyer, C. B. (1989) A History o Mathematics (rev. U. C. Mertzbach). New York. Canaan, . (1937–8) ‘Te decipherment o Arabic talismans, Part ’, Berytus 4: 69–110; ‘Part II’, Berytus 5: 141–51. (Reprinted in ed. E. Savage-Smith (2004) Magic and Divination in Early Islam. rowbridge: 125–77.) Chemla, K. (2003) ‘Generality above abstraction: the general expressed in terms o the paradigmatic in mathematics in ancient China’, Science in Context 16: 413–58.
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Groningen, B. A. van (1955) Short Manual o Greek Palaeography. Leiden. Heath, . (1885) Diophantus o Alexandria: A Study in the History o Greek Algebra. Cambridge. Hultsch, F. (1894) ‘Diophanti Alexandrini Opera, ed. annery, vol. [Rezension]’, Berliner Philologische Wochenschrif14(26): 801–7. Klein, J. (1934–6) Greek Mathematical Tought and the Origins o Algebra . Cambridge, MA. (Reprinted 1968.) Knorr, W. R. (1986) Te Ancient radition o Geometric Problems. Boston, MA. (1993) ‘“Arithmetike stoicheiosis”: on Diophantus and Hero o Alexandria ’, Historia Mathematica20: 180–92. McNamee, K. (1981) Abbreviations in Greek Literary Papyri and Ostraca. Chico, CA. Nesselmann, G. H. F. (1842) Versuch einer kritischen Geschichte der Algebra . Berlin. Netz, R. (2000) ‘Why did Greek mathematicians publish their analyses?’, in Ancient and Medieval raditions in the Exact Sciences, ed. P. Suppes, J. M. Moravcsik and H. Mendell. Stanord, CA: 139–57. (2004) Te Works o Archimedes, vol. . Cambridge. Otte, M. and Panza, M. (1997) Analysis and Synthesis in Mathematics. Dordrecht. Rashed, R. (1984) Diophante / Les Arithmétiques. Paris. Sesiano, J. (1982) Books to o Diophantus’Arithmetica in the Arabic ranslation attributed to Qusta Ibn Lūqā. New York. Tomaidis, Y. (2005) ‘A ramework or de ning the generality o Diophantos’ methods in Arithmetica’, Archive or History o Exact Sciences 59: 591–641. Wittgenstein, L. (1922) ractatus Logico-Philosophicus. London.
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Mathematical justi cation as non-conceptualized practice: the Babylonian example
Speaking about and doing – doing without spea king about it
Greek philosophy, at least its Platonic and Aristotelian branches, spoke much about demonstrated knowledge as something undamentally different rom opinion; ofen, it took mathematical knowledge as the archetype or demonstrated and hence certain knowledge – in its scepticist period, the Academy went so ar as to regard mathematical knowledge asthe only kind 1 o knowledge that could really be based on demonstrated certainty. Not least in quarters close to Neopythagoreanism, the notion o mathematical demonstration may seem not to correspond to our understanding o the matter; applying our own standards we may judge the homage to demonstration to be little more than lip service. Aristotle, however, discusses the problem o nding principles and proving propositions rom in a Even way that comes airly close to mathematical the actual practice o Euclid andthese his kin. though Euclid himsel only practises demonstration and does not discuss it we can thereore be sure that he was not only making demonstrations but also explicitly aware o doing so in agreement with established standards. Te preace to Archimedes’ Method is direct evidence that its author knew demonstration according to established norms to be a cardinal virtue – the alleged or real heterodoxy consisting solely in his claim that discovery without strict proo was also valuable. Philosophical commentators like Proclus, nally, show beyond doubt that they too saw the mathematicians’ demonstrations in the perspective o the philosophers’ discussions. As to Diophantus and Hero we may nd that their actual practice is not quite in agreement with the philosophical prescriptions, but there is no doubt that even their presentation o mathematical matters was
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A preprint version o this article appeared in HPM 2004: History and Pedagogy o Mathematics, Fourth Summer University History and Epistemology o Mathematics, ICME 10 Satellite Meeting, Uppsala 12–17 July 2004. Proceedings Uppsala: Universitetstryckeriet, 2004. I thank Karine Chemla or questions and commentaries which made me clariy the nal text on various points. See, e.g., Cicero, Academica .116–17 (ed. Rackham 1933).
Mathematical justi cation: the Babylonian example
meant to agree with such norms as are re ected in the philosophical prescriptions. Justi cation unproclaimed – or absent
But is it not likely that mathematical demonstration has developed as a practice in the same process as created the norms, and thus beore such norms crystallized and were hypostasized by philosophers? And is it not possible that mathematical demonstration – or, to use a word which is less loaded by our reading o Aristotle and Euclid, justi cation – developed in other mathematical cultures without being hypostasized? A good starting point or the search or a mathematical culture o this kind might be that o the Babylonian scribes – i only or the polemical reason that ‘hellenophile’ historians o mathematics tend to deny the existence o mathematical demonstration in this area. In Morris Kline’s (relatively moderate) words,2 written at a moment when non-specialists tended to rely on selective or not too attentive reading o popularizations like Neugebauer’s Science in Antiquity (1957) and Vorgriechische Mathematik (1934) or van der Waerden’sErwachende Wissenschaf (1956): Mathematics as an organized, independent, and reasoned discipline did not exist beore the classical Greeks o the period rom 600 to 300 . . entered upon the scene. Tere were, however, prior civilizations in which the beginnings or rudiments o mathematics were created ... Te question arises as to what extent the Babylonians employed mathematical proo. Tey did solve by correct systematic procedures rather complicated equations involving unknowns. However, they gave verbal instructions only on the steps to be made and offered no justi cation o the steps. Almost surely, the arithmetic and algebraic processes and the geometrical rules were the end result o physical evidence, trial and error, and insight.
Te only opening toward any kind o demonstration beyond the observation that a sequence o operations gives the right result is the word ‘insight’, which is not discussed any urther. Given the vicinity o ‘physical evidence’ and ‘trial and error’ we may suppose that Kline reers to the kind o insight which makes us understand in a glimpse that the area o a right-angled triangle must be the hal o that o the corresponding rectangle. 2
Kline 1972: 3, 14.
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2
3
Figure 11.1
Te con guration o VA 8390 #1.
Evident validity
In order to see how much must be put into the notion o ‘insight’ i Kline’s characterization is to be deended we may look at some texts.3 I shall start by problem 1 rom the Old Babylonian tablet VA 8390 (seeFigure 11.1) (as also in ollowing examples, an explanatory commentary ollows the translation): 4 Obv. 1. 2. 3
4 5
[Length and width] I have made hold:5 10` the surace.6 [Te length t]o itsel I have made hold:
I use the translations rom H2002 with minor corrections, leaving out the interlinear transliterated text and explaining key operations and concepts in notes at their rst occurrence – drawing or this latter purpose on the results described in the same book. In order to acilitate checks I have not straightened the very literal (‘conormal’) translations. Te rst text (VA 8390 #1) is translated and discussed on pp. 61–4. Te Old Babylonian period covers the centuries rom 2000 to 1600 (according to the ‘middle chronology’). Te mathematical texts belong to the second hal o the period. o make the linesa and b ‘hold’ or ‘hold each other’ (with urther variations o the phrase in the present text) means to construct (‘build’) the rectangular surace (a,b) which they contain. I only one lines is involved, the square (s) is built. I ollow Tureau-Dangin’s system or the transliteration o sexagesimal place value numbers, where `, ``, . . . indicate increasing and ´, ´´, . . . decreasing sexagesimal order o magnitude,
6
Mathematical justi cation: the Babylonian example
3. 4. 5. 6. 7. 8. 9. 10.
[a surace] I have built. [So] much as the length over the width went beyond 7 I have made hold, to 9 I have repeated: 8 as much as that surace which the length by itsel was [ma]de hold. Te length and the width what? 10` the surace posit, 9 and 9 (to) which he has repeated posit:
11. 12.
Te equalside 10 o 9 (to) which he has repeated what? 3. 3 to the length posit 13. 3 t[o the w]idth posit. 14. Since ‘so [much as the length] over the width went beyond 15. I have made hold’, he has said 16. 1 rom |3 which t]o the width you have posited 17. tea[r out:] 2 you leave. 18. 2 which yo[u have l]ef to the width posit. 19. 3 which to the length you have posited 20. to 2 which 〈to〉 the width you have posited raise, 11 6.
7
8 9
10 11
and where ‘order zero’ when needed is marked ° (I omit it when a number o ‘order zero’ 1 + 2·600 + 10·60–1. It stands alone, thus writing 7 instead o 7°). 5`2°10´ thus stands or 5·60 should be kept in mind that absolute order o magnitude is not indicated in the text, and that `, ´ and ° correspond to the merely mental awareness o order o magnitude without which the calculators could not have made as ew errors as actually ound in the texts. Te present problem is homogeneous, and thereore does not enorce a particular order o magnitude. I have chosen the one which allows us to distinguish the area o the surace (10`) rom the number 1/6 (10´). Te text makes use o two different ‘subtractive’ operations. One, ‘by excess’, observes how much one quantity A goes beyond another quantity B; the other, ‘by removal’, nds how much remains when a quantity a is ‘torn out’ (in other texts sometimes ‘cut off’, etc.) rom a quantity A. As suggested by the terminology, the latter operation can only be used i a is part o A. ‘Repetition to/untiln’ is concrete, and produces n copies o the object o the operation. n is always small enough to make the process transparent, 1 < n < 10. ‘Positing’ a number means to take note o it by some material means, perhaps in isolation on a clay pad, perhaps in the adequate place in a diagram made outside the tablet. ‘Positing n to’ a line (obv. 12, etc.) is likely to correspond to the latter possibility. Te ‘equalside’s o an area Q is the side o this area when it is laid out as a square (the ‘squaring side’ o Greek mathematics). Other texts tell that s ‘is equal by’ Q. ‘Raising’ is a multiplication that corresponds to a consideration o proportionality; its etymological srcin is in volume determination, where a prismatic volume with height h cubits is ound by ‘raising’ the base rom the implicit ‘deault thickness’ o 1 cubit to the real height h. It also serves to determine the areas o rectangles which were constructed previously (lines 20 and 7), in which case, e.g., the ‘deault breadth’ (1 ‘rod’, c. 6 m) o the length is ‘raised’ to the real width. In the case where a rectangular area is constructed (‘made hold’), the arithmetical determination o the area is normally regarded as implicit in the operation, and the value is stated immediately without any intervening ‘raising’ (thus lines 7 and 10).
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366 Igi 6 12 detach: 10´. 22. 10´ to 10` the surace raise, 1`40. 23. Te equalside o 1`40 what? 10. 21.
Obv. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
10 to 3 wh[ich to the length you have posited] raise, 30 the length. 10 to 2 which to the width you have po[sited] raise, 20 the width. I 30 the length, 20 the width, the surace what? 30 the length to 20 the width raise, 10` the surace. 30 the length together with 30 make hold: 15`. 30 the length over 20 the width what goes beyond? 10 it goes beyond. 10 together with [10 ma]ke hold: 1`40. 1`40 to 9 repeat: 15` the surace. 15` the surace, as much as 15` the surace which the length by itsel was made hold.
Tis problem about a rectangle exempli es a characteristic o numerous Old Babylonian mathematical texts, namely that the description o the procedure already makes its adequacy evident. In Obv. 4–5 we are told to construct the square on the excess o the length o the rectangle over its width and to take 9 copies o it, in lines 6–7 that these can ll out the square on the length. Tereore, these small squares must be arranged in square, as in Figure 11.1, in a 3×3 pattern (lines 11–13). But since the side o the small square was de ned in the statement to be the excess o length over width ( 14–15, an explicit quotation), removal o one o three rows will leave the srcinal rectangle, whose width will be 2 small squares. 13 In this unit, the area o the rectangle is 2·3 = 6 ( 18–20); since the rectangle is already there, there is no need or a ‘holding’ operation. Because the area measured in standard units (square ‘rods’) was 10`, each small square must be 1⁄6 . 10` = 1`40 and its side √1`40 = √100 = 10 ( 21–23). From this it ollows that the length must be 3·10 = 30 and the width 2·10 = 20 ( 1–3). 12
13
‘Igi n’ designates the reciprocal o n. o ‘detach igin’, that is, to nd it, probably reers to the splitting out o one on parts o unity. ‘Raisinga to igi n’ means nding a ⋅ 1/n, that is, to divide a by n. In our understanding, 2 times the side o the small square. However, the Babylonian term or a square con guration (mithartum, literally ‘[situation characterized by a] conrontation ˇ [between equals]’), was numerically identi ed by and hence with its side – a Babylonian square (primarily thought o as a square rame) ‘was’ its side and ‘had’ an area, whereas ours (primarily thought o as a square-shaped area) ‘has’ a side and ‘is’ an area.
Mathematical justi cation: the Babylonian example
Te one who ollows the procedure on the diagram and keeps the exact (geometrical) meaning and use o all terms in mind will eel no more need or an explicit demonstration than when conronted with a modern stepby-step solution o an algebraic equation,14 in particular because numbers are always concretely identi ed by their role (‘3 which to the length you have posited’, etc.). Te only place where doubts might arise is why 1 has to be subtracted in 16–17, but the meaning o this step is then duly explained by a quotation rom the statement (a routine device). Tere should be no doubt that the solutionmust be correct. None the less a check ollows, showing that the solution is valid ( 5 onwards). Tis check is very detailed, no mere numerical control but an appeal to the same kind o understanding as the preceding procedure: as we see, the rectangle is supposed to be already present, its area being ound by ‘raising’; the large and small squares, however, are derived entities and thereore have to be constructed (the tablet contains a strictly parallel problem that ollows the same pattern, or which reason we may be con dent that the choice o operations is not accidental). A similar instance o evident validity is offered by problem 1 o the text BM 13901 (Figure 11.2),15 the simplest o all mixed second-degree problems (and by numerous other texts, which however present us with the inconvenience that they are longer): Obv. 1.
2. 14
Te sura[ce] and my conrontation16 I have accu[mulated]:17 45´ is it. 1, the projection,18 you posit. Te moiety19 o 1 you break, [3]0´ and 30´ you make hold.
For instance, 3x + 2 =17 ⇒ 3x = 17 − 2 = 15 ⇒ x = 1⁄3 ⋅ 15 = 5.
15 16 17
18
ranslation and discussion in H2002: 50–2. Te mithartum or ‘[situation characterized by the] conrontation [o equals]’, as we remember rom n.ˇ13, is the square con guration parametrized by its side. ‘o accumulate’ is an additive operation which concerns or may concern the measuring numbers o the quantities to be added. It thus allows the addition o lengths and areas, as here, in line 1, and o areas and volumes or o bricks, men and working days in other texts. Another addition (‘appending’) is concrete. It serves when a quantitya is joined to another quantityA, augmenting thereby the measure o the latter without changing its identity (as when interest, Babylonian ‘the appended’, is joined tomy bank account while leaving it as mine). Te ‘projection’ (wās.ītum, literally something which protrudes or sticks out) designates a line o length 1 which, when applied orthogonally to another lineL as width, transorms it into a rectangle (L,1) without changing its measure. Te ‘moiety’ o an entity is its ‘necessary’ or ‘natural’ hal, a hal that could be no other raction – as the circular radius is by necessity the exact hal o the diameter, and the area o a triangle is
19
367
368 1 2
1 2
S
1
S
1 2
1 2
1 2
Figure 11.2
Te procedure o BM 13901 #1, in slightly distorted proportions.
Mathematical justi cation: the Babylonian example
3. 4.
15´ to 45´ you append: |by] 1, 1 is equal. 30´ which you have made hold in the inside o 1 you tear out: 30´ the conrontation.
Te problem deals with a ‘conrontation’, a square con guration identi ed by its side s and possessing an area. Te sum o (the measures o) these is told to be 45´. Te procedure can be ollowed in Figure 11.2: the lef side s o the shaded square is provided with a ‘projection’ ( 1). Tereby a rectangle (s,1) is produced, whose area equals the length o the side s; this
rectangle, together with the shaded square area, must thereore also equal 45´. ‘Breaking’ the ‘projection 1’ (together with the adjacent rectangle) and moving the outer ‘moiety’ so as to make the two parts ‘hold’ a small square (30´) does not change the area ( 2), but completing the resulting gnomon by ‘appending’ the small square results in a large square, whose area must be 45´ + 15´ = 1 ( 3). Tereore, the side o the large square must also be 1 ( 3). ‘earing out’ that part o the rectangle which was moved so as to make it ‘hold’ leaves 1–30´ or the ‘conrontation’, [the side o] the square con guration. As in the previous case, once the meaning o the terms and the nature o the operations is understood, no explanation beyond the description o the steps seems to be needed. In order to understandwhy we may compare to the analogous solution o a second-degree equation: x2 + 1⋅x = ¾ ⇔ x2 + 1⋅x + (½)2 = ¾ + (½)2 ⇔ x2 + 1⋅x + (½)2 = ¾ + ¼ = 1 ⇔ (x + ½)2 = 1 ⇔ x + ½ = √1 = 1 ⇔ x = 1–½ = ½ We notice that the numerical steps are the same as those o the Babylonian text, and this kind o correspondence was indeed what led to the discovery that the Babylonians possessed an ‘algebra’. At the same time, the terminology was interpreted rom the numbers – or instance, since ‘making ½ and ½ hold’ produces ¼, this operation was identi ed with a numerical multiplication; since ‘raising’ and ‘repeating’20 were interpreted in the same way, it was impossible to distinguish them. Similarly, the two additive
20
ound by raising exactly the hal o the base to the height. It is ound by ‘breaking’, a term which is used in no other unction in the mathematical texts. Actually, both Neugebauer and Tureau-Dangin knew that this was not the whole truth: none o them ever uses a wrong operation when reconstructing a damaged text. On one occasion Neugebauer (1935–7: 180) even observes that the scribe uses a wrong multiplication. However,
369
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operations were con ated, etc. All in all, the text was thus interpreted as a numerical algorithm: Halve 1: ½. Multiply ½ and ½: ¼. Add ¼ to ½: 1. ake the square root o 1: 1. Subtract ½ rom 1: ½.
A similar interpretation as a mere algorithm results rom a reading o the symbolic solution i the lef-hand side o all equations is eliminated. It is indeed this lef-hand side which establishes the identity o the numbers appearing to the right, and thereby makes it obvious that the operations are justi ed and lead to the solution. In the same way, the geometric reerence o the operational terms in the Babylonian text is what establishes the meaning o the numbers and thereby the pertinence o the steps. Didactical explanations
Kline wrote at a moment when the meaning o the terms and the nature o the operations yet understood where the text washis thereore usually read as a was merenotprescription o a and numerical algorithm; opinion is thereore explainable (we shall return to the act that this opinion o his also re ects deeply rooted post-Renaissance scienti c ideology). How this understanding developed concerns the history o modern historical scholarship.21 But how did Old Babylonian students come to understand these matters? (Even we needed some explanations and some training beore we came to consider algebraic transormations as sel-explanatory.) Neugebauer, ully aware that the complexity o many o the problems solved in the Old Babylonian texts presupposes deep understanding and not mere glimpses o insight, supposed that the explanations were given in oral teaching. In general this will certainly have been the case, but afer Neugebauer’s work on Babylonian mathematics (which stopped in the late 1940s) a ew texts have been published which turn out to contain exactly the kind o explanations we are looking or.
21
they never made this insight explicit, or which reason less brilliant successors did not get the point. For instance, Bruins and Rutten 1961 abounds in wrong choices (even when Sumerian word signs are translated into Akkadian). See Høyrup 1996 or what evidently cannot avoid being a partisan view.
Mathematical justi cation: the Babylonian example
Most explicit are some texts rom late Old Babylonian Susa: MS , MS , MS .22 Since MS is closely related to the problem we have just dealt with, whereas MS in vestigates non-determinate linear problems and MS the transormation o linear equations, we shall begin by discussing MS (Figures 11.3 and 11.4). It alls in three sections, o which the rst two run as ollows: #1 1. 2. 3. 4. 5. 6. 7. 8. 9.
Te surace and 1 length accumulated, 4[0´. ¿30, the length,? 20´ the width.]23 As 1 length to 10´ |the surace, has been appended,] or 1 (as) base to 20´, [the width, has been appended,] or 1°20´ [¿is posited?] to the width which together[with the length ¿holds?] 40´ or 1°20´ toge 〈ther〉 with 30´ the length hol[ds], 40´ (is) [its] name. Since so, to 20´ the width, which is said to you, 1 is appended: 1°20´ you see. Out rom here you ask. 40´ the surace, 1°20´ the width, the length what? [30´ the length. ]hus the procedure.
#2 10. 11.
12. 13. 14. 15. 16. 17. 18.
[Surace, length, and width accu]mulated, 1. By the Akkadian (method). [1 to the length append.] 1 to the width append. Since 1 to the length is appended, [1 is app]ended, 1 andwidth 1 make 1 you see. 2 you see. [1 to to the the width accumulation o length,] andhold, surace append, [o 20´ the width, 1 appe]nd, 1°20´. o 30´ the length, 1 append, 1°30´.24 [¿Since? a sur]ace, that o 1°20´ the width, that o 1°30´ the length, [¿the length together with? the wi]dth, are made hold, what is its name? 2 the surace. Tus the Akkadian (method).
Section 1 explains how to deal with an equation stating that the sum o a rectangular area (l,w) and the length l is given, reerring to the situation that the length is 30´ and the width 20´. Tese numbers are used as identiers, ul lling thus the same role as our letters l and w. Line 2 repeats the
22
23
24
All were rst published by Bruins and Rutten 1961 who, however, did not understand their character. Revised transliterations and translations as well as analyses can be ound in H2002: 181–8, 89–95 and 85–9 (only part 1), respectively. A ull treatment o MS is ound in Høyrup 1990: 299–302. As elsewhere, passages in plain square brackets are reconstructions o damaged passages that can be considered certain; superscript and subscript square brackets indicate that only the lower or upper part respectively o the signs close to that bracket is missing. Passages within ¿ . . . ? are reasonable reconstructions which however may not correspond to the exact ormulation that was once on the tablet. My restitutions olines 14–16 are somewhat tentative, even though themathematical substance is airly well established by a parallel passage in lines 28–31.
371
372 30
′
20
′
1
Figure 11.3
Te con guration discussed in MS
30
′
1
′
20
1
Figure 11.4
Te con guration o MS
#2.
#1.
Mathematical justi cation: the Babylonian example
statement but identiying the area as 10´. In line 3, this is told to be equivalent to adding ‘a base’ 1 to the width, as shown inFigure 11.3 – in symbols, (l,w) + l = (l,w) + (l,1) = (l,w + 1); the ‘base’ evidently ul ls the same role as the ‘projection’ o BM 13901. Line 4 tells us that this means that we get a (new) width 1°20´, and line 5 checks that the rectangle contained by this new width and the srcinal length 30´ is indeed 40´, as it should be. Lines 6–9 sum up. Section 2 again reers to a rectangle with known dimensions – once more l = 30´, w = 20´. Tis time the situation is that both sides are added to the area, the sum being 1. Te trick to be applied in the transormation is identi ed as the ‘Akkadian method’. Tis time, both length and width are augmented by 1 (line 11); however, the resulting rectangle (l + 1,w + 1) contains more than it should (c.Figure 11.4), namely beyond a quasi-gnomon representing the given sum (consisting o the srcinal area (l,w), a rectangle (l,1) whose measure is the same as that o l, and a rectangle (1,w) = w), also a quadratic completion (1,1) = 1 (line 12). Tereore, the area o the new rectangle should be 1 + 1 = 2 (line 13). And so it is: the new length will be 1°30´, the new width will be 1°20´, and the area which they contain will be 1°30´·1°20´ = 2 (lines 15–17). Since extension also occurs in section 1, the ‘Akkadian method’ is likely
to reer to the quadratic (thisthe conclusi on iscontext). supported by urther arguments which do notcompletion belong within present Afer these two didactical explanations ollows a problem in the proper sense. In symbolic orm it can be expressed as ollows: (l,w) + l + w = 1 , 1⁄17(3l + 4w) + w = 30′
Te rst equation is the one whose transormation into (λ,ω) = 2
(λ = l + 1, ω = w + 1) was just explained in Section 2. Te second is multiplied by 17, thus becoming
and urther transormed into
3l + 21w = 8°30′.
3λ + 21ω = 32°30, whereas the area equation is transormed into (3λ,21ω) = 2′6.
373
374 1/
1
1
W
4W
W
′
45
Figure 11.5
Te situation o MS
#1.
Tereby, the problem has been reduced to a standard rectangle problem (known area and sum o sides), and it is solved accordingly (by a method similar to that o BM 13901 #1). Te present text does not explain the transormation o the equation 1/17 (3l + 4w) + w = 30′, but a similar transormation is the object o Section 1 o MS (Figure 11.5): [Te 4th o the width, rom] the length and thewidth to tear out, 45´. You, 45´ [to 4 raise, 3 you] see. 3, what is that? 4 and 1 posit, 3. [50´ and] 5´, to tear out, |posit|. 5´ to 4 raise, 1 width. 20´ to 4 raise, 4. 1°20´ you 〈see〉, 4 widths. 30´ to 4 raise, 2 you 〈see〉, 4 lengths. 20´, 1 width, to tear out, 5. rom 1°20´, 4 widths, tear out, 1 you see. 2, the lengths, and 1, 3 widths, accumulate, 3 you see. 6. Igi 4 de[ta]ch, 15´ you see. 15´ to 2, the lengths, raise, [3]0´ you 〈see〉, 30´ the length. 7. 15´ to 1 raise, [1]5´ the contribution o the width. 30´ and 15´ hold.25 8. Since ‘Te 4th o the width, to tear out’, it is said to you, rom 4, 1 tear out, 3 you see. 9. Igi 4 de 〈tach〉, 15´ you see, 15´ to 3 raise, 45´ you 〈see〉, 45´ as much as (there is) o [widths]. 10. 1 as much as (there is) o lengths posit. 20, the true width take, 20 to 1´ raise, 20´ you see. 11. 20´ to 45´ raise, 15´ you see. 15´ rom 3015´ [tear out], 12. 30´ you see, 30´ the length. 1. 2.
Even this explanation deals ormally with the sides l and w o a rectangle, although the rectangle itsel is wholly immaterial to the discussion. In symbolic translation we are told that (l + w) − ¼w = 45′. 25
Tis ‘hold’ is an ellipsis or ‘make your head hold’, the standard phrase or retaining in memory.
Mathematical justi cation: the Babylonian example
50 1
(30 ) ′
′
(20 ) ′
5
2
1 20 °
′
′
4 1
Figure 11.6
Te transormations o MS
20
′
#1.
Te dimensions o the rectangle are not stated directly, but rom the numbers in line 3 we see that they are presupposed to be known and to be the as me as beore, 50´ being the value ol + w, 5´ that o 1⁄4w – c. Figure 11.6. Te rst operation to perorm is a multiplication by 4. 4 times 45´ gives 3, and the text then asks or an explanation o this number (line 2). Te subsequent explanation can be ollowed onFigure 11.6, which certainly is a modern reconstruction but which is likely to correspond in some way to what is meant by the explanation. Te proportionals 1 and 4 are taken note o (‘posited’), 1 corresponding o course to the srcinal equation, 4 to the outcome o theomultiplication. 50´‘torn (the out’) total are o length 5´ (the ourth the width thatNext is to be taken plus note width) o (lineand 3), and the multiplied counterparts o the components o the srcinal equation (the part to be torn out, the width, and the length) are calculated and described in terms o lengths and widths (lines 3–4); nally it isshown that the outcome (consisting o the components 1 = 4w–1w and 2 = 4l) explains the number 3 that resulted rom the srcinal multiplication (lines 4–5). Now the text reverses the move and multiplies the multiplied equation that was just analysed by ¼. Multiplication o 2 (= 4l) gives 30´, the length; multiplication o 1 gives 15´, which is explained to be the ‘contribution o the width’; both contributions are to be retained in memory (lines 6–7). Next the contributions are to be explained; using an argument o alse position (‘i one ourth o 4 was torn out, 3 would remain; now, since it is torn out o 1, the remainder is 3 ⋅ ¼’), the coefficient o the width (‘as much as (there is) o widths’) is ound to be 45´. Te coefficient o the length is seen immediately to be 1 (lines 1–10). Next (line 10) ollows a step whose meaning is not certain; the text distinguishes between the ‘true length’ and the ‘length’ simpliciter, writing however the value o both in identical ways. One possible explanation (in my opinion quite plausible, and hence used in the translation) is that the ‘true width’
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is the width o an imagined ‘real’ eld, which could be 20 rods (120 m), whereas the widthsimpliciter is that o a model eld that can be drawn in the school yard (2 m); indeed, the normal dimensions o the elds dealt with in second-degree problems (which are school problems without any practical use) are 30´ and 20´ rods, 3 and 2 m, much too small or real elds but quite convenient in school. In any case, multiplication o the value othe width by its coefficient gives us the corresponding contribution once more (line 11), which indeed has the value that was assigned to memory. Subtracting it rom the total (which is written in an unconventional way that already shows the splitting) leaves the length, as indeed it should (lines 11–12). Detailed didactical explanations such as these have only been ound in Susa; once they have been understood, however, we may recognize in other texts rudiments o similar explanations, which must have been given in their ull orm orally,26 as once supposed by Neugebauer. Tese explanations are certainly meant to impartunderstanding, and in this sense they are demonstrations. But their character differs undamentally rom that o Euclidean demonstrations (which, indeed, were ofen reproached or their opacity during the centuries where theElements were used as a school book). Euclidean demonstrations proceed in a linear way, and end up with a conclusion which readers must acknowledge to be unavoidable an error) butBabylonian which may didactical leave themtexts, wondering where the(unless rabbitthey camend rom. Te Old in contrast, aim at building up a tightly knit conceptual network in the mind o the student. However, conceptual connections can be o different kinds. Pierre de la Ramée when rewriting Euclid replaced the ‘super uous’ demonstrations by explanations o the practical uses o the propositions. Numerology (in a general sense including alsoanalogous approaches to geometry) links mathematical concepts to non-mathematical notions and doctrines; to this genre belong not only writings like the ps-IamblicheanTeologoumena arithmeticae but also or some o their aspects, Liu Hui’s commentaries toTe Nine Chapters on Mathematical Procedures, which cannot be understood in isolation rom the Book o Changes.27 Within this spectrum, the Old Babylonian expositions belong in the vicinity o Euclid, ar away rom Ramism as well as numerology: the connections that they establish all belong strictly within the same mathematical domain as the object they discuss. 26
27
Worth mentioning are the unpublished text IM 43993, which I know about through Jöran Friberg and Farouk al-Rawi (personal communication), and YBC 8633, analysed rom this perspective in H2002: 254–7. According to Chemla 1997.
Mathematical justi cation: the Babylonian example
Justi ability and critique
Whoever has tried regularly to give didactical explanations o mathematical procedures is likely to have encountered the situation where a rst explanation turns out on second thoughts – maybe provoked by questions or lacking success o the explanation – not to be justi able without adjustment. While didactical explanation is no doubt one o the sources o mathematical demonstration, the scrutiny o theconditions under which and the reasons or which the explanations given hold true is certainly another source. Te latter undertaking is what Kant termed critique, and its central role in Greek mathematical demonstration is obvious. In Old Babylonian mathematics, critique is less important. I read as demonstrations, explanations oriented toward the establishment o conceptual networks tend to produce circular reasoning, in the likeness o those persons reerred to by Aristotle ‘who . . . think that they are drawing parallel lines; or they do not realize that they are making assumptions which cannot be proved unless the parallel lines exist’.28 In their case, Aristotle told the way out – namely to ‘take as an axiom’ ἀ( ξιόω) that which is proposed. Tis is indeed what is done in the Elements, whose fh postulate can thus be seen to answer metatheoretical critique. However, though less mathematics. important than in instance Greek geometry, critique is text not absent rom Babylonian One is illustrated by the YBC 6967,29 a problem dealing with two numbers igûm and igibûm, ‘the reciprocal and its reciprocal’, the product o which, however, is supposed to be 1` (that is, 60), not 1: . [Te igib]ûm over the igûm, 7 it goes beyond [igûm] and igibûm what? Yo[u], 7 which the igibûm over the igûm goes beyond to two break: 3°30´; 3°30´ together with 3°30´ make hold: 12°15´.
Obv 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
o 12°15´ which comes up or you [1` the sur]ace append: 1`12°15´. [Te equalside o 1`]12°15´ what? 8°30´. [8°30´ and] 8°30´, its counterpart,30 lay down.31
28
Prior Analytics , 64b34–65a9, trans. redennick 1938: 485–7.
29
ransliterated, translated and analysed in H2002: 55–8. Te ‘counterpart’ o an equalside is ‘the other side’ meeting it in a common corner. Namely, lay down in writing or drawing.
30 31
377
378 igibûm
m û g i
igûm
7
6
0
121/4
3
1 2
8
1 2
5 1 2 8 1 2 3
12
Figure 11.7
Te procedure o YBC 6967.
. 3°30´, the made-hold, 2. rom one tear out, 3. to one append. 4. Te rst is 12, the second is 5. 5. 12 is the igibûm, 5 is the igûm.
Rev 1.
Te procedure can be ollowed in Figure 11.7; the text is another instance o sel-evident validity, and only differs rom those discussed under this perspective in having the sides and the area o the rectangle represent numbers and not just themselves. Te interesting point is ound in Rev. 2–3. In cases where there is no constraint on the order, the Babylonians always speak o addition beore subtraction. Here, however, the 3°30´ that is to be added to the lef o the gnomon (that is, to be put
Mathematical justi cation: the Babylonian example
back in its srcinal position) must rst be at disposition, that is, it must already have been torn out below. Tis compliance with a request o concrete meaningulness should not be read as evidence o some ‘primitive mode o thought still bound to the concrete and un t or abstraction’; this is clear rom the way early Old Babylonian texts present the same step in analogous problems, ofen in a shortened phrase ‘append and tear out’ and indicating the two resulting numbers immediately aferwards, in any case never respecting the norm o concreteness. Tis norm thus appears to have been introduced precisely in order to make the procedure justi able – corresponding to the introduction in Greek theoretical arithmetic o the norm that ractions and unity could be no numbers in consequence o the explanation o number as a ‘collection o units’.32 But the norm o concreteness is not the only evidence o Old Babylonian mathematical critique. Above, we have encountered the ‘projection’ and the ‘base’, devices that allow the addition o lines and suraces in a way that does not violate homogeneity, and the related distinction between ‘accumulation’ and ‘appending’. Even these stratagems turn out to be secondary developments. A text like AO 8862 (probably rom the early phase o Old Babylonian mathematics, at least within Larsa, its local area) does not make use o them. Its rst problem starts thus: 1. 2. 3. 4. 5. 6. 7.
Length, width. 33 Length and width I have made hold: A surace have I built. I turned around (it). As much as length over width went beyond, to inside the surace I have appended: 3`3. I turned back. Length and width I have accumulated: 27. Length, width, and surace w[h]at?
As we see, a line (the excess o length over width) is ‘appended’ to the area; ‘accumulation’ also occurs, but the reason or this is that ‘appending’ or example the length to the width would produce an irrelevant increased width and no symmetrical sum (c. the beginning o MS , above, which rst creates aosymmetrical and next removes o it).is absurd. Tis ‘appending’ a line to an sum area does not mean thatpart the text In order to see that we must understand that it operates with a notion o ‘broad lines’, lines that carry an inherent virtual breadth. Tough not made 32 33
See Høyrup 2004: 148. Tat is, the object o problem is told to be the simplest con guration determined solely by a length and a width – namely, according to Babylonian habits, a rectangle.
379
380 34 it explicit, this notion underlies the determination o areas by ‘raising’; is widespread in pre-modern practical mensuration, in which ‘everybody’ (locally) would measure in the same unit, or which reason it could be presupposed tacitly35 – land being bought and sold in consequence just as we are used to buying and selling cloth, by the yard and not the square yard. However, once didactical explanation in school has taken its beginning (and once it is no longer obvious which o several metrological units should serve as standard breadth), a line which at the same time is ‘with breadth’ and ‘without breadth’ becomes awkward. In consequence, critique appears to have outlawed the‘appending’ o lines to areas and to have introduced devices like the ‘projection’ – the latter in close parallel to the way Viète established homogeneity and circumvented the use o broad lines o Renaissance algebra.36 All in all, mathematical demonstration was thus not absent rom Old Babylonian mathematics. Procedures were described in a way which, once the terminology and its use have been decoded, turns out to be as transparent as the sel-evident transormations o modern equation algebra and in no need o urther explicit arguing in order to convince; teaching involved didactical explanations which aimed at providing students with a corresponding understanding o the terminology and the operations; and math-
ematical and o procedures were transormed critically so as to allow coherentconcepts explanation points that may initially have seemed problematic or paradoxical. No surviving texts suggest, however, that all this was ever part o an explicitly ormulated programme, nor do the texts we know point to any thinking about demonstration as a particular activity. All seems to have come as naturally as speaking in prose to Molière’s Monsieur Jourdain, as consequences o the situations and environments in which mathematics was practised. Mathematical Taylorism: practically dubious but an effective ideology
eachers, in the Bronze Age just as in modern times, may have gone beyond what was needed in the ‘real’ practice o their uture students, blinded by the act that the practice they themselves knew best was that o their own 34 35 36
C n. 11 above. See Høyrup 1995. Namely the ‘roots’, explained by Nuñez 1567: os. 6r, 232r to be rectangles whose breadth is ‘la unidad lineal’.
Mathematical justi cation: the Babylonian example
trade, the teaching o mathematics. None the less, the social raison d’êtreo Old Babylonian mathematics was the training o uture scribes in practical computation, and not deeper insight into the principles and metaphysics o mathematics. Why should this involve demonstration? Would it not be enough to teach precisely those rules or algorithms which earlier workers have ound in the texts and which (in the shape o paradigmatic cases) also constitute the bulk o so many other pre-modern mathematical handbooks? And would it not be better to teach them precisely as rules to be obeyed without distracting re ection on problems o validity? Tat ‘the hand’ should be governed in the interest o efficiency by a ‘brain’ located in a different person but should in itsel behave like a mindless machine is the central idea o Frederick aylor’s ‘scienti c management’ – ‘hand’ and ‘brain’ being, respectively, the worker and the planning engineer. In the pre-modern world, where craf knowledge tended to constitute an autonomous body, and where (with rare exceptions) practice was not derived rom theory, aylorist ideas could never ourish.37 In many though not all elds,autonomous practical knowledge survived well into the nineteenth, sometimes the twentieth century; however, the idea that practice should be governed by theory (and the ideology that practice is derived rom the insights o theory) can be traced back to the early modern epoch. New Atlantis Already beore its appearance Francis Bacon’s nd something very similar orceullyinexpressed in Vesalius’De humaniwecorporis abrica, according to which the art o healing had suffered immensely rom being split into three independent practices: that o the theoretically schooled physicians, that o the pharmacists, and that o vulgar barbers supposed to possess no instruction at all; instead, Vesalius argues, all three bodies o knowledge should be carried by the same person, and that person should be the theoretically schooled physician. In many elds, the suggestion that material practice should be the task o the theoretically schooled would seem inane; even in surveying, a eld which was totally reshaped by theoreticians in the eighteenth century, the scholars o the Académie des Sciences (and later Wessel and Gauss), even when working in the eld, would mostly instruct others in how to perorm the actual work and control they did well. Such circumstances avoured the development o views close to those o aylorism – why should those who merely made the single observations or straightened the chains be bothered 37
Aristotle certainly thought thatmaster artisans had insight into ‘principles’ and common workers not (Metaphysics , 981b1–5), and that slaves were living instrumentsPolitics ( .4); but reading o the context o these amous passages will reveal that they do not add up to anything like aylorism.
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by explanations o the reasons or what they were asked to do? I the rules used by practitioners were regarded in this perspective, it also lay close at hand to view these as ‘merely empirical’ i not recognizably derived rom the insights o theoreticians. Such opinions, and their ailing in situations where practitioners have to work on their own, are discussed in Christian Wolff’sMathematisches Lexikon: It is true that perorming mathematics can be learned without reasoning mathematics; but then one remains blind in all affairs, achieves nothing with suitable precision and in the best way, at times it may occur that one does not nd one’s way at all. Not to mention that it is easy to orget what one has learned, and that that which one has orgotten is not so easily retrieved, because everything depends only on memory.38
Wolff certainly identi ed ‘reasoning mathematics’ (also calledMathesis ‘ theorica’ or ‘speculativa’) with established theoretical mathematics, but none the less he probably hit the point not only in his own context but also i we look at the conditions o pre-modern mathematical practitioners: without insight into the reasons why their procedures worked they were likely to err except in the execution o tasks that recurred so ofen that their details could not be orgotten.39 Even the teaching o practitioners’ mathematics through paradigmatic cases exempliying rules that were or were not stated explicitly will always have involved some level o explanation and thus o demonstration – and certainly, as in the Babylonian case, internal mathematical rather than philosophical or otherwise ‘numerological’ explanation. Whether critique would also be involved probably depended on the level o proessionalization o the teaching institution itsel. But those mathematicians and historians who were not themselves involved in the teaching o practitioners were not orced to discover such subtleties. For them, it was all too convenient to accept aylorist ideologies (whether ante litteram or post) and to magniy their own intellectual standing by identiying the appearance o explicit or implicit rules with mindless rote learning (i derived rom supposedly real mathematics) or blind 38 39
Wolff 1716: 867 (my translation). Te ‘rule o three’, with its intermediate product deprived o concrete meaning, only turns up in environments where the problems to which it applies were really the routine o every working day – notwithstanding the obvious computational advantage o letting multiplication precede division. Its extensions into ‘rule o ve’ and ‘rule o seven’ never gained similar currency. A more recent example, directly inspired by Adam Smith’s theory o the division o labour, is Prony’s use o ‘several hundred men who knew only the elementary rules o arithmetic’ in the calculation o logarithmic and trigonometric tables (McKeon 1975 ).
Mathematical justi cation: the Babylonian example
experimentation (i not to be linked to recognizable theory). Such ideologies did not make opinions such as Kline’s necessary and did not engender them directly, but they shaped the intellectual climate within which he and his mental kin grew up as mathematicians and as historians. Bibliography Bruins, E. M., and Rutten, M. (1961) extes mathématiques de Suse. Mémoires de la Mission Archéologique en Iran vol. . Paris. Chemla, K. (1997) ‘What is at stake in mathematical proos rom third-century China’, Science in Context 10: 227–51. Høyrup, J. (1990) ‘Algebra and naive geometry: an investigation o some basic aspects o Old Babylonian mathematical thought’, Altorientalische Forschungen 17: 27–69 and 262–354. (1995) ‘Linee larghe: Un’ambiguità geometrica dimenticata’,Bollettino di Storia delle Scienze Matematiche15: 3–14. (1996) ‘Changing trends in the historiography o Mesopotamian mathematics: an insider’s view’, History o Science 34: 1–32. (2004) ‘Conceptual divergence – canons and taboos – and critique: re ections on explanatory categories’, Historia Mathematica31: 129–47. Mathematical Tought rom Ancient to Modern imes Kline, M. R. (1972) McKeon, M. (1975) ‘Prony, Gaspard-François-Clair-Marie Riche de’., New in York. Dictionary o Scienti c Biography, vol. . New York: 163–6. Neugebauer, O. (1934) Vorlesungen über Geschichte der antiken mathematischen Wissenschafen, vol. , Vorgriechische Mathematik. Berlin. (1935–7) Mathematische Keilschrif-exte, vols. – . Berlin. (1957) Te Exact Sciences in Antiquity, 2 edn. Providence, RI. Nuñez, P. (1567) Libro de Algebra en Arithmetica y Geometria. Anvers. Rackham, H. (ed. and trans.) (1933) Cicero: De natura deorum. Cambridge, MA. redennick, H. (ed. and trans.) (1938) Aristotle: Prior Analytics, in Aristotle: Te Categories, On Interpretation and Prior Analytics , ed. and trans. H. P. Cook and H. redennick. Cambridge, MA. van der Waerden, B. L. (1956) Erwachende Wissenschaf: Ägyptische, babylonische und griechische Mathematik. Basel. Wolff, C. (1716) Mathematisches Lexicon. Leipzig.
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12
Interpretation o reverse algorithms in several Mesopotamian texts ,
Is it possible to discuss proos in texts which contain only numbers and no verbal element? I propose to analyse a Mesopotamian tablet containing a long series o reciprocal calculations, written as numeric data in sexagesimal place value notation. Te provenance o this tablet, which today is conserved at the University Museum in Philadelphia under the number CBS 1215, is not documented, but there are numerous parallels rom the scribal schools o southern Mesopotamia, notably Nippur and Ur, all rom the Old Babylonian period (beginning two millennia beore the Christian era). Tus, one might suppose that it shares in the scribal tradition inherited rom the southern Sumero-Akkadian culture.1 Te text is composed o only two graphemes: vertical wedges (ones) and Winkelhaken (tens).2 Te limited number o graphemes is clearly not due to the limited knowledge o writing possessed by the author o the text. Te tablet was composed at the time when ‘the scribal art’ (nam-dub-sar, in Sumerian) achieved its most re ned developments, not only in the domains o mathematics and Sumerian or Akkadian literature, but also in the consideration o writing, language and grammar.3 Hence, my hypothesis is that this text contains an srcinal mathematical contemplation and that a close analysis o the tablet and its context yields the keys to understanding the text.4 Purely numeric texts are not rare among cuneiorm documentation, but, with the exception o the amous tablet Plimpton 322 which has inspired an abundant literature, such texts have drawn relatively little attention rom historians.5 Indeed, the numeric tablets do not contain inormation written in
1
2 3 4
5
384
According to A. Sachs who published it, the tablet CBS 1215 is part o a collection called ‘Khabaza 2’, purchased at Baghdad in 1889. He thought it hardly possible that it came rom Nippur, making reerence to the intervening disputes among the team o archaeologists at Nippur (Sachs 1947: 230 and n. 14). See the copy by Robson 2000: 23, and an extract o this copy in able 12.3 below. Cavigneaux 1989. I thank all those who, in the course o seminars or through critical readings, have participated in the collective work o which this article is the result, beginning with Karine Chemla, whose remarks have truly improved the present version o the text. On the subject o Plimpton 322, a tablet probably rom the Old Babylonian period perhaps rom Larsa, which presents a list o feen Pythagorean triplets in the orm o atable, see
Reverse algorithms in several Mesopotamian texts verbal style (in Sumerian or in Akkadian language) and then numeric tablets are less explicit than other types o tablets in the intentions and the methods o their authors. It is generally admitted that numeric tablets are some sort o collection o exercises destined or pedagogical purposes. However, the content and context o the tablets show that the purposes o a text such as that o tablet CBS 1215 were greater than simple pedagogy. In particular, I would like to show in this chapter that the text is organized in order to stress the operation o the reciprocal algorithm and to show why the series o steps on which it is ounded leads effectively to the desired reciprocal. Beore I go too ar into the analysis, let me give a brie description o the tablet. Te text is composed o 21 sections. (See the transcription in Appendix 1.) Te entries o the sections are successively 2.5, 4.10, 8.20, …, 10.6.48.53.20, namely the rst 21 terms o a geometric progression or an initial number 2.5 with a common ratio o 2. (Details on the cuneiorm notation o numbers and their transcription appear later.) Other than the absence o any verbal element in its writing, the text possesses some obviously remarkable properties (see able 12.3 and Appendix 1). (1) In each section, the numbers are set out in two or three columns. Tus, the spatial arrangement o the numeric data is an important element o the text. (2) Te sections are increasingly long and, as will be seen, the result appears to be the application o iterations. (3) In each section, the last number is identical to the rst. Te pr ocedure progresses in such a ashion that its point o arrival corresponds exactly with its point o departure. Te text, thereore, reveals the phenomena o reciprocity. What do these three properties (spatial arrangement, iteration and reciprocity) reveal to us? Do they disclose the thoughts o the ancient scribes about the mathematical methods which constitute the reciprocal algorithm, particularly about the topic o its validity? In order to respond to these questions, it will be necessary not only to analyse the text in detail, but also to compare and contrast it with other texts. Reciprocal algorithms are not known only by their numeric orm. In particular, a related tablet, VA 6505, contains a list o instructions composed notably Robson 2001a; Friberg 2007: Appendix 7; Brittonet al. 2011. Among the other analyses o numeric texts, outside that which bears upon the tablet studied here, one may cite those which concern the tables rom the rst millennium, such asthe large table o reciprocals rom the Seleucid period AO 6456 – or example Bruins 1969 and Friberg 1983, and several other tables rom the same period (Britton 1991–3; Friberg 2007: Appendix 8).
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Table 12.1 Principal texts studied here
A B C D
Museum number
Provenance
Contents
Style
CBS 1215 VA 6505 UE 6/2 222 IM 54472
Unknown Unknown Ur Unknown
Reciprocal Reciprocal Square Root Square Root
Numeric Verbal Numeric Verbal
in Akkadian. Sachs has shown that these instructions reer to calculations ound in the numeric tablet CBS 1215.6 Tus, we have both a numeric text and a verbal text related to the same algorithm. Tese two texts both reer to the reciprocal algorithm in widely different manners. Tey neither employ the same means o expression, nor do they deliver exactly the same type o inormation. Tus there is a shif between the different texts and the practices o calculation to which they reer. In addition, some properties o the tablet CBS 1215, notably those which concern spatial arrangement and reciprocity, are likewise maniested in calculations o square roots. Such is notably the case or the tablet UE 6/2 222, which is an Old Babylonian school exercise rom Ur (see able 12.1). Also, in the case o the square root algorithm just as or the reciprocal algorithm, both numeric and verbal texts are attested. In act, J. Friberg has shown that tablet IM 54472, composed in Akkadian, contains instructions which relate to calculations ound in the numeric tablet UE 6/2 222. 7 In order to acilitate the reading o the ollowing sections, which alternate between different tablets, I have designated the tablets by the letters A to D. Te concordance between these letters, their museum numbers and provenance is presented inable 12.1.8 In addition, many other parallels to ablet A exist. In some cases, entire sections o the text are identically reproduced. Such reproductions and citations occur principally in the texts rom the scribal schools which operated 6 7 8
Sachs 1947. Friberg 2000: 108–12. Te tablets o able 12.1 have been published in the ollowing articles and works. A = CBS 1215 in Sachs 1947 or the transliteration and interpretation; Robson 2000: 14, 23–4 or the hand copy and several joins ; B = VA 6505 in Neugebauer 1935 –7: 270, pl.14, 43; C = UE 6/2 222 in Gadd and Kramer 1966: 248; D = IM 54472 in Bruins 1954. Other than the tablet rom Ur, the tablets come rom illicit excavations. VA 6505 may come rom the north because o its orthographic and grammatical properties (H2002: 331, n. 383); according to Friberg 2000: 106, 159–60, it may come rom Sippar. IM 54472 likewise may come rom the north, perhaps rom Shaduppum (Friberg 2000: 110).
Reverse algorithms in several Mesopotamian texts in Nippur, Ur and elsewhere.9 Te school texts yield precious inormation about the context o the use o the reciprocal algorithm and will be used on a case-by-case basis to supplement the small, essential body o texts presented in able 12.1.10 Te historical problem posed by relationships that may have existed between the authors o different texts is difficult to resolve because the provenance is usually unknown and the dating is uncertain. Some available inormation seem to indicate that the numeric texts and their pedagogical parallels may pertain to the southern tradition (Ur, Uruk, Nippur), and the verbal texts, notably ablet B which may come rom Sippar, belong to the northern tradition o Old Babylonian Mesopotamia.11 Te possible historic opposition between the north and the south, however, did not exclude certain orms o communication, since the two traditions were not isolated and the scribes rom different regions had contact with one another through numerous exchanges, notably circulating schoolmasters.12 Even though uncertainty about the sources does not permit the establishment o a clear geographic distribution, it is entirely possible that the two types o texts existed in the same contexts. Regardless o the relationship between the authors o these different styles o texts, it is possible to hypothesize two points o view about the same algorithm. Te important point, whether or not these two points o view emanate rom the same scribal context, is that they clearly have different objectives. Te verbal texts are series o instructions, which appear to have been intended to help someone execute the algorithm. Some portion o the numeric tables are school exercises intended or the training o student scribes. Te unction o ablet A seems to have been o a different nature. ablet A does not conorm to the typology o a school tablet, even though it was used in an educational context, as was probably the case with all the mathematical texts o the Old Babylonian period. Trough a comparison o ablet A with parallel or similar texts, I would like to provide more detailed 9
10
11
12
What are called ‘school tablets’ in Assyriology are the products o students in scribal schools. Tese tablets generally have a standardized appearance and contents, and because o this act are easily recognizable, at least in the case o those that date rom the Old Babylonian period. Tis documentation may be speci ed urther: the list o parallels with A is presented in able 12.6; the other tablets containing reciprocals are assembled in able 12.7; those which contain calculations o square and cubic roots are in able 12.8. Te provenances o different tablets and their parallels are detailed in the notes relative to ables 12.1, 12.6, 12.7 and 12.8. In the case o Mari, it is interesting to note that the tablets rom this northerly site seem more akin to the tablets o the south than those o the north. Tus, i different scribal traditions were con rmed, they would clearly reveal complex trans-regional phenomena o communication, and not only local peculiarities. Charpin 1992; Charpin and Joannès 1992.
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responses to the questions concerning its unction. Is ablet A only a collection o exercises, rom which the school exercises were extracted? What is the tablet‘s relationship to pedagogical practice? How does the inormation differ rom the inormation presented in the verbal texts? What speci c signi cance may be determined rom its structure or its layout? Tese questions, as will be seen, are connected in the way that ablet A corresponds with the operation o the reciprocal algorithm and with its justi cation.
Place-value notation and reciprocals Since numeric texts are constructed o numbers written in the sexagesimal place value notation characteristic o Mesopotamian mathematical texts, let us review the key principals o this notation. With the base being 60, there are 59 ‘digits’. (Zero is not ound in the Old Babylonian period.) Tese 59 digits are represented by the repetition o the signs 1 (a vertical wedge) and 10 (the Winkelhaken) as many times as necessary.13 Examples: (2) (13) (20) According to the positional principle, each unit in a given place represents 60 units o the preceding place (at its right). For the transcription o numbers, I have ollowed the modern notation proposed by F. TureauDangin, wherein the sexagesimal digits are separated by dots.14 Example: is rendered as 2.13.20 In cuneiorm texts, no place is marked as being that o the units, thus the numbers have no value; they are determined to a actor 60n (where n is some whole positive or negative number), which, afer a ashion, resembles ‘ oating decimal’ notation. For example, the numbers 1, 60, 602 and 1/60 are all written in the same way, with a vertical wedge: the scribes did not make use o any special signs such as commas or zeros in the nal places similar to those we use in modern Indo-Arabic numerals. In the texts studied here, the operations perormed on the numbers are multiplications and the determination o reciprocals and square roots, namely operations which do not require that the magnitudes o the numbers be xed. In the transcriptions, translations and interpretations presented here, I have thereore not 13
14
Te word ‘digit’ here indicates each sexagesimal place. Tese ‘digits’ are written in additive decimal notation. Other authors preer to separate the sexagesimal places by a blank space or a comma (such is the case o Sachs, as will be seen later).
Reverse algorithms in several Mesopotamian texts restored the orders o magnitude, in keeping with the indeterminacy o the value in the cuneiorm writing. However, in these circumstances, might it be possible to establish ‘equalities’ between numbers, although their values are not speci ed? Even though the sign ‘=’ might be considered an abuse o language (and an anachronism), I use it in the commentary. Tis convenience seems acceptable to me insoar as we bear in mind that the sign ‘=’ denotes not a relationship o equality between quantities, but rather an equivalence between notations. For example, 2 × 30 = 1 signi es that the product o 2 and 30 is noted as 1. How were these sexagesimal numbers used in calculations? Te great number o school tablets discovered in the reuse heaps o the scribal schools present relatively accurate inormation about both the way in which place-value notation was introduced in education in the Old Babylonian period and also its use. Te course o the scribes’ mathematical education is particularly well documented at Nippur, the principal centre o teaching in Mesopotamia.15 At Nippur, and undoubtedly in the other schools, the rst stage o mathematical apprenticeship consisted o memorizing many lists and tables: metrological lists (enumerations o measures o capacities, weights, areas and lengths), metrological tables (tables o correspondence between different measures and numbers in place-value notation) and 16
numerical tables (reciprocals, multiplications and squares). Afer having memorized these lists, the apprentice scribes used these tables in calculation exercises which chie y concerned multiplication, the determination o reciprocals and the calculation o areas. Documentation shows that placevalue notation came at precise moments in the educational curriculum. Place-value notation does not occur among the expression o measurements which appeal to other numerations, based on the additive principle. Tey appear in the metrological tables, where each measure (a value written in additive numeration ollowed by a unit o measure) is placed in relation to an abstract number (a number in place-value notation, not ollowed by a unit o measure). Moreover, the abstract numbers are ound exclusively in the numeric tables and in exercises or multiplication and advanced calculations o reciprocals.17 Te calculation o areas necessitates the transormation o measures into abstract numbers and back again, transormations assured by the metrological tables.18 15 16 17
18
Robson 2001b; Robson 2002; Proust 2007. Tese tables are described in detail in Neugebauer 1935–7: ch. I. In the ollowing pages, ‘abstract numbers’ will reer to the numbers written in sexagesimal place value notation. For more details about these mechanisms, see Proust 2008.
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Table 12.2 Standard reciprocal table N 2 3 4 5 6 8 9 10 12
inv(N)
N
inv(N)
N
inv(N)
30 20 15 12 10
15 16 18 20 24
4 3.45 3.20 3 2.30
36 40 45 48 50
1.40 1.30 1.20 1.15 1.12
25 27 30 32
2.24 2.13.20 2 1.52.30
54 1.4 1.21
7.30 6.40 6 5
1.6.40 56.15 44.26.40
Let us return to the topic o the determination o reciprocals, which is the subject o ablet A. A small list o reciprocal pairs was memorized by the apprentice scribes in the course o their elementary education. Tese pairs orm a standard table, ound in numerous sources at Nippur and also in the majority o Mesopotamian educational centres. Tat table is as shown in able 12.2. Obviously, the entries o the standard reciprocal table are the reciprocals o regular sexagesimal single-place numbers, plus two reciprocals or numbers in two places (1.4 and 1.21). 19 Te determination o a reciprocal is an important operation or the scribes because the operation that corresponds with our division was effected through multiplication by the reciprocal. wo consequences result rom this conceptualization o ‘division’. First, it privileges the regular numbers, which, in act, are omnipresent in the school texts. Next, division is not properly identi ed as an operation. In order to effect a division, rst a reciprocal is ound, then a multiplication is made. 20 In this way, division has 19
20
wo numbers orm a reciprocal pair i their product is written as 1. A regular number in base-60 is a number or which the reciprocal permits a nite sexagesimal expression (numbers which may be decomposed into the product o actors 2, 3 or 5, the prime divisors o the base). Te oldest reciprocal tables contain not only the regular numbers, but also the complete series o numbers in single place (1 to 59). In these tables, the irregular numbers are ollowed by a negation: ‘igi 7 nu’, meaning ‘7 has no reciprocal’; see or example the two Neo-Sumerian reciprocal tables known rom Nippur, HS 201 in Oelsner 2001 and Ni 374 in Proust 2007 : § 5.2.2. It may be said that although the Sumerian language contains no speci c term to indicate the regular numbers, it nonetheless contains an expression or the irregular numbers: igi‘ … nu’. Te concept o division presented here is that which was taught in the scribal schools and the one used most ofen in mathematical texts, particularly in those texts discussed in the present chapter. However, this is not the only extant conceptualization. For example, divisions by irregular numbers occur sometimes, but they are ormulated as problems: nd the number, which, when multiplied by some number, returns some other number (H2002: 29). Likewise, among the mathematical texts, there exist slightly different usages o ‘reciprocals’, somewhat closer to our concept o ractions. In certain texts, the goal is to take the raction 1/7 or 1/11
Reverse algorithms in several Mesopotamian texts no name in Sumerian, contrary to the determination o a reciprocal (igi, in Sumerian) and multiplication (a-ra2, in Sumerian). Te determination o the reciprocal o a regular number is thus a undamental objective o Babylonian positional calculation. Te standard tables urnish the reciprocals o the ordinary regular numbers. In what ollows, I call the numbers that appear in able 12.2 ‘elementary regular actors’. For the other regular numbers which do not appear in the standard table, the scribes had recourse to a reciprocal algorithm, which is precisely what ablet A addresses. Sachs identi ed the reciprocal algorithm thanks to the verbal text o ablet B (VA 6505).21 First, I present the way in which Sachs understood this algorithm and described it in an algebraic ormula. Ten, I will analyse the way in which ablets A and B both reer to the same algorithm and the ways in which they differ. Tis contrast will indirectly permit some o the particular objectives pursued in ablet A to be clari ed.
Sachs’ formula Te colophon o ablet B indicates that the text is composed o twelve sections. Te entries are the rst twelve terms o a geometric progression or an initial number 2.5 with a common ratio o 2 – the same terms which constitute the beginning o ablet A. In act, only ve sections are even partially preserved but these remains allowed Sachs to reconstitute the entirety o the srcinal text. Te well-preserved entry o the seventh section is 2.13.20, that is 2.5 afer six doublings. Te text may be translated as ollows: 22 1. 2,[13],20 is the igûm.[What is the igibûm?] 2. [As or you, when you] perorm (the operations), 3. take the reciprocal o 3,20; [you will nd 18] 4. Multiply 18 by 2,10; [you will nd 39] 5. Add 1; you will nd 40. 6. ake the reciprocal o 40; [you will nd] 1,30. 7. Multiply 1,30 by 18, 8. you will nd27. Te igibûm is 27. 9. Such is the procedure.
21 22
o a number (see, or example, the series o problems such as A 24194). Finally, in rare cases, approximations or the reciprocals o irregular numbers are ound (H2002: 29, n. 50). Sachs 1947. B Section 7, translation by Sachs 1947: 226. Damaged portions o text are placed in square brackets. igûm and igibûm are Akkadian words or pairs o reciprocals.
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According to Sachs, whose notations I have reproduced,23 the algorithm is based on the decomposition o the initial number c as the sum a + b, this decomposition is summarized by the ollowing ormula (in which the reciprocal o a number n is denoted by nˉ ): ca= + = ba⋅
+
(1
ba
)
Applied to the data in B Section 7, this ormula leads to the ollowing reconstruction:24 c = 2,13;20 c = a+ =b 3;20 + 2,10 a = 3;20
=
0;18
ab = 0;18 × 2,10 = 39 1 + ab = 1 + 399 = 40 1 + ab
=
40 = 0;1, 30
ca= ×+ 1= ab× 0;18= 0;1,30 0;0,27 On the one hand, the ‘Sachs ormula’ allows us to ollow the sequence o calculations by the scribe and on the other hand it establishes or us the validity o the algorithm according to modern algebra. Moreover, it provides historians with a key to understanding ablet A and its numerous parallels. In act, as indicated above, the rst twelve sections o ablet A contain the same numeric data as their analogues in ablet B. For example, the transcription o Section 7 o ablet A is as ollows: [2.]13.20 40 [27]
18 1.30 2.13.20
In ablet A Section 7 are ound, in the same order, the numbers which appear in the corresponding section o ablet B. Clearly, the numeric ablet A reers to the same algorithm as the verbal text o ablet B. Until now, the ‘Sachs ormula’ has provided a suitable explanation o the reciprocal algorithm. Tis ormula is generally reproduced by specialists in order to explain texts reerring to this algorithm in numeric versions (ablet A and its school 23
24
In translations, like Neugebauer, Sachs used commas to separate sexagesimal digits, but unlike Neugebauer, he did not use ‘zeros’ and semicolons to indicate the order o magnitude o the numbers. He used these marks only in the mathematical commentaries and interpretations o the sources. Sachs 1947: 227.
Reverse algorithms in several Mesopotamian texts parallels) as well as in verbal version (ablet B) (seeables 12.6, 12.7 and 12.8 below). However, in my estimation, this ormula does not permit us to explain the differences between the ablets A and B, nor to grasp speci c objectives pursued by them in reerring to the algorithm. Te principal shifs that I note between the ‘Sachs ormula’ and the texts that it supposedly describes are the ollowing: (1) Te tools employed by Sachs in his interpretation (algebraic notation, using semicolons and zeros) are not those used by the Old Babylonian scribes. Te ‘Sachs ormula’ leaves unclear the actual practices o calculation to which the texts o ablets A and B make reerence. (2) Te text o ablet B, just like the remains o ablet A, does not reer to the algorithm in an abstract manner but in a precise manner, with a series o particular numbers, namely 2.5 and its successive doublings. Te algebraic ormula does not explain the choice o these particular numbers. (3) None o the properties o ablet A (spatial arrangement, iteration and reciprocity) are ound in ablet B. Te ‘Sachs ormula’ does not allow the stylistic differences that separate ablets A and B to be described or interpreted. I would like to draw attention to the act that ablet A tells us much more than an algebraic ormula in modern language can convey. What inormation is conveyed by the text o ablet A but not contained by the ‘Sachs ormula?’ Answering this question will help us understand the srcinal process o the ancient scribes and their methods o reasoning. In that attempt, I will concentrate or now on the particular properties o the text o ablet A, then on the particular numbers ound therein.
Spatial arrangement Using Sachs’ interpretation as a starting point, I am ready to detail the algorithm o determining a reciprocal to which ablet A reers. I rely on the numeric data in ablet A Section 7, which are presented above and in Appendix 1: – the number 2.13.20 terminates with 3.20, which appears in the reciprocal table, thus 3.20 is an elementary regular actor25 o 2.13.20; – the reciprocal o 3.20 is 18; 18 is set out on the right; 25
As indicated above, I call any actor which appears in the standard reciprocal table (that is, able 12.2) an ‘elementary regular actor’.
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– the product o 2.13.20 by 18 is 40; 40 is thereore a second actor and it is regular; 40 is set out on the lef and its reciprocal 1.30 is set out on the right; – the number 2.13.20 is thereore actored into the product o two elementary regular actors: 3.20 and 40; – the reciprocal o 2.13.20 is the product o the reciprocals o these two actors, namely the numbers set out on the right: 1.30 and 18; – the product o 1.30 by 18 is 27 – 27 is the desired reciprocal. Ten, the reciprocal o this result is ound, leading back to 2.13.20, the same number as the initial data. For the time being, let us put aside this last step in order to comment on the reciprocal algorithm, as I have reconstituted it in the steps above. Essentially, the algorithm is based on two rules. On the one hand, a regular number can always be decomposed into the product o elementary regular actors – that is, into the product o numbers appearing in the standard reciprocal table.26 On the other hand, the reciprocal o a product is the product o reciprocals. Tese rules correspond to the spatial arrangement o the numbers into two columns. Te actorization o 2.13.20 appears in the lef column: 2.13.20 = 3.20 × 40 Te actorization o the reciprocal appears in the right column: 18 × 1.30 = 27
Let us note an interesting difference between ablets A and B in their manner o executing the procedure. No addition appears in ablet A, but one instance appears in ablet B (line 5). Tis addition may be interpreted as being a step in the multiplication o 2.13.20 by 18. Te number 2.13.20 is decomposed into the summation o 2.10 and 3.20. Ten each term is multiplied separately by 18, and nally the two partial products are added. Tis method o multiplication is economical. With one o the partial products being obvious (3.20 × 18 is equal to 1 by construction), the multiplication is reduced to 2.10 × 18. Tis decomposition o multiplication may draw on the practices o mental calculation or the use o an abacus. It thereore seems that the instructions o text B reer not only to the steps o the algorithm, but also to the execution o multiplications. ext A, on the contrary, makes reerence only to the steps o the algorithm. Te execution o multiplication 26
Naturally, this decomposition is not unique. Te choices made by the scribes will be analysed later.
Reverse algorithms in several Mesopotamian texts seems to be outside the domain o text A. I will return later to this external aspect o multiplication in relation to the analysis o errors. Finally, let us underscore that the spatial arrangement o the text on ablet A does not correspond to the normal rules o ormatting tablets in the scribal tradition. When the scribes wrote on tablets, they were accustomed to starting the line as ar lef as possible and ending it as ar right as possible, even i it meant introducing large spaces into the line itsel. Tis method o managing the space on the tablet is ound in all genres o texts – administrative, literary and mathematical. Te example on the obverse o tablet Ni 10241 (see the copy in Appendix 2) is a good illustration o this. In this tablet, the last digit o the number contained in each line is displaced to the right and a large space separates the digits 26 and 40 in the number 4.26.40. Te same happens with the digits 13 and 30 in the number 13.30. Tis space has no mathematical value. It corresponds to nothing save the rules o ormatting. Te management o spaces in ablet A, and likewise the reverse o ablet Ni 10241, is different. Te spaces there have a mathematical meaning, since they allow columns o numbers to appear. Te areas o writing to the lef, centre and right have a unction with respect to the algorithm. Tus in ablet A appear the principles o the spatial arrangement o numbers which have a precise meaning in relation to the execution o the reciprocal algorithm. In each section, certain numbers (the actors o the number or which the reciprocal is sought) are placed to the lef; others (the actors o the reciprocal) are set out on the right; and still others (the products o the actors) are located in the central position. A simple description o these principles o spatial arrangement suffi ces to account or the basic rules on which it is based. Every regular number may be decomposed into products o elementary regular actors, and the reciprocal o a product is the product o the reciprocals. More than an algebraic ormula, this explanation o the principles o spatial arrangement allows us to understand the working o the algorithm and to reveal some elements o what might have been the actual practices o calculation. Te calculations to which the different results appearing in the columns correspond are multiplications. Tere is, in this text, a close relationship between the oating place-value notation and multiplication, just as in the body o school documentation. However, i the text records the results o multiplications, it bears no trace o the actual execution o these operations, whereas such traces seem detectable in the verbal text o ablet B as said above. In ablet A, in contrast, the steps o the algorithm and the execution o multiplication are dissociated.
395
396
Table 12.3 ranscription and copy o Section 20 Line 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
ranscription 5.3.24.26.40 45.30.40 1.8.16 4.16 16
14.3.45 5[2.44].3.45 1.19.6.5.37.30 11.51.54.50.37.30 23.43.49.41.15 1.34.55.18.45* 25.18.45* 6.45 9
Copy Robson 2000: 23 [9] 1.30 3.45 3.45 3.45
2 4 16 16 1.20 6.[40]
8.53.20 2.22.13. 20 37.55.33.20 2.31.42.13.20 5.3.24.26.40
Even though the texts o the ablets A and B reer to the same algorithm, some eatures distinguish them. In the rst case, the text is two-dimensional: the spatial arrangement o the numbers plays a critical role, reerring to the steps o calculation but not to the manner o carrying out the multiplications. In the second case, the text concerns a linear continuation o the instructions, which reer not only to the algorithm, but also to the execution o the multiplications. Another difference appears in Section 5. When the numbers or which the reciprocal is determined reach a certain size, the phenomenon o iteration appears in ablet A, but not in ablet B (so ar as the preserved portion allows us to judge).
Iteration Let us consider Section 20, o which the transcription and the copy are given in able 12.3. (Te bold type and underscoring have been added.) First, I will explain the rst part o the section, concerning the reciprocal o 5.3.24.26.40 (lines 1 to 9). Te idea o determining the reciprocal through actorization is used with more orce here. Te number or which the reciprocal is sought is
Reverse algorithms in several Mesopotamian texts 5.3.24.26.40. Te rst actor chosen is 6.40, the last part o the number. Its reciprocal is 9 (written to the right). Te product o 5.3.24.26.40 and 9 is 45.30.40 (written to the lef). Te reciprocal o this number is not given in the standard reciprocal tables, thus once again the same sub-routine is applied. Te process continues until an elementary regular number is obtained. In the ourth iteration, 16 is nallyobtained. With the reciprocals having been written down in the right-hand column at each step, it suffices to multiply these numbers to arrive at the desired reciprocal. Te multiplication is carried out term by term,27 in the order o the group o intermediate products in the central column. In other words, 3.45 is multiplied by 3.45. Te result (14.3.45) is multiplied by 3.45. Ten that result is multiplied by 1.30; and that result is multiplied by 9. Tus or 11.51.54.50.37.30 the desired reciprocal is obtained. In modern terms, the algorithm may be explained by two products: Te actorization o 5.3.24.26.40 appears in the lef-hand column (or, more precisely, in the last part o the numbers in the lef-hand column): 5.3.24.26.40 = 6.40 × 40 × 16 × 16 × 16. Likewise, the actorization o the reciprocal appears in the right-hand column: 9 × 1.30 × 3.45 × 3.45 × 3.45 = 11.51.54.50.37.30.
Since the sub-routine is repeated, the useulness o the rules or spatial arrangement o the text becomes clear. Te actors o a number or which the reciprocal is sought are on the lef. Te actors o the reciprocal are on the right and the partial products are in the centre. Te spatial arrangement o the text probably corresponds with a practice allowing an automatic execution o the sequence o operations. Such an arrangement displays the power o the algorithm and demonstrates possibilities o the spatial organization o the writing – possibilities that the linear arrangement o a verbal text like ablet B does not include.
Reverse algorithms Now let us consider the entirety o Section 20 o ablet A ( able 12.3 above). Lines 1–9 show step by step that the reciprocal o 5.3.24.26.40 is 11.51.54.50.37.30. Tis number, in turn, is set out on the lef and subjected to the same algorithm: 11.51.54.50.37.30 ends with 30; the reciprocal o 30, which is 2, is set out on the right, etc. As in the other examples, the number 27
In the cuneiorm mathematical texts, multiplication is an operation which has no more than two arguments.
397
398
11.51.54.50.37.30 is decomposed into the product o elementary regular actors. Te reciprocals o these actors are set out on the right, and nally the reciprocal is obtained by multiplying term by term the actors set out on the right. Te result is, naturally, the initial number, 5.3.24.26.40. It is the same in all the sections: afer having ‘released’28 the reciprocal in terms o a quite long calculation, the scribe undertakes the determination o the reciprocal o the reciprocal by the same method and returns to the point o srcin. Each section is thus composed o two sequences: the rst sequence, which I will call the direct sequence, and the second sequence, the reverse o the rst (in the sense that it returns to the point o departure). In what way did this scribe execute the algorithm in the reverse sequence? What interest did he have in systematically undoing what he had done? o execute the reverse sequence, the scribe would have been able to use the results o the direct sequence, which provided him with decomposition into elementary regular actors. It was enough or him to consider the actors set out on the right in the rst part o thealgorithm. For example in Section 20, to nd the reciprocal o 11.51.54.50.37.30, he was able to select the actors 3.45, 3.45, 3.45, 1.30 and 9 which appeared in the rst part, but this simple repetition o actors was not what he did. He applied the algorithm in its entirety, and as in the direct sequence, the actors were provided by the nal part o the number. (In 11.51.54.50.37.30, the rst elementary actor is 30, then 15, etc.) Tis same algorithmic method is applied in the direct sequence and in the reverse sequence o each section. I will elaborate on this point later, particularly when analysing the selection o actors in the whole text. Already this remark suggests a rst response to the question o the unction o the reverse sequence. It might be supposed that the reverse sequence is intended to veriy the results o the direct sequence, but i such were the case, it would be expected that the scribe would choose the most expedient method, and the most economic in terms o calculations. Clearly, he did not search or a short cut. He did not use the results provided rom his previous calculations, which could have been done in several ways. As has just been seen, he could have used the actors already identi ed in the direct sequence. It would also have been simple or him to use the reciprocal pairs calculated in the preceding section. Section 19 establishes that the reciprocal o 2.31.42.13.20 is 23.43.49.41.15. However, several texts attest to the act that the scribes knew perectly well that when doubling a number, the reciprocal is divided by 2 (or, more exactly, its reciprocal is multiplied 28
Te Sumerian verb which designates the act o calculating a reciprocal is du8 (release) and the corresponding Akkadian verb is pat.ārum; F. Tureau-Dangin translates this verb as ‘dénouer,’ and J. Høyrup as ‘to detach’.
Reverse algorithms in several Mesopotamian texts by 30).29 In veriying the result o Section 20, it was thereore sufficient to multiply 23.43.49.41.15 by 30. Proceeding in another way, the scribe could have multiplied together the initial number and its reciprocal in order to veriy the act that the product was equal to 1. Tese simple methods show that it was unnecessary to reapply the reciprocal algorithm. In act, the reverse sequence does not seem to have had the veri cation o the result o the direct sequence as a primary purpose. Te act that, in the second part, the algorithm was used in its entirety provokes speculation that i it were a veri cation, it concerns the algorithmic method itsel and not merely the results that it produced. Another important aspect o the algorithm is the selection o particular numbers. Tis aspect appears in comparison between ablets A and B. Both use the same geometric progression. Te particular role o this series, omnipresent in all Mesopotamian school exercises o the Old Babylonian period, is one o the rst points that ought to be made clearer. A second point is connected to the algorithm itsel. Given that the decomposition into the product o elementary regular actors is not unique, one wonders i some rule governed the scribes’ choice o one actor over another. Tis question invokes another question, even more interesting in light o the questions discussed in this article: did the scribes apply different rules to select actors in the direct and reverse sequences? Does this selection clariy the unction o the reverse sequences?
Numeric repertory As has been seen, the entries in the sections o ablet A, as with those o B, are the terms o the geometric progression or an initial number 2.5 with a common ratio o 2. What inormation did the scribe obtain in each o these sections? Afer the reciprocal o 2.5 has been obtained by actorization, it is possible to nd all the other reciprocals by more direct means, as has been explained above. For example, in each section, the reverse sequence could repeat the calculations o the direct sequence, since it leads back to the point o departure, but this is not the case. Te repeated application o the reciprocal algorithm does not produce any new result (other than the reciprocal o 2.5). From the perspective o an extension o the list o reciprocal pairs, 29
Some texts containing lists o reciprocal pairs ounded on this principle are known: beginning with a number and its reciprocal, they give the ollowing doublings and halvings. For example the tablet rom Nippur N 3958 gives the series o doublings/halvings o 2.5 / 28.48 (Sachs 1947: 228).
399
400
this text is useless. Tus, what is the unction o the repetition o the same algorithm orty-two times (in 21 sections, each one containing a direct sequence and a reverse sequence), since it returns results already seen? First o all, why has the scribe chosen the number 2.5, the cube o 5, as the initial number o the text? Tis selection undoubtedly has some importance, because the entry 2.5 and the terms o the dyadic series which result provide the majority o numeric data in exercises ound in the school archives o Mesopotamia. An initial explanation could be drawn rom the arithmetic properties o this number. It has been seen previously that the list o entries in the standard reciprocal table ( able 12.2) is composed o regular numbers in a single place, ollowed by two more numbers in two places, 1.4 and 1.21. However, we note that 1.4, 1.21 and 2.5 are respectively powers o 2, o 3 and o 5 (1.4 = 26; 1.21 = 34; 2.5 = 53). Better yet, i the list o all the regular numbers in two places is set in the lexicographic order, 30 the rst number is the rst power o 2, that is, 1.4; the rst power o 3, that is, 1.21, comes next, and the rst power o 5, 2.5, comes thereafer. Tus, in some ways, 2.5 is the logical successor in the series 1.4, 1.21. Even i this explanation is thought too speculative, one must admit the privileged place accorded to the numbers 1.4, 1.21 and 2.5. Te importance o the powers o 2, o 3 and o 5 perhaps indicates the manner by which the list o regular numbers (and their reciprocals) were obtained. Beginning with the rst reciprocal pairs, the other pairs can be generated by multiplications by 2, by 3 and by 5 (and their reciprocals by multiplication by 30, 20 and 12 respectively). Tis process theoretically would allow the entire list o regular numbers in base-60 and their reciprocals to be obtained.31 Te importance o the series o doublings o 2.5 in the school documentation could also be explained by its pedagogical advantages. I will return to this point later. For now, let us try to draw some conclusions by analysing the selection o actors in the actorization procedure. Te execution o the actorization depends, at each step, on the determination o the actors or the number or which the reciprocal is sought. Does the selection o these actors correspond to xed rules? First o all, let us note that in all o ablet A, the same choices o the actors correspond to identical numbers. For example, the number 1.34.55.18.45 appears several times, and in each case, the actor chosen is 3.45. Let us now examine these selections, by distinguishing between the case o the direct sequences (able 12.4) and the reverse 30
31
Te numbers cannot be arranged according to magnitude, since this is not de ned. Te school documentation shows that in some cases the scribes used a lexicographical order. See or example the list o multiplication tables. Here, reerence is made to this lexicographical order. Te numbers are set out in increasing order by the lef-most digit, then ollowing, etc. I think that reciprocal tables such as the one ound in the large Seleucid tablet AO 6456 were constructed in this way. A similar idea is developed by Bruins 1969.
Reverse algorithms in several Mesopotamian texts Table 12.4 Selection o actors in the direct sequences Number to actor
Section
Factor chosen
Reciprocal o actor
2.5 4.10 4.16
1 2 18, 20, 21
5 10 16
12 6 3.45
1.8.16 8.20 10.40 2.50.40 45.30.40 42.40 13, 14 11.2218 .40 3.2.2.40 21 33.20 5 2.13.20 7 8.53.20 9 35.33.20 11 2.22.13.20 13 9.28.53.20 15 10.6.48.5 213.20
20, 21 3 11, 12 15 20
16 20 40 40 40
3.45 3 1.30 1.30 1.30
16.40 1.6.40 4.26.40 8 17.46.40 10 1.11.6.40 4.44.26.40 18.57.416 6.40 1.15.51.6.40 5.3.24.26.40
4 6
2.40 2.40 2.40 3.20 3.20 3.20 3.20 3.20 3.20 3.20
22.30 22.30 22.30 18 18 18 18 18 18 18 6.40 6.40
6.40 6.40 12 14
(1)
9 9 9 9
6.40 6.40 6.40
18 20
40 40 40
Largest elementary regular actor
9 9 9
6.40 6.40
9 9
sequences (able 12.5). Te actorizations that present irregularities (in a meaning to be speci ed later) are shown in grey and numbered at the right o the tables. Te actorizations are ordered according to column 3, which contains the actors chosen in the different decompositions. Column 5 gives the largest elementary regular actor i it is different rom the actor chosen by the scribe. Column 2 speci es the section to which the appropriate decomposition belongs (I considered only sections well enough preserved to permit a sae reconstitution o the text). ables 12.4 and 12.5 show that the chosen actor is determined by the last digits o the number to be actored. In so doing, the scribes made use o an arithmetical property o the base-60 place value notation – that is, the numbers to be actorized are all regular and thus they always end with
401
402
Table 12.5 Selection o actors in the reverse sequences
Section
Factor chosen
Reciprocal o actor
Largest elementary regular actor
7.12 1.41.15 23.43.49.41.15
3 11 19
12 15 15
5 4 4
1.15 1.15
2.15 14.24 3.22.30 12.39.22.30 47.27.39.22.30 50.37.30 11.51.54.50.37.30 13.30 3.36 6.45 1.48 28.48 25.18.45 1.34.55.18.45 5.55.57.25.18.45
5 2 10 14 18 12 20 8 4 9 5 1 13 17 21
15 24 30 30 30 30 30 30 36 45 48 48 3.45 3.45 3.45
4 2.30 2 2 2 2 2 2 1.40 1.20 1.15 1.15 16 16 16
Number to actor
(2) ′
(2 ) 2.30 2.30 2.30 7.30 7.30
(3)
′
(3 )
45 45 45
(4)
a sequence o digits which orm a regular number.32 All that is needed is to adjust or a suitable sequence. (In the case o 2.13.20, we may take 20, or 3.20, or even 13.20.) In practice, the nal part, insoar as it is an elementary regular number, is likely to be a actor. (For 2.13.20, the actor might be 20, or 3.20.) In the majority o cases, the scribe chose, rom among the possible actors, the ‘largest’ (3.20 rather than 20), in order to render the algorithm aster.33 Tus, in general, the selected actor is the largest elementary 32
33
Tis property is the result o a more general rule: or a given base, the divisibility o an integer by the divisors o the base is seen in the last digits o the number. For a discussion o the particular problems resulting rom divisibility in ‘ oating’ base-60 cuneiorm notation, a sys tem in which there is no difference between whole numbers and sexagesimal ractions, see Proust 2007 : §6.2. Te word ‘large’ has nothing to do with themagnitude o the abstract numbers, since magnitude is not de ned, but with their size. A two-place number is ‘larger’ than a singleplace number; or numbers with the same number o digits, the ‘larger’ number is the last in the lexicographical order. Te speed o the algorithm depends on the size o the numbers thus de ned: the ‘larger’ the actors are, the ewer actors there will be and thus ewer iterations. Let us speciy that the order according to the size o the numbers is different rom the lexicographical order mentioned above. Te two orders appear in cuneiorm sources. Te order according to the size appears in the Old Babylonian reciprocal tables, and the lexicographical order occurs in the Seleucid reciprocal tables such as AO 6456, as well as in the arrangement o the multiplication tables in the Old Babylonian numerical tables.
Reverse algorithms in several Mesopotamian texts regular number ormed by the terminal part o the number.34 Nevertheless, this rule allows our exceptions (cases numbered in the last column o ables 12.4 and 12.5), that need to be considered. (1) Te selected actor, 2.40, does not appear in the standard reciprocal tables, and it is the actor 40 which ought to have been chosen. Nonetheless let us note that the reciprocal o 2.40 is 22.30, which is a common number that gures among the principal numbers o the standard multiplication tables. (Te table o 22.30 is one o those learned by heart in the primary level o education, especially at Nippur.) Tus, 2.40 is ‘nearly’ elementary, and its reciprocal was undoubtedly committed to memory – so case (1) does not truly constitute an irregularity. (2) and (3) In case (2), the largest elementary actor is 1.15, but the actor 15, the entry o (2 ), is used instead. In case (3), the selected actor could be either 2.30 or 7.30, but the actor 30, the entry o (3 ), is used instead. Tis choice occurred as i the scribe sought to restrict the actors used in the calculation. Te general rule o the ‘largest elementary regular actor’, regularly applied in the direct sequences, is, in the reverse sequences, opposed by another rule restricting the numeric repertory. (4) In this case, the actor might have been 45, but the scribe has obviously tried to use a larger actor. However, the numbers derived rom the last two sexagesimal places (18.45 or 8.45) are not regular. Tus, 8 is decomposed into the summation 5+3, and the nal part o the number selected as a actor is 3.45. ′
′
Several general conclusions may be drawn rom these observations. First, the number o actors occurring in the decompositions is limited. Tey are principally 3.20 and 6.40 (less requently 10, 16, 25, 40 and 22.30) or the direct sequences and principally 30 (less requently 12, 15, 24, 36 and 45, 48 and 3.45) or the reverse sequences. Tis limited number o actors is explained by the way in which the list o entries was constructed – namely, 2.5, a power o 5, is multiplied by 2 repeatedly, giving rise to a series o numbers or which the nal sequences describe regular cycles. However, the scribes’ choices intervene. On the one hand, the direct sequences obey the ‘greatest elementary regular actor’ rule. On the other hand, the reverse sequences present numerous irregularities in regard to this rule. Te number o actors used in the calculations is reduced. Finally, an interesting point to emphasize is that although the direct and reverse sequences 34
For this reason, Friberg 2000: 103–5 designates this procedure the ‘trailing part algorithm’.
403
404
reer to the same algorithm, they do not seem to share in the same way the liberty permitted by the act that the decomposition o numbers into regular actors is not unique. How do these two different ways o choosing the decomposition clariy the unction o the reverse algorithm or us? Part o the answer is ound in the school documentation. I will return to this question afer analysing the parallels with ablet A. Te observation o errors appearing in this tablet brings something else to light. Te act that these errors are not numerous shows the high degree o erudition o the author o the text. Appearing in the transcription o A. Sachs and the copy o E. Robson, these errors are as ollows: Section 4: the scribe has written 15.40 in place o 16.40. Section 5: the scribe has written 9 in place o 8. Section 11: the scribe has written 35.33.20 in place o 3 6.23.20. Section 19: the scribe has written 19 in place o 18. Te errors are all o the same type: orgotten or super uous signs. Te absence o a vertical wedge in certain instances, or example in Section 4, may be the result o the deterioration o the surace o the tablet, not an error. In act, in clay documents, signs are requently hidden by particles o dirt or salt crystals, or akes o clay have been broken off due to both 35
ancient and modern handling. Whatever the case may be, i the errors exist, they are not the result o errors in calculation, but simple aults in writing. Moreover, and this detail has great signi cance, the errors are not propagated in the ollowing sequence o calculations.36 Te arithmetic operations themselves, namely the multiplications, are then carried out in another medium in which the error had not occurred. Te text proceeds as i it does nothing but receive and organize the results o calculations computed in this external medium. For example, the act that, in the number 36.23.20 o Section 11, the scribe has transormed one ten in the middle place into a unit in the lef-hand place may be explained as an error in transerring a result rom some sort o abacus. Quite probably, some o the multiplications, particularly those which appear in the last sections and involve big numbers, required outside assistance, probably in the orm o a physical instrument (such as an abacus). 35 36
See the description o the state o this tablet by Sachs 1947: 230. It is not always the case in this genre o text. For example, in the tablet MLC 651, a school tablet in which the reciprocal is determined o 1.20.54.31.6.40 (a term rom the series o doublings o 2.5; see able 12.4), an error appears in the beginning o the algorithm and propagates throughout the ollowing text. Te error is a real error in calculation, which arose in the course o the execution o one o the multiplications.
Reverse algorithms in several Mesopotamian texts
Computing reciprocals in school texts ablet A possesses numerous parallels, nearly all o which appear in the characteristic orm o tablets called ype by Assyriologists. Scribes used these ype tablets to train in numeric calculation. Te copy presented in Appendix 2 is typical o these small lenticular or square tablets. Consideration o these parallels allows us to establish our tablet in the context o the scribal schools. Tis corpus in particular will allow us to determine the elements that relate directly to the school education to be detected, as well as those which do not seem to be connected to purely pedagogical purposes. From these comparisons, hypotheses about the unction o the tablet, the reciprocal algorithm, and most notably the direct and reverse sequences may be put orth. Let us consider all the known Old Babylonian tablets containing nonelementary reciprocal pairs (other than those which gure in the standard tables). o my knowledge, this set comprises a small group o about thirty tablets, listed in ables 12.6 and 12.7 below.37 In the rst table, I have gathered the parallels o ablet A. In the second table are ound the other texts; they also contain reciprocal pairs extracted rom geometric progression. Te dierent columns o the tables provide inormation about the ollowing points: (1) Te inventory number and type o school tablet. (2) Te provenance. (3) Reciprocal pairs contained in the tablet; when there are several pairs, the entries are always the terms o a geometric progression with a common ratio o 2; I have indicated only the number o pairs and the rst pair. (4) Te ormat o the text, indicated by numbers: (1) i the text appears as a simple list o reciprocal pairs; (2) i the presence o a actorization algorithm is noted; (3) i the presence o direct and reverse sequences o the actorization algorithm is noted.38 (5) In able 12.6, a supplementary column indicates the corresponding section o ablet A. Sections which have more than twenty doublings 37
38
Te tablets cited in the ables 12.4and 12.5 have been published in the ollowing articles and works. CBS 10201 in Hilprecht 1906: no. 25; N 3891 in Sachs 1947: 234; 2N- 500 in Robson 2000: 20; 3N- 362 in Robson 2000: 22; Ni 10241 in Proust 2007: §6.3.2; UE 6/2 295 in Friberg 2000: 101; MLC 651 in Sachs 1947: 233; YBC 1839 in Sachs 1947: 232; VA 5457 in Sachs 1947: 234; H99-192, H99-196, H99-584, H99-304a are unedited tablets, soon to be published by A. Cavigneaux et al.; MS 2730, MS 2793, MS 2732, MS 2799 in Friberg 2007: $1.4. (Note: among the tablets o the Sch yen Collection published in this last work are ound other reciprocal pairs, but their reading presents some uncertainty.) For example, ormat (1) is ound onthe obverse o the tablet Ni 10241, and ormat (2) on its reverse (see the Appendix).
405
4 n io t ec
8 n io t ec
51 n o it ec
.6 2 1 e l b a T
8 1– n o it ec
9 n io t ec
01 n o it ec
1 n io t ec
12 n o it ec
S S S S S S S S S S S
ta m ro F
)2 ( d n a )1 )1 )1 )1 )1 ( ( ( ( (
C
lesw l alr a P
8 n io t ec
01 n o it ec
A
ts n et n o
A te l b a T h it
81 n o it ec
0 22.3. 93 7. .72 4 / 40. 6 15. 51. .1
) ce en u q se es re erv ( 02 n o i cet S
)y tlc e ir d n (i 3 n o it ec S
1 n io t ec
)2 )3 ( ( d d n n a a )1 2) )1 )3 2) )1 )1 )1 ( ( ( ( ( ( ( (
45. .02 8 3 25.1. 48.5. 75 6. .5 01 .55 / / 54 02 .81 .3 .5 .5 .2 4.8 5.7 6. 55 01 .5
03 .3 1 / 04 .6 2. 4
03 .2 2. 3 / 4.0 64 .7 1
.c t ,8e 4. 82 / 5 2:. sr ai p 8
54 .6 / 02 3.5 .8
03 .2 2. 3 / 4.0 64 .7 1
r u p ip N
r u k p ip r u r N U U
,1 98 3 N
26, 3 T N 3
s iar p acl ro icp e R
6 .33 / 04 .6 1
03 31. / 04 .6 2. 4
51 1. .94 1. 6 / 02 35. 8.2 .9
ec n a n ev o r P
r u p ip N
r u p ip N
r u p ip N
r u p ip N
r u p ip N
r u p ip N
r u p ip N
e ytp ,r e b m u N
,6 94 T N 2
,5 06 T N 3
,5 11 T N 2
,4 42 01 i N
,1 42 01 i N
,0 05 T N 2
10 20 1 S B C
4.8 82 / 5. 2
,5 92 2/ 6 T E U
× 0 .82 = 04 62. . .9 e:1t o N 4 .02 5. 15 / 0 .64 .29 .1
.3 # A f yo rt n e e th si 02 .8 d n a 02 .8
31 – 1 s n io tc e
) n ito al o p ar tx (e 42 n ito ce
S S S S
8 n o it ce S S
1)( ()1 )1( ()2
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84 .8 2 / .25 se:r ev e ;rb r veo r :ep sr e vb O
ir a M
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n w o n k n U
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,2 19 -9T 9 H T
3,8 12 P L F
,y a3 4 617
31 n o it ec
) n io atl o p a trx (e 32 n o it ec
54 81. .5 2 / 02 31. .2 2. 2
.c t ,8e 4. 82 / 5 2.: sr ia p 31 e;l atb icr e m u N
03 . 3.7 05 9. )n 0.34 tio .9 lau 2. cl a 44 c / n i 4.0 ro 6. rr 13 e 45. n a 0.2 th i .1 w (
03 .3 1 / 04 .6 2. 4
.105 .4 1. 22 / 02 .3 1. .92 4. 14 .2
n w o n k n U
n w o n k n U
n w o n k n U
n w o n k n U
n w o n k n U
,2 08 01 C B Y
05 ,1 10 65 8 C M L B M
,9 38 1 C B Y
,9 97 2 S M
5 .84 1. 55 .9
ta m ro F
) (1
)1 )1 )2 )2 )1 )2 ( ( ( ( ( ( 32)
2
)² 2 × .14 = 61 4.(
st n te n o C
A etl b a T in g irn ae p p a to n se si rce elx ca o r p eci R 7 . 2 1 e l b a T
.c te 30 22. / 04 .2 s: iar p 6
.c et 20. 35 8. .44 1 / 3. 4: sr ai p 9
) 14 2 × 4. 1 = 16. 61 15. 4.(
.1 4× = 03 32. 84 .0 3 52. 14. (
03 . .87 2. 03 .5 .63 / 8. 6. 9
45. .43 .47 3. 3.4 12 .2 1 / 6 .61 1. 15 .4
> 03 .2 .51 < . 22. 1 .41 3. .512 .41 5. 62 .1 / 03 23 .8 .04 3. 52 .1 4
5. 25 / 21 .7 .9 .1
9)
2 × 4. 1 = .68 9.(
01 )
2 × .43 = 21 . .97 .1 (
sr ia p acl ro p ic e R
.c te 48 82. / 5. 2: s iar p 30
ec an en vo r P
r u p p i N
n w o ir ir n a a kn M M U
n w o n k n U
n w o kn n U
n w o kn n U
re b m u N
201 – 31 – 92 M U
,4 85 T -9 9 H T
a,4 03 T -9 9 H T
,0 37 2 S M
,3 97 2 S M
,2 37 2 S M
.c te 36 / 04 .1 :s iar p 01
. tec 5 .61 5 / .14 :s iar p 8
51 .6 5 / 4. 1
5 .34 .4 1 / 16. 4
,7 54 5 T A V
408
o 2.5 are called ‘extrapolations’. Since ablet A is limited to twenty doublings, these sections do not appear there. ables 12.6 and 12.7 show that a strong relation exists between ablet A and the school texts. Nearly all the direct parallels (able 12.6) or indirect parallels (able 12.7) are ype school tablets. Each concerns a single reciprocal calculation. Te tablets that are not o ype contain lists o reciprocals, all like ablet A. Tese tablets are UM 29–13–021 and CBS 10201, rom Nippur, as well as BM 80150, o unknown srcin. Te majority o school exercises use the data ound in ablet A. wo exercises rom Nippur are reproductions identical to Sections 9 and 10 o ablet A, including the reverse sequence. When the actorization method is employed in the exercises, it uses the actors chosen in ablet A, except in one case.39 Te tablets in able 12.7 that do not use the geometric progression with a common ratio o 2 and an initial number 2.5 still have links with ablet A. Speci cally, they use a geometric progression with a common ratio o 2, but with an initial term o 1.4 (and in one case 4.3), as ound or example in the tablet UM 29–13–021 rom Nippur. Tese observations could indicate that the tablets such as ablet A and the other tablets which are not ype school exercises (CBS 10201, UM 29–13–021, BM 80150) were the work o schoolmasters and that one o the purposes o their authors was the collection o exercises or the education o scribes. Te link between ablet A and teaching is incontestable, but does this signiy that ablet A is a ‘teacher’s textbook’ rom which the exercises were drawn? Several arguments t with this hypothesis, but it also raises serious objections. Beginning with what is now known about the school context and proceeding more speci cally to ablet A and its parallels I will present arguments or and against this text‘s being a ‘teacher’s textbook’. Te structure o school documents o an elementary level speaks in avour o the hypothesis. Lists o exercises can be considered a ‘teacher’s textbook’ i we consider them only on this level. Exercises rom the elementary level are extracts o texts written on tablets o a particular type, called ype by Assyriologists.40 Tis relationship between a ‘teacher’s textbook’ and pedagogical extracts appearsasboth orthe theadvanced mathematical or the lexical texts. However, ar as schooltexts textsand are also concerned, 39
40
In tablet CBS 1020, the actorization o 16.40 uses the actor 40 in place o 6.40. It is not, however, a ype school text, but atablet containing a list o eight reciprocals, the unction o which is closer to the unction o ablet A. Some authors think that the ype tablets rom Nippur are perhaps the product o students who have nishedtheir elementary education, undergoing some type o examination (Veldhuis 1997: 29–31).
Reverse algorithms in several Mesopotamian texts whether they are lexical or mathematical like the reciprocal exercises, the situation is different and ar rom simple. Te exercises are not ormulaic like those o an elementary level. I the documentation regarding the elementary level is composed o numerous duplicata, the documentation at an advanced level is composed only o unique instances, and this is true or the lexical texts and or the mathematical texts. Duplicata occur neither among the advanced school exercises nor among the most erudite texts to which they are connected. Te school documentation at an advanced level thus does not present as clear and regular a structure as that at an elementary level, and it cannot be relied on to identiy the nature o the relationship that connects ablet A with the school exercises. Nevertheless, the important act remains that ablet A has a large number o pedagogical parallels. Moreover, the known school exercises about reciprocal calculations all bear upon a number connected with the data in ablet A, whether directly (one o the terms o the series o doublings o 2.5), or indirectly (one o the terms o the series o doublings o another number such as 1.4 or 4.3). Tese instances have a unique relationship with the direct sequences on ablet A. On the other hand, reverse sequences are rarely ound in the school exercises. Tey appear only in two tablets rom Nippur, which reproduce exactly Sections 9 and 10 o ablet A, and in a tablet rom Mari (H99-196). Again, in the two cases rom Nippur, the reverse sequences are not isolated, but associated with the direct sequences. Tus, it is not the data rom the reverse sequences that provide the material or the school exercises, but rather the data rom the direct sequences. In general, the reverse sequences provide a very small contribution to the prospective ‘collection o exercises’ or teaching, and yet they constitute hal the text o tablet A. Te pedagogical interest in the series o doublings o 2.5 must also be considered because this series allows the repetition o the same algorithm many times, under conditions where it provides only results known in advance, with a gradually increasing level o difficulty. In act, this argument relates to the educational value o the geometric progression with a common ratio o 2 and an initial term o 2.5, not to ablet A in its entirety. ablet A is constructed around the idea o reciprocity, a notion clearly undamental to its author and hardly present in the ordinary exercises about reciprocal calculations. Tese considerations lead to the notion that it is possible that the relationship between ablet A and the school exercises is exactly the opposite o what is usually believed. ablet A does not seem to be the source o school exercises: rather it seems derived rom the school materials with which the
409
410
scribes o the Old Babylonian period were amiliar. In this case, the material was developed, systematized and reorganized with different objectives than the construction o a set o exercises.41 Te unction o the reverse sequence seems to be the key to understanding the whole text. It has been suggested above that the reverse sequence might play a role in relation to the unctional veri cation o the algorithm. Te question that arises concerns, more precisely, the nature o the relationship between the direct sequence and the reverse sequence. In order to advance this inquiry, we turn to other cases in the cuneiorm documentation which present direct and reverse sequences. As emphasized in the introduction, these cases appear in several tablets containing calculations o square roots. Tus let us examine these calculations.
Square roots Sources presently known to contain calculations o square roots are not so numerous as those concerning reciprocals. Nonetheless, they present interesting analogies with what we have just considered. First o all, texts in both a numeric and verbal style are ound or the same algorithm. Additionally, the undamental elements o the reciprocal algorithm – actorization, spatial arrangement in columns (in the case o the numeric texts) and the presence o reciprocity – appear in these texts. Tis small collection o texts allows us to consider some o the problems raised above rom other angles: the nature o the reciprocal algorithm, the connections between the direct and reverse sequences, the speci city o numeric texts with respect to verbal texts and the nature o the links that the different types o texts have with education. able 12.8 gives the list o tablets containing the calculations o square roots (I recall in column 1 the letters indicated in able 12.1).42 I have likewise included those which contain calculations o cube roots, though 41
42
Tis process may be compared to that described by Friberg or the various Mesopotamian and Egyptian texts under the name o ‘recombination texts’. For him, this type o compilation is tightly connected with educational activity (Friberg 2005: 94). Te tablets o able 12.8 have been published in the ollowing articles and works: C = UE 6/2 222 in Gadd and Kramer 1966: no. 222 – see able 12.1; YBC 6295 in Neugebauer and Sachs 1945: 42; VA 8547 in Sachs 1952: 153; D = IM 54472 in Bruins 1954: 56 – see able 12.1; H99-3 is an unedited tablet, soon to be published by A. Cavigneaux et al.; Si 428 in Neugebauer 1935–7: 80; HS 231 in Friberg 1983: 83; 3N- 611 in Robson 2002: 354; YBC 6295 in Neugebauer and Sachs 1945: text Aa, this tablet is believed to have come rom Uruk, in the south o Mesopotamia according to Neugebauer 1935–7: 149 and to H2002: 333–7; VA 8547 in Sachs 1952: 153.
Reverse algorithms in several Mesopotamian texts Table 12.8 Calculations o square and cube roots ablet
Number, type
Provenance
Calculation
Style
C
UE 6/2 222,
Ur
Numeric
3N- 611,
Nippur
HS 231,
Nippur
Square root o 1.7.44.3.45 (result: 1.3.45) Square root o 4.37.46.40 (result: 16.40) Square root o 1.46.40
D
H99-3,
Mari
Si 428,
Sippar
IM 54472
Unknown
YBC 6295
Unknown
VA 8547
Unknown
(result: 1.20) (uncertain reading) Square root o 2.6.33.45 (result: 11.15) Square root o 2.2.2.2.5.5.4 (result: 1.25.34.8) Square root o 26.0.15 (result: 39.30) Cube root o 3.22.30 (result: 1.30) Cube roots o 27, 1.4, 2.5 and 3.36 (results: 3, 4, 5, 6 respectively)
Numeric Numeric
Numeric Numeric Verbal Verbal Verbal
no numeric version occurs with cube root calculations. Tis absence poses an interesting question: is this the result o chance in preservation or a signi cant act? ablet D, o unknown srcin, contains a text composed in Akkadian which concerns the procedure o calculating the square root o 26.0.15. For a detailed analysis, see the various publications on the subject o this text.43 wo interesting points should be highlighted here. Te rst is the presence o the actorization algorithm, in the orm o instructions wherein the terms are quite similar to those in ablet B regarding reciprocals. Te second is the last phrase: ‘39.30 is the side o your square. 26.0.15 is the result (o the product o 39.30 by 39.30).’ Te tablet thus ends with a veri cation o the result. ablet C is a small lenticular school tablet, the transcription and copy o 44
which are shown in able 12.9. Te process o calculation by actorization occurs in the case o ablet C, as Friberg has remarked. Te number 1.7.44.3.45 ends with 3.45, which is selected as an elementary regular actor. 43
44
Chemla 1994: 21; Muroi 1999: 127; Friberg 2000: 110. Because no copy o the text has yet been published, it is not known i the presence o zero in the middle place is indicated on the tablet by a blank space, as sometimes happens in cuneiorm texts, particularly those o the rst millennium. Copy: Gadd and Kramer 1966; transcription: Friberg 2000: 108. See also Robson 1999: 252.
411
412
Table 12.9 ablet C ranscription
15 15 17
Calculations
1.3.45 1.3.45 1.7.44.3.45 18.3.45 4.49 3.45 1.3.45
Copy
1.3.45 × 1.3.45 = 1.7.44.3.45 16 16
inv(3.45) = 16; sq.rt.(3.45) = 15 inv(3.45) = 16; sq.rt.(3.45) = 15 sq.rt.(4.49) = 17 15 × 15 = 3.45 3.45 × 17 = 1.3.45
Te number 16, its reciprocal, is set out on the right; on the same line, the number 15, its square root, is set out on the lef; the product o 1.7.44.3.45 by 16 (which gives a second actor) is placed on the centre o the ollowing line. Te process is repeated until a number or which the square root is given by the standard tables is ound.45 Te desired square root is the product o the numbers recorded on the lef. It should be noted that this small text, like those ound in the sections o ablet A, begins and ends with the same number, and as beore, the calculation orms a loop. It starts with an arithmetical operation (the square o 1.3.45), then it proceeds by a sequence which carries out the reverse operation (the square root o the resulting number, 1.7.44.3.45). Here, the direct sequence and the reverse sequence rely on algorithms o a different nature, even though in the cases involving reciprocals, they rely on the same algorithm. Could it be said that the calculation o the square o 1.3.45 is a simple veri cation o the result o the calculation o the square root? In this case, it would be logical that the veri cation should come at the end o the calculation (as is the case in the verbal ablet D) and not at the beginning. Te text thus illustrates something else, which seems to relate to the act that the square and the square root are reciprocal operations. Tis ‘something else’ is perhaps akin to what the author o ablet A illustrated with the reverse sequences. Te algorithm or calculating square roots is based on the same mechanism o actorization as that or determining the reciprocal. In the numeric versions, the rules concerning the layout are analogous: the actors are 45
As in the case o the reciprocals, the calculations o the squares and square roots rely on a small stock o basic results memorized by the scribes during their elementary education. Te tables o squares and square roots are largely ound in the school archives. See, notably, Neugebauer 1935–7: ch. I.
Reverse algorithms in several Mesopotamian texts placed in the central column; the reciprocals o these are placed to the right; a supplementary column appears on the lef, in which are placed the square roots o the actors. Tis supplementary column shows us that the algorithm in act has two components: a actorization (right-hand column) and square root (lef-hand column). In the case o the reciprocal’s algorithm, the right-hand column provides the actors which serve all at once as the actorization and the determination o the reciprocals. Tus the two components merge. However, the method o application o the actorizations presents a particular mathematical problem or the square roots. In effect, the algorithm or nding areciprocal is, by de nition, applied to the regular numbers. Te actorizations are always possible, and lead mechanically to the result. Alternately, perect squares can quite easily be the product o irregular numbers, and in this case, actorization by the standard method is impossible. Te important point to note is that, even though the algorithms or the determination o the reciprocal and the extraction o a square root diverge rom one another in their components and even though they present different mathematical problems as their topic, they are presented in the texts in a parallel ashion. Te speci city o the numeric texts with regard to the verbal texts thus appears more clearly. For the square roots, the layout o the numeric texts observes the same rules regarding arrangement in columns as or the determination o reciprocals. Tis spatial arrangement acilitates control o the calculation. In act, it is enough, when nding the desired number, to multiply all those that are set out on the right in the case o reciprocals, and those on the lef in the case o roots. It is notable that, in the case o reciprocals as well as square and cube roots, the verbal versions contain only numbers o a small size, which do not demand recourse to iteration. Te numeric versions contain numbers o large size, and the arrangement in columns shows that it is possible to develop the iterations without limit, which coners power on the process. Te verbal and numeric versions o the calculations o square roots reer nonetheless to the same algorithms. In act, the verbal texts contain instructions which detail how to ‘place’ certain numbers ‘beneath’ others, in a way which corresponds with the spatial arrangement o the numeric texts. What is the place o square roots in the education o the scribes? Te ormat o the tablets containing the calculation o square roots, which are all o ype or the numeric versions, shows that they were school exercises. However, in this case the exercises are much less standardized than the calculation o reciprocals. For square roots, the numeric repertory offers no regularity, whereas or the reciprocals, the repertory is homogeneous (as
413
414
seen above, it is based principally on the doublings o 2.5). Moreover, the group o tablets containing the calculations o square roots is small, whereas the group o exercises o the calculation o reciprocals is numerically important. Te great requency o calculations o reciprocals is undoubtedly explained by the importance o this technique in calculation, but another reason may be postulated. In the reciprocal, the two components (actorization and the determination o reciprocals) are superimposed. Te algorithm or the determination o a reciprocal by actorization puts the mechanism o actorization rst. Te determination o a reciprocal by actorization is thus a undamental procedure,46 essential to other algorithms, even though it is applied in a less general way or the roots than or the reciprocals. Consequently, the reciprocal exercises probably occupy a more elementary educational level than those that contain square roots. Te calculations o square roots may be situated between the work o beginning scribes and works o scholars, in a grey area that has lef us ew traces. What, then, o the cube roots? Tey appear in two verbal texts, wherein they are treated in a manner identical to the square roots, except or the veri cations, which do not appear in either case.47 No numeric version is known or these calculations. It cannot be excluded that the absence o a numeric version o the calculation o a cube root is due to the chances o preservation but other explanations are possible. Indeed, tables o squares, square roots and cube roots are known to us rom the preserved numeric tablets, but tables o cubes are unknown. Te absence o a table o cubes is undoubtedly linked to the act already mentioned that multiplication is an operation with two arguments. Consequently, the cube root has no reverse operation in the Mesopotamian mathematical tradition. Tis act would explain why it has not been ound in a numeric ormat, which is ounded on the notion o reciprocity. Tis analysis o the calculation o square roots also emphasizes by contrast the act that the reciprocal algorithm is a combination o two different components (actorization and the determination o a reciprocal). In addition, it may be seen that the numeric texts have an approach 46
47
Te Akkadian term maks.arum probably has some link with the process o actorization. It appears in two texts, in slightly different senses: it appears in the incipit o tablet YBC 6295 cited in able 12.6 ([ma]-ak-s. a-ru-um šaba-si = the maks.arum o the cube root); it designates an enlargement in tablet YBC 8633. Note also the ollowing curious detail: in VA 8547, all the entries appear in the standard tables o cube roots, and the application o the reciprocal algorithm to these numbers leads to a complication o the situation. Tus, 27 is decomposed according to a somewhat arti cial manner as the product o 7.30 and 3.36. It is clear that in this case, as in that o ablet A, the purpose is not to obtain a new result.
Reverse algorithms in several Mesopotamian texts relatively uni ed with that o the reciprocal algorithm. Te unction o the reverse algorithm seems the same in all cases. It does not enact a veri cation o the result, or even a veri cation o the algorithm itsel in the case o the square roots, since the direct and reverse sequences do not rely on the same algorithm. Teir presence seems to indicate something else with respect to the nature o the operations themselves. It stresses the act that the reverse operation o a square is the square root, and the reverse operation o the reciprocal is the reciprocal itsel.
Conclusion I can now reconsider several questions lef aside rom the preceding discussion. Te unction o the tablet is at the heart o these questions, and I will treat these questions beore returning to the ways o reasoning we can detect in the text. It has been seen that the content o ablet A is connected with the context o teaching but that it cannot be interpreted as a simple collection o data intended to provide exercises or the education o young scribes. I have suggested that its relationship with the school exercises could be the reverse o what is generally supposed. It might not be a ‘teacher’s textbook’ rom which the school exercises were taken but rather a text constructed and developed rom existing school material. Indeed the relationships between school exercises and scholarly texts were probably not so unidirectional and the two relations could well be combined. However, the point which interests us here is that ablet A appears in the orm o an srcinal inquiry and its purpose seems to have been communication between erudite scribes. Seen rom this perspective, the same piece o text takes on another dimension. Te way in which the text is organized and arranged, and the repertory o numeric data on which it is built, are essential components o the text. In a certain way, these components constitute the means o expression by which ablet A reers to the reciprocal algorithm. But what is the relationship between ablet A and the algorithm or reciprocals? Is it a practical text in the sense that the text executes concretely the operations necessary or the determination o a reciprocal? It is not certain that the writing o a text was essential to the execution o the algorithm, since the known texts obviously record only part o the series o actions that allow the result to be obtained. On the one hand, the multiplications are probably executed elsewhere. On the other hand, by the standards o school practices, the written traces are incomplete. Tey ofen state only the rst
415
416
step o the process o actorization, as is notably the case in the tablets o the Schøyen Collection published by Friberg listed in able 12.7. Te tablet does not reer to all the steps necessary to execute the algorithm. ablet A is not a simple set o instructions or execution o the reciprocal algorithm. What does tablet A say about this algorithm and how? First o all, the author o ablet A expresses himsel by means o numbers arranged in a precise way, not by means o a linear continuation o the instructions, as is done in the verbal texts. Te numeric texts reer to the same algorithms as the verbal texts, but they do it in a different way. Te spatial arrangement o the writing has its own properties and emphasizes certain unctions o the algorithm. Te arrangement into columns renders the process o determining a reciprocal transparent. Indeed, to nd the desired number, it is enough to multiply the numbers on the right in the case o the reciprocals and the numbers on the lef in the case o the roots. Te arrangement into columns certainly recalls the practices o calculation external to the text, but the act that this arrangement was set in writing clearly emphasizes the principles o the unction o the algorithm – that is, the act that it is possible to actorize the regular numbers into the product o regular numbers and the act that the reciprocal o a product is the product o the reciprocals. Moreover, the spatial arrangement o the text underscores the power o the procedure o developing the iterations without limitation. On this topic, let us recall the striking act that the recourse to iteration does not appear in the verbal texts, which limit themselves to numbers o a small size, whereas the iteration expands in a rather spectacular way in ablet A, and in a more modest way in the numeric versions o the calculations o the square roots. For the ancient reader, the spatial arrangement o the numbers in ablet A serves the unctions that Sachs’ ormula does or the modern reader: it shows why the algorithm works. Te layout says more than the ormula in showing not only why, but also how it operates and what its power is. ablet A is constructed on the repetition o the doublings o 2.5. Te educational value o this series in the instruction o the actorization algorithm has been underscored above, but perhaps the essence lies elsewhere. Te act that the scribes limited themselves to the geometric progression with an initial number 2.5 and a common actor o 2 guarantees the regularity o the entries. Tis series assures the calculator that the result remains in the domain o regular sexagesimal numbers, a condition necessary or the existence o a sexagesimal reciprocal (with nite expression) and or the operation o the algorithm. It undoubtedly did not escape the scribes that it was possible to choose other series (in tablet UM 29–13–021 are ound series based on other initial terms, such as 2.40, 1.40, 4.3). However, the series o doublings o 2.5 is a typical example which allows the scribes to
Reverse algorithms in several Mesopotamian texts reer to the algorithm by speci c numeric data. In other words, this series plays the role o a paradigm. It is possible that the choice o 2.5 comes rom the previously noted act that this number is a logical continuation o the standard reciprocal tables in which the last entries are 1.4 and 1.21. Fundamentally, ablet A is built on reciprocity. What expresses the systematic presence o the reverse sequences? It has been shown that the purpose was not the veri cation o the results because such a matter could have taken a much simpler orm. It could have had a role in the veri cation o the algorithm itsel and thus ensured the validity o the mechanism. However, as suggested above, the signi cance o the reverse sequences could have been above all to express a mathematical rule: ‘Te reverse o the reverse is itsel.’ Whatever the case may be, it is clear that in the reverse sequences, the author abandons the stereotypical patterns ound in the direct sequences o the text (and ound also in the school exercises) and plays with the reedom remaining to him in the choice o actors or the decomposition into elementary regular actors. Te reverse sequences thus highlight another important mathematical aspect: the multiplicity o decompositions. Te purpose o the text on ablet A is thus clearly the algorithm itsel, its operation and its justi cation. Te text reers to the algorithm not in a verbal manner, but by an interpretable spatial arrangement, the exploitation o a paradigm well known to the scribes, and the recourse to the reverse sequences in a systematic way. ablet A thereore bears witness to the re ection o the ancient Mesopotamian scribes on some o the undamental principles o numeric calculation: the possibility o decomposing the regular numbers into two or more (through iteration) elementary regular actors, the reedom which the multiple valid decompositions offer to the calculator (given that the direct and reverse sequences show two different strategies or the selection o actors), the stability o the multiplication or reciprocal (the reciprocal o a product is the product o reciprocals o the actors) and the involutive character o the determination o a reciprocal (given the act that this operation is its own reverse operation).
Appendix
ablet A (CBS 1215)
Sachs 1947: 237; Robson 2000: 23. Te asterisks reer to the remark which ollows the transcription. I have added the elements o the appearance to acilitate the reading: the nal part o the number which plays a role as a actor is set in bold; the nal result o the calculation is underlined; the ormat reproduces the layout o the tablet.
417
418
Obverse Column 1–8
Column 9–13
2.5
12 2.24 1.15 1.40
25
28.48 36
8.53.20
4.10
6.45 9
14.24 36
8.53.20 6 2.24 2.30
25
1 .40 4.10
8.20 7.12 36
8.20 16.40
9 24 [1.40] 10
2.30
3.[36] 6
17.46.40 2.40
3.22.30 6.45 8.53.20 17.46.40 36sic.2sic3.20 10.40 [16] 5.37.30 [1.41.1]5 [6.45] [9]
18 6 1.15 4 6.40
10
1.48 2.15 8sic
26.40 33.20 1.6.40 54
9 6 1.6.40
[2].13.20 [40] [27]
18 1.30 2.13.20
10
4.26.40
9 1.30 2 2.13.20
40
13.30 27
4.26.40
18 1.[30] 3.4[5] 4 1.20 6.40
[8.53].20 [35.33].20
15sic.40 33.20
9 22.30 2 1.20 6.40
9
3 2.24 5 1.40
25
18 22.30 1.20 6.40
2.40
2.5
Column
[1].11.6.[40] 10.40 16
9 1.[30] 3.4[5]
5.37.30 50.37.30 1.41.15 6.4[5] 9
2 4 1.20 6.40 [8.5]3.[20] 35.33.20 1.11.6.40
2.22.13.20 42.40 16
[18] 22.30 3.45
1.24.22.30 25.18.45* [16]
13–16
6.45
1.20 [6.40]
9
8.53.20 [2.2]2.13.[20] 4.44.26.40 42.40 16
[9] 2[2.30] 3.[45]
1.24.22.30 [12.3]9.22.30 [2] [25.18].45* [16] [6.45] [1.20] [9] [6.40] [8].53.20 [2.22.13. 20] [4.44.26.40] [9.28].53.[20] 2.50.40 [4.16] [16]
[18] [1.30] [3.45] [3.45]
14.3.[45] [2]1.5.3[7.30] [6.19.4]1.15 [4] [25.18.45]* [16] [6.45] [1.20] [9] [6.40] [8.53.20] 2.[22.13.20] 9.[28.53.20] 18.57.[46.40] [2.50.40] 4.[16] 16 [14].3. [45]
[9] [1.30] [3.45] [3.45]
[21.5.37.30] [3.9.50.37.30] [2] [6.19.41.15] [4] (continued on the reverse)
Reverse algorithms in several Mesopotamian texts Reverse (on the reverse o the tablet, the columns run rom right to lef, as
is customary) Column 21 10.6.48.53.20 3.2.2.40 1.8.16 4.16
Column 19–20 18 22.[30] 3.4[5] 3.[45]
3.[45] 1[4.3.4]5 52.44.[3.4]5 19.46.31.24.22.[30] 5.55.57.25.18.4[5] 16 1.34.55.18.45* 16 25.18.45* [16] 6.45 [1.20] 9 [6.40] 8.53.20 2.22.13. 20 37.55.33.20 10.6.48.53.20 16
[2.31.42.13.20 [45.30.40 [1.8.16 [4.16
Column 16–18 18] 1.30] 3.45] 3.45]
[3.45] 14.[3.45] 5[2.44.3.45] 1.18sic.6.[5.37.30] 23.43.49.[41.15] [4] 1.[3]4.55.18.45* [16] [25].18.45* 1[6] [6].45 1.[20] [9] 6.40 8.53.20 2.22.13.20 37.55.3[3.20] 2.31.42.13.[20] 5.3.24.26.40
[9]
40 45.30. 1.8. 16 4.16
1.30 3.45 3.45 3.45
14.3.45 5[2.44].3.45 1.19.6.5.37.30 11.51.54.50.37.30 2 23.43.49.41.15 4 1.34.55.18.45* 16 25.18.45* 16 6.45 1.20 9 6.[40] 8.53.20 2.22.13. 20 37.55.33.20 2.31.42.13.20 5.3.24.26.40
16] 1.20] 6.40]
[8.53.20] [2.22.13.20] [9.28.53.20] [18.57.46.40]
16
16
Notes are on p. 420
(continued) [25.18.45* [6.45 [9
[37.55.33.20 18] [11.22.40 22.30] [4.16 3.45] [16 3.45] [14.3.45] [5.16.24.22.30] [1.34.55.18.45* 16] [25.18.45* 16] [6.45 1.20] 9 [6.40] [8.53.20] 2.22.13.[20] 37.55.33.[20]
1.15.51.6.40 11.22.40 4.16 16
9 22.30 3.45 [3.45]
14.[3.45] 5.16.[24.22.30] 47.27.[39.22.30 2] [1.34.55.18.45* 16] [25.18.45* 16] [6.45 1.20] [9 6.40] 8.[53.20] 2.2[2.13. 20] 37.55.[33.20] 1.15.51.[6.40]
419
420
Notes to pp. 418–19 Section 4: Read 16.40 in place o 15.40. Section 5: Read 9 in place o 8. Section 11: Read 35.33.20 in place o 36.23.20. Section 19: Read 19 in place o 18. *Section 13 to Section 21: Te actor chosen is 3.45 (rom the reciprocal 16). I could not set it in bold type because it does not obviously constitute the nalpart o the number, as in the other cases. However, i 8 is decomposed into the sum 5+3, the actor 3.45 is scarcely hidden. (For more precise details, see the part o the article devoted to the analysis o the entirety o this text.)
Appendix
Ni 10241
Old Babylonian school tablet rom Nippur, conserved in Istanbul, copy Proust 2007. Obverse
4.26.[40] its reciprocal 13.30
Reverse
4.26.40
9
41sic
1.30 13.30
Bibliography Britton, J. P. (1991–3) ‘A table o 4th powers and related texts rom Seleucid Babylon’, Journal o Cuneiorm Studies 43–5: 71–87. Britton, J. P., Proust, C. and Schnider, S. (2011) ‘Plimpton 322: a review and a different perspective’, Archive or History o Exact Sciences 65: 519–66.
Reverse algorithms in several Mesopotamian texts Bruins, E. (1954) ‘Some mathematical texts’, Sumer 10: 55–61. (1969) ‘La construction de la grande table de valeurs réciproques AO 6456’, Proceedings o the 17thRencontre Assyriologique Internationale.Bruxelles: 99–115. Cavigneaux, A. (1989) ‘L‘écriture et la ré exion linguistique en Mésopotamie’, in Histoire des idées linguistiques, vol. , La naissance des métalangages en Orient et en Occident, ed. S. Auroux. Liège: 99–118. Charpin, D. (1992) ‘Les malheurs d‘un scribe ou de l’inutilité du sumérien loin de Nippur’, in Nippur at the Centennial, Papers read at the 35th Rencontre Assyriologique Internationale, Philadelphia, 1988, ed. M. deJong Ellis. Philadelphia, PA: 7–27. Charpin, D., and Joannès, F. (eds.) (1992) La circulation des biens, des personnes et des idées dans le Proche-Orient ancien, Proceedings o the 38th Rencontre Assyriologique Internationale, Paris. Chemla, K. (1994) ‘Nombres, opérations et équations en divers onctionnements’, in Nombres, astres, plantes et viscères, sept essais sur l’histoire des sciences et des techniques en Asie Orientale, ed. I. Ang and P.-E. Will. Paris: 1–36. Friberg, J. (1983) ‘On the big 6-place tables o reciprocals and squares rom Seleucid Babylon and Uruk and their Old Babylonian and Sumerian predecessors’, Sumer 42: 81–7. (2000) ‘Mathematics at Ur in the Old Babylonian period’, Revue d’Assyriologie 94: 98–188. (2005) Unexpected Links Between Egyptian and Babylonian Mathematics . Singapore. (2007) A Remarkable Collection o Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection – Cuneiorm Texts, vol. . New York. Gadd, C. J., and Kramer, S. N. (1966) Literary and Religious Texts, Second part. Ur Excavations exts vol. /2. London. Hilprecht, H. V. (1906) Mathematical, Metrological and Chronological Tablets rom the Temple Library o Nippur. Babylonian Expedition vol. /1. Philadelphia, PA. Muroi, K. (1999) ‘Extraction o square roots in Babylonian mathematics’, Historia Scientiarum 9: 127–32. Neugebauer, O. (1935–7) Mathematische Keilschriftexte, vols. – . Berlin. Neugebauer, O., and Sachs, A. J. (1945) Mathematical Cuneiorm Texts. New Haven, C. ‘Eine Reziprokentabelle der Ur -Zeit’, inChanging Views on Oelsner, J. (2001) . Berlin: Ancient Near Eastern Mathematics, ed. J. Høyrup and P. Damerow 53–8. Proust, C. (2007) Tablettes mathématiques de Nippur, Part , Reconstitution du cursus scolaire, Part , Édition des tablettes conservées à Istanbul. Istanbul. (2008) ‘Quanti er et calculer: usages des nombres à Nippur’, Revue d’histoire des mathématiques 14: 143–209.
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Robson, E. (1999) Mesopotamian Mathematics, 2100–1600 : Technical Constants in Bureaucracy and Education. Oxord. (2000) ‘Mathematical cuneiorm tablets in Philadelphia. Part 1 : problems and calculations’, SCIAMVS 1: 11–48. (2001a) ‘Neither Sherlock Holmes nor Babylon: a reassessment o Plimpton 322’, Historia Mathematica28: 167–206. (2001b) ‘Te ablet House: a scribal school in Old Babylonian Nippur’, Revue d’Assyriologie95: 39–66. (2002) ‘More than metrology: mathematics education in an Old Babylonian scribal school’, in Under One Sky: Astronomy and Mathematics in the Ancient Near East, ed. J. M. Steele and A. Imhausen. Münster: 325–65. Sachs, A. J. (1947) ‘Babylonian mathematical texts ’, Journal o Cuneiorm Studies 1: 219–40. (1952) ‘Babylonian mathematical texts : Approximations o reciprocals o irregular numbers; : Te problem o nding the cube root o a number’, Journal o Cuneiorm Studies 6: 151–6. Veldhuis, N. (1997) ‘Elementary education at Nippur: the lists o trees and wooden objects’, PhD thesis, University o Groningen.
13
Reading proos in Chinese commentaries: algebraic proos in an algorithmic context
Te Chinese text devoted to Te mathematics that has been handed downearliest through the written tradition, Nine Chapters on Mathematical Procedures (Jiuzhang suanshu), was probably compiled on the basis o older documents and completed in the orm in which we have it today in the rst century .1 Until recently, there was no evidence indicating the nature o the documents that may have been used in composing Te Nine Chapters. However, in 1984, in a tomb that had been sealed c. 186 at Zhangjiashan (today in the Hubei Province), archaeologists ound a text entitled the Book o Mathematical Procedures (Suanshushu) which may have been used or this purpose.2 Like this book that was brought to light thanks to archaeological excavations but did not survive through written transmission, Te Nine Chapters is mainly composed o particular problems and algorithms or solving them, without displaying any apparent interest in establishing 1
2
In what ollows, the title is abbreviated as Te Nine Chapters. Te ull title would be more accurately translated as ‘Mathematical procedures in nine chapters/patterns’. However, to avoid conusion with titles o other Chinese mathematical books, the English translation o which is quite close to that o Te Nine Chapters, I give a translation that does not diverge rom the usual English title given to the book. In this volume, A. Volkov (see Chapter 15, Appendix 2) chooses to translate the title as Computational Procedures o Nine Categories . In the earliest document that was handed down and that outlines the history o Te Nine Chapters as a book, i.e. the third-century commentator Liu Hui’s preace, the process o compilation is sketched and mentioned as having lasted more than a century. In the introduction to Chapter 6 in CG2004, I gather the evidence on the basis o which I consider the book to have been completed in the rst century . In this chapter, unless otherwise stated, I ollow the critical edition o Te Nine Chaptersgiven in CG2004. Te reader can nd in this book a complete French translation o the Classic and its traditional commentaries; see below. Other translations o the same texts have appeared in recent years: some into modern Chinese (Shen Kangshen 1997; Guo Shuchun 1998; Li Jimin 1998), one other into English (Shen, Crossley and Lun 1999, based on Shen 1997). It is impossible, within the ramework o this chapter, to comment on all the differences between the translation given here and these other translations. Te interested reader can compare the various interpretations. Te rst critical edition o this text can beound in Peng Hao 2001 . wo translations into English have already appeared (Cullen 2004; Dauben 2008). Te Nine Chapters and the Book o Mathematical Procedures have a number o similarities. For example, they deal with the same concept o ractions, conceived o as composed o a numerator and a denominator. Moreover, they contain similar algorithms to compute with ractions. In addition to testiying to the act that these elements o mathematical knowledge existed in China beore 186 , the Book o Mathematical Procedures provides additional inormation that will prove useul or us below.
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the correctness o the algorithms provided.3 However, soon afer its completion, the book became a ‘Classic’ (jing) and retained this status in the subsequent centuries, which accounts or the speci c ate it had not only in China, but also in Korea and Japan. On the one hand, as is clear rom the reerences made to it, the book remained a key reerence work or practitioners o mathematics in China until at least the ourteenth century, and this act most probably explains why it is the earliest extant text to have been handed down through the written tradition. On the other hand, commentaries on it were regularly composed, two o which were perceived as so essential to the reading o the text that they were handed down with the Classic itsel. In act, no ancient edition o Te Nine Chapters has survived that does not contain the commentary completed by Liu Hui in 263 and the explanations added to it by a group o scholars under the supervision o Li Chuneng.4 Tis detail o textual preservation indicates how closely linked to each other these texts became, to the extent that, at some point in history, they constituted, or Chinese readers, an integrated set o texts that were no longer dissociated. As a consequence, i we, as contemporary exegetes, are to understand how Te Nine Chapters was approached in ancient China, it is important that we, like Chinese readers, read the text o the Classic in relation to that o its commentaries. Tis relationship proves important in several respects. On the one hand, through the commentaries, one can establish that even though the problems contained in Te Nine Chapters all appear to be particular statements, they were read by the earliest readers whom we can observe as general statements. Te commentators exhibit the expectation that the algorithm linked to a problem should solve not simply this problem, but the category o problems or which the problem, taken as paradigm, stood.5 On the other hand, the commentators make explicit some theoretical dimensions that 3
4
5
In Chemla 1991 and 1997/8, I have given several hints indicating that the situation is not so simple. However, since the ocus o this chapter lies elsewhere, I shall not dwell on this question. Te reason why this issue is crucial or us here will become clear in Part o this chapter. Let us stress that the title o Te Nine Chapterscontains the character shu ‘procedure’ which introduces the statement o the algorithms contained in both books. Below, or the sake o simplicity, we reer to this layer o the text by the expression o ‘Li Chuneng’s commentary’. In act, the situation is less simple than is presented here. Tere are problems in distinguishing between the two layers o commentaries (I have summarized the state o our present knowledge on the topic in CG2004: 472–3). In the present chapter, I have attempted to deal with my topic in a way that is not jeopardized by this diffi culty. In act, this presentation o Te Nine Chaptersis simpli ed. An algorithm can be given afer a set o problems. Moreover, there are cases when an algorithm is given outside the context o any problem, or constitutes an instantiation o such an algorithm. However, this does not invalidate the main thesis.
Reading proos in Chinese commentaries
were driving the inquiry into mathematics in ancient China. For instance, they reveal that generality was a key theoretical value and that nding out the most general operations was an aim pursued by the practitioners o mathematics.6 However, a crucially important act or us lies elsewhere: afer the description o virtually every algorithm presented in Te Nine Chapters, or between the sentences prescribing its successive operations, the commentators set out to prove its correctness. Tese texts thus provide the earliest evidence available today regarding the practice o mathematical proo in ancient China, and this is the reason why, in this chapter, we shall concentrate on them. In contrast to what can be ound in ancient Greek geometrical sources, where statements are proved to be true, the Chinese commentators system7 atically strove to establish the correctness o algorithms. It can hence be assumed that the commentaries bear witness to a practice o mathematical proo that, as a practice, developed independently rom what early Greek sources demonstrate. However, weshall not dwell on this issue here. Instead, and as a prerequisite to tackling this question in the uture, we shall aim at better understanding this practice o proo. Tereby, we may hope to cast light more generally on some o the undamental operations required when proving the correctness o algorithms – a section o the history o mathematical proo that, to my knowledge, has been so ar almost entirely neglected. Even though it constitutes an oversimpli cation to be re ned later, let us say, or the present, that an algorithm consists o a list o operations that can be applied to some data in order to yield a desired magnitude. In this context, proving that such an algorithm is correct involves establishing that the obtained result corresponds to the desired magnitude. It can be shown that, when ul lling this task, the commentators systematically made use o some key operations. Moreover, they employed specialized terms to reer to concepts related to these operations.8 Tese acts disclose that, ar rom being ad-hoc developments, these proos complied with norms amiliar to the actors, since they devised technical terms related to them. Te way in 6
7
8
Chemla 2003 establishes these points. Below, we shall nd additional evidence supporting these theses. It can be shown that this is how the commentators themselves conceive o the aim o their reasonings. See Chapter A in CG2004: 26–8. I do not come back to this point here. Note that the commentators leave some o the most basic algorithms without proo. Guo 1992 : 301–20 stressed this act, emphasizing that this eature meant that the commentators were shaping an architecture o algorithms, the proos o which depended on algorithms proved previously. From another angle, one can argue that reduction to undamental algorithms, and not to simple problems, is also a key point at stake in the proos carried out by the commentators. Chapter A o CG2004: 26–39 sketches these points.
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which the re ection about proo developed in ancient China still awaits urther study. In this chapter, I shall ocus on urther highlighting and analysing two key operations that are undamental constituents o the practice o proo documented by our commentators. Te rst partpresents in some detail an example illustrating the two eatures on which we shall concentrate: on the one hand, determining the ‘meaning’ o a computation or o a sub-procedure; on the other hand, carrying out what I called an ‘algebraic proo within an algorithmic context’ – what I mean by this expression will become clear with the example. In the case o the ormer eature, our analysis will provide an opportunity to examine the modalities according to which the ‘meaning’ o a sequence o computations can be determined. As or the latter eature, afer having brought to light undamental transormations characteristic o this part o the proo, I shall present evidence in avour o the hypothesis that there existed an interest in ancient China regarding what could guarantee the validity o these transormations. In particular, in Part o this chapter, I shall explain why the commentaries on the algorithms carrying out the arithmetical operations on ractions can be read as related to this concern. Tis explanation will lead us to examine the algorithms that Te Nine Chapters contains or multiplying and dividing ractions. Beyond the act that the proo o their correctness urther illustrates how the commentators proceeded in their proos, we shall show why they can be considered as belonging to the set o undamental ingredients grounding the ‘algebraic proo in an algorithmic context’. Bringing this point to light will require that we view algorithms rom the two distinct perspectives by which they were worked out in ancient China. Not only should we read algorithms, as the commentators did, as pure sequences o operations yielding a magnitude, but we should also consider them as prescriptions o computations, carried out on the surace, on which the calculations were executed, and yielding a value. 9 In conclusion, we shall be in a p osition to raise some questions on the nature and history o algebraic proo.
I wo key operations for proving the correctness of algorithms Te serng and the rst key components of the proofs Te main example in the ramework o which we shall ollow the thirdcentury commentator Liu Hui in his proo o the correctness o an algorithm deals with the volume o the truncated pyramid with circular 9
On this opposition, see Chemla 2005.
Reading proos in Chinese commentaries
base (see Figure 13.1 below).10 Te problem in which Te Nine Chapters introduces this topic reads as ollows:11 (5.11) S
,
-
, , .O
.A
:
/
.
Note the numerical values attached to the particular solid considered: the circumference of the circle forming the base is 3zhang. Tis detail will prove important below. Let us stress the fact thatTe Nine Chapters uses throughout the ratio of 3 to 1 for that of the circumference of a circle to its diameter. Liu Hui opens his commentary by putting forward the hypothesis that these were also the values used when the examined procedure was shaped. He states: ‘Tis procedure presupposes that the circumference is 3 when the diameter is 1.’ Elsewhere, the commentator designates such values as lüs, thereby indicating that they can be multiplied or divided by a same number without their relative meaning, which is to represent a relationship between the circumerence and the diameter o the circle, being affected. We shall meet this concept again below. o go back to problem 5.11 inTe Nine Chapters, 10
11
I translate the Chinese term yuanting as ‘truncated pyramid with a circular base’ on the basis o an analysis o the structure o a system o terms designating solids in Te Nine Chapters. In the terminology o solids, three pairs o names work in a similar ashion: each o these pairs contains two terms ormed by pre xing either ang (square, rectangle) or yuan (circle) to the name o a given body. Te designated solids correspond to each other, in that they belong to the same genus. Tey differ only in that they have, respectively, either square or circular sections. Te relation between the terms in Chinese expressed a relation between the designated solids. I hence translated these pairs as such, reproducing, in English, the structure o the terminology o the Chinese. Tis leads to an interpretation o the second term as designating a general kind o solid, two species o which are considered: the one with square base and the one with circular base. Since angting designates the ‘truncated pyramid with square base’,yuanting was translated as ‘truncated pyramid with circular base’. For more details, see Chapter D in CG2004: 103–4. On previous occasions (Chemla 1997/8; Chapter A in CG2004: 36–8), I have already discussed this passage o Te Nine Chapters and the commentaries. Te critical edition and the translation into French can be ound in CG2004: 424–7. I come back to it again in this chapter to cast light on the proo rom a new angle. LD1987: 73, Li Jimin 1990: 327–8 and Guo 1992: 137–8 present an outline o Liu Hui’s proo. A problem oTe Nine Chaptersis indexed by a pair o numbers: the rst number indicates the chapter in the Classic in which the problem is placed. Te second number indicates its position in the sequence o problems o the chapter. We shall always translate the text o the Classic in upper-case letters, in contrast to the commentaries, which are translated in lowercase ones. In addition to indicating clearly to which part o the text a given passage belongs, this convention imitates the way in which the different types o text are presented in the earliest extant documents.
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Cs
h
Ci
Figure 13.1
Te truncated pyramid with circular base.
i such is the case, as a consequence, the diameter o the lower circle o the solid to be considered is consequently equal to its height. Te truncated pyramid dealt with can thus be inscribed into a cube. In the Classic, the outline o the problem is immediately ollowed by an algorithm allowing the reader to rely on the data provided to determine the desired volume. It reads as ollows: ,
,
(
);
36.
o expound the argument on proo that I have in view, I shall need to make use o a representation o the algorithm as list o operations. o this end, let us note, as on Figure 13.1, Cs(resp. Ci) the circumerence o the upper (resp. lower) circle and h the height o the solid. With these notations, the algorithm can be represented in a synoptic way, as ollows:
Multiplications Multiplication Division sum by h by 36 > CiCs + Ci2 + Cs2 > (CiCs + Ci2 + Cs2)h Ci Cs
> (CiCs + Ci2 + Cs2)h/36
In what ollows, I shall regularly employ such representations or lists o operations. Immediately afer the statement o the algorithm as given by the Classic, in the rst section o his exegesis, the commentator sets out to establish its correctness within the ramework o the hypothesis that Te Nine Chapters made use o a ratio between the circumerence and the diameter o the circle equivalent to taking π = 3. His proo proceeds along three interwoven
Reading proos in Chinese commentaries
lines o argumentation. Te rst line consists o establishing an algorithm, or which Liu Hui proves that it yields the desired volume. Te second line amounts to transorming this algorithm as such into the algorithm the correctness o which is to be proved. For this, Liu Hui applies valid transormations to the algorithm taken as list o operations, thereby modiying it progressively into other lists o operations, without affecting its result. In the ollowing, we shall make clear what such transormations can be. Tird, in doing so, the commentator simultaneously accounts or the orm o the algorithm as ound in Te Nine Chapters, by making explicit the motivations he lends to its author or not stating the algorithm as he or she most probably rst obtained it, but insteadchanging it. Tis whole process provides an analysis o the reasons underlying the algorithm. Te analysis is not developed merely or its own sake. It also yields a basis on which the commentators devise new algorithms or determining the volume o the truncated pyramid with circular base. Accordingly, in a second shorter section o his exegesis, Liu Hui can make use o the values he employs or the relationship between the circumerence and the diameter o the circle (314 and 100) to offer new algorithms. Later on, Li Chuneng will similarly rely on the values he selects or these magnitudes to do the same. However, our analysis will concentrate on the rst section o Liu Hui’s commentary. Interestingly enough, a reasoning that has exactly the same structure and the same wording is developed to account or the algorithm that Te Nine Chapters gives or the volume o the cone, afer problem 5.25. On the one hand, this similarity indicates that the text o the commentary analysed here is reliable. On the other hand, such a act shows that the proos o the correctness were established by the commentators in relation to other proos and not developed independently. Other phenomena lead to the same conclusion.12 Tis similarity relates to the act that the proo had a certain kind o generality – an issue to which we shall come back later. Let us or now concentrate on how Liu Hui deals with the truncated pyramid with circular base. Te rst step in Liu Hui’s reasoning is to make use o an algorithm or which the correctness was established in the section placed immediately beore this one. Provided afer problem 5.10, this algorithm allows the computation o the volume o the truncated pyramid with square base when one knows the sides o the upper square (Ds) and lower square (Di) as well as the height h (see Figure 13.2).13 12 13
See Chemla 1991 and 1992, or example. Te proo is analysed in Li Jimin 1990: 304ff., Chemla 1991 and Guo Shuchun 1992: 132–5.
429
430
Ds
Di
Figure 13.2
Te truncated pyramid with square base.
Using the same notations or algorithms as above, it can be represented as ollows:
Multiplications Division sum by 3 Multiplication by h Di > (DiDs + Di2 + Ds2)h Ds
> (DiDs + Di2 + Ds2)h/3
On this basis, Liu Hui states a rst algorithm (algorithm 1) which determines the volume o the truncated pyramid with square base circumscribed to the truncated pyramid with circular base which is considered. Quoting the algorithm o the Classic verbatim – a act that I indicate by using quotation marks in the translation – his commentary reads: ′
Tis procedure presupposes (yi ) that the circumerence is 3 when the diameter is 1. One must hence divide by 3 the circumerences o the upper and lower circles to make the upper and lower diameters respectively. ‘Multiplying them by one another, then multiplying each o them by itsel’, adding, ‘multiplying this by the height and dividing by 3’ makes the volume o the truncated pyramid with square base.
Te only transormation (transormation 1) needed to make use o the algorithm quoted in this new context is to pre x its text with two divisions by 3. Tese operations change the given circumerences into the corresponding diameters, the lengths o which are respectively equal to the lengths o the sides o the upper and lower circumscribed squares.14 Algorithm 1 can be represented as ollows: 14
Incidentally, this proposition is stated in the Book o Mathematical Procedures(slips 194–5, Peng Hao 2001: 111).
Reading proos in Chinese commentaries
Divisions by 3 Multiplications, sum, Multiplication by h
Ci
> Di = Ci/3
Cs
Ds = Cs/3
C C C > i s + () i 3 3 3
2
Division by 3
C + () s 2 .h 3
C C C > i s + () i 3 3 3
2
C + () s 2 .h/3 3
o determine the ‘meaning’ o the result, that is, that one obtains the volume o the truncated pyramid with square base, Liu Hui has to rely on both the algorithm established earlier and values corresponding to a value o π. Such an operation o ‘interpretation’ corresponds to a key concept used by the commentators in the course o proving the correctness o algorithms: they reer to the ‘intention’ o an operation or a procedure, or its ‘meaning’, by the speci c term oyi. In what ollows, we shall pay particular attention to the ways in which such a ‘meaning’ is determined. Te rst step in Liu Hui’s proo o the correctness o the investigated algorithm belonged to what I have called above the ‘ rst line o argumentation’. Te next step goes along both the second and the third lines. Tis step makes us encounter the aspect o proo that is the main ocus in this chapter. I shall hence examine it in great detail. Afer having obtained the algorithm just examined, Liu Hui considers a case: Suppose that, when one simpli es the circumerences o the upper and lower circles by 3, none o the two is exhausted, . . .
Here, as is the rule elsewhere, the term ‘simpliying’ has to be interpreted as meaning ‘dividing’.15 In all extant mathematical documents rom ancient China, the result o a division is given in the orm o an integer to which, i the dividend is not ‘exhausted’ by the operation, a raction is appended. Te numerator and denominator consist o the remainder o the dividend and the divisor, respectively, both possibly simpli ed when this was possible. As a consequence, more generally, in these texts, ractions are always smaller than 1. With respect to the algorithm he has just established, Liu Hui then considers the case in which, afer dividing the circumerences by 3,neither o them yields an integer. In such cases, the next step o the algorithm would lead to multiplying quantities composed o an integer and a raction with each 15
o obtain evidence supporting this claim, the reader is reerred to the glossary o Chinese terms I composed (CG2004: 897–1035). Unless otherwise mentioned, all glosses o technical terms rely on the evidence published in this glossary.
431
432
other. Tis operation implies inserting at this point the algorithm thatTe Nine Chapters gave or multiplying not only such quantities, but also any two quantities – integers, ractions, integers with ractions – the correctness o which has been established in the rst oTe Nine Chapters. Let us examine this algorithm in detail beore considering the modalities o its insertion.
Te general procedure for multiplying Tis algorithm, like the others, has two aces. On the one hand, it is a list o operations, the text o which is recorded in Te Nine Chapters. On the other hand, the operations it prescribes were carried out on a surace on which quantities were represented with counting rods in ancient China.16 For the sake o my argument, it will prove useul to have some knowledge about the way in which computations were physically handled on this surace. At rst sight, it may seem strange that such details are necessary, since we deal with proos and not with actual computations. However, the relation between the two will become clearer below. On the surace, the execution o division and multiplication started rom the basis o a xed layout o their operands, which evolved throughout the ow o computations. At the beginning o a multiplication, the multiplicand was set in the lower row o the space in which the operation was executed, while the multiplier was placed in its upper row. At the end o the computation, the multiplier had disappeared, leaving the result in the middle row o the surace and the multiplicand in the lower row. In contrast, division started with the dividend placed in the middle row, in opposition to the divisor, put in the lower row. At the end o the computation, the quotient had been obtained in the upper row. Under the quotient, either the place o the dividend had been lef empty, which indicated that the result was an integer, or there was its remainder, in which case the result had to be read as integer (upper row) plus numerator (middle row) over denominator (lower row). Let us illustrate this description by what the computations or the algorithm yielding the volume o the circumscribed truncated pyramid must have looked like. Figure 13.3 shows a sequence o three successive states o the surace or computing. We indicate a separation between the rows or the sake o clarity. Inact, we have no idea whether or not there were marks on this surace. In the rst state, on the lef-hand side, the circumer16
Although they do reer to the act that computations were carried out on such a surace, the earliest extant texts discussed in this chapter contain very little inormation regarding how these computations were handled. Te argumentation supporting the way in which I suggest recovering them is provided in Chemla 1996.
Reading proos in Chinese commentaries
Cs
Cs Dividend 3 Divisor
as integer bs numerator 3 denominator
Dividing by 3 ai Ci Figure 13.3 ractions.
Ci Dividend 3 Divisor
integer
bi numerator 3 denominator
Te layout o the algorithm up to the point o the multiplication o
ences o the upper and lower bases were displayed, respectively in the upper and lower rows o the surace. Te reason or this is that numbers derived rom them would soon enter into a multiplication. Beore that multiplication, the algorithm prescribes that both circumerences be divided by 3. Tese divisions were to be set up and carried out in the upper and lower spaces, with the row in which the numbers had been placed becoming in turn a space in which a computation was executed according to the same rules o presentation. For instance, the upper row was split into three subrows, with the dividend Cs occupying the middle sub-row and the divisor 3 the lower sub-row (second state o the surace in Figure 13.3).17 In the situations considered by Liu Hui, once the divisions were completed, none o the dividends in the upper and lower spaces would have vanished, the result o each division being o the orm o an integer increased by a raction (third state in Figure 13.3). Tese, then, are the quantities to be multiplied according to the next step o the algorithm (‘Multiplying them by one another, then multiplying each o them by itsel ’). Tis eature o hierarchical organization, according to which a space in which a number is placed can become a sub-space, in which an operation is perormed according to the same rules at any level, is, in my view, one o the most important characteristics o this system o computation. Tis eature ensures that the successive computations required by an algorithm will be articulated with each other spatially, instead o being dissociated and carried out independently o each other. Te right-hand part o Figure 13.3 shows the state o the surace or computing, at the point where the algorithm requires inserting the algorithm or multiplying quantities that consist o an integer and a raction. Let us 17
In LD1987: 16–18, the reader can nd descrip tions o how the computations o a multiplication and a division were carried out on the surace or computing.
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as integer bs numerator 3as bs 3 denominator 3
3as + bs 3
‘parts o the product’
3
(3as + bs).(3ai + bi) (3as + bs).(3ai + bi) dividend 9 divisor ai bi 3
integer numerator 3ai denominator
bi 3
3ai + bi
3ai + bi
3ai + bi
3
3
3
Figure 13.4 Te execution o the multiplication o ractions on the surace or computing.
read what is called in Te Nine Chapters the ‘ ,’ which ul ls this task. P
:
(
);
; [
]
-
. .O
.
I we represent the successive states o the surace or computing when this sequence o operations is used rom lef to right, we obtain the result shown in Figure 13.4.18 Te same algorithm can be ound in the Book o Mathematical Procedures. Te description here, while slightly more speci c regarding the display o the arrays o numbers on the surace, can be interpreted along the same lines. Liu Hui’s commentary on the rststep o the procedure contains two elements that prove quite interesting or our purpose. Te rst element relates to the conception o the movements effected on the surace by the computations. Liu Hui offers a slight rewriting o the way in which the rst step should be carried out: the products o the denominators by the corresponding integers are, in his words, ‘made to enter the (corresponding) numerators’. Tis does not change anything in the resulting con guration (column 3). However, this rst sequence o operations prescribed by the ‘procedure or the eld with the greatest generality’ 18
Perhaps, the layout o the rst step should berestored in a different way. Te middle row o the upper and lower spaces could be divided into two sub-rows: one in which the result o the multiplication would be placed – that is, in the middle as usual – and a second one in which the numerator would remain. Tereafer, the two sub-rows would again use into a unique row, with the numerator joining the product.
Reading proos in Chinese commentaries
thereby appears as an operation o multiplication carried out on the three lines that are the array o numbers yielded by the previous division. Te operation multiplies the content o the upper row by that o the lower row, progressively adding the results to the middle row, where, in the end, the nal result is to be read. Tis point is quite important. First, it reveals the continuity between an array o positions read as a quantity (a + b/3) and the con guration on which a computation is carried out on the surace. In the same vein, an array o two lines will regularly be considered as a quantity (a raction) or as an operation (a division). We shall come back to this eature on several occasions below. Second, this point shows the material articulation between the operations o multiplication and division on the surace or computing. Each o the operations can be applied to the con guration at which the other operation ends. Te management o positions on the surace hence appears to be highly sophisticated and careully planned to allow orms o articulation between the different computations. It is rom this point o view that we can best understand Li Chuneng’s interpretation o the name o the operation carried out by the procedure: ‘Field with the greatest generality’.19 What explains such a name, in his view, is that, in contrast to previous algorithms, this procedure uni es the three algorithms or multiplying either integers, or ractions, or even quantities composed o integers and ractions. I we interpret integers as being numbers o the type a + 0/n (or any number n), ractions as o the type 0 + b/n, the ‘procedure or the eldwith the greatest generality’ can be uniormly applied to multiply any type o numbers. Furthermore, the ‘procedure or multiplying ractions’ is embedded in it. Note that the procedure is quite complex in the case o multiplying integers. However, uniormity, as stressed by Li Chuneng, seems to be preerred over simplicity.20 Tese remarks will prove useul below. In case the procedure Liu Hui devised or
19
20
In act, Li Chuneng explains the name ‘the greatest generality’, which is actually the name given to the same operation in the Book o Mathematical Procedures. It may well be the case that the srcinal name o the procedure in Te Nine Chapters was ‘the greatest generality’. We shall see that the generality o the procedure is precisely the key point Li Chuneng stresses in his comment. Te critical edition and the translation o this piece o commentary can be ound in CG2004: 172–3. It is rom this angle that one may understand why the description o an algorithm given in the introduction o this chapter is oversimpli ed. An algorithm may cover several types o cases and include branchings to deal with them. In relation to this, practitioners o mathematics in ancient China seem to have valued generality in algorithms, which led to writing algorithms o which the text may be less straightorward than our rst description at the beginning o this chapter. See Chemla 2003.
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the circumscribed truncated pyramid dealt only with integers or ractions, other procedures could be used to multiply. However, given the act that there are cases in which ‘none o the circumerences is exhausted’ by the division by 3, the most general procedure must be used. Te second element important or us in Liu Hui’s commentary on the rst step o the ‘procedure or the eld with the greatest generality’ is the intention he reads in the act that the operation be used. Multiplying an integer by the corresponding denominator, as he interprets, intends to ‘make’ the integers ‘communicate’ t(ong) with the numerators. In other words, the units o the integer a and those o the numerator (expressed by the denominator) are made equal, which allows adding up the transormed integer and the numerator. As is ofen the case, the reason brought to light or employing an operation is expressed in the orm o an operation (‘make communicate’). Te ormer operation can be prescribed by directly making use o the latter name, which thus reers to both the operation to be carried out and the intention motivating its use. Te result, in our case 3a + b, is designated as the ‘parts o the product’ (jien). It is ‘parts’, here a number o ‘thirds’, in that it is composed o units, the size o which is de ned by a denominator. In what ollows, we shall meet with these terms again.21 We are now in a position to go back to the list o operations established by Liu Hui or computing the volume o the truncated pyramid circumscribed to the one considered in problem 5.11.
Inserting an algorithm: a key operation for proof As Liu Hui envisaged, it is possible that none o the upper and lower circumerences is ‘exhausted’ by the division by 3. Tus, in order to carry out the various multiplications required by algorithm 1, one needs to make use o the ‘procedure or the eld with the greatest generality’. Te insertion o this procedure in algorithm 1 (transormation 2) yields algorithm 2, which, qua list o operations, can be represented by the ollowing list o operations:
21
For the interpretation o the terms, see my glossary (CG2004). In act, jien ‘parts o the product’ reers to the numerator in our sense, when its value is greater than that o the denominator. One may view the numerator as a dividend, when looking at it rom an operational point o view, and as ‘parts o the product’, when considering it as constituting a quantity. o be more precise, the commentator introduces the expression o ‘parts o the product’ (jien) in relation to the operation o ‘making communicate’, when the latter is rst used in Te Nine Chapters, that is, when commenting on the procedure or dividing between quantities with ractions. We shall analyse this operation and the commentary on it below.
Reading proos in Chinese commentaries
Divisions by 3 Multiplying integers by corresponding denominator, incorporating the numerator ai
Ci
> Di= bi [[[ 3
as
>
3ai + bi
Multiplications, sums,
>
(3ai + bi )2 + (3ai + bi ) (3as + bs ) +
Ds= bs [[[ 3as + bs (3as + bs )2 3 Multiplying denominators, dividing by the result, 9, ]]], multiplying by h, dividing by 3 Cs
Te way in which Liu Hui describes this process is highly interesting or our purpose. Here is how his text reads (my emphasis): Suppose that, when one simpli es the circumerences o the upper and lower circles by 3, none o the two is exhausted, then, backtracking, one makes them communi. cate, as a consequence they are taken respectively as upper and lower diameters
In terms o computation, the rst operation or multiplying quantities with is prescribed –bywhich means o therespectively, operation expressing tion –ractions ‘make communicate’ yields, 3ai + bi andits3ainten+ bs. s However, in this context, Liu Hui states, this computation carries out a backtracking. Tis term captures two nuances. First, it reers to the act that one goes in a direction opposite to the one just ollowed. Second, it implies that one goes back to the starting point: 3ai + bi restores Ci, whereas 3as + bs restores Cs. wo acts allow this conclusion. On the one hand, ‘making communicate’ turns out to be the operation inverse to the division by 3, carried out just beore – and we saw how that was displayed on the counting surace. On the other hand, since the results o division are given in the orm o an integer increased by a raction, they are exact. Tis is a key act or ensuring that the application o the multiplication opposite to a given division restores the srcinal numbers – and even restores the srcinal set-up o the division as column 3 in Figure 13.4 shows.22 We meet with the importance o this key act here or the rst time. We shall stress its relevance or our topic on several occasions below.
22
Te act that the divisor is 3 is important to ensure that one goes back to the numbers one started with. I simpli cation o the remaining raction in the result could occur, the operation o ‘making communicate’ would not amount to applying the inverse operation.
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Why backtrack, one may ask, when discussing these two operations, i it leads us to start rom where, in any case, our starting point already was? Liu Hui’s next sentence makes clear where the relevance or this ‘detour’ lies. Indeed i the value obtained is the same, the sequence o two opposed operations provides it with a new meaning (yi): Ci and Cs no longer represent the circumerences, but as results o the operation o ‘making communicate’, they are now interpreted as representing the diameters, disregarding denominators, that is, with reerence to other algorithms. Tis passage reveals the importance the commentator grants to interpreting the meaning o operations.
Cancelling opposed operations: another key operation for proof Let us now consider the consequences o these remarks or algorithm 2 when considered as a list o operations. What was just analysed implies that the rst section o the list o operations can be transormed (transormation 3): Division by 3
Ci
Make communicate ai > Di = bi 3
Multiplications, sums, etc. >
3ai +–(…) bi = Ci
>
as Ds = bs 3 is transormed into: Multiplications, sums, etc. (…………) Ci Cs
3as + bs= Cs
Cs
>
Te rst two operations cancel each other, since their sequence amounts to returning to the srcinal values – and to the srcinal set-up. Deleting both operations rom the list o operations does not change the value yielded by algorithm 2, nor does this transormation change the meaning o the nal result. Tis is the rst transormation o a list o operations qua list that we encounter and it belongs to what I called the second line o argumentation. We shall meet with other transormations o this kind below. Tis particular transormation is valid or the reasons stressed above. aken as a whole, algorithm 2, which computed the volume o the truncated pyramid
Reading proos in Chinese commentaries
with square base circumscribed to one with a circular base in case quantities with ractions occurred, can hence be transormed into algorithm 2 , without altering the result: ′
Multiplications Sums > Ci2 + Ci·Cs + Cs2 Ci
Cs
Multiplying the denominators, dividing by the result 9, multiplying the result by h, dividing by 3 ′
Te essential point now is that algorithm 2 shares the same initial list o operations with the algorithm or the truncated pyramid with circular base as described in Te Nine Chapters. Te reason why this act is important is that the arguments outlined above allow the interpretation o the ‘meaning’, namely, the ‘intention’ (yi) o the rst part o the algorithm, the correctness o which is to be established. Liu Hui writes (my emphasis): I one multiplies by one another the upper and lower diameters, then multiplies each by itsel respectively, then adds these and multiplies by the height, this gives the parts o the product (jien) o 3 truncated pyramids with square base.
Again, this statement is worth analysing in detail. Note, rst, that Liu Hui reers toCi and Cs as ‘diameters’. Tis is the meaning o the initial values entered in the algorithm that was established by bringing to light the pair o deleted, opposed operations. Tese values are diameters, with respect to the denominators. Such an analysis corresponds to the act that the result o the rst section o the lagorithm is interpreted as ‘parts o the product’ in reerence to the ‘procedure or the eld with the greatest generality’. More generally, it is by reerence to algorithm 2 , itsel obtained rom a combination o three algorithms, that the interpretation o the result o the rst part o the algorithm is made explicit. Algorithm 2 has been shown to yield the volume o the circumscribed truncated pyramid. o state the meaning o the result o its rst part as the ‘parts o the product ( jien) o 3 truncated ′
′
pyramids with square base’, two o its nal computations had to be dropped (dividing the result by 9 and dividing by 3). Each computation relates to a different algorithm among the algorithms that are combined, and the structure o the statement highlights the different statuses o the actors which are lef out. Te proo o the correctness o the algorithm or the truncated pyramid with square base had established that the rst part o its computations yielded the value o 3 pyramids. Te proo o the correctness
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o the ‘procedure or the eld withthe greatest generality’ shows that, beore dividing by the product o the denominators, the resulting ‘dividend’ corresponds to the ‘parts o the product’.23 Note, however, that the order o the division by the product o the denominators and the multiplication by the height was implicitly inverted so that the meaning o the result could be stated in this way. Tis transormation is valid. Its validity again rests on the act that the results o divisions are exact. Here too, this transormation is one that may be applied to the list o operations as such in order to change it into another list. In other passages, Liu Hui brings to light and comments on this inversion, which he calls by the name o ‘an’ (inversion). However, here the inversion is carried out tacitly. We shall come back to it later. In conclusion, we see the operations involved here in determining the ‘meaning’ (yi) o the result o the rst part o the lagorithm, the correctness o which is to be established. Tey depend in an essential way on relying on the meaning o previously established algorithms. Te discussion above highlights an interesting act. I we concentrate on the rst section o the algorithm determining the volume o the truncated pyramid with circular base, we can view it rom two angles. When seeking to uncover its ‘meaning’, it is necessary to restore the opposed operations that cancel each other and consider algorithm 2. However, when using the section as a list o operations or computing, it is more rational to delete the unnecessary operations, as in algorithm 2 . Although both algorithms yield the same result, the algorithm or computing differs rom the algorithm or shaping the meaning (yi) o the result. Tis is a crucial act or proving the correctness o procedures. Sometimes, the two algorithms coincide, in which case the algorithm is transparent concerning the reasons or which it is correct. Te main reason or which it may not be transparent is due precisely to the very transormations that are applied to the list o operations as such, and which interest us in relation to the second line o argumentation. At this point o our argument, several remarks can be made on the way in which Liu Hui deals with the algorithms ound in Te Nine Chapters. First, ′
23
Here, an element o argumentation can be retrospectively added to what was said earlier. Te ‘procedure or the eld with thegreatest generality’ is not reerred to by the name o the operation in the commentary we are analysing. Tree elements lead us, nevertheless, to the conclusion that such is the procedure that is inserted. First, the situation described is exactly the one or which the procedure was made: multiplying in general and multiplying integers increased by ractions in particular. Tis is clearly the case envisaged by Liu Hui. In addition, the list o operations to be ollowed corresponds exactly to that o the ‘procedure or the eld with the greatest generality’. However, other procedures could be used, as is demonstrated by the ‘procedure or more preciselü’ (CG2004: 194–7). Lastly, the termstong ‘make communicate’ andjien ‘parts o the product’ are speci cally attached to the arithmetical procedures given in Te Nine Chaptersto deal with integers increased by r actions.
Reading proos in Chinese commentaries
the commentator aims at accounting or the algorithm as described in the Classic – this is part o what we called the third line o argumentation and is interwoven with the rst two lines. For instance, in this case, he seems to be attempting to account or the reason why the algorithm does not begin with a division by 3, or, more directly, or why the algorithm is not transparent, in the sense just introduced.24 Tis question will lead him to ormulate motivations which explain the transormation o the algorithm he obtained into the algorithm actually provided by Te Nine Chapters, which yields the same result. Second, the reason Liu Hui adduces or that is the possibility that the division by 3 may introduce results with ractions. Here this detail reveals a key dimension in his expectations towards Te Nine Chapters. I we recall the data o problem 5.11, the circumerence o the lower circle is 3 zhang. However, the case Liu Hui considers, to reconstitute the motivations o the author(s) o the procedure, is one in which ‘none’ o the two circumerences is ‘exhausted’ by the division by 3. Tis indicates that he believes the authors considered other cases than that o the problem in Te Nine Chapters in order to shape the procedure. Hence the commentator does not imagine that the Classic provides algorithms or solving only the particular problem afer which they are given. He expects the algorithm to have been generally established and consequently he accounts or the correctness o the general algorithm as well as its orm.25 o be more precise, Liu Hui seems to be considering that, in their shaping o the procedure, the author(s) o the procedure took into account all cases in which the data or the circumerences would be integers. His reasoning would otherwise have been ormulated in a different way. Such hints regarding the types o numbers that may constitute data or a given algorithm would be extremely important to gather i we want to understand better what generality meant in ancient China and how the possibility o covering cases with different types o numbers was handled. It would be all the more important in the context o the argument I want to make in this chapter, or establishing a link between the ‘algebraic proo in an algorithmic context’ and the re ection about numbers. Tird, it appears that the commentator believes that, when possible, the author(s) o procedures avoided unnecessarily complex computations, in particular computations with ractions. He regularly repeats this hypothesis 24
25
On the basis o additional evidence, Chemla 1991 argues in avour o the hypothesis that Liu Hui seeks to read reasons accounting or its correctness in the statement o an algorithm. He succeeds in doing so or the algorithm which computes the volume o the truncated pyramid with square base. Tis is also what is shown by other passages o his commentary; see Chemla 2003.
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about their motivations, when he accounts or why the order o a division and a multiplication was inverted with regard to the order given by the reasoning he offered. Te rewriting o lists o operations that the author(s) o procedures undertook may hence be motivated, in his view, by the actual handling o computations. Tis is how Liu Hui explains the orm o the beginning o the procedure. As we shall discuss below, several speci c eatures o the mathematics o ancient China can be correlated with this concern. In our case, the act which the commentator brings to light in this respect is that the procedure offered by Te Nine Chapters has the property o working uniormly or all the data. As mentioned above, this property was stressed by Li Chuneng as characterizing the ‘procedure or the eld with the greatest generality’. It would then be transerred to the algorithm or determining the volume o the examined truncated pyramid. Note that, in contrast to the ormer, or which uniormity was obtained at the expense o simplicity, in the latter case, no arti cial step is necessary to guarantee a uniorm treatment o all the possible data. It is to be noted, however, that uniormity is not a property shared by all the algorithms in Te Nine Chapters. Te procedure given or dividing between quantities having ractions, which will be discussed below, is a counterexample, in which the latter cases are reduced to the ormer ones. Tese remarks lead to an observation that is essential or the argument made in this chapter. I we observe the transormation between the rst part o algorithm 1 and that o algorithm 2 , what was carried out was an inversion in the order o divisions and multiplications. Tis transormation, accomplished in the algorithm as a list o operations, was actually carried out and accounted or through a procedure dealing with quantities with ractions. A link is thereby established between a transormation that operates on lists o operations as such and an algorithm or executing arithmetical operations on quantities with ractions. Tis link will be more generally the ocus o Part o this chapter. Furthermore, as has already been stressed, this decomposition o the transormation that leads rom the rst section o algorithm 1 to that o algorithm 2 highlighted the necessity o relying on the possibility o cancelling two opposed operations that were ′
′
placed one afer the other. Tis is how the transormation appears to be carried out in Liu Hui’s view. In Part , we shall also come back to this point. Without entering into all the details, let us give a sense o what the ow o computations on the surace or computing looks like or the algorithms considered. We can represent the main structure o the initial section o algorithm 1 – which amounts to that o algorithm 2 – as the ollowing sequence o states (Figure 13.5).
Reading proos in Chinese commentaries
Dividing
Multiplyingbyoneanother (Procedure or the eld withthe greatest generality)
as bs 3
Cs 3
Cs 3
3 Cs . Ci
Cs . Ci
Dividing
9 as bs 3
Cs 3
Cs 3
Cs 3
Cs 3
Figure 13.5 Te basic structure o algorithms 1 and 2, or the truncated pyramid with square base. Multiplying by one another
Cs
Ci
Dividing
3 Cs . Ci
Cs . Ci
Ci 3
9 Ci 3
Dividing
′
Figure 13.6 Te basic structure o algorithm 2 , which begins the computation o the volume sought or. ′
Te beginning o algorithm 2 would instead yield Figure 13.6.
Post xing operations to an algorithm within the context of the proof Let us now return to Liu Hui’s commentary on the algorithm determining the volume o the truncated pyramid with circular base and read its ollowing section. Te commentator’s interpretation o the result o the rst section o the algorithm as ‘parts o the product (jien) o 3 truncated pyramids with square base’ produces a oundation upon which his reasoning can be built. He writes (transormation 4): Here, one must multiply the denominators, 3, by one another – hence one obtains 9 – to make the divisor, and divide by this. I, in addition to this, one divides by 3, one obtains the volume o the truncated pyramid with square base.
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Te rst division ends the ‘procedure or the eld with the greatest generality’. Te reason underlying its correctness is not mentioned here. Te second division ends the algorithm or computing the volume o the truncated pyramid with square base. Mentioning the two divisions in succession allows making sense o the operations step by step, and hence, globally, o the result. Moreover, this will prove important or the ollowing part o the reasoning.26 As a consequence, by successive transormations o algorithm 1, the ollowing algorithm (algorithm 3) is obtained or determining the volume o the truncated pyramid with square base circumscribed to the desired truncated pyramid with circular base: Multiplications Sum
by 9 Multiplication by h
Ci Cs
> (CiCs + Ci2 + Cs2)h
Division
Division by 3
> (CiCs + Ci2 + Cs2)h/9
>[(CiCs + Ci2 + Cs2)h/9]/3
Te appending o two operations to yield algorithm 3 belonged to the rst line o argumentation, as does the next transormation to be effected. Indeed, once he has obtained an algorithm or the truncated pyramid with square base, Liu Hui turns to considering how to derive the volume o the truncated pyramid with circular base on the basis o the volume o the circumscribed pyramid. It is by a fh transormation o the obtained list o operations that he achieves this goal: operations are to be post xed to the ormer sequence to get an algorithm yielding the volume o the truncated pyramid with circular base inscribed in the obtained pyramid with square base. Liu Hui rst makes a geometrical statement (my emphasis): o look or the volume o the truncated pyramid with circular base, when knowing the truncated pyramid with square base, is also like to look or the surace o the circle at the centre o the surace o the square.
wo words deserve some attention here, which is why I emphasized them. Te rst one is ‘to look or’ (qiu). It regularly introduces the task that
26
Below, we shall meet with cases in which Liu Hui combines two divisions that ollow each other. Te act that he does in some cases and does not in others relates clearly to the argument he is making. Tis eature highlights how careully the relationship between shaping a procedure and arguing or the correctness o a procedure is handled.
Reading proos in Chinese commentaries
the outline o a problem asks to ul l. Tis detail indicates that, in ancient China, algorithms may have been conceived as composed by combining a sequence o algorithms which carry out a sequence o tasks, the completion o which was identi ed as leading to the solution o a given kind o problem. Tis corresponds quite well to the kind o reasoning Liu Hui has been developing so ar in the commentary we are reading. Te second word to be stressed is ‘also’. It reers to the act that the same argument was given earlier in the commentary, afer problem 5.9, when Liu Hui was deriving the algorithm or the volume o the cylinder rom that o the volume o the parallelepiped. Tis ‘also’ thus indicates that the proos are not carried out in isolation rom each other, but rather in parallel with each other – a act that we have already stressed above. In act, afer problem 1.33, devoted to computing the area o a circle, Liu Hui had derived the values o 3 to 4 as corresponding lüs or the area o the circle and that o the circumscribed square, respectively, rom the values o 3 to 1 or expressing the relationship between the circumerence o the circle and its diameter. In the commentary on problem 5.9, these values were declared to allow the transormation o the volume o a cylinder into that o the circumscribed parallelepiped. Te same statement is made here, and the geometrical assertion is ollowed by its translation into algorithms (transormation 5): the same multiplication by 3 and division by 4 ensure the transormation rom the truncated pyramid with square base into the truncated pyramid with circular base. As Liu Hui puts it: Hence, i one multiplies by the lü o the circle, 3, and divides by the lü o the square, 4, one obtains the volume o the truncated pyramid with circular base.
As a consequence, at this point o his commentary, Liu Hui has determined a correct algorithm yielding the volume o the desired truncated pyramid, which ends the rst lineo argumentation. Algorithm 4 correctly yields the value o the desired magnitude.
Multiplications Sum Multiplication by h Ci Cs
> (CiCs + Ci2 + Cs2)h
Division by 9 > [(
Division by 3
Multiplication by 3
CiCs + Ci2 + Cs2)h/9]/3
> [[(CiCs + Ci2 + Cs2)h/9]/3].3
Division by 4 > [[[(CiCs + Ci2 + Cs2)h/9]/3].3]/4
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ransforming algorithms as lists of operations Te goal, rom this point onwards, is the transormation o this algorithm 4, qua algorithm, into the one or which the correctness is to be established, that is, the one provided by Te Nine Chapters or the volume o the truncated pyramid with circular base. Liu Hui hence resumes reasoning along the second line o argumentation. Considering the list o operations obtained by the last transormation (5), he remarks: But, earlier, in order to look or the volume o the truncated pyramid with square base, we had divided by 3. Now, in order to look or the volume o the truncated pyramid with circular base, one must also multiply by 3. Since the two denominators are equal, hence they compensate each other.
Beore clariying the italicized terms, let us observe the argument made here. Te commentator clearly considers the operations that ollow each other as a list and carries out a transormation o this list as such. Te algorithm yielding the circumscribed truncated pyramid with square base, he remarks, ended by a division by 3, whereas transormation 4 rst appended to it a multiplication by 3. 27 Liu Hui thus suggests deleting both rom the list o operations, thereby carrying out transormation 6. It can be represented as ollows (Figure 13.7): Multiplications Sum
by 9 Multiplication by h Ci > (Ci Cs + Ci2 + Cs2)h Cs
Division
Division by 3
> [(Ci Cs + Ci2 + Cs2)h/9]/3
Multiplications by 3
> [(Ci Cs + Ci2 + Cs2)h/9]/3.3
Division by 4 > [(Ci Cs + Ci2 + Cs2)h/9]/4
Figure 13.7
Algorithm 5: cancelling opposed multiplication and division.
ransormation 6 modi es the list o operations without altering the meaning or the value o the result. We meet here with the same phenomenon as above. Bringing to the light the opposed and division was crucial to interpreting meaning (yi) omultiplication the result. However, when viewing the list o operations as a means or computing, the two operations appear unnecessary. Tis is how Liu Hui progressively accounts or the shape o the algorithm ound in the Classic. 27
Let us stress, in the previous quotation, the use o the same term when reerring to the two algorithms: ‘to look or’ (qiu). Tis con rms the part played by problems in decomposing the task to be ul lled into sub-tasks conceived o as problems.
Reading proos in Chinese commentaries
Although the transormation seems comparable to transormation 3 discussed earlier, it is worth noticing that Liu Hui reers to the two in dierent terms. Earlier, the commentator spoke o ‘backtracking’ and in correlation with this he stressed the act that the values o the circumerences had been restored while their meaning had changed. In contrast to this, Liu Hui stresses here the act that the two operations ‘compensate each other’ (xiang zhunzhe). Te emphasis is placed on the cancellation o their effects as operations. Tis gives a hint o the subtlety o the ormulation o the reasoning. Te validity o this transormation is not to be taken or granted. It is again guaranteed by the act that, in ancient China, the result o a division was given exactly, that is, as an integer increased by a raction. We shall show below that the commentator links these two acts. Te quoted sentence makes use o another expression, which requires urther analysis: the argument given or establishing the conclusion that the two operations ‘compensate each other’ is ormulated in the orm that ‘the two denominators are equal’. Why is the word ‘denominator’ mu ( ) used here? Tere appears no reason explaining in which sense the ‘3’ with which one multiplies can be considered as a ‘denominator’. Let us stress that, in the other passage in which the same reasoning is developed, afer problem 5.25, the same term recurs, which indicates that this is not due to an error in the transmission o the text. Tese occurrences seem to imply that this term mu has another technical meaning that I was unable to elucidate. Tis is why, beore it is ound out, I translate the term in the usual way. However, consequently, a very striking act must be noted: in the commentaries, there is only one other occurrence o this term with exactly this same use, and this usage is ound in the commentary establishing the correctness o the algorithm or multiplying ractions.28 Tis hint again links the line o argumentation we are examining with the algorithms or carrying out arithmetical computations with ractions. Te point is worth noting, in relation to the argument to be developed in Part o this chapter. Another detail casts some light on the way in which Liu Hui operates. I we observe the list o operations that Liu Hui is transorming, we can see that it rstenumerates a division by 9, where the ‘3’s’ involved stand or π; second, a division by 3 corresponding to the computation o the volume o the circumscribed pyramid; and, thirdly, a multiplication by 3, where the ‘3’ again stands or π. One might have expected that the proo would cancel a multiplication and a division by 3 that would both be linked to π.29 Te 28 29
See mu ‘denominator’ in my glossary, CG2004. I am indebted to Anne Michel-Pajus or this remark.
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expectation is all the more natural when we know that in a second part o his commentary, Liu Hui relies on his proo to yield a new algorithm that makes use o his own values or π. However, such is not the case. Te commentator cancels operations that ollow each other. Tis seems to indicate that he takes care not to modiy arbitrarily the order in which the reasoning led to establishing the operations constituting an algorithm. Such a detail reinorces the hypothesis that he is working on lists o operations as such, being careul to make explicit the transormations applied to them and the motivations or using them.30 Tere is, however, another way o accounting or this detail, i.e. that Liu Hui thinks that he recovers the reasoning ollowed by the author(s) o the Classic. By transormation 6, a list o operations was remodelled into another list, equivalent in that it yielded the same result. ransormation 7 continues along the second line o argumentation, even though it consists o applying a different operation to algorithm 5. Liu Hui goes on as ollows: We thus only multiply the lü o the square, 4, by the denominator 9, hence we obtain 36, and we divide at a stroke.
Liu Hui designates the two actors by which one should still divide to end algorithm 5, i.e. 4 and 9, by the part they were shown to play in the reasoning (lü o the square, denominator). Instead o carrying out the divisions successively, transormation 7 suggests ‘dividing in combination’lianchu ( ), which I translated as ‘dividing at a stroke’. Tis implies transorming the end o algorithm 5 into the multiplication o the two divisors by each other and dividing by the product. With the expression o ‘dividing at a stroke’, we meet with a technical term that recurs regularly in the commentaries but is not to be ound in Te Nine Chapters. We may account or this by noticing that it is a designation o the division typical o the mode o proving the correctness o algorithms on which the chapter concentrates. wo successive divisions were accounted or, each being shown to be necessary or its own reasons. As above, Liu Hui had to dissociate them to bring to light the meaning o the result o the algorithm he shaped. However, viewing the list o operations as a means or computing leads to modiying the way o carrying them out, namely, by transorming the end o algorithm 5. Liu Hui thereby accounts or the orm o the algorithm given by Te Nine Chapters, by highlighting that the two operations were 30
Tis conclusion should be nuanced by the remark made above concerning the change in the order o the multiplication byh and the division by 9.
Reading proos in Chinese commentaries
grouped into a unique division. Te technical term chosen or this division reers to the motivation o the effected transormation. As a consequence, algorithm 5 Ci, Cs > (Ci Cs + Ci2 + Cs2)h > (Ci Cs + Ci2 + Cs2)h/9 > [(Ci Cs+ Ci2 + Cs2)h/9]/4
is transormed into the algorithm 2
2
2
2
Ci, Cs > (Ci Cs + Ci + Cs )h > (Ci Cs + Ci + Cs )h/36 which is equivalent to it and identical to the desired algorithm. Tis was what was to be obtained: the correctness o the procedure provided by Te Nine Chapters is established. Te way in which the proo was conducted highlights in the best way possible how the activities o shaping an algorithm and proving the correctness are intertwined. Such is the type o proo that I suggest designating as ‘algebraic proo in an algorithmic context’. It is characterized by the articulation o the three lines o argumentation I distinguished. However, clearly, the second line o argumentation is the one that is speci c to it. Several points need to be made clear to explain the expression by which I suggest reerring to this kind o proo. the act speak here oaan ‘algorithmic context’,terms. it will be First, useultotojustiy compare whatthat we Ianalysed with translation in modern Te reasoning we ollowed can be rewritten as the ollowing sequence o steps:
⎡ C C ⎛ C ⎞2 ⎛ C ⎞2 ⎤ ⎢ i s + ⎜ i ⎟ + ⎜ s ⎟ ⎥ .h ⎢ 3 3 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎥⎦ 3 V=⎣ . 3 4 ⎡ Ci Cs + Ci 2 + C s2 ⎤ ⎢ ⎥ .h 9 ⎢⎣ ⎥⎦ 3 . = 3 4 ⎡Ci C s + Ci 2 + C s2 ⎤ .h 1 ⎦ . =⎣ ⎡Ci C s + C9i 2 + C s2 ⎤ .h 4 ⎦ =⎣ 36 Te rst line encapsulates the rst line o reasoning, which establishes an algorithm ul lling the task required by the terms o the problem. In the ollowing lines, corresponding to the second line o argumentation,
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equalities are reshaped, whereas, in the commentaries, what is rewritten are instead algorithms.31 In correlation with this, in the latter case, intermediary sequences o operations are provided with an interpretation Second, why do I speak o an ‘algebraic proo’? I take it as a typical element o this kind o proo that it involves transorming lists o operations as such – the second line o argumentation – and that the validity o these transormations should be addressed. I we observe the transormations leading rom one line to the next one in the modern version o the reasoning, sequences o operations are reshaped, with complete generality, and this leads to transorming a correct equality in a correct way into an equality that is equivalent and was desired. I claim that, although in a different orm, the same mathematical work is carried out on the basis o algorithms in the commentary we analysed. Tis is the element that I recognize to be present in the ancient Chinese text and or which I retain the expression under discussion. Tis interpretation implies a use o the term ‘algebraic’ in relation to operating on the operations themselves. Let us, at this point, recapitulate the transormations that we identi ed by means o our analysis and that were carried out on a list o operations. We had: • . Eliminating inverse operations that ollow each other Division by 3
Make communicate
Multiplications, sums, etc.
ai Ci
> Di = bi
>
3ai + bi = Ci
(…………)
>
3
as Cs
Ds = bs 3
3as + bs = Cs has been transormed into
Multiplications, sums, etc. (…………) > Ci Cs
31
In an algebraic proo o a more general type, transormations can be applied to both sides o the sign o equality in parallel, that is, to two lists o operations simultaneously. Te ormulas used recall those stated by Li Ye in his Sea-Mirror o the Circle Measurements (1248), where ormulas express the act that different operations on different entities lead to the same result.
Reading proos in Chinese commentaries
•
. Inverting the order o divisions and multiplications
Ci, Cs
Dividing by 9 Multiplying by h (…) > (CiCs + Ci2 + Cs2)/9 > [(CiCs + Ci2 + Cs2) /9]·h has been transormed into
Multiplying by h Dividing by 9 Ci, Cs (…) > (CiCs + Ci2 + Cs2)h
> (CiCs + Ci2 + Cs2)h/9
We saw that this very inversion had been carried out tacitly in the commentary we examined but it is made explicit in other commentaries and reerred to by the technical term an.32 Moreover, I underlined the act that the transormation between algorithm 1 and algorithm 2 could be conceived o as belonging to this type. ′
•
. Combining divisions
Dividing by 9 Dividing by 4 (CiCs + Ci2 + Cs2)h > (CiCs + Ci2 + Cs2)h/9 > [(CiCs + Ci2 + Cs2)h/9]/4 has been transormed into Dividing by 36 (CiCs + Ci2 + Cs2)h > (CiCs + Ci2 + Cs2)h/36
Now, several questions present themselves with respect to these transormations, which appear to be the undamental transormations needed to argue along the line o argumentation examined. First, how were they conceived o? Moreover, what guaranteed their validity? Furthermore, did the commentators consider this question and in which ways? Addressing these issues is essential to determine in which sense, in these commentaries, we may have an ‘algebraic proo in an algorithmic context’. As announced in the introduction, I shall argue that a link was established in ancient China between the validity o these undamental transormations and the kind o one operated. in provided what ollows, to numbers show thatwith the which commentaries on theMoreover, algorithms byTeI intend Nine Chapters or carrying out arithmetical operations on numbers containing ractions can be interpreted as addressing the question o the validity o the undamental transormations, in the ways in which these transormations 32
See the commentaries on the ‘procedure o suppose’ (rule o three), at the beginning o Chapter 2; the procedure or unequal sharing, at the beginning o Chapter 3; the procedures ollowing problems 5.21 and 5.22.
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were conceived. Tese suggestions seem to be natural on the basis o the previous discussion. Indeed, on several occasions, we observed the connection between transormations applied to a list o operations and algorithms carrying out arithmetical operations on quantities with ractions. We now need to ocus on the latter procedures to analyse this connection systematically.
II Grounding the validity of the fundamental transformations of lists of operations Te rst hint that the commentators link the validity o the undamental transormations to the kinds o numbers used in them is ound when Liu Hui accounts or why, in his view,Te Nine Chapters introduces quadratic irrationals. We shall hence ollow him in his argumentation.
Eliminating inverse operations that follow each other Afer problem 4.16, Te Nine Chapters describes a general and abstract ‘procedure or extracting the square root’.33 In a rst part o the procedure, an algorithm is provided or determining the root o an integer digit by digit. It is ollowed, in a second part, by a procedure dealing with quantities containing ractions, which reduces the problem to the case dealt with in the rst part. Te commentary in which we are interested discusses a statement that concludes the rst part o the procedure and asserts: I ,
,
( (
‘
) )
,
, ,
(i.e., the number)
’.
Tree historians, independent rom each other, have established that, here, Te Nine Chapters was addressing the case when the number N, the root o which is sought, was not exhausted when one had reached the digit or the units in the square root. All concluded that Te Nine Chapters was prescribing, or such cases, that the result be given as ‘side o N’, which is to be interpreted as meaning ‘square root o N’.34 33
34
It relies on a numeration system that is place-valued and decimal. Te introduction to Chapter 4 in CG2004: 322–35 analyses its main eatures. Te critical edition and the translation o the piece o commentary discussed can be ound in CG2004: 364–6. Volkov 1985 ; Li Jimin 1990. As or me, reerences can be ound in Chemla 1997/8 or CG2004. Note that the Classic states, without providing any argument in avour o this assertion, that in these cases the extraction cannot be carried out.
Reading proos in Chinese commentaries
Te reason Liu Hui adduces or explaining why it was necessary to give the result in the orm o quadratic irrationals, when necessary, is undamental or our purpose. Te commentator rst considers a way o providin g the result as a quantity o the type integer increased by a raction but discards it as impossible to use. Tis leads him to make explicit the constraints that, in his view, the result should satisy. He writes (my emphasis): Every time one extracts the root o a number-product35 to make the side o a square, the multiplication o this side by itsel must in return (huan) restore (u) (this number-product).
Tis sentence is essential: the kind o result to be used is the one that guarantees a property or a sequence o opposed operations. A link is thereby established between the kinds o numbers to be used as results and the possibility o transorming a sequence o two opposed operations. More precisely, the result o the square root extraction must ensure that the sequence o two opposed operations annihilates their effects and restores the srcinal data: their sequence can thereby be deleted. Why is this important? o suggest answers to this question, one may observe how the results o actual extractions are given in the commentaries. It turns out that, when a commentator is seeking to establish a value, the 36
results square root extraction as approximations. However, the actothat the operation inverseare togiven a square root extraction restores (u) the srcinal number and the meaning o the magnitude to which the extraction was applied is used precisely in the context o an ‘algebraic proo in an algorithmic context’.37 Tis con rms the link we suggested between the
35
36
37
Te type o number or which one can extract the square root is a number that, rom a conceptual point o view, is a ‘product’. Tis corresponds to a speci c concept in Chinese, ji, which can designate a number-product, an area, or a volume. Tis is the case when the commentator discusses new values or expressing the relationship between the circumerence o the circle and its diameter. See the commentary afer problem 1.33, CG2004: 178–85. However, this statement must be nuanced. Tere is a context in which Liu Hui uses quadratic irrationals as such in computations. Tis is in act the passage that allows interpretation o the obscure sentence by which Te Nine Chaptersintroduces quadratic irrationals. In it, the commentator seeks to assess with precision the ratio between the sphere and the circumscribed cube that Zhang Heng (78–142) derived rom his approximation or π, which states that the square on the circumerence is to the square on the diameter as 10 is to 1. As I suggested, the use o the irrationals here is driven by the aim o highlighting that Zhang Heng’s algorithms were worse than that oTe Nine Chapters. In the end, Liu Hui introduces an approximation o a square root in the orm o an integer to conclude the evaluation. See Chemla and Keller 2002. Te text in question, that is, the commentary afer problem 5.28, is discussed in Chemla 1997/8. An outline is provided below, in note 39.
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introduction o certain kinds o numbers and the line o proo that made use o transormations carried out on lists o operations. In act, the commentary urther bears witness to the act that the link is not merely established or such quantities. Once Liu Hui has introduced the constraint that the result o a square root extraction must satisy or the cases in which the number N is not exhausted, he examines more closely two results or root extraction in the orm o an integer increased by a raction – one by deect and one by excess. It is revealing that his analysis o the values concerns how they behave when one applies the inverse operation to them but this is not what is most important or us here. Te statement by which he concludes his investigation is essential or the comparison it establishes. Liu Hui writes: . Tereore, it is only when One cannot determine its value (i.e. the value o the root) ‘one names it (i.e. the number N) with “side” ’ that one does not make any mistake (or, that there is no error). Tis is analogous to the act, when one divides 10 by 3, to take its rest as being 1/3, one is hence again able to restore (u) its value . (My emphasis)
Te mention o this other ‘restoring’ in the context o the commentary on square root extraction reveals that or quantities o the type o an integer increased by a raction, it was a property that was also deemed essential. Indeed, the comparison made here between square root extraction and division urther con rms the link I seek to document. In his commentary, Liu Hui maniests his understanding that, as kinds o numbers, quadratic irrationals and integers with ractions differ.38 However, he stresses here the analogy between them precisely rom the point o view that introducing them as results in both cases allows two opposed operations applied in succession to cancel their effects. In Part o this chapter, we saw how this cancelling led to deleting such a sequence o operations rom the algorithm that was being shaped. It is hence tempting to conclude that, as with quadratic irrationals, Liu Hui linked the introduction o ractions to possibilities o transorming lists o operations as such. Tis hypothesis is supported by the act that the ‘restoring’ made possible by the introduction o ractions is also evoked and used within the context o ‘algebraic proos’ o the type we study. Tis is easily established by noticing that the concept o u ‘restoring’ introduced here occurs only in such contexts. Tis act con rms, i it were necessary, the correlation between this property shared by various kinds o numbers and the conduct o such
38
See Chemla and Keller 2002.
Reading proos in Chinese commentaries
types o proo.39 Te introduction o such quantities is hence related to a speci c perspective on lists o operations as such. In conclusion, Liu Hui interprets the necessity o introducing ractions and quadratic irrationals as deriving rom the necessity o restoring the srcinal value when applying the inverse operation to the result o an operation – this is the only motivation he brings orward. In other words, or the results o divisions or square root extractions – which are conceived 39
Compare the discussion o the commentary placed afer problem 5.28, mentioned above. In it, the commentator successively applies the operation inverse to the last o the operations to the results o a sequence o algorithms. Tis operation, he states, restores the meaning and value o the last intermediary step. I we represent the sequence o operations as above, we have the ollowing pattern o reasoning. Te algorithm known to be correct is the ollowing one: C > C2 multiplying multiplying h by itsel by
> C2h dividing by 12
>V
Te question is to determine the meaning o the ollowing sequence o operations applied V to: V > multiplying dividing by 12 by
>
>? extracting the h square root
Te meaning o the result o the rst two steps can bedetermined as ollows: C multiplying by itsel
> C2 multiplying h by
> C2h dividing by 12
> C 2h
>V multiplying by 12
then C multiplying by itsel
> C2 multiplying by
> C2h dividing h
> C2 by h
Tis is correct, because multiplying by 12 restores that to which the division by 12 had been applied. Tereafer, dividing byh restores that to which multiplying byh had been applied. Now, because o the property o square root discussed, we have > C2 >C multiplying extracting the by itsel square root C
and the meaning o the result o the ollowing algorithm is established V
>
multiplying by 12
dividing by
> C2 h
>
12V h
=C
extracting the square root
Tis is how the correctness o the inverse algorithm is established. In the case o problem 5.28, the inverse operations successively applied are a multiplication, a division and a squaring. At each step, the commentator stresses that ‘restoring’ was achieved. Note that the reasoning implicitly put into play to express the meaning o the rst parto algorithm 2 as ‘the parts o the product o 3 truncated pyramids with square base’ in the passage discussed above can be seen as similar to the one just described. Tese examples show the relationship o the property o numbers which permits restoration and the conduct o the second line o argumentation with the operation o establishing the meaning ( yi) o the result o a list o operations. ′
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as a kind o division – the act that they are exact guarantees that inverse operations which ollow each other can be deleted rom an algorithm.40 Yielding exact results perhaps matters less to computations than to proos: it grounds the validity o one o our three undamental transormations. Such is the link that is established between the numbers with which one works and the transormations that can be applied to sequences o operations. Because the evidence relating to quadratic irrationals is ar less abundant than the evidence involving ractions, or the remaining part o my argumentation, I shall hence ocus mainly on the latter. So ar, we can establish that the commentator Liu Hui ascribes the motivation in question to Te Nine Chapters, thereby demonstrating that he himsel makes the connection between the use o some quantities and the validity o a transormation. Can we ollow Liu Hui and attribute the same idea to the author(s) o the Classic? Te argumentation is delicate and dicult to conclude with certainty. It is true that quantities such as ractions and quadratic irrationals date to the time when Te Nine Chapters was compiled. In act, only ractions occur in the Book o Mathematical Procedures. As or using such quantities in relation to proos, so ar, our terminus ante quem is 263, when Liu Hui completed his commentary. Te occurrences o the term ‘restoring’ or ‘returning to’ u ( ) the srcinal value provide interesting clues. Te concept is not to be ound in Te Nine Chapters. However, it is attested to in the Book o Mathematical Procedures, in contexts where similar concerns can be perceived. Interestingly enough, there, u occurs only afer the statement o an algorithm or carrying out division or root extraction. Afer these algorithms, a procedure is then prescribed that aims at ‘returning to’ the srcinal value. By contrast, u never occurs in a procedure solving a problem. It is always appended to another algorithm and carries out the inverse operation. Tis is complementary to the idea one may derive rom the commentaries on Te Nine Chapters that there is a link between the way in which the results o division and root extraction are given and an interest in the possibility o restoring the srcinal value. 41 Even 40
41
Note that, so ar, the link has been established only or multiplications and divisions by integers. Te more general case still awaits consideration. See u in my glossary (CG2004: 924–5). In the Book o Mathematical Procedures, one occurrence o u is to be ound in the context o the operation o ‘detaching the length’, which asks to determine the length o a rectangle when its area and its width are given (slips 160–3, Peng Hao 2001: 114). Tere, the rst procedure deals with the casewhen both the area o the rectangular eld and its width are integers. Te inverse procedure distinguishes the case when the result is an integer rom the one in which it has a raction. A second procedure considers the case when both data are pure ractions. Te algorithm that returns to the srcinal value is that o multiplying ractions. When the width consists o an integer increased by a set o ractions, the operation called ‘small width’ is carried out by a general procedure,
Reading proos in Chinese commentaries
more interesting is that, although in the Book o Mathematical Procedures the aim o restoring is achieved or division, the results o which are always exact, this requirement is not ul lled or root extraction. Te procedure provided or the latter operation gives only approximate results. In other words, we reach an interesting conclusion: the concern or ‘restoring’, which is explicit or both division and root extraction in the Book o Mathematical Procedures, that is, already as early as the second century , apparently existed beore the solution satisying it did or root extraction. Tis seems to indicate that the need or ‘restoring’ motivated the introduction o a new algorithm or root extraction and the introduction o quantities that would ensure that the result be always exact, as we nd them in Te Nine Chapters, and not the converse. Tese remarks thus lend support to Liu Hui’s thesis that, in Te Nine Chapters, the introduction o quadratic irrationals and ractions aimed at ensuring that opposed operations cancel each other. We see how the evidence rom the Book o Mathematical Procedureshelps to avoid misinterpreting the act that neither the concept o ‘returning to’ (u) the srcinal value nor the related one o ‘backtracking’ ( huan) occur in Te Nine Chapters. Tis absence cannot be explained by the act that these concerns appear only at a later date. Nor, in act, should the absence be explained by the hypothesis that Te Nine Chapters was merely a set o recipes without any interest in accounting or the correctness o the algorithms. I have already alluded to the act that the commentator regularly maniests his expectation that the procedures given by Te Nine Chapters be transparent on the reasons underlying them.42 In addition to this, with respect to the point under discussion, i the term u ‘restoring’ does not occur in Te Nine Chapters, the Classic makes use o a technical expression that clearly belongs to a set o cognate terms and betrays the same concern: baochu ‘dividing in return’.43 For a division to be prescribed in this way indicates the reason why it is carried out: the expression points out the
42 43
again ollowed by an algorithm explicitly aiming at ‘returning to’ the srcinal value. In this context, there are several occurrences o u (slips 165–6, Peng Hao 2001: 116). However, the text o the procedure or doing so appears to be corrupted. Te last occurrence o u is the most interesting or us. It is to be ound in the Book o Mathematical Procedures, afer a procedure giving approximations or extracting square roots (slips 185–6, Peng Hao 2001: 124–5). Te case considered in the paradigm to which the procedure is attached is that o an integer that is not a perect square. Te result is given as an approximation by an integer increased by a raction. However, it is asked to return to the srcinal value. Te end o the slip reads: ‘one restores it like in the procedure or detaching the width’. In other words, not only is the concern o u common to the two contexts o division and root extraction, but also the procedures or carrying it out. See notes 3 and 24. See, or instance, the second part o the algorithm or square root extraction.
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act that a value was used earlier in the ow o computations, that it was interpreted as having been expanded by an unnecessary actor, and that the ‘division in return’ compensates or this by cancelling the actor. In dealing with the proo o the correctness, the commentary usually brings to light a pattern in the way in which the algorithm is accounted or, thereby echoing the ormulation o the procedure in the Classic. Such divisions highlight an interesting point, suggesting a hypothesis to account or why u does not occur in Te Nine Chapters. So ar, we have shown that Liu Hui establishes a link between the introduction o kinds o numbers expressing the results o divisions and root extractions, on the one hand, and the act that the sequence o a division and the multiplication inverse to it restored the srcinal value, on the other hand. Tis link coordinated perectly with situations we met in the example analysed in Part o this chapter, where this property was twice used to explain why pairs o operations were deleted rom the nal algorithm. However, situations in which one ‘divides in return’ reveal other ways in which the annihilation o the effects o a pair o two opposed operations by each other can be put into play in an algorithm. In such cases, the two operations do not both disappear rom the algorithm. Tis is precisely why, when prescribing one o them, Te Nine Chapters can reer to the reason or using it. By contrast, since the operation o ‘restoring’ is disclosed when one accounts or an algorithm but not when one describes it, the act may explain why the term u does not occur in the Classic.
Establishing the validity of fundamental operations and the arithmetical operations on parts In act, one o the divisions examined in Part o thechapter is o the kind o a ‘division in return’. When, in algorithm 3, a division by 9 is prescribed, it echoes the act that earlier in the computations, instead o multiplying diameters, the algorithm multiplied their triple.44 Liu Hui does not use speci c terminology that would indicate its nature as a ‘division in return’. Like Te Nine Chapters, he more generally indicates the point only occasionally. However, in this case, the division by 9 is part o the ‘procedure or the eld 44
Perhaps the distinction between the two types o situation is grasped by the distinction which Liu Hui introduces between ‘backtracking’ (huan) and ‘compensating each other’ (xiang zhunzhe). I this is the case, a relation would be introduced between various types o cancellation o opposed multiplication and division. In any event, although the distinction is important, the undamental reason underlying the act that the effects o the operations eliminate each other is the same: it relies on the premiss that the exact results o division are given.
Reading proos in Chinese commentaries
with the greatest generality’. And the nature o the division as being ‘in return’ is highlighted in the commentaries, precisely when they establish the correctness o this other algorithm. Tis brings us back to the thesis that we aim at establishing here: that is, that the reasoning which accounts or the validity o the undamental transormations identi ed in Part may have to beread rom the commentaries on the procedures or carrying out arithmetical operations on numbers with ractions. We saw that the simple act o introducing ractions was essential to accounting or the validity o the rst undamental transormation. Computing with ractions proves essential or the validity o the other two transormations. When introducing transormation , I already stressed the link between transorming sequences o operations (in that case, inverting the order o division and multiplication) and describing algorithms or computing with quantities having ractions (inserting the ‘procedure or the eld with ull generality’). In the remaining part o this chapter, I shall argue or my main thesis by showing that the validity o transormations and can beinterpreted as being treated in the commentaries dealing with the correctness o algorithms given or multiplying and dividing between quantities having ractions, respectively. o do so, we shall discuss them in the order in which they are presented in Te Nine Chapters, since, interestingly enough, it appears to be also the relevant order o the underpinning reasons. We shall hence deal rst withdivision in relation to transormation , and then turn to multiplication in relation to transormation . Note that all the procedures that allow the execution o arithmetical operations with ractions are systematically provided in Chapter 1 o Te Nine Chapters. One point will appear to be central in this discussion: the relationship between the pair numerator and denominator and the pair dividend and divisor.45 Let us then examine urther this relationship as a preliminary to the ollowing subsections o this chapter. In Part , we recalled that, in ancient China, ractions, conceived o as a pair o a numerator and a denominator, were introduced as the result o division. As we showed in Figure 13.3, dividend and divisor were arranged in an orderly ashion on the surace or computing and, at the end o the division, what remained in the position o the dividend and the divisor were read, respectively, as numerator and denominator. Te continuity between the two pairs o objects is hence maniest rom the point o view o the surace or computing. One can choose to read the two lower lines on the surace either as the 45
In his discussion on ractions, Li Jimin 1990: 62–91 stresses this relationship and discusses the algorithms or dividing and multiplying that we analyse below.
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dividend and divisor o a division to be carried out, or as the numerator and denominator o a completed division. Both interpretations will be used in the commentaries examined below. Te act that the operation o division and the expression o a raction are set up in the same way evokes the identity o their representations in modern notations. However, two differences should be stressed. First, in ancient China, the undamental concept o quantity was not that o a general raction – a rational number, i you will – but that o an integer increased by a raction smaller than 1, which is precisely the result o a division on the surace. Fractions were just a component o them. Second, in our case, we do not have, on the surace, notations or ‘objects’, but rather ‘operational notations’, i.e. notations on which operations are carried out. Te continuity just emphasized derives rom the act that, ollowing the ow o a division, we go rom one to the other and back again. Indeed, the application o the inverse multiplication to the nalcon guration o a division restores the division one started with, exactly as it was srcinally set up (see Figure 13.4). But, in the case o adding up ractions, the corresponding numerator and denominator are placed on the same line horizontally, in such a way that, in the end, the result o the addition is yielded in three lines consisting o an integer, a numerator and a denominator.46 Seen rom another angle, a numerator and a denominator compose a quantity and are essentially dependent on each other. In ancient China, they were both conceived o as constituted o the same ‘parts’ en o a unit, which could either be abstract or not.47 Te size o the part was determined by the denominator, which amounted to the number o parts into which the unit was cut. As or the numerator, it was understood as consisting o a multiplicity o such parts. In contrast to this, a dividend and a divisor are, to start with, separate entities, which happen to be brought into relation when they become unctions in the same operation o division. Tis operation o bringing entities into relation with each other seems to have been deemed essential in ancient China, as we shall see below. As regards the entities considered, at that point, they become linked in a way that makes them share properties with the pair o a numerator and a denominator. Tis parallel is regularly stressed by the commentaries. Te rst example o this kind is ound in the commentary glossing the name o the operation o ‘simpliying parts’ – the rst operation on 46
47
Compare Li Jimin 1982b: 204–5, especially; Chemla 1996, where I reconstruct operational notations in a different way. When the raction was appended to an integer, its numerator and denominator were made o parts o the smallest unit used in the expression o the integer.
Reading proos in Chinese commentaries
ractions discussed in Te Nine Chapters. Tere, the commentary discusses the reasons why, once ractions are introduced, it is a valid operation to divide – or to multiply – both the numerator and the denominator by the same number to transorm the expression o the raction. Tis property is required in order or the ‘procedure or simpliying parts’ to be correct. Te validity o the operation is approached rom the perspective that the numerator and the denominator are constituted o parts o the same size. Multiplying them by the same number n is interpreted as a dissection o each part into n ner and equal parts – a process called ‘complexi cation’, and the operation opposite to the ‘simpli cation’ that the commentator introduces. Conversely, a simultaneous division o the numerator and the denominator by n leads to uniting the parts composing them, n at a time, and getting coarser parts. Tis does not change the quantity as such, but just its inner structuring and its expression. Tus the commentary can conclude: ‘Although, hence, their expressions differ, when it comes to making a quantity, this amounts to the same.’48 Note that, rom the point o view o the operations involved, the reasoning establishes the validity o another mode o inserting a multiplication and a division opposed to each other in the course o an algorithm. What is important is that, immediately aferwards, this question o multiplying and dividing conjointly numerator and denominator is extended to the case o dividend and divisor. Te commentator writes: ‘Dividend and divisor are deduced one rom the other.’ Once the two entities are placed in relation to each other, as unctions o a division, the same reasoning then applies. One can break up or assemble units in the same way. However, the difference between the case o the raction and the general case is that dividend and divisor ‘ofen have (parts) that are o different size’. Te dividend, or instance, may have an integer and a raction. Its expression would then include at least two types o units. Both terms o the division may also have different ractions. ‘Tis is why’, the commentator concludes, ‘those who make a procedure (a procedure generalizing simpli cation?) rst deal with all the parts.’ Tis will require a technique, introduced immediately aferwards, related to the adding up o ractions. On this basis, the question will be taken up again in the context o dividing between quantities having ractions, or which all the necessary ingredients will be available. Tereby, the parallel between the pair o numerator and denominator and the pair o dividend and divisor will be completed. 48
o be precise, part o the above discussion is held in the commentary on the algorithm ollowing the ‘procedure or simpliying parts’, i.e. the ‘procedure or gathering parts’, which allows adding up ractions. Compare, respectively, CG2004: 156–7, 158–61.
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We can now turn to examining in greater detail the relationship between proving the correctness o procedures dealing with ractions and establishing the validity o transormations and . o do so, we shall have to analyse new samples o proo contained in our Chinese sources. Tis will give us the opportunity to describe urther the speci cities o the practice o proo to which our documents bear witness.
Proving the correctness of the general algorithm for division Let us examine the way in which, in his commentary, Liu Hui establishes that the ‘procedure or directly sharing’ is correct, beore considering why this argument can be interpreted as related to the validity o transormation .49 Te Nine Chapters introduces the algorithm or dividing between quantities with ractions afer the two ollowing problems: (1.17) S A (1.18) S A
/ .O : , / :
.
/
/ / .O /
. . .
In the rst problem, the quantity that is to become the dividend contains one raction, whereas the second problem leads to both the dividend and the divisor having ractions. Te act that the dividend even contains two ractions is remarkable. Interestingly enough, such quantities, in which an integer is ollowed by a sequence o ractions, occur only in problems related to similar divisions.50 We shall see that this is linked to the act that Liu Hui uses the operations introduced in his commentary on the addition o ractions or his proo. Tese two problems are in act the rst ones in Chapter 1 or which the data are neither pure integers nor pure ractions. Moreover, they are the rst problems in which the ractions derive rom sharing a unit that is not abstract. Furthermore, problem 1.18 mixes together ractions o different
49
50
Te critical edition and the translation o this piece o commentary can be ound in CG2004: 166–9. In addition to the situation examined here, this also designates problems linked to the ‘procedure or the small width’, which opens Chapter.4Te procedure provides another way o carrying out the division. For comparison, I reer the reader to the introduction to Chapter 4 in CG2004. Te interpretation o the ‘procedure or directly sharing’ requires an argumentation that I developed in Chemla 1992 (I do not repeat the bibliography given in this earlier publication).
Reading proos in Chinese commentaries
kinds o units – cash and persons. In correlation with these changes, the algorithm described is o a type that breaks with previous procedures.51 Let us translate how it reads beore providing an interpretation: D P I
. :O
, . . (here comes a
,
commentary by Liu Hui that we shall analyse below) I , part o Liu Hui’s commentary)
. (second
Te procedure hence presents itsel as one that covers all possible (rational) cases or the data. Te organization o the set o problems distinguishes between cases when the data are both integers (case 1), cases when they both contain only one type o denominator (case 2), and cases where there appear several distinct denominators (case 3; problem 1.18 illustrates which situations may occur in this case). Te undamental case is case 1. It is solved by the rst operation prescribed by the procedure: a simple division. For problems alling in the category o case 2 (that o problem 1.17), the data can be o the type either (a + b/c) and d, or (a + b/c) and (d + e/c). In the second case, the procedure suggests applying the operation o ‘making communicate’. Let us stress that the operation o ‘making communicate’ is used here or the rst time by Te Nine Chapters. In Part o this chapter, we encountered the operation in the context o Liu Hui’s commentary. Tere, we saw that this operation was applied to quantities such as (a + b/c) and ensured that a and b shared the same units, thus transorming (a + b/c) into ac + b. For the case considered here, it transorms the units o the two integers a and d accordingly, so that the number o units obtained (ac and dc, respectively) share the same size as the corresponding numerators. Te quantities (a + b/c) and either d or (d + e/c) are thereby transormed into ac + b and cd (or cd + e). Te problem is thus reduced to the rst case, and the procedure is concluded by a division. In modern symbolism, the procedure can be represented as ollows: b (a ) / d c b e (a ) / (d ) c c 51
(ac b) / dc (ac b) / (dc e )
Te previous procedures all prescribed operations involving numerators and denominators to yield the result. Clearly the description o the procedure to come is o a different style.
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One should not orget, however, that the modern symbolism erases the act that the two ractions are ractions o units that are o a different nature. Te nal case (case 3) is, in turn, reduced to the previous one by the operation o ‘equalizing’, tong’.52 Tis operation relates to the ractional parts o the quantities, making them share the same denominator (‘equalizing them’ in terms o parts). Te resulting transormation or cases such as that o problem 1.18 can be represented as ollows: +
(a + b ) / (d + e + g ) = (a + bh ) / (d + ech gc c h ch ch Once all ractions share the same denominator, we are brought back to case 2, and the problem is solved as above, by ‘making’ integers and ractions ‘communicate’. Such is the complete procedure, the correctness o which Liu Hui sets out to establish in his commentary. Note that the procedure or solving case 3 contains that or solving case 2 which, in turn, embeds that or solving the undamental case. Liu Hui develops the proo with respect to the whole procedure, that is, the one solving case 3, addressing the operations in the order in which they are carried out in this case. In the rst section, Liu Hui thus addresses the operation that occurs last in the text, i.e. that o ‘equalizing’. He does so by reerence to the algorithm or adding up ractions, which he has discussed previously (afer problem 1.9). Te commentator quotes the rst steps o this other procedure or computing bf, ech, gc, on the one hand, and cf on the other hand, thereby providing a translation o ‘equalizing’ into operational terms. It thus appears that, to divide in case 3, the operations to be applied rst are the same as those by which one starts adding up ractions. In parallel, Liu Hui recalls his interpretation o the ‘meaning’ o these steps: he had shown that the latter computed a denominator equal or all ractions whereas the ormer homogenized the numerators so that the value o the srcinal ractions might be preserved. Liu Hui thereby reers the discussion or establishing the ‘meaning’ o the operation that Te Nine Chapters calls here ‘equalizing’ to this other commentary o his, where he showed how the corresponding steps ensured that one ‘makes’ parts corresponding to different denominators ‘communicate’. Te algorithms or adding up ractions, on the one hand, and dividing in case 3, 52
o make things simpler, Imark the transcription o the term inpinyin with an apostrophe, to distinguish it rom the term that has the same pronunciation tong ‘make communicate’. For all these terms, I reer the reader to my glossary in CG2004. I argue there that the operation to which ‘equalizing’ corresponds differs slightly, whether one considers Te Nine Chapters or its commentaries.
Reading proos in Chinese commentaries
on the other hand, share a common sub-procedure and, in the context o division, which comes second in Te Nine Chapters, the commentator states the conclusions o his previous analysis without developing the reasoning again. Tis stands in contrast to the luxury o details with which Liu Hui discusses the second operation to be considered within the context o division in the ollowing sentences. Te Nine Chapters prescribes this operation with the same term o ‘making them communicate’ as the one we discussed above. Te term is encountered here or the rst time inthe Classic proper. However, although the name is the same as the term already discussed, it corresponds here to the prescription o different computations. Following Liu Hui in his analysis, we shall be able to make clear which prescription is meant and why the same term can reer to different operations according to the context. As above, Liu Hui translates what, in this context, ‘making them communicate’ amounts to in operational terms. He then brings to light the ‘meaning’ o the operation in terms o parts. He writes: With the help o the denominator53 ‘making them communicate’ is multiplying by the denominator o the parts the integers (or: integral parts o the quantities) and incorporating these (the results) into the numerators. By multiplying, one disaggregates the integers, thus making the parts o the product ( jien). Te parts o the product and the numerators hence communicate with each other, this is why one can make them join each other.
Liu Hui hence makes explicit what the operation o ‘making them communicate’ means or the quantities at hand. In terms o computations, (a + b/c) and (d + e/c) are transormed into (ac + b) and (dc + e), respectively. We recognize the result o the operation as indicated in Part othis chapter. However, in contrast to that previous occurrence, here the commentator decomposes this transormation into elementary operations and interprets their effects in such a way that he brings to light why Te Nine Chapters may reer to it as ‘make communicate’. Te rst operation consists o multiplying the integersa and d by c, thereby transorming them intoac and dc. Tese quantities are what is rst desig54 nated here as ‘parts o the product’, or parts yielded by a multiplication. 53
54
Note that, whether one is within the context o case 2 or afer the equalization in case 3, only a single value remains or all denominators. We have already discussed the expression jien in Part o this chapter. In this new context, jien could also be understood as ‘accumulated parts’, which would give ji an ordinary meaning. As we suggested above, jien may be interpreted as reerring to what, or us, would be a numerator, in a situation in which the numerator is larger than the denominator. In the
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Whatever the literal interpretation o this expression may be, there is no doubt that the result is understood as being o the kind o ‘parts’, that is, as sharing the same identity as the numerator and the denominator – both o which are a collection o ‘parts’. Tis identi cation derives rom interpreting the ‘meaning’ o the multiplication, in terms o the situation in which it is applied, as a disaggregation. As we alluded to above, Liu Hui had already discussed the link between multiplying and disaggregating parts in the context o the addition o ractions. Tere, afer the numerator and denominator were both multiplied by the same number n – an operation he called a ‘complexi cation’ – the raction obtained was interpreted as composed o parts that weren times ner. Moreover, in this other context, different ‘sets o parts’a/b ( , c/d, . . .) were ‘complexi ed’ jointly, that is, in correlation with each other, in such a way that their denominators became equal to bd ( . . .) and the parts composing them were identical. Liu Hui interpreted this joint transormation as ‘making the parts communicate’ and thereby allowing them to be added to each other. Te same link between multiplication and disaggregation recurs here, but in a slightly different way. Trough the multiplication, the units composing the integers are interpreted to be dissociated into parts o the same size as the ractional parts. Tis dissymmetric transormation o the integers alone ensures that the parts orming the two elements o a quantity o the type a + b/c are ‘made to communicate’ and can be added to each other. It will prove interesting to distinguish here two dimensions in the interpretation o the effect o the operation. On the one hand, with the disaggregation, Liu Hui brings to light a ‘material meaning’ o the multiplication. On the other hand, he recognizes in this transormation the operation o ‘making entities communicate’. In different contexts, the way in which this ormal result is achieved may differ. However, rom a ormal point o view, the action is the same. Tis is what accounts or the act that the same name can be used to reer to different actual computations. In act, so ar, the commentator has considered the operation o ‘making entities communicate’, prescribed by the Classic or case 2 o the division, only rom the point o view o each quantity o the type a + b/c taken separately. As above, each quantity is transormed by the operation into an integral number o parts. However, in case 2 o the ‘procedure or directly case under discussion, the numerator consists o an accumulation o layers o parts equal in number to the denominator, in contrast to the state in which, afer the division is carried out, these layers are each transormed into a unit. Te glossary in CG2004 discusses why the technical term jien can reer, in some circumstances, to ac and, in others, to ac + b.
Reading proos in Chinese commentaries
sharing’, this transormation is carried out on the dividend and the divisor jointly and the denominators c will both be orgotten. Te correctness o the procedure can only be established afer this other aspect has been accounted or. In the next section o the commentary, Liu Hui turns to address the transormation. Again, it will be dealt with in terms o ‘making communicate’ and this expression will take on new concrete meanings. Indeed, the argument will show why communication is established not only between components o the same quantity (a and b/c; d and e/c) but also between the dividend and the divisor, contained in the middle and lower parts o the surace or computing. Tis is how the procedure can be concluded by a division between integers. Tis remark suggests that, in so doing, Liu Hui is still deploying his interpretation o the meanings he reads in the term ‘making communicate’, a phrase used here byTe Nine Chapters. Te correlative transormation o the dividend and the divisor recalls the commentary on the ‘procedure or simpliying parts’. We noted above that, in this commentary, Liu Hui had compared the two situations rom the point o view that rstnumerator and denominator, and then dividend and divisor could be transormed in relation to each other. Pointing out a contrast between the two pairs, the commentator had stressed that dividend and divisor could involve ‘parts’ o different size. We highlighted the act that, in the context o the addition o ractions, he showed how to transorm distinct ractional parts into parts o the same size. However, one aspect o the difference between the two situations has not yet been discussed. We meet it here or the rst time, and it appears that it is precisely this difference that Liu Hui addresses now. Following him, we can explain the difference as ollows. Te simpli cation or complexi cation o a raction implies considering the quantities expressed with respect to the same part jointly. However, the terms o a division can have parts that differ not only in size, but also in nature. In our case, the dividend contains parts o cash while the divisor has parts o persons. It is interesting that, or the operations discussed previously (adding up, subtracting, comparing, computing the average), the data o the problems were all abstract and, in correlation with this, these operations can be applied only to terms that are homogeneous with each other in this respect – they only need to be homogenized with regard to their size. For division, in contrast, the terms can urthermore be o a different nature.55 Tis is what the problem shows and what is dealt with rom a theoretical and, most importantly, general point o view now. 55
Li Chuneng’s commentary on the name o the operation, ‘directly sharing’, may address this difference. We shall come back to it below. Note that the same remark holds true or ‘multiplying parts’.
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Te key point that Liu Hui stresses is that by the very act that these quantities are taken as dividend and divisor, they are ‘put in relation’. By this act, a relationship is established between them, which has operational consequences. Here, the commentator rst introduces the concept o lü which precisely characterizes the situation created: ‘Whenever quantities are given/put in relation with each other, one calls them lü.’ In the case we examine, dividend and divisor are ‘put’ in relation, as quantities o given, but distinct, units. It is the context o an operation that shapes this relationship. Te values expressing the relation between the circumerence and the diameter o a circle are also lüs. However, by contrast to the ormer quantities, they are rather ‘given’ in relation with each other. In this case, it is a situation that brings them into relation. Liu Hui, meeting here with a phenomenon that, rom a ormal point o view, will turn out to be quite widespread, discusses it rom a much more general angle, which will thus prove useul and relevant in several other sections o his commentary. Tis is a recurring and important eature o the commentator’s proos and one that makes them difficult to interpret: he systematically brings to a given context a more general outlook rom which to address the correctness o a given operation, and thereby introduces a concept and an argument that will be shown to recur in different contexts.56 In act, the concept o lü had already been introduced by Te Nine Chapters in relation to the prescription o the rule o three, at the beginning o Chapter 2. Te commentary will regularly, and more generally, bring to light in all kinds o mathematical situations that quantities are lüs and use this property or establishing the correctness o a procedure. Once the concept is introduced, Liu Hui states the consequence or the entities that it quali es: ‘Lüs, being by nature in relation to each other, communicate.’ We hence meet with a second occurrence o the term ‘communicate’ in the context o the commentary on the ‘procedure or directly sharing’, an occurrence which echoes the wording o the procedure itsel. Tis time, it reers to the act that the dividend and divisor are brought into communication, even though this operation is grasped rom a more general point o view.
56
On this eature o proos, see Chemla 1991. Te same phenomenon is shown to happen or the operations o ‘homogenizing’ and ‘equalizing’, which are introduced in the commentary on adding up ractions. We saw above another dimension o the relationship between the conduct o a proo and the search or generality when we stressed the parallel between the proos o the correctness o the algorithms or the truncated pyramid with circular base and the cone, respectively. On the concept o lü, see Li Jimin 1982a, Guo Shuchun 1984, Li Jimin 1990: 136–61, Guo Shuchun 1992: 142–99, and the entry in the glossary in CG2004.
Reading proos in Chinese commentaries
Te consequences o such a state are made explicit in the commentary ollowing problem 3.17, in which Liu Hui asserts: Every time one obtains lüs, that is that, since when one re nes (the units in which they are expressed), one re nes them all and, when one makes them coarser, one makes them all coarser, the two quantities are transormed in relation to each other (literally, interact with each other) and that is all.57
Once the relationship is set, or instance, in our case, by the act that ‘dividend’ and ‘divisor’ are ‘put in relation to each other’ as quantities o given units, any modi cation o the value o one that comes rom a systematic dissection o its units – or a reunion o them – must be re ected in a dissection – or reunion – or the units o the other or the relationship to be maintained.58 Tis is where the property o numerator and denominator is seen in a more general perspective. Tis is also the point where a parallel is established between the commentary on the ‘procedure or simpliying parts’ and our context. Te next sentence o Liu Hui’s commentary on the ‘procedure or directly sharing’ states the same property with respect to lüs: ‘I there are parts, one can disaggregate; i parts are reiterated superpositions, one simpli es.’ However, in contrast to the ormer statement, this quote makes precise in which circumstances may nd is, it useul to ‘disaggregate’ the units o both terms, or ‘simpliy’one them – that carry out a systematic aggregation o their units. Te disaggregation is to be used when the values put in communication have ‘parts’, that is, contain ractions. Previously, being in communication allowed the integer and the ractions to enter together into the same operation o addition. Here, being in communication urther implies that, when modi ed, the values are transormed simultaneously. Tis latter property is used to transorm the values o thelüs into integers
57
58
See CG2004: 306–7, 797, n. 73. In that case, the commentary brings to light that, in order to account or the procedure, one must understand that the lüs chosen to express the relationship between two different kinds o silk are given in different units o weight. By virtue o their quality o being lüs, they nevertheless change in relation to each other. Note that there can be more than two quantities, the set o which constitutes lüs. In Chemla 2006, I discuss source material rom the Book o Mathematical Procedureswhich documents the process o introduction o the concept o lü, as encapsulating parallel sequences o computations carried out on quantities that occur within a dividend and a divisor. Te way in which the transormations encapsulated are described echoes in many ways Liu Hui’s commentary here. Since the lüs express this relationship, the nature o the units o the quantities involved can be orgotten, even though this is by no means mandatory. Tis corresponds to what is ound in the text, where in most cases, the values o lüs are expressed by abstract numbers. In some sense, introducing the concept o lü is a way o addressing the p ossibility o carrying out an abstraction with respect to units.
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in correlation with each other. In the case o a division, by a simultaneous dissection o the units o the dividend and the divisor, one may get rid o the ractions. Just as in the context o problem 3.17, Liu Hui approaches the correlative transormation o the values olü with ull generality, introducing the disaggregation o the basic units in parallel with the opposite operation, i.e. aggregating units. Te circumstances in which the latter operation can be used are reerred to by the technical expression o ‘reiterated superpositions’, which had been introduced earlier, in the commentary on the simpli cation o ractions. Tere, it designated the possibility that the numerator and denominator could be represented as rectangular arrangements o units – ‘parts’ in this case – having a side o the same length, equal to their greatest common 59 divisor, or their ‘equal number’ in the terminology o ancient China. As a consequence, dividing both by the ‘equal number’ amounted to expressing the raction in terms o parts coarser than the srcinal ones by a actor equal to that number. In the context o the general discussion aboutlüs in the commentary on ‘directly sharing’, disaggregation has been introduced. Te next sentence then reers to the units as ‘parts’, even though they may be o a different nature, and states: ‘I parts are reiterated superpositions, one simpli es.’ Te concept o ‘reiterated parts’ and the operation o simpli cation that it helps justiy are thus imported into a new and more general context. Once the general considerations have been developed ully, the commentary applies them to the case under discussion, namely, dividend and divisor. In a rst step, ollowing on the last statement, Liu Hui introduces the new concept o ‘lüs put in relation with each other’, precisely when he identi es the rst instance or it: ‘Divisor and dividend, divided by the equal number (i.e. their greatest common divisor), are lüs put in relation with each other.’ In a second step, Liu Hui translates the properties o lüs discussed above or the speci c case examined in this context. Dividend and divisor having both parts, one disaggregates repeatedly their units in parallel, which amounts to multiplying. Te commentator writes with ull generality: ‘Tereore, i one disaggregates the parts, one necessarily makes the two denominators o the parts both multiply divisor and dividend.’ Te general prescription o disaggregating (ormulated at the level o reasons) leads, within our speci c context, to speci c operations (at the level o computations), namely, two multiplications. Tinking o the process in terms o disaggregating and joining, the procedure amounts to 59
On these terms, see the glossary in CG2004.
Reading proos in Chinese commentaries
b e ec (a + ) / (d + ) = (ca + b) / (cd + ) = ( ca + b) / ( cd + ec ) c ( rst multiplication) (second multiplication) which is equivalent to the algorithm as provided in Te Nine Chapters: b e b ec (a + ) / ( d + ) = (a + ) / (d + ) = ( ca + b ) / ( ccd c c c
+
ec )
(equalizing) (multiplying by the two denominators) Tese are the operations applied to the divisor and the dividend, and this is what is meant by the prescription o ‘making them communicate’, i we ollow Liu Hui’s interpretation. Te values o the dividend and the divisor are transormed correspondingly, and they both become integers, without their relationship being altered. Here the analysis o the operation o ‘making communicate’ is completed, and the correctness o the procedure or ‘directly sharing’ is established.60 From the previous discussion, three points are worth stressing. Te rst two are important or a description o the practice o proo. First, as we already emphasized, the proo is carried out in such a way as to approach the phenomena with the greatest generality possible. In our case, this leads to the introduction o some key abstract concepts such as lü. Second, through the analysis that is conducted during the proo, a simpli cation o the algorithm is hinted at, since it is shown that dividend and divisor can be simpli ed beore a division is to be carried out. Again this is a recurrent eature in the commentators’ proos: they offer a basis on which to develop new algorithms. Tird, and more importantly or our purpose, the concept o lü that is introduced is intimately related to the theme o this chapter. Tis is the point where we go back to the main thesis or which we argue here.
Combining divisions that follow each other In act, identiying, in a given context, the property o entities to be lüs with respect to each other is a way o establishing the validity o introducing into the ow o computations multiplications and divisions that compensate 60
In a last paragraph, the commentator describes another procedure that articulates the different cases possible in a different way; see Chemla 1992 or a discussion.
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a b c
wo readings: Dividend/Division o ac + b by c
ac + b
Dividend
d
Divisor
cd
Dividing by the product
Figure 13.8 computing.
Te division between quantities with ractions on the surace or
each other.61 In the context o dividing between quantities with ractions, the last analysed sentence o Liu Hui’s commentary shows how the commentator links the proposed transormation o units, the correctness o which was established, to the application to both the dividend and the divisor – both, in this case, themselves the results o a previous division – o the same sequence o multiplications. In other cases, the concept o lü is brought into play when accounting or an inversion in the order o a multiplication and a division is at stake. 62 Tis brings us back to the main question o this subsection: what is the relationship between this development o Liu Hui’s and the validity o our undamental transormation ? o bring the link to light, let us consider one o the cases to which the ‘procedure or directly sharing’ can be applied: b (a + ) / d = (ac + b) / dc c and let us look at this rom the point o view o the surace or computing (Figure 13.8). Te set-up o the dividend (column 1) shows in which ways it can be considered as the result o the division o ac + b by c (column 2). Te algorithm thus amounts to dividing by d the result o a division by c. On the one hand, ac + b is that to which one returns when ‘making communicate’ the integer a and the numerator b – this property is guaranteed, as Liu Hui stressed, by the act that the results o division are given as 61
62
Above, the introduction o speci c quantities such as ractions or quadratic irrationals was justi ed by the necessity o having inverse operations cancel each other. Here, it is the introduction o a concept, that o lü, that is to account or cancelling opposed multiplication and division. See, or instance, the second proo o correctness o the ‘procedure or multiplying parts’ or the proo o the correctness o the ‘rule o three’ in CG2004: 170–1, 224–5.
Reading proos in Chinese commentaries
exact. On the other hand, as Liu Hui shows, the ‘procedure or directly sharing’ amounts precisely to multiplying c by d to divide ac + b by both o them at a stroke (column 3).63 Such a reasoning would be only the observation o an equivalence, were it not indicated by precisely the name given to the operation o division between any two quantities in Te Nine Chapters, i.e. jingen . I suggest understanding that the srcinal meaning o this name was ‘directly sharing’. Tere are two pieces o evidence to support this interpretation. First, the procedure or carrying out the same division in the Book o Mathematical Procedures has the same name, except or the act that the character jing is written with a homophone that means ‘directly’.64 Secondly, when the seventh-century commentator Li Chuneng comments on the name o the procedure in Te Nine Chapters, his interpretation is in conormity with how the name is written in the Book o Mathematical Procedures. Since this interpretation is quite important or our purpose, let us read it: D . Your servant, Chuneng, and the others comment respectully: 65 (the procedures) all As or ‘Directly sharing’, rom ‘Gathering parts’ onwards, made the (quantity o) parts homogeneous with each other, but this one directly seeks the part o one person.66One shares that which is shared by the number o persons, this is why one says ‘Directly sharing’.
Te most important statement or us here is the one I italicized: the operation is interpreted as dividing a quantity that is understood as itsel being yielded by a ‘sharing’ or, in other terms, a division. Li Chuneng thus also reads the operation as we suggest doing, that is, as dealing with the succession o two divisions. He thereby links, on the one hand, dealing with operations that ollow each other, and, on the other hand, how arithmetical operations are carried out on quantities having ractions. In doing so, Li Chuneng probably seeks to account not only or the name o the operation, but also or why the style o the algorithm breaks with the description o all the others beore it. However, this interpretation ts with what the Book 63
64 65
66
It is interesting that theoperation o m ‘ aking communicate’ thatTe Nine Chaptersprescribes is, or one part, the very operation that restores what can be interpreted as the srcinal dividend. For another part, this reading provides an interpretation o ‘dividing at a stroke’ in terms o ‘making communicate’, which can be shown to be meaningul. Compare slip 26, Peng Hao 2001: 48. Tat is, all the procedures or adding up ractions, subtracting them, comparing them and determining their average. Tese procedures are all interpreted by the commentators as making the number o parts, that is, the numerators, homogeneous to each other, beore applying the operation in question. Compare CG2004: 166–7. One may understand that the division is prescribed directly, without having made the ractions rst homogeneous in any respect.
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o Mathematical Procedures contains in relation to this operation, which provides a hint that this was how the situation was understood even beore Te Nine Chapters was compiled. It is important in this respect that one o the prerequisites or this interpretation – namely, that the multiplication inverse to a division restores the srcinal number divided – appears to be a concern documented in the Book o Mathematical Procedures, as shown above. Tis completes our argument in this case. An additional remark should be made. So ar, in our argument, we have only considered the validity o transormation with respect to integers. What about establishing its validity more generally? wo points should be added in this respect. On the one hand, i we observe the contexts in which ‘dividing at a stroke’, or its synonym, ‘dividing together’ bingchu ( ), are used in the commentaries, it turns out that the two divisions that are joined are usually both divisions by integers. On the other hand, i this is the case, in some situations this relates to the act that the property o entities to be lü was put into play.67 Similarly, i the capacity o the quantities involved to be transormed into integers is employed, transormation is to be used in contexts in which they were already turned into integers. Tat such may have been the idea is plausible: more generally, Te Nine Chapters exhibits a way o carrying out computations that grants a predominant part to integers, and the introduction o the concept o lü can be interpreted as one technique among several devised to ul l this aim. Several hints can be given in avour o these hypotheses. First, the commentators regularly interpret the choice o describing a procedure in a given way in Te Nine Chapters as derived rom the motivation o the authors to avoid generating ractions in the midst o computations. Tis is how, or instance, Liu Hui accounts or why, in the rule o three, the multiplication is prescribed beore the division, and not afer.68 Te commentators thus attribute to Te Nine Chapters the intention o computing with integers wherever possible. Second, the way in which division between quantities containing ractions is dealt with in the general case amounts precisely to getting rid o ractions. Liu Hui reads this way o proceeding as made possible by the status o the dividend and divisor as lüs. Tird, in the procedure o Te Nine Chapters in the context o which the concept o lü is introduced, that 67
68
See, or instance, how Liu Hui interprets the algorithm provided afer problem 6.10 (CG2004: 514–15). Te validity o this operation is discussed in the next subsection.
Reading proos in Chinese commentaries
is, the rule o three, it guarantees precisely that the number by which one multiplies and divides be an integer.69 Tis leads to mixing together integers and non-integers in the computations in a dissymmetric way that is quite speci c to the procedure or the rule o three described in Te Nine Chapters.70 Te predominant role given to integers can be read in the way in which algorithms are composed and in the speci c concepts that are introduced in correlation with this. o establish whether this eature actually plays a part in the proos o correctness, as we suggested above, we would have to observe how the concept o lü is actually put into play in the commentaries, an issue that we leave or another publication. 71 Let us turn instead to the relationship between transormation and multiplying between quantities containing ractions.
Inverting the order of a division and a multiplication that follow each other We already hinted at the reasons or linking the ‘procedure or the eldwith the greatest generality’ and transormation . It is hence natural to seek, in the commentary o the ormer, a proo o the validity o the latter. As in the previous subsection, we shall rst examine how the correctness o the algorithm or multiplying quantities o the type a + b/c is established. While doing so, we shall naturally be led to connecting this proo to that o the validity o transormation . Let us recall the procedure given by Te Nine Chapters, which was already discussed in Part o the chapter: P
:
(
);
; . .O .
Liu Hui establishes the correctness o the procedure in two steps, each o which relates to a step in the procedure. Te commentary on the rst seto operations reads as ollows: 69 70
71
Incidentally, it also allows that these numbers be prime with respect to each other. Such is not the case or the rule o three given by the Book o Mathematical Procedures. Discussing this difference exceeds the scope o this chapter and I shall deal with it elsewhere. As already indicated above, the nature o the data to which the operations o the various algorithms are applied should also be systematically observed, i we were to be more precise regarding the extension o the algorithms or which correctness is established.
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I ‘the denominators o the parts respectively multiply the integer corresponding to them and the numerators o the parts join these (the results)’, one makes the bu72 that are integral communicate and be incorporated in the numerator o the parts. In this way, denominators and numerators all (contribute to) make the dividends.
Above, we already alluded to the main elements o this commentary. Let us add only two remarks. First, we now see how the operation o ‘making communicate’ that is used in this proo is precisely one that was analysed in the commentary on ‘directly sharing’. Second, in the transormation o b into ac + b, the latter is designated as ‘dividend’. Tis is one o the several c signs o the continuity, which we already stressed, between quantities o the type a + b/c and division, rom both a conceptual and a notational point o view. Tis point will prove important below. As a commentary on the remaining part o the procedure, Liu Hui states: Tis is like ‘multiplying parts’.
In other words, he asserts that the algorithm is, rom this point onwards, analogous to the procedure or multiplying between ‘pure’ ractions, which, in Te Nine Chapters, is placed just beore it. As was observed above, the commentator reers the interpretation o some steps o the procedure to his previous commentary.73 Tree points are worth noting. First, in the same way as we showed previously how the procedure or the truncated pyramid with circular base embedded, among other algorithms, the ‘procedure or the eldwith the greatest generality’, the latter is now shown to embed another procedure. Tis embedding is, however, to be distinguished rom the one which accounts or the name o the operation, discussed in Part o the chapter. Te latter embedding related to the act that the ‘procedure or the eld with the greatest generality’ uni ed three procedures or multiplying different types o numbers: it reerred to the algorithm as a list o operations. Te new embedding maniests itsel in the proo: it brings to light that, among the three cases covered by the algorithm, one o them is, in terms o reasons, more undamental in that the correctness o the general procedure relies on its correctness. Tese two cases show that algorithms may be built by making use o other algorithms 72
73
Te commentary reers to the data o the problems afer which the procedure is given. Tey are all lengths expressed with respect to the unit o measure bu. Tis conclusion is reinorced by the commentary placed afer the procedure, which repeats one o the arguments given to account or the correctness o the algorithm or multiplying ractions.
Reading proos in Chinese commentaries
a b c
integer numerator denominator
ca + b c
c ′
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′
′
′
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(ca + b).(c a + b ) (ca + b). (c a + b ) cc ′
′
a b
′
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c
integer numerator denominator
Figure 13.9 computing.
′
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c a +b c
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c a +b
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c a +b
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c
c
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Dividend Divisor: Te order o the operations was inverted
Te multiplication between quantities with ractions on the surace or
in various ways, and the proo o the correctness o the ormer may as well incorporate the proo or the latter according to different modalities. Second, interestingly enough, in their proos, the commentaries regularly reer to the proos o algorithms placed just beore in the Classic. 74 Tis seems to possibly provide an interpretation o the reasons why the algorithms are presented in this order in Te Nine Chapters. Tird, i we look at Figure 13.9, we see that the part o the algorithm that is applied to the elements placed on the surace or computations afer the rst step, that is, when divisions are restored, can be considered similar to the algorithm applied to ractions: this is an essential prerequisite or the proo o this section o the algorithm to be reerred to that o the ‘procedure or multiplying parts’. Tis yields yet another hint o the act that practitioners o mathematics in ancient China saw continuity between the notation o quantities and the set-up o operations. Te commentary on ‘multiplying parts’, to which we shall now turn, starts by discussing precisely this point. Te algorithm reerred to reads as ollows: M P
: ; .O
.
Te opening sentence o the commentary relates the pair o a numerator and a denominator to that o a dividend and a divisor. Liu Hui writes: 74
Te second proo o the correctness o the ‘procedure or multiplying parts’ reers explicitly to ‘directly sharing’. See CG2004: 170–1. We shall show below that the rst proo also needs to rely on ‘directly sharing’.
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a b Figure 13.10
Te layout o a division or a raction on the surace or computing.
In each o the cases when a dividend does not ll up a divisor, they hence have the names o denominator and numerator.
In other words, one may choose to read an array o two lines on the surace or computing, as in Figure 13.10, in two ways. On the one hand, the array is the layout o an operation o division, which we shall represent as a : b. On the other hand, when a is smaller than b, which is precisely the case ‘when a dividend does not ll up a divisor’, it can be read as the quantity resulting rom carrying out the operation, that is, the raction a/b. Tese dual points o view allow Liu Hui to link the raction and the numerator operationally. Placing himsel at the most general level, as we have seen him ofen do in proos, he writes: I there are parts (i.e. ractions), and i, when expanding the corresponding dividend by multiplication, then, correlatively, it (the dividend produced by the multiplication)75 lls up the divisor, the (division) hence only yields an integer.
Te application o this remark that appears relevant in the context in which it is ormulated is that the sequence o a multiplication and a division like (b · a) : b yields a. Seen rom the other point o view, this remark leads to stating that the multiplication b · a/b yields a as its result. Te numerator can thereby be seen as a quantity that is b times larger than the raction. I, urthermore, one multiplies something by the numerator, the denominator must consequently divide (the product) in return (baochu). Dividing in return is ‘dividing the dividend by the divisor’.
Tis is the point where Liu Hui introduces the operation o ‘dividing in return’, which we already mentioned above and which occurs only later in the text o Te Nine Chapters. In terms o operations, ‘dividing in return’ is a simple division. However, the expression by which it is prescribed indi75
Te name o ‘dividend’ designates what is in the position o the ‘dividend’ on the surace or computing, at the moment when it is used. Tis is the assignment o variables typical o the description o algorithms in Te Nine Chapters.
Reading proos in Chinese commentaries
cates the reason why it must be used: earlier in the ow o computations, one multiplied by a magnitude which was n times larger than it ought to be – in most cases, by a numerator instead o the corresponding raction – thereore a division by n is needed to cancel this unwarranted dilation.76 In our case, Liu Hui’s statement is an answer to the question o determining the product o a/b by ‘something’ (let us call this ‘something’ X) – one may note the generality o the question considered. Te reasoning appears to be that, since a·X is equivalent to [(b · a) : b]·X or b·a/b·X, then a/b.X is hence equivalent to a·X : b. I we pause a moment here, we can observe that what is dealt with is precisely our transormation . A division ollowed by a multiplication, that is, a/b·X, which Liu Hui emphasized as equivalent to (a : b)·X, has been replaced by a multiplication ollowed by a division, a·X : b. Te way in which the commentator discusses the issue highlights the link he reads between multiplying ractions (multiplying the result n/c by X) and what we called transormation – transorming the sequence (n : c)·X into nX : c.77 In addition, the discussion has not yet speci ed the quantity X. Its result holds or any such quantity. Tis is yet another case where the proo does not limit itsel to the context in which it is developed, but highlights the most general phenomenon possible. In relation to the context in which Liu Hui develops this discussion, the next step turns to the consideration o a speci c value or X, that is, the numerator c o the raction c/d to be multiplied bya/b. He writes: ‘Now, “the numerators are multiplied by one another”, hence the denominators must each divide in return.’
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In all observed cases, the ‘division in return’ eliminates a actor that is an integer. Note that the beginning o Liu Hui’s commentary can be read as addressing the validity o such a division: dividing, by a actor, a quantity that resulted rom a multiplication by this very actor eliminates rom it this actor. Te commentary on the procedure solving problem 6.3 also stresses that the sequence o multiplying bya and dividing by b can be car ried out as multiplying by a/b, that is, multiplying bya and dividing in return by b. Te commentary on the procedure o ‘suppose’, at the beginning o Chapter 2, establishes the correctness o the algorithm carrying out the rule o three in two ways. On the one hand, afer having shown that a sequence o a division and a multiplication yields the correct result, the commentator ‘inverts their order’ ( an) to obtain the algorithm as described in Te Nine Chapters. On the other hand, he transorms the lüs expressing the relationship between the things to be changed one into the other, the ormer into 1 and the latter into a raction, by which the reasoning shows one must multiply to carry out the task required. Tis, says Liu Hui, corresponds to ‘with the numerator, multiplying and with the denominator, dividing in return’. A link is thereby established between the operation o ‘inverting the order’ an o a division and a multiplication and that o multiplying by a raction. Note how using the concept o lü and its operational properties is essential or bringing this link to light. Te commentary on the procedure solving problem 6.10 puts into play all the elements examined so ar.
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Te problem in Te Nine Chapters asks to compute the product o a/b by c/d. On the basis o the previous observation, this operation is shown to amount to a·c/d: b, which, in its turn and or the same reasons, amounts to ac : d : b. Liu Hui can hence interpret the ‘meaning’ o the rst prescribed operation (computing ac) and can establish that it must be ollowed by two divisions or the desired result to be obtained. Te commentator has thus produced an algorithm yielding the result required by the Classic. Te last step needed to prove the correctness o the procedure given by Te Nine Chapters is to transorm the algorithm obtained (a·c : d : b) into the one or which the correctness is to be proved. Such a transormation comes under the rubric o the second line o argumentation in an ‘algebraic proo in an algorithmic context’, which we introduced in Part o the chapter. Liu Hui concludes his proo by transorming the ormer algorithm into the latter, as ollows: ‘Consequently, one makes “the denominators multiply each other” and one divides at a stroke (by their product) (lianchu).’ In other words, the commentator here applies transormation , the validity o which was, as I argued above, dealt with in the commentary on ‘directly sharing’.
Conclusions Te analysis developed in this chapter invites drawing conclusions on several levels. First, the passages examined illustrate how the earliest known commentators on Te Nine Chapters ul lled the task o establishing the correctness o algorithms. As we suggested in the introduction, this branch o the history o mathematical proo has not yet been deeply explored. We see how the Chinese source material calls or its development. wo issues are at stake here. We need to understand the part played by proving the correctness o algorithms in the overall history o mathematical proo, and in particular in the history o algebraic proo. Moreover, on this basis, we must determine how we should locate Chinese sources in a world history o mathematical proo. Whatever conclusion we may reach in this latter respect, it remains true that Liu Hui’s and Li Chuneng’s commentaries provide source material or the analysis o the undamental operations involved in proving the correctness o an algorithm not only in ancient China but also in general. In our limited survey o proos rom the Chinese source material, several undamental operations appeared.
Reading proos in Chinese commentaries
We saw how proos relied on algorithms, which had already been established as correct, and how proos articulated these algorithms as a basis or establishing the correctness o other procedures. Most importantly, the algorithms, together with the situations in relation to which they were introduced, provided means or determining the ‘meaning’ o an operation or a sequence o operations. Tis appears to be a key act or proving the correctness o algorithms, and it is noteworthy that a term (yi ‘meaning’) seems to have been specialized to designate it in ancient China. Furthermore, as was stressed above on several occasions, the evidence provided by the commentaries seems to maniest a link – perhaps speci c to ancient China – between the way in which the proo o the correctness o algorithms was conducted and a systematic interest in the dimension o generality o the situations and concepts encountered.78 Te act that proos ofen relate to each other, as we emphasized several times, can be correlated to this speci city. However, it will be only when historical studies o such proos develop that we will be in a reasonable position to conclude whether this eature is characteristic o Chinese sources or intrinsic to proving the correctness o algorithms in general. Finally, the second key operation in the activity o proving the correctness o algorithms that is documented in ancient China, and on which we ocused in this chapter, was what I called the ‘algebraic proo in an algorithmic context’. So ar, I can locate it only in ancient Chinese source material, as ar as ancient mathematical traditions are concerned. But again this conclusion may have to be revised in the uture. Again, whatever the case may be, what can we learn rom this occurrence regarding algebraic proo in general? I we recapitulate our analysis in this respect, we have seen that several technical terms were introduced in relation to this dimension o proo: u ‘restoring’,huan ‘b acktracking’,79 baochu ‘dividing in return’,80 an ‘inverting’,
78
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80
I have dealt with this issue on several occasions, rom Chemla 1991 onwards. However, given the complexity o this link, I cannot ully discuss it within the ramework o this chapter. I plan to revisit the issue in another publication that would be entirely devoted to it. Note, however, that, here again, the commentators introduced a technical term in relation to this acet o the problem. In my glossary, I transcribed it as yi’ ‘meaning, signi cation’, to distinguish it romyi, and the reader will ndin these two entries partial discussion o the problem. Yi’ designates a ‘meaning’ that captures the undamental procedures that proos disclose to be at stake within each algorithm dealt with. A variant or this operation is huan yuan ‘return to the srcin’. On all these terms, the reader is reerred to my glossary in CG2004. A variant or this concept is the pair o terms ru ‘enter’/chu ‘go out’. See the glossary in CG2004.
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lianchu ‘dividing at a stroke’,bingchu ‘dividing together’.81 Tese terms reer to the three undamental transormations (those we designated by , and ) involved in the ‘algebraic proo in an algorithmic context’ as carried out or establishing the correctness o the algorithms presented in Te Nine Chapters. In act, the validity o these transormations rests on the act that the results o divisions and extractions o square root are given as exact. We have seen that Liu Hui explicitly related the validity o the rst undamental transormation to this act. Beore we go urther in concluding about the two other transormations, let us introduce the general remark regarding algebraic proo to which this act leads us. Such a type o proo can be characterized by the act that it carries out transormations on sequences o operations as such. What appears here is that the validity o such transormations rests on the structural properties o the set o quantities to which the variables and constants involved in the ormulas transormed may reer. As soon as it is stated, the remark sounds obvious. My claim is that it can be documented that a rst version o this act came to be understood in ancient China, in relation with the conduct o ‘algebraic proo in an algorithmic context’. Tis claim, in turn, raises a historical question regarding this range o issues on which I shall conclude the chapter: how was the relationship between the validity o algebraic proo and structural properties o the set o magnitudes on which it operated historically discussed? It is clear that inquiring into this question should elucidate a undamental dimension o the history o algebraic proo. Te second level on which I would like to ocus in concluding relates to my argument regarding the validity o transormations and . In the chapter, I argued that there was an interest, in ancient China, in illuminating the grounds on which this validity rested. Moreover, I suggested that the question was dealt with in the commentaries on the algorithms or dividing and multiplying quantities o the type a + b/c . It is to be noted that ractions conceived as a pair o a numerator and a denominator, as well as quantities a + b/c, appeared in Asia, in the earliest known Chinese and Indian books. In China, the rst extant document attesting to the arithmetic with such numbers, that is, the Book o Mathematical Procedures, also exhibits a concern or the problem o ‘restoring’ (u) the srcinal quantity that was divided, when applying the inverse operation. Te main point, however, is that, to my knowledge, the pages that Chinese commentators devoted to establishing the correctness o algorithms carrying out arithmetical 81
Note that, although multiplications also happen to be joined – or instance, in the commentary ollowing problem 6.10 – no speci c term was coined or this transormation. Tis dissymmetry between multiplication and division is remarkable.
Reading proos in Chinese commentaries
operations with quantities containing ractions are unique to China, by contrast to other ancient traditions. I this were con rmed, there would appear to be a correlation between the latter proos, on the one hand, and the use o ‘algebraic proos in an algorithmic context’, on the other hand. In Part o this chapter, however, my argument was based only on internal considerations. One o the most important acts that grounded the argument was the continuity o concepts and notations on the surace or computing, such as operations like division or multiplication on the one hand and quantities such as ractions or numbers o the type a + b/c, on the other. Te same con guration o numbers on the surace or computing could be read as the set-up o a division, or the result o a division, that is, a raction. Moreover, applying a multiplication byc to the con guration in three lines representing a + b/c – hence read as the set-up o a multiplication – could restore the division that had yielded it. Te key element or this continuity is that o a position on the surace in which one could place and operate on a component o a quantity or a unction o an operation. Te surace served as a medium articulating these mathematical objects. In this way, arithmetical operations on ractions were transormed into sequences o operations, and the algorithms carrying them out were established on the basis o interpretations and transormations o these sequences o 82
operations. A link was thereby established between transorming lists o operations and operating on ractions. I suggested reasons or considering that this was the way in which the commentators understood it. On the one hand, in the commentary ollowing problem 5.11, we saw how the inversion o the order o a division and a multiplication was carried out by making use o the ‘procedure o the eld with the greatest generality’. On the other hand, when Li Chuneng interprets the name o the operation or dividing between quantities o the type a + b/c, he reers to the division o a quantity itsel yielded by a division. Tere is, however, another angle rom which to consider the relationship between the undamental transormations , , and the proos o thecorrectness o algorithms or arithmetical operations on quantities o the type a + b/c. Most o the technical terms listed above, by which the commentators reer to these transormations, are introduced precisely in relation to commentaries discussing the necessity o using quantities like ractions or quadratic irrationals (u), on the one hand, and establishing the algorithms 82
One example or this is how, i there are parts in the dividend and the divisor, ‘directly sharing’ is explained to be equivalent to ‘multiplying’ both quantities by the two denominators.
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operating on quantities such as a + b/c (huan, baochu, lianchu) on the other hand. In that way, these terms are introduced at the beginning o the book.83 Tese concluding remarks lead to a whole range o questions, which we shall ormulate as a conclusion to the chapter. How was the correctness o algorithms or multiplying and dividing quantities with ractions approached elsewhere, and what connections did this concern have with the kind o ‘algebraic proo in an algorithmic context’ discussed here? Is there a historical relationship between the proos we examined and the overall history o algebraic proos? I there exists some relationship, did the proos that were devised in ancient China actually play a historical part in this process? It is clear, I believe, that the history o mathematical proo still has many new territories to explore.
Acknowledgements Tis chapter has bene ted rom the discussions o the working group ‘History and historiography o mathematical proo in ancient traditions’, which the research group REHSEIS convened at Columbia University in Paris, at Reid Hall, between March and June 2002 thanks to the generosity o the Maison des sciences de l’homme and Columbia University. It is a pleasure to express my gratitude to all the participants in this workshop and the sponsors who made it possible.
Bibliography Chemla, K. (1991) ‘Teoretical aspects o the Chinese algorithmic tradition ( rst to third century)’, Historia Scientiarum42: 75–98 (+ errata in the ollowing issue). (1992) ‘Les ractions comme modèle ormel en Chine ancienne’, in Histoire de ractions, ractions d’histoire, ed. P. Benoit and J. Ritter. Basel: 188–207. (1996) ‘Positions et changements en mathématiques à partir de textes chinois des dynasties Han à Song-Yuan. Quelques remarques’, Extrême-Orient, Extrême-Occident 18: 115–47. (1997/8) ‘Fractions and irrationals between algorithm and proo in ancient China’, Studies in History o Medicine and Sciencen.s. 15: 31–54.
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Only an will be introduced at the beginning o Chapter 2.
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(2003) ‘Generality above abstraction: the general expressed in terms o the paradigmatic in mathematics in ancient China’, Science in Context16: 413–58. (2005) ‘Te interplay between proo and algorithm in 3rd century China: the operation as prescription o computation and the operation as argument’, in Visualization, Explanation and Reasoning Styles in Mathematics , ed. P. Mancosu, K. F. Jorgensen and S. A. Pedersen.Dordrecht: 123–45. (2006) ‘Documenting a process o abstraction in the mathematics o ancient China’, in Studies in Chinese Language and Culture: Festschri in Honor o , ed. C. Anderl Christoph Harbsmeier on the Occasion o his 60th Birthday and H. Eiring. Oslo: 169–94. Chemla, K., and Keller, A. (2002) ‘Te Sanskrit karanis, and the Chinese mian (side): computations with quadratic irrationals in ancient China and India’, in , ed. Y. DoldFrom China to Paris: 2000 Years o Mathematical ransmission Samplonius, J. W. Dauben, M. Folkerts and B. van Dalen. Stuttgart: 87–132. Cullen, C. (2004) Te Suan shu shu ‘Writings on reckoning’: A ranslation o a Chinese Mathematical Collection o the Second Century , with Explanatory Commentary. Needham Research Institute Working Papers vol. . Cambridge. Dauben, J. W. (2008) ‘. Suan Shu Shu (A Book on Numbers and Computations): English translation with commentary’, Archive or History o Exact Sciences 62: 91–178. Guo Shuchun (1984) ‘Analysis o the concept and uses o lü in Te Nine Chapters on Mathematical Proceduresand Liu Hui’s commentary’, Kejishi Jikan (Journal or the history o science and technology) 11: 21–36. (In Chinese.) (1992) Gudai Shijie Shuxue aidou Liu Hui(Liu Hui, a leading gure o ancient world mathematics). Jinan. (In Chinese.) (1998) Yizhu Jiuzhang suanshu (ranslation and commentary on Te Nine Chapters on Mathematical Procedures). Shenyang. (In Chinese.) Li Jimin (1982a) ‘“Jiuzhang suanshu” zhong de bilü lilun(Ratio theory in Te Nine Chapters on Mathematical Procedures)’, In Jiuzhang Suanshu yu Liu Hui (Te Nine chapters on mathematical procedures and Liu Hui), ed. Wu Wenjun. Beijing: 228–45. (In Chinese.) (1982b) ‘Zhongguo gudai de enshu lilun(Te theory o ractions in ancient China)’, in Jiuzhang Suanshuyu Liu Hui (Te Nine Chapters on Mathematical (In Chinese.) Procedures and Liu Hui), ed. Wu Wenjun. Beijing: 190–209. (1990) Dongang Shuxue DianjiJiuzhang suanshu ji qi Liu Hui Zhu Yanjiu (Research on the Oriental mathematical ClassicTe Nine Chapters on Mathematical Procedures and on its Commentary by Liu Hui). Xi’an. (In Chinese.) (1998) Jiuzhang suanshu daodu yu yizhu (Guidebook and annotated translation o Te Nine Chapters on Mathematical Procedures). Xi’an. (In Chinese.)
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Peng Hao (2001) Zhangjiashan Hanjian «Suanshushu» Zhushi (Commentary on the Book on Bamboo Rods rom the Han dynasty Found at Zhangjiashan: the Book o Mathematical Procedures). Beijing. (In Chinese.) Shen Kangshen (1997) Jiuzhang Suanshu Daodu ( Guidebook or ReadingTe Nine Chapters on Mathematical Procedures). Hankou. (In Chinese.) Shen, K., Crossley, J. N., and Lun, A. W.-C. (1999) Te Nine Chapters on the Mathematical Art: Companion and Commentary. Beijing. Volkov, A. K. (1985) ‘Ob odnom driévniékitaïskom matiématitchiéskom tiérminié’ (‘On an old Chinese mathematical term’), in iézisy Kon ériéntsii Aspirantov i Molodykh Naoutchnykh Sotroudnikov AN SSSR(Workshop o the PhD Students and Young Researchers o the Institute or Oriental Studies o the Academy o Science o the Soviet Union), vol. /1. Summaries: 18–22. Moscow. (In Russian.)
14
Dispelling mathematical doubts: assessing mathematical correctness o algorithms in Bhāskara’s commentary on the mathematical chapter o the Āryabhat. īya
Introduction
Contrary to the perception prevalent at the beginning o the twentieth century, a concern or the mathematical correctness o algorithms existed in the mathematical tradition in Sanskrit. Re ections on the systematic upapattis o Kr.s. n.a’s ( . c. 1600–25) commentary on the Bījagan.ita, the explorations o the Mādhava school (ourteenth–sixteenth century) or other traditions o mathematical validity have already been published. 1 Still, the variations among this tradition o justi cation and explanation need to be studied. . ya o Bhāskara In the ollowing sections, the Āryabhat (BAB) is analysed with regard to its reasoning and.īyabhās vocabulary. Te second chapter o Āryabhat. īya (Ab) – an astronomical siddhānta composed in verse at the end o the fh century – treats mathematics gan ( . ita). Respecting the requirements o the genre, these aphoristicāryas usually provide the gist o a procedure, such as an essential relationship or the main steps o an algorithm. Te BAB is not only the earliest known commentary on this treatise but also the oldest known text o mathematics in Sanskrit prose that has been handed down to us. Te BAB thus gives us a glimpse into the reasonings used in the scholarly mathematical tradition in Sanskrit at the beginning o the seventh century.2 Very little is known about who practised scholarly mathematics in classical India, and why scholarly texts were elaborated. Te BAB provides inormation on the intellectual context in
which both the Ab and the BAB were composed. First, the commentator’s deence o Āryabhat. a’s treatise (and the commentator’s own interpretations
1 2
I would like to thank Karine Chemla and Micah Ross or their attentive and helpul scrutinizing o this article. Ikeyama 2003; Jain 2001; Patte 2004; Srinivas 1990. Some o Kr.s. n. a’s demonstrations are noted, among others, in the ootnotes o C1817. Keller 2006: Introduction.
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o the verses) will provide a backdrop or re ections on the mathematical correctness o procedures. Next, the arguments behind the algorithms o mathematical justi cation will be clari ed. Aferwards, Bhāskara’s vocabulary including explanations, proos and veri cations will be more precisely characterized.
1 Defending the treatise
Bhāskara’s commentary, a prolix prose text, gives us a glimpse into the intellectual world o scholarly astronomers and mathematicians. Te commentary records their intellectual debates. For the opening verse in which the author o the treatise mentions his name, Bhāskara’s commentary explains: . . . as a heroic man on battle elds, whose arms have been copiously lacerated by the strength o vile swords, having entered publicly a battle with enemies, who proclaims the ollowing, as he kills: ‘Tis Yajñadatta here ascended, a descendant o the Aditis, having undaunted courage in battle elds, now strikes. I someone has power, let him strike back!’ In the same way, this master also, who has reached the other side o the ocean o excessive knowledge about Mathematics, ime-reckoning and the Sphere, having entered an assembly o wise men, has declared: ‘Āryabhat.a tells three: Mathematics, ime-reckoning, the Sphere.’3
Within this hostile atmosphere, Bhāskara’s commentary attempts to convince the reader o the coherence and validity o Āryabhat . a’s treatise. o this end, the commentary dispels ‘doubts’ s(andeha) that arise in the explanations o Āryabhat.a’s verses. Tus, the analysis provides reutations (parihāra) to objections and establishes (sādhya, siddha) Bhāskara’s readings o Āryabhat.a’s verse. Tis commentary presents mainly syntactical and grammatical discussions which debate the interpretation o a given word in the treatise. More ofen than not, the discussion o the meaning and use o a word de nes and characterizes the mathematical objects in question. (Are squares all equal sided quadrilaterals? Do all triangles have equally halving 3
. . . yastejasvī purus.ah. samares.u nikr. .st.āsitejovitānacchuritabāhuś śatrusa˙ nghātam prakāśam . praviśya praharan evam āha ‘ayam asāv udito ’ditikulaprasūtah. samares.v anivāritavīryo yajñadattah. praharati / yadi kasyacicchaktih . pratipraharatvi’ti / evam asāo apy ācāryo gan.itakālakriyāgolātiśayajñānodadhipārago vitsabhām avagāhya ‘āryabhat . as trīn . i gadati gan.itam . kālakriyām . golam’ iti uktavān /. (Shukla 1976: 5).
Unless otherwise speci ed, the text ollows the critical edition published in Shukla 1976. I would like to thank . Kusuba, . Hayashi and M. Yano or the help they provided in translating this paragraph, during my stay in Kyoto in 1997.
Algorithms in Bhāskara’s commentary on Āryabhat īya .
heights? etc.)4 Bhāskara’s commentary adopts technical words, and the specialized readings o the verses show that the Ab cannot be understood in a straightorward way. Te verses need interpretation and the interpretation should be the correct one. Te search or the proper interpretation thus de nes the commentator’s task. Te importance o interpretation becomes especially clear when Bhāskara criticizes Prabhākara’s exegesis o the Ab.5 For instance, in his comment on the rule or the computation o sines, Bhāskara explains that the expression samavr. tta reers to a circle, not a disk as Prabhakara understood it.6 More crucially, through his understanding o the wordagra (remainder) as a synonym osa˙nkhyā (number), Bhāskara provides a new interpretation o the rule given in BAB.2.32–33:7 the verse giving the rule or a ‘pulverizer with remainder’ s(āgrakut.t. akāra) can now be read as giving 8 a rule or the ‘pulverizer without remainder’ niragrakut ( . t. akāra). Tis peculiar reading o the word agra is an extreme example o the technical and inventive devices commentators use or their interpretations. Outside the syntactical discussion o a verse, Bhāskara sometimes considers the mathematical content o the procedure directly. Deending Āryabhat. a’s approximation oπ against those o competing schools, he undertakes a reutation (parihāra) o the jaina value o √10 (daśakaran.ī), claiming that the value rests only on tradition and not on proo. In this case also, it is just a tradition (āgama) and not a proo (upapatti) . . . But this also should be established (sādhya).9
Te above statement should not induce a romantic vision o an enlightened Bhāskara using reason to overthrow prejudices transmitted through (religious) traditions. Although here he criticizes the reasoning which cites ‘tradition’ to justiy a rule, in other cases Bhāskara accepts this very argument as evidence o the correctness o a mathematical statement.10 Te question nonetheless is raised: Bhāskara argues that the procedures o the Āryabhat. īya are correct, but how does Bhāskara ‘establish’ a rule? Moreover, what does Bhāskara consider a ‘proo’? Te answer to these questions 4 5 6 7 8
9 10
Keller 2006: Introduction. Keller 2006; BAB.2.11; BAB.2.12. Shukla 1976: 77; Keller 2006: : 57. Shukla 1976: 77; Keller 2006: 132–3. Both rules are mathematically equivalent but do not ollow the same pattern. Furthermore, the second reading also involves omitting the last quarter o verse 33. See Keller 2006: , Appendix on BAB.2.32–3. atrāpi kevala evāgamah . naivopapatih . / . . . cetad api sādyam eva.(Shukla 1976: 72). Keller 2006: Introduction.
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presents difficulties. Indeed, the rationale behind the ragmentary arguments that BAB sets orth is at times hard to grasp. Te aim o this chapter is to show that two speci c commentarial techniques, the ‘reinterpretation’ o procedures and establishing an alternative independent procedure, were used to ground the Ab’s rules. o establish this point, a characterization o these commentarial techniques will be necessary. Tis characterization will be ollowed by a description o the different ways Bhāskara explicitly tries to establish the mathematical validity o Ab’s rules. 2 ‘Reinterpretation’ of procedures
Bhāskara, in an attempt to elucidate Āryabhat .a’s rules, gives interpretations o Āryabhat.a’s verses. He thus makes clear what are the different steps required to carry out a procedure, or the word used to de ne a mathematical object. In certain cases, having put orth such an interpretation, Bhāskara reinvests his understanding o the rule with an additional meaning. Tis is what I call a ‘reinterpretation’. A ‘reinterpretation’ does not invalidate a previous interpretation. It is somehow like the poetic process ośles.a which reads several meanings in the same compound, creating thus a poetic aura. A ‘reinterpretation’ adds a layer oprovides meaning,a new givesmathematical depth to the context interpretation o a rule. A ‘reinterpretation’ or the different steps o a procedure which is not modi ed. Another name or this commentarial technique could be ‘rereading’ a procedure. Te next section describes how an ‘explanation’, a ‘proo’ or a ‘veri cation’ consisted o providing either an alternative independent procedure or a ‘reinterpretation’ o a given procedure via the Rule o Tree or the Pythagorean theorem. In both cases, these arguments would provide a mathematical justi cation or what alone could appear as an arbitrary succession o operations. Beore examining ‘reinterpretations’ o procedures in Bhāskara’s commentary, the expression o the Rule o Tree and the Pythagorean Teorem in BAB must be explained. 2.1 Rule of Tree
Te Rule o Tree (trairāśika11) appears in verse Ab.2.26. Now, when one has multiplied that ruit quantity o the Rule o Tree by the desire quantity| 11
For a general overview on the Rule o Tree in India see Sarma 2002.
Algorithms in Bhāskara’s commentary on Āryabhat īya .
Te quotient o that divided by the measure should be this ruit o desire|| 12
In other words, i M (the measure) produces a ruit FM, and D is a desire or which the ruit FD is sought, the verse may be expressed in modern algebraic notation as: FD
=
FM
×D
M
(1)
Obviously, this expression can also be understood as a statement that the ratios are equal: FD D
=
FM
(2)
M
Te procedure given in the verse provides an order or the different operations to be carried out. First, the desire is multiplied by the ruit. Next, the result is divided by the measure. Tis order o operations causes the procedure to appear as an arbitrary set o operations.13 Bhāskara provides a standard expression to de ne the kind o problem which the Rule o Tree solves. When the commentator thinks that a situation involves proportional quantities and thus the Rule o Tree is (or can be) applied, he brings this act to light by using a verbal ormulation (vāco yukti) o the Rule o Tree. Tis verbal ormulation is a syntactically rigid question which reads as ollows: I the measure produces the ruit, then with the desire what is produced? Te ruit o desire is produced.
Tis question, when it appears, shows that Bhāskara thinks that the Rule o Tree can be applied. I believe that or Bhāskara the Rule o Tree invokes proportionality. 2.2 Te Pythagorean Teorem
Bhāskara, like other medieval Sanskrit mathematicians, does not use the concept o angles. In his trigonometry, Bhāskara uses lengths o arcs. As or right-angled triangles, Bhāskara distinguishes them rom ordinary triangles by giving to each side a speci c name. Whereas scalene, isosceles 12 13
trairāśikaphalarāśim . tam athecchārāśinā hatam . kr. tvā| labdham . pramān.abhajitam . tasmād icchāphalamidam. syāt|| (Shukla 1976: 115–223).
I the division was made rst(resulting in the ‘ruit’ o one measure) and then the multiplication, the computation would have had a step-by-step meaning, but this is not the order adopted by Ab.
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karna ·
koti ·
bhuja¯
Figure 14.1
Names o the sides o a right-angled triangle.
and equilateral triangles have sides (aśra, or all sides), anks (pārśva, a synonym) and sometimes earths (bhū, or the base), right-angled triangles have a ‘base’ (bhujā), an ‘upright side’ (kot. i) and a ‘hypotenuse’ (karn.a), as shown in Figure 14.1. In the rst hal o Ab.2.17, Bhāskara states the Pythagorean Teorem: Tat which precisely is the square o the base and the square o the upright side is the square o the hypotenuse.14
Tereore, in order to indicate that a situation involves a right-angled triangle, Bhāskara gives the names o the sides o a right-angled triangle to the segments concerned by his reasoning. wo examples o Bhāskara’s ‘reinterpretation’ will demonstrate how he employed this theorem. 2.3 ‘Reinterpretation’ with gnomons
Te section devoted to gnomons (śa˙nku) contains two illuminating cases. 2.3.1 A gnomon and a source of light
Te standard situation is as ollows: a gnomon ś(a˙nku, DE) casts a shadow (EC), produced by a source o light (A), as illustrated inFigure 14.2. First, consider the procedure given in Ab.2.15: Te distance between the gnomon and the base, with the gnomon or multiplier, divided by the difference o the gnomon and the base.| 14
yaś caiva bhujāvargah. kot. īvargaś ca karn . avargah. sah. (Shukla 1976: 96).
Algorithms in Bhāskara’s commentary on Āryabhat īya .
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A
D F
B E
Figure 14.2
C
A schematized gnomon and light.
Its computation should be known indeed as the shadow o the gnomon〈measured〉 rom its oot.||15
Tisnotation: procedure involves a multiplication and a division. In modern algebraic EC =
BE × DE AF
Te procedure given in the verse appears to be an arbitrary set o operations. Bhāskara begins with a general gloss. Ten, as in all his verse commentaries, Bhāskara’s commentary provides a list o solved examples. Tese examples have a standard structure: rst comes a versi ed problem, then a ‘setting down’ (nyāsah.) section, and nallya resolution ( karan.a). Tus, in his ‘reinterpretation’ o the above procedure afer a solved example, Bhāskara writes: Tis computation is the Rule o Tree. How? I rom the top o the base which is greater than the gnomon [AF], the size o the space between the gnomon and the base, which is a shadow, [FD = BE] is obtained, then, what is obtained with the gnomon [DE]? Te shadow [EC] is obtained.16 15 16
śa˙nkugun.am . śa˙nkubhujāvivaram . śa˙nkubhujayor viśes . ahr. tam| yal labdam . sā chāyā jñeyā śa˙nkoh. svamūlād hi|| (Shukla 1976: 90). etatkarma trairāśikam/ katham ? sa˙ nkuto ’dhikāyā uparibhujāyā yadi śa˙ nkubhujāntarālapramān . am . chāyā labhyate tadā śa˙nkunā keti chāyā labhyate/(Shukla 1976: 92.)
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Te standard ormulation o the Rule o Tree, applied to the similar triangles AFD and DEC, can be recognized here. Te standard expression o the Rule o Tree provides the proportional elements on which the computation is based. Here the rule indicates that the ratio o AF to FD is equal to the ratio o DE to EC. Te ‘reinterpretation’ o the rule thus gives the arbitrary set o operations a mathematical signi cance. Rather than just a list o operations, the rule in Ab.2.15 becomes a Rule o Tree. 2.3.2 A gnomon in relation to the celestial sphere
In the previous commented verse (BAB.2.14), Bhāskara sets out two procedures. Both rest on the proportionality o the right-angled triangle ormed by the gnomon and its midday shadow with the right-angled triangle composed by the Rsine o the altitude and the zenithal distance. In the present example, one procedure uses only the Rule o Tree, while the other uses the Rule o Tree with the Pythagorean Teorem. Both procedures compute the same results. Consider Figure 14.3. Here, GO represents a gnomon and OC indicates its midday shadow. Te circle o radius OSu (Su symbolizing the sun) represents the celestial meridian. Te radius is thusoequal to the radius o the celestial sphere. S′u designates the OSu projection the sun onto the horizon. Te segment SuS′u illustrates the Rsine o altitude. Bhāskara I notes that the triangle SuS′uO is similar to GOC. Tereore the segment S′uO (that is, the Rsine o the zenithal distance) is proportional to the shadow o the gnomon at noon and the Rsine o the altitude is proportional to the length o the gnomon. Tis proportionality is urther illustrated in Figure 14.4. In modern algebraic notation, SuS′u GO
=
S′uO SuO = OC GC
Te mathematical key to this situation is the relationship between the celestial sphere and the plane occupied by the gnomon, which Bhāskara and Āryabhat. a call ‘one’s own circle’ (svavr. tta). Tis relationship is highlighted here by a set o puns. Tus, the gnomon and the Rsine o the altitude have the same name ( śa˙nku), as do the shadow o the gnomon and the Rsine o zenith distance (chāya). GC is the ‘hal-diameter o one’s own circle’.
Algorithms in Bhāskara’s commentary on Āryabhat īya .
Su
G
′
S u
Figure 14.3
O
C
Proportional astronomical triangles.
17 Bhāskara states this relationship by considering the Rule o Tree:
In order to establish the Rule o Tree – ‘I or the hal-diameter o one’s own circle both the gnomon and the shadow〈are obtained〉, then or the hal-diameter o the 〈celestial〉 sphere, what are the two 〈quantities obtained〉?’ In that way the Rsine o altitude and the Rsine o the zenith distance are obtained.
He also adds:18 Precisely these two [i.e. the Rsine o the sun’s altitude and the Rsine o the sun’s zenith distance] on an equinoctial day are said to be the Rsine o colatitude (avalambaka) and the Rsine o the latitude (aks.ajyā).
Indeed, as illustrated inFigure 14.5, on the equinoxes the sun is on the celestial equator. At noon, the sun occupies the intersection o the celestial equator and the celestial meridian. At that moment, the zenithal distance z equals the latitude o the gnomon φ and the altitude (a) becomes co-latitude (90 − φ˚). Once again, the( )similarity o SuS′uO and OGCthe is
underlined by a certain number o puns. Here, the Rsine o latitude (SuSu ′) is called ‘perpendicular’ (avalambaka). 17 18
trairāśikaprasiddhyartham – yady asya svavr . ttavis. kambhārdhasya ete śan . kuc chāye tadā golavis. kambhārdhasya ke iti śan . kuc chāye labhyete(Shukla 1976: 89). tāv eva vis. uvati avalambakāks . ajye ity ucyete/ (Shukla 1976: 89).
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496 Z
Su
Z R G
C
a
N
S O ′
S u
Figure 14.4
Altitude and zenith.
Now, Bhāskara considers an example or an equinox in which OG = 13, OC = 5 and the radius o the celestial sphere (SuO) is the customary 1348. Bhāskara writes:19 When computing the Rsine o latitude (aks.ajyā) the Rule o Tree is set down: 13, 5, 3438. What is obtained is the Rsine o latitude, 1322.20 Tat is the base (bhujā) the hal-diameter is the hypotenuse (karn. a); the root o the difference o the squares o the base and the hypotenuse is the Rsine o co-latitude ( avalambaka), 3174.21
In this case, Bhāskara uses the act that the triangles are both right and similar. Bhāskara then uses this similarity to compute SuS ′u. Bhāskara 19 20 21
aks. ajyā “nayane trairāśikasthāpanā- 13/ 5/ 3438/ labdham aks . ajyā 1322/ es. ā bhujā, vyāsārdham . karn.ah. , bhujākarn . avargaviśes. amūlam avalambakah . 3174. (Shukla 1976: 90).
Tis is an approximate value. For more on this value, see Keller 2006 : BAB.2.14. Tis value is also an approximation.
Algorithms in Bhāskara’s commentary on Āryabhat īya . Z
P
′
Q = Su
G
ϕ 90 – ϕ
N
S C
O
Su
′
Q
P
Z
Figure 14.5
′
′
Latitude and co-latitude on an equinoctial day.
employs the Pythagorean Teorem to compute OS′u. In order to identiy the right-angled triangle, Bhāskara renames the Rsine o latitude (aksajyā, SuS′u) as the base o a right-angled triangle (bhujā) and he identi es the radius o the celestial sphere as the hypotenuse. Tus, the Rsine is identied with the upright side o a right-angled triangle. Tis identi cation implicitly explains how the computation is carried out. However, Bhāskara immediately adds:22 With the Rule o Tree also 13, 12, 3438; what has been obtained is the Rsine o the colatitude, 3174.23
In this way, Bhāskara again computes OS′u by using the similarity o OSuSu′ and OGC. Bhāskara thus computes the same value twice, using two different methods. Te most likely explanation is that he veri es the results obtained with one algorithm by using another independent process. 22 23
trairāśikenāpi 13/ 12/ 3438/ labdham avalambakah . 3174/ (Shukla 1976: 90).
Tis value is an approximation again.
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Te mathematical key to both these computations is the prior relationship between the gnomon and the celestial sphere. A syntactical connection establishes the relationship between these two spaces. Te invocation o the Rule o Tree begins with a standard question. Te naming o two o its segments identi es a right-angled triangle. Tis identi cation not only indicates one o the mathematical properties underpinning the procedure but also maps the speci c astronomical problem onto a more general and abstract mathematical situation. (Tat is, Rsines o altitudes and zenithal distances become the legs o a simple right-angled triangle.) Since this mathematical interpretation is linked to a set o operations ( rst multiplication and division, then squaring the lengths with subsequent additions or subtractions o the results), the unexplained steps o the procedure are given a mathematical grounding that may serve as a justi cation o the algorithm itsel. Tis analysis thus brings to light two kinds o reasoning: the con rmation o a result by using two independent procedures and the mathematical grounding o a set o operations via their ‘reinterpretation’ according to the Rule o Tree and/or the Pythagorean Teorem. Tese kinds o mathematical reasoning are also ound in the parts o BAB which explicitly have a persuasive aim, attempting to convince the reader that the algorithms o the Ab are correct. 3 Explanations, veri cations and proofs
Bhāskara uses speci c names when reerring to a number o arguments: ‘explanations’, ‘proos’ (upapatti) and ‘veri cations’ (pratyāyakaran.a). Tese arguments do not appear systematically in each verse commentary and – as will be seen below – are always ragmentary. Te ollowing description o explanations, proos and veri cations will attempt to highlight how they are structured and the different interpretations they can be subject to. 3.1 Explanations
Bhāskara’s commentary on verse 8 o the mathematical chapter o the Ab presents an example o explanation. Verse 8 describes two computations concerning a trapezoid (see Figure 14.6). Te rst calculation evaluates the length o two segments (svapātalekha, EF and FG) o the height o a trapezoid. In this case, the height is bisected at the point o intersection
Algorithms in Bhāskara’s commentary on Āryabhat īya . A
E
B
F
C
H
Figure 14.6
G
I
D
Inner segments and elds in a trapezoid.
or the diagonals. Te procedure is made o a multiplication ollowed by a division:24 Ab.2.8. Te two sides, multiplied by the height 〈and〉 divided by their sum, are the ‘two lines on their own allings’.| When the height is multiplied by hal the sum o both widths, one will know the area.||
In other words, with the labels used in Figure 14.6, we have: AB × EG ; AB + CD CD × EG FG = . AB + CD EF =
Likewise, the area is: (AB + CD) 2 On the rst part o the verse, Bhāskara comments:25 A = EG ×
Te size o the ‘lines on their own allings’ should be explained pratipādayitavya ( ) with a computation o the Rule o Tree on a gure drawn by〈a person〉 properly instructed. Ten, by means o just the Rule o Tree with both sides, a computation o〈the lines whose top is〉 the intersection o the diagonals and a perpendicular〈is perormed〉.
Tis explanation consists o ‘reinterpreting’ the procedure – which is a multiplication ollowed by a division – according to the Rule o Tree. Te 24 25
āyāmagun.e pārśve tadyogahr . te svapātalekhe te| vistarayogārdhagun . e jñeyam . ks. etraphalam āyāme|| (Shukla 1976: 63). samyagādis.t.ena (rather than samyaganādis.t.ena o the printed edition) ālikhite ks.etre svapātalekhāpraman . am . trairāśikagan . itena pratipādayitavyam/ tathā trairāśikenaivobhaya pārśve karn.āvalambakasampātānayanam/(Shukla 1976: 63).
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explanation contains two steps. Te rst step considers the proportionality in a diagram, then ‘reinterprets’ the set o operations o the algorithm as the application o the Rule o Tree. As previously, the seemingly arbitrary set o operations is endowed with a mathematical meaning. Te second computation in verse 8 determines the area o the trapezoid. As shown in Figure 14.6, the area o the trapezoid can be broken into the summation o the areas o several triangles. Alternately, the trapezoid can be decomposed into a rectangle and two triangles. Although no gure is explicitly drawn, Bhāskara seems to have such a diagram in mind. Indeed, he seems to reer to such a drawing when he writes:26 Here, with a previous rule [Ab.2.6.ab] the area o isosceles and uneven trilaterals should be shown/explained (darśayitavya). Or, with a rule which will be stated [Ab.2.9] the computation o the area o the inner rectangular eld 〈should be perormed〉;
Even though it has not survived, such a gure shows how areas can be added to give the area o the trapezoid. Tis time, a collection o already known procedures, those computing the area o triangles and rectangles, is mobilized. We do not know i they are used to ‘reinterpret’ the procedure or to establish an alternative independent procedure. Te procedures o Ab.2.9 will be analysed below. Both o Bhāskara’s explanations in BAB.2.8 consist o: (1) an explanation o a diagram, and (2) either a ‘re-interpretation’o the procedure or exposing anindependent alternative procedure. Tis ‘re-interpretation’ either con rms or veries the reasoning by looking at a diagram. Tree words reer to an explanation: vyākhyāna, pradarśan.a and pratipādita. Te word vyākhyāna indicates that the commentary gives an explanation, but it is also used or an argument connected with a diagram:27 Or else, all the procedures 〈used〉 in the production o chords are in the realm o a diagram, and a diagram is intelligible 〈only〉 with an explanation (vyakhyāna). Tereore it has not28 been put orth (pratipādita) 〈by Āryabhat. a in theĀryabhat. īya〉. 26 27 28
pūrvasūtren.ātra dvisamavis . amatryaśraks . etraphalam . darśayitavyam/ vaks.yamān.asūtre-sn . āntarāyatacaturaśraks . etraphalānayanam (. . .) vā/(Shukla 1976: 63). athavā jyotpattau yatkaran . am . tatsarvam . chedyakavis.ayam . , chedyakam . ca vyākhyānagamyamiti [na] pratipāditam/(Shukla 1976: 78). na has been added by the editor, K. S. Shukla, and is not ound in the manuscripts. Another
possible interpretation o the sentence reads: ‘Tereore it has been put orth’ 〈by Bhāskara in his commentary〉.
Algorithms in Bhāskara’s commentary on Āryabhat īya .
Note that this passage emphasizes that explanations belong to the genre o commentary and, at least according to Bhāskara, should not be exposed in a treatise. Te word pradarśan.a is derived rom the verbal root dr. s.-, ‘to see’. It has a similar range o meaning as the English verb ‘to show’. It is ofen hard to distinguish i the word reers to the visual part o an explanation or to the entirety o the explanation. For instance, in BAB.2.11, Bhāskara uses a diagram and writes:29 In the eld drawn in this way all is to be shown/explained (pradarśayitavya).
Finally, the word pratipādita is more technical and straightorward. It commonly appears in lists o solved examples ound in most o the commented verses in the mathematical part o BAB. In the illustrations o explanations presented above, the commentator ‘reinterprets’ geometrical procedures according to the Rule o Tree or the Pythagorean Teorem. Only geometrical procedures receive such arguments. Each time, the commentary omits a diagram to which the text seemingly reers. Among the geometrical processes, explanations are ‘seen’, as will be seen in the only example rom the BAB in which the word ‘proo’ occurs. 3.2 Te only two occurrences of the word ‘proof ’
Te Sanskrit word upapatti reers directly to a logical argument. Tis word 30 Te is used twice in Bhāskara’s commentary, as noted by akao Hayashi. gender o this word is eminine and it is derived rom the verbal root upaPAD-, meaning ‘to reach’. Tus, anupapatti is literally ‘what is reached’ and has consequently been translated as ‘proo’. In both instances, some ambiguity surrounds this word, and the meaning o the word is not certain. One occurrence has been quoted above, wherein proos (upapatti) are described as opposed to tradition. Te other instance reers to the reasoning whereby the height o a regular tetrahedron is determined rom its sides. In this case, Bhāskara understands Āryabhat . a’s rule in the second hal o verse 6 o the mathematical chapter as the computation o the volume o a regular tetrahedron. Such a situation is described in Figure 14.7. Given a regular tetrahedron ABCD, AH is the line through A perpendicular to the plane de ned by the triangle BDC. AH is called the ‘upward side’ (ūrdhvabhujā). AC is called karn.a (literally, ‘ear’) because it is the 29 30
evam ālikhite ks . etre sarvam . pradarśayitavyam (Shukla 1976: 79).
H1995: 75–6.
501
502 A
B
C
H
D
Figure 14.7
An equilateral pyramid with a triangular base.
hypotenuse o AHC. Bhāskara explains how to compute the upright side by using the Pythagorean Teorem and the Rule o Tree. Te determination o CH, rom which the upright side AH may be computed, rests upon the proportional properties o similar triangles, illustrated in Figure 14.8. Te triangles BB′C and B′CH are similar: BB′ : CB = CB′ : CH. From this relationship it is known that: ′
CH CBBBCB . Bhāskara expressed this relationship as the Rule o Tree. Te text does not give a precise argument, but it alludes to the properties as being clear rom a diagram. It is in this context that the wordupapatti appears:31 ′
31
trairāśikopapattipradarśanārtham . ks. etranyāsah . – (Shukla 1976: 59).
Algorithms in Bhāskara’s commentary on Āryabhat īya . C
B′
H
C′
B
D
Figure 14.8
Te proportional properties o similar triangles.
In order to show the proo (upapatti) o 〈that〉 Rule o Tree, a eldsi set down.
Te argument implied by this word depends on the diagram. As in the case o the explanations, the proo must have been presented orally. Tis situation differs rom the acts o ‘reinterpretation’ seen above. In the present case, an argument is created, and there is no pre-existing algorithm to ‘reinterpret’. However, the oundations o this new argument are set out in a diagram. Furthermore, the procedure used is the Rule o Tree, as in the ‘explanations’ seen above. Another type o argument concerns the correctness o algorithms: veri cation. 3.3 Veri cation
Veri cations are distinguished rom explanations and proos by their name, pratyayakaran.a. Indeed, pratyaya has an etymological root in a verb meaning ‘to come back’, which has connotations o conviction. Pratyayakaran . a thus means ‘enabling to come back’ or ‘producing conviction’. Historians o Indian mathematics usually understand this word as a type o veri cation and translate it accordingly.32 A veri cation resembles an explanation in that a veri cation ‘reinterprets’ a given procedure according to another rule and establishes a mathematical grounding. Te arguments that the commentator labels ‘veri cations’ sometimes present difficulties, and currently our understanding o them is not at all certain. Below are set out several hypotheses about how these veri cations can be understood. 32
H1995: 73–4.
503
504
3.3.1 Veri cation of an arithmetical computation
Bhāskara states a veri cation by the Rule o Five or the rule given in Ab.2.25. Āryabhat. a states the rule in Ab.2.25 as ollows:33 Te interest on the capital, together with the interest 〈on the interest〉, with the time and capital or multiplier, increased by the square o hal the capital| Te square root o that, decreased by hal the capital and divided by the time, is the interest on one’s own capital||
Tis passage can be ormalized as ollows. Let m (mūla) be capital; let p1 (phala) be the interest on m during a unit o time, k1 = 1 (kāla), usually a month. Let p2 be the interest on p1 at the same rate or a period o time k2. I p1 + p2, m, and k2 are known, the rule can be expressed in modern mathematical notation as:
mk2 ( p1 + p2 ) + p1 =
( m2 )
k2
2 −
m
2
.
Tis rule is derived rom a constant ratio: p1
m p1
=
2 p2 k .
Te Rule o Five, described in BAB.2.26–27.ab, rests on the same ratio as the rule given in Ab.2.25. In the ormer instance though, k1 may be a number other than 1: m p1
k1
=
p1 p2
k2 .
Te Rule o Five indicates an expression equal in value top2: 2
p2
=
p1 k2 mk1
Te Rule o Five may thereore be used in the opposite direction to nd a value or p1. Bhāskara gives an example:34 In BAB.2.25 33 34
mūlaphalam . saphalam . kālamūlagun . am ardhamūlakr . tiyuktam| tanmūlam . mūlārdhonam . kālahr. tam . svamūlaphalam|| (Shukla 1976: 114). jānāmi śatasya phalam . na ca kintu śatasya yatphalam . saphalam | māsaiś caturbhir āptam . s.ad. vada vr. ddhim . śatasya māsotthām||(Shukla 1976: 114).
Algorithms in Bhāskara’s commentary on Āryabhat īya .
1. I do not know the 〈monthly-〉 interest on a hundred. However, the〈monthly-〉 〈interest on a hundred increased by the interest〉| Obtained in our months is six. State the interest o a hundred produced within a month||
Tis example states a case in which: m = 100 k2 = 4 p1 + p2 = 6
By the procedure given in Ab.2.25, the value op1 is 5. Bhāskara then adds:35 Veri cation (pratyayakaran.a) with the Rule o Five: ‘I the monthly interest (vr. ddhi)36 on a hundred is ve, then what is the interest o the interest [o value (dhana)- ve] on a hundred, inour months?’ 1 4 Setting down: 100 5 Te result is one. Tis increased by the 5 0 〈monthly〉 interest on the capital is six rūpas, 6.
Simply stated, the veri cation consists o knowingm, p1 and k2, nding pp2 + and con rming that its value increased byp1 will give the same value or 1 p2 as stated in the problem. Te Rule o Five, as seen above, returns the value op2. Tis procedure
does not deliver the same result but gives a method o inverting the procedure to check independently that the result makes sense. In this case, an independent procedure is established. Te use o the Rule o Five, which Bhāskara describes as a combination o two Rules o Tree, also imbues the computation with a mathematical basis in proportionality. 3.3.2 Veri cation of the area of plane gures
Bhāskara interprets the rst hal o Ab.2.9 as a way to veriy procedures or areas given by Āryabhat.a in the previous verses. 35
36
pratyayakaran. am pañcarāśikena-yadi śatasya māsikī .vrddhin pañca tadā caturbhir māsaih . 1 4 śatavr. ddheh. [pañcadhanasya] kā vr. ddhih. iti/ nyāsar - 100 5 labdham . 1 / etatsahitā śatavr. ddhih. s.ad. rūpān.i 6/ (Shukla 1976: 114–15). 5 0
From now on, unless otherwise stated this is the word translated as ‘interest’.
505
506 For all elds, when one hasacquired the two sides, the area is their product |
37
Bhāskara endows the verse with the goal o ‘veri cation’ – agoal nowhere explicitly appearing in the verse itsel. wo steps can be distinguished in the veri cations o this verse commentary, each corresponding to a diagram. Te rst step constructs a diagram o the gure or which an area is veried. Te length and width o a rectangle with the same area as the gure are identi ed. Tis ‘length’ and ‘width’ are usually values rom Āryabhat . a’s procedure or which veri cation is sought. For instance, to veriy the area o a triangle, the length o the corresponding rectangle is identi ed as the height o the triangle, while the width o the rectangle is hal the base o the triangle. Precisely, the area o a triangle is given elsewhere by Ab (in the rst hal o verse 6) as the product o hal the base by the height o a triangle. Te second step o the argument presents a diagram o the rectangle and computes the multiplication. How should this argument be understood? According to one means o understanding, this argument is a ormal interpretation. Te reasoning would consist o considering the rule one seeks to veriy as the multiplication o two quantities. Each quantity is then interpreted geometrically as either the length or width o a rectangle with the same area as the initial gure. In this way, Bhāskara calculates the length and height o the rectangle, as required by verse 9. Another way o understanding the argument begins with the act that the veri cation or a given gure produces a rectangle o the same area as the given gure. Te act that all gures have a rectangle with the same area would then become an implicit assumption o Sanskrit plane geom38 Te etry. akao Hayashi has interpreted this argument in such a manner. reasoning would produce a rectangle and veriy that its area is equal to the area o the gure. A third approach relies on the ‘setting down’ parts which contain diagrams. Such a veri cation consists o constructing a rectangle with the same area rom a given gure. For instance, in the second step o the veri cation o the area o a triangle, Bhāskara speci es that when the parts o the yasta), they area a triangle are rearranged produce the rectangle whichoissuch drawn. Te construction o av(rectangle rom the srcinal gure is not described in Bhāskara’s commentary. However, such constructions could have been known, as shown by the methods exposed in BAB.2.13. Furthermore, this process recalls the algorithms rom the śulbasūtras, the 37 38
sarves.ām. ks.etrān.ām. prasādhya pārśvephalam. tadabhyāsah . | (Shukla 1976: 66).
H1995: 73.
Algorithms in Bhāskara’s commentary on Āryabhat īya .
earliest known texts o Sanskrit geometry. Tese algorithms produce a construction which, although not described in the text, corresponds with the discussion contained in the text. With just such a diagram, the argument in the text would arithmetically veriy that the construction is correct. Tese three interpretations can be combined i a veri cation is allowed to be simultaneously geometrical and arithmetical. Bhāskara relies on a geometrical strategy to produce a rectangle with the appropriate area, showing that he knows how to construct the corresponding rectangle rom the initial gure. Because the construction is obvious, it would not be detailed, and only the lengths o the rectangle would be given. From an arithmetical perspective, this ‘reinterpretation’ provides a new understanding o the rule given by Āryabhat. a. Trough his arithmetical ‘veri cation’, Bhāskara explains the geometrical veri cation. Bhāskara explains the link between the sides o the initial gure and the lengths and widths o the rectangle with the same area as the initial gure. Regardless o which interpretation is accepted, the veri cation either ‘reinterprets’ a rst algorithm (BAB.2.9) and produces a new understanding o the procedure, or it produces a new procedure that gives the same result (BAB.2.25). In either case, the so-called ‘veri cation’ con rms the numerical results and places the procedure in a secure mathematical context. Tus, afer steps.veri cation, the calculations do not appear to be a set o arbitrary Conclusion
Tis survey o the BAB has brought to light two kinds o reasonings checking the Ab rules and seeking to convince readers o their validity. One argument exhibits an independent alternative procedure. In one case the procedure exhibited arrives at the same result as the opposite direction procedure. Te second type o reasoning, which we have called ‘reinterpretation’, uses the Rule o Tree and the Pythagorean Teorem to provide a new outlook onto the arbitrary steps o the procedure. How should the Rule o Tree and the so-called Pythagorean Teorem be described in this context? Tey are mathematical tools which enable astronomical situations or speci c problems to be ‘reinterpreted’ as abstract and general cases, involving right-angled triangles and proportionalities. Te arbitrary steps o the procedure are thus given a mathematical explanation. Nonetheless, the methods o reasoning are hard to understand and pin down. Tis difficulty may arise rom their oral nature, o which Bhāskara’s
507
508
written text preserves only a portion. For instance, the unction o diagrams in these reasonings still remains mysterious. Further detailed explorations o how Sanskrit texts explain, proveand veriy mathematical algorithms will advance understanding about how the mathematical correctness o algorithms was conceptualized by mathematicians in the Indian subcontinent.
Bibliography Ikeyama, S. (2003) ‘Brāhmasphut.asiddhānta (ch. 21) o Brahmagupta with Commentary o Pr.thudaka, critical edition with English translation and notes’, Indian Journal of History of Science38: S1–308. Jain, P. K. (2001) Te Sūryaprakāśa of Sūryadasa: A Commentary on Bhāskarācarya’s Bijagan.ita, vol. : A Critical Edition, English ranslation and Commentary for the Chapters Upodghata, Sadvidhaprakaran . a and Kuttakadhikara. Vadaroda. Keller, A. (2006) Expounding the Mathematical Seed: Bhāskara and the Mathematical Chapter of the Āryabhat.īya, 2 vols. Basel. Sarma, S. R. (2002) ‘Rule o Tree and its variation in India’, in From China to Paris: 2000 years of ransmission of Mathematical Ideas , ed. Y. Dold-
Samplonius, J. W. Dauben, M. Folkerts and B. Van Dalen. Stuttgart: 133–56. Shukla, K. S. (1976) Āryabhat. īya of Āryabhat. a, with the Commentary of Bhāskara I and Someśvara. Delhi. Srinivas, M. D. (1990) ‘Te methodology o Indian mathematics and its contemporary relevance’, in History of Science and echnology in India: Mathematics and Astronomy, vol. , ed. G. Kuppuram and K. Kumudamani. Delhi: 29–86.
15
Argumentation or state examinations: demonstration in traditional Chinese and Vietnamese mathematics
Introduction Recently a number o authors have argued, once again, that a historical study o mathematical texts conducted without taking into consideration the circumstances o their production and use could be undamentally awed. For instance, E. Robson claimed that a large number o cuneiorm Babylonian mathematical tablets were produced in the process o mathematical instruction, either by students or instructors, and thereore their interpretation as ‘purely mathematical texts’ would be inadequate. 1 Robson’s taking into consideration the educational unction o the cuneiorm mathematical tablets provided additional arguments in support o a somewhat unorthodox interpretation o the mathematical tablet Plimpton 322, hitherto believed to be one o the best-studied Babylonian mathematical texts (this interpretation was srcinally suggested by Bruins in 1940s and 1950s and reiterated by other authors in the early 1980s). In conventional historiography o Chinese mathematics the mathematical treatises compiled prior to the end o the rst millennium were ofen tacitly assumed to be mathematical texts per se rather than mathematical textbooks; this assumption to a large extent shaped the approaches to their interpretation. Te characteristic eatures o textbooks (i.e. texts composed as collections o problems ofen containing groups o generic problems and detailed descriptions o elementary arithmetical operations without explanations or justi cations o the provided algorithms) were not allotted much attention; instead, historians ofen ocused on singular ‘mathematically signi cant’ methods and results (such as the calculation o the value o π and the algorithm or solution o simultaneous linear equations, or instance) thus reinorcing the image o the received Chinese mathematical treatises as ‘research monographs’ rather than ‘textbooks’. However, even in modern mathematics a research paper can be used as teaching material, and, conversely, a mathematical statement rom a 1
Robson 2001: 171.
509
510
textbook can become the starting point o a proessional mathematical inquiry. Similarly, it well may be possible that in a given mathematical tradition there was no wall separating texts o the two types rom each other, and a special investigation o the social circumstances o the use o given mathematical texts has to be provided each time in order to avoid historiographic distortions. Unortunately, even the most outstanding modern historians have ofen presented Chinese mathematical treatises as i they were research this approach to Chinese mathematical is ound alreadymonographs; in Mikami (1913) and certainly in Yushkevich (1955)texts and Needham (1959), not to mention their numerous Chinese counterparts. An attempt to classiy the mathematical problems ound in Chinese treatises was recently made by Martzloff, 2 yet his classi cation apparently re ected the seeming heterogeneity o Chinese mathematical treatises as perceived by modern historians solely on the basis o the contents o individual problems rather than the way in which mathematical treatises containing them were actually read and used in traditional China. Presumably, there may have existed social settings in which one and the same problem was treated as belonging to different categories. It can be demonstrated that the majority o the extant treatises o the late rst millennium to the rst millennium were used as mathematical textbooks in state educational institutions or several centuries,3 unlike the mathematical treatises o the Song (960–1279), Yuan (1279–1368) and Ming (1368–1644) dynasties o which the circumstances o use are ofen unknown. Unortunately, all the attempts to offer a plausible reconstruction o the unctioning o these texts in educational context have been thwarted by the lack o data concerning mathematics instruction in traditional China in the late rst to early second millennium , and, in particular, by the lack o the srcinal examination papers. o circumvent this difficulty, in what ollows I will use 2
3
Martzloff 1997: 54 suggests that the mathematical problems in Chinese treatises belonged to the our ollowing categories: (1) ‘real problems’ (applicable in real-lie situations); (2) ‘pseudo-real problems’ (‘neither plausible nor directly usable’); (3) ‘recreational problems’; (4) ‘speculative or purely mathematical problems’. Only problems o category (2) thus may have been used in mathematical instruction, while problems o type (4) represented ‘pure mathematics’. Martzloff himsel 1( 997: 58) played down the applicability o his classi cation when stating that the problems o category (1) also belonged to category (4). Te circumstances o the use o the recently unearthed mathematical treatise Suan shu shu (Writing on computations with counting rods) as well as the mathematical treatises and ragments ound in Dunhuang caves remain unknown. Here and below I use the pinyin transliteration o the Chinese characters which nowadays has become a de facto standard in continental European sinology. I use my own translations o the titles o Chinese mathematical treatises; or the reader who may be conused by these translations I provide a list o them in Appendix II together with the translations o the titles as ound in Martzloff 1997.
Demonstration in Chinese and Vietnamese mathematics
a ‘model examination paper’ ound in a nineteenth-century Vietnamese mathematical treatise that turned out to be instrumental in reconstructing the role played by the commentaries on mathematical texts in the context o institutionalised mathematical instruction in traditional China and Vietnam.
Mathematics education in traditional China In Western historiography the part played by Chinese mathematics education arguably remains underestimated, probably due to a particular stand adopted by the nineteenth-century European authors and perpetuated in the publications o in uential historians o the twentieth century. A highly negative (as much as inaccurate) evaluation o mathematics education in traditional China was offered by the French sinologist Édouard Biot (1803–50) who presented mathematics education in the Mathematical College (Suan xue ) as ollows:4 ‘. . . to call it a “mathematics school” would mean to praise too high the studies in this elementary [educational] institution’.5 In this chapter I will not investigate reasons or this surprisingly low evaluation o the mathematical education in China – to do so, one probably would need to study the history o the image o China in Europe, in particular in France, created by various individuals and institutions beginning with the Jesuits.6 Certainly, at the time when Biot was writing his lines, not much was known about the history o Chinese mathematics; Biot himsel never systematically worked on Chinese mathematics and had only a partial access to the srcinal texts. 7 It is interesting to note that Biot (mistakenly) believed that the Jiu zhang suan shu (Computational 4
5
6 7
In this chapter I use both the characters suan and suan even though in modern editions o historical materials the ormer is ofen changed to the latter, since their srcinal meaning, as the dictionary Shuo wen jie zi by Xu Shen (55?–149? ) speci es, was not the same: the character suan meant the counting rods, and suan , the operations perormed with the instrument. In this chapter I use suan i it occurred in a title o a book or in a name o an institution at least once in an edition o the quoted source. ‘. . . le nom d’école des mathématiques donnerait une trop haute idée des études de cet établissement élémentaire. . .’ (Biot 1847:257, n. 1). In this chapter the translations rom French and Chinese are mine, unless stated otherwise. Biot 1847: v–ix. Biot was amiliar with three o the twelve books used or mathematics instruction in seventh-century China, namely, with the mathematical treatises Qi gu suan jing (Computational treatise on the continuation o [traditions o ] ancient [mathematicians]) and Sun zi suan jing (Computational treatise o Master Sun), as well as the astronomical treatise Zhou bi suan jing (Computational treatise on the gnomon o Zhou [dynasty]). He was unable to identiy correctly the titles o the other treatises (p. 261), and the
511
512
procedures o nine categories) compiled no later than the rst century contained the Pascal triangle (reerred to by Biot as ‘binomial expansions up to the sixth degree’,8 which could hardly be seen as ‘elementary’, and yet argued or the ineriority o the Chinese mathematical treatises. Te ollowing phrase o Biot seemingly explains his reasons: ‘[Te treatises] are collections o problems, the most part o them elementary, with the solutions given without demonstrations’.9 Te word ‘demonstrations’ might make one think that Biot meant a comparison with the European textbooks o his time written in ‘Euclidean’ style, as lists o theorems accompanied by proos. Tis conjecture, however, lacks any supporting evidence; on the contrary, an anti-Euclidean trend was rather powerul among French educators at the moment when Biot was writing his lines, as the ollowing quotation shows: Whoever wishing rom now on to put geometry within the reach o mind and to teach it in a rational way should, I think, present it as we just have seen it [above] and remove all that is no more than just a vague expression and pure hassle. Tis bothering equipment o de nitions, principles, axioms, theorems, lemmas, scholia, corollaries, should be completely eliminated, as well as all other utile particularities [o the same kind], the only effect o which is that they put too heavy a burden on the [human] spirit and make it tired in its progress.10
Moreover, a cursory analysis o the contemporaneous French arithmetical textbooks suggests that by ‘demonstrations’ Biot most likely meant step-by-step explanations o numerical solutions ound in a large number o French textbooks published by the mid nineteenth century, and not
8 9
10
way he approached the documents transpires rom his remark on the Zhou bi suan jing: ‘Te Zhou bi, which has in China an immense reputation, presents several exact notions concerning the movement o the sun and the moon surrounded by strange absurdities’ (Le cheou-pei, qui a une réputation immense en Chine, présente, au milieu d’étranges absurdités, quelques notions exactes sur les mouvements du soleil et de la lune) (p. 262). Moreover, Biot did not have access to the Jiu zhang suan shu (Computational procedures o nine categories), the cornerstone o the mathematical curriculum, and made his judgement solely on the basis o the Suan fa tong zong (Summarized undamentals o computational methods, 1592) by Cheng Dawei the contents o which he believed to be identical with that o the Jiu zhang suan shu (ibid.). Biot 1847: 262. ‘[Les ouvrages] sont des collections de questions qui sont, pour la plupart, élémentaires, et dont la solution est donnée sans démonstration’ (Biot 1847: 262). ‘Quiconque voudra désormais mettre la géométrie à la portée des intelligences et l’enseigner d’une manière rationnelle, devra, je crois, la présenter telle que nous venons de la voir et en écarter tout ce qui n’est que vague expression et pure en ure. Cet attirail embarrassant de dé nitions, de principes, d’axiomes, de théorèmes, de lemmes, de scolies, de corollaires, doit être mis complètement de côté, ainsi que les autres distinctions utiles qui n’ont d’autre effet que de surcharger l’esprit et de le atiguer dans sa marche’ (Bailly 1857: 11–12).
Demonstration in Chinese and Vietnamese mathematics
deductions perormed in an axiomatic system.11 Te statement o Biot as well as his reasons to claim the ineriority o Chinese textbooks certainly deserve a urther investigation which, unortunately, would lead us ar beyond the scope o the present chapter. A detailed description o mathematics instruction (once again, in the ramework o a general outline o the state education in China o the ang dynasty) was offered almost a century later by Robert des Rotours (1891– 1980) who, unlike Biot, avoided any critical remarks concerning the contents and the level o the mathematical instruction in China. 12 Te critique o Chinese mathematics education was back in 1959 when Needham energetically accused Ming Conucian scholarship o ‘con n[ing] mathematics to the back rooms o provincial yamens’ and the ‘deadening in uence’ o the examination system.13 Yet his accusations missed the target, since the Song dynasty (960–1279) algebra he praised in the same paragraph had vanished some sixty years prior to the beginning o the Ming dynasty (1368–1644) and thus certainly well beore the introduction o the examination system eaturing the ormalized way o writing examination papers known as ‘eight-legged essays’ he reerred to.14 Chinese mathematics education was once again judged unsatisactory by U. Libbrecht and J.-C. Martzloff.15 In turn, M.-K. Siu and A. Volkov brie y addressed the critique o the latter authors, yet a ull analysis o the role o the state 16mathematics education in traditional China remains a challenging task. In this chapter I will not discuss general issues such as whether the state examinations system impeded or boosted the development o mathematics in China, 17 but shall ocus instead on the changes in the interpretation and understanding o mathematical treatises which might have happened as the result o their embedding into the curriculum o the state educational institutions in the seventh century .
11
12 13 14 15 16 17
See, or instance, P.-N. Collin, Manuel d’arithmétique démontrée. . ., Paris, 1828 (7th edn), which, as its very title suggests, was supposed to provide ‘demonstrations’. Te ormat o this textbook is similar to that o a large number o contemporaneous French textbooks, such as the anonymous Abrégé d’arithmétique, à l’usage des écoles chrétiennes (Rouen, 1810), Abrégé d’arithmétique à l’usage des écoles primaires(Paris, 1850), Abrégé d’arithmétique décimale. . . (Perpignan, 1855, actual printing 1856), among many others. Des Rotours 1932. Needham 1959: 153–4; esp. see n. f on p. 153. Lee 2000: 143–4. Libbrecht 1973: 5; Martzloff 1997: 79–82. Siu and Volkov 1999. See, among others, interesting observations o Wong 2004 on the role o the ‘Conucian’ context in modern mathematics education.
513
514
Chinese mathematical instruction of the rst millennium It remains unclear when and where mathematical subjects were introduced into the curriculum o Chinese state educational institutions. Sun claims that the Mathematical College (Suan xue ) was established during the Northern Zhou dynasty (557–81) in the capital o this state, Chang’an (modern Xi’an);18 the students o the College were called suan fa sheng , literally, ‘students o computational methods’. Lee reports that he was unable to nd any evidence con rming that the Mathematical College was indeed established under the Northern Wei dynasty (386–534), as Sun suggested.19 However, Lee agrees that the subject had been taught officially in the North or a long time even beore the Northern Wei, in particular by official historians, who excelled in calendar calculation. Te system o state mathematics education established by the early seventh century in China united under the rule o the Sui (581–618) and ang (618–907) dynasties comprised two elements: (1) the state mathematics examinations held on a regular basis, and (2) the Mathematical College operating under the control o the governmental agency called ‘Supervisorate o National Youth’Guo ( zi jian );20 the latter was metaphorically reerred to by some modern authors as the ‘State University’. In Song dynasty China the Mathematical College returned under the authority o the Supervisorate o National Youth or a relatively short period o time, 1104–1131;21 the College unctioned beore and afer this period o time under the auspices o other governmental agencies.22 Tis explains why ten out o twelve mathematical treatises used as textbooks during the ang dynasty (618–907) were re-edited and reprinted with educational purposes in 1084 and 1200–1213. Mathematical courses also constituted a part o the curricula o the uture astronomers and calendar experts instructed at the courts o the non-Chinese Jin dynasty (1115–1234), and, later, Yuan (1271–1368).23 Tere exist several descriptions o the instruction in the Mathematical College (Suan xue ) during the ang dynasty; the descriptions speciy the number o students, a list o the textbooks, the periods o time allotted to the study o each book, as well as other details. 24 Te textbooks and the 18 19 20
21 22 23 24
Sun 2000: 138. Lee 2000: 515, n. 230. Rendered ‘Directorate oEducation’ by Hucker 1985: 299 and ‘Directorate o National Youth’ by Lee 2000. Li 1977: 271–9; Lee 2000: 519–20. Hucker 1985: 461. Lee 2000: 520–3. Te descriptions are ound in the ang liu dian (Te six codes o the ang [dynasty]), compiled in 738, see LD 21: 10b and in the Xin ang shu (Te New History o the
Demonstration in Chinese and Vietnamese mathematics able 15.1. Te mathematical curriculum o the ang State University Number
itle
1
Sun zi (Master Sun)
2 3
Wu cao (Five departments) Jiu zhang (Nine categories)
4 5
Hai(Sea dao island) Zhang Qiujian ( [Master] Zhang Qiujian) Xiahou Yang ([Master] Xiahou Yang) Zhou bi (Te gnomon o the Zhou [dynasty]) Wu jing suan (Computations in the ve classical books) Zhui shu (Mending procedures)b Qi gu (Continuation [o traditions] o ancient [mathematicians])
6 7 8
9 10
11 12
Ji yi (Records o [things] lef behind or posterity) San deng shu (Numbers o three ranks)
Duration o study
Programmea
One year or two Regular treatises together Regular Tree years or two Regular treatises together One year
Regular
Regular
One year
Regular
One year or two treatises together
Regular Regular
Four years
Advanced
Tree years
Advanced
Not speci ed
Supplementary
Not speci ed
Supplementary
Notes: a Te terms ‘regular’ and ‘advanced’ are not ound in the srcinal descriptions; they are added or the convenience o the reader. For the explanation o these terms, see below. b Te meaning o the title remains unclear; see Yan 2000 : 125–32.
duration o their study as speci ed in the Xin ang shu (Te New History o the ang [dynasty]) are listed in able 15.1. Te order o the books in able 15.1 is that adopted in the Xin ang shu; it remains unclear why the list begins with the treatises Sun zi and Wu cao, certainly less important than the treatises under numbers 3, 7 and 9, as suggested by an inspection o their extant versions listed in able 15.2 ang [dynasty]), compiled in 1060, see XS 44: 2a. Te lists o the books and the duration o their study speci ed in these two sources are identical. For a translation o the description ound in the Xin ang shu, see des Rotours 1932: 139–42, 154–5; see also Siu 1995: 226; Siu and Volkov 1999.
515
516 able 15.2. Conventional identi cation o the ang dynasty textbooks
with the extant mathematical treatises reatises listed in Number the Xin ang shu 1
Sun zi (Master Sun)
Identi ed as the ollowing extant treatises
Author
Sun zi suan jing
Unknowna
Date o compilation C. 400 (?)b
(Computational treatise o Master Sun) 2
3
Wu cao Wu cao suan jing Unknownc (Five departments) (Computational treatise o ve departments)
Not earlier than 386
Jiu zhang (Nine categories)
Unknowne
Prior to the mid rst century f
C. 263
Jiu zhang suan shu
(Computational procedures o nine categories) 4
5
6
7
d
Hai dao
Hai dao suan jing
Liu Hui
(Sea island)
( . 263)
Zhang Qiujian ([Master] Zhang Qiujian)
Zhang Qiujian suan jing (Computational treatise o Zhang Qiujian)
Zhang Qiujian Mid fh century
Xiahou Yang ([Master] Xiahou Yang)
Xiahou Yang suan jing (Computational treatise o Xiahou Yang)
Han Yan
Zhou bi (Te gnomon o Zhou [dynasty])
Zhou bi suan jing
Unknown
(Computational treatise [beginning with a problem] about a sea island)
(Computational treatise on the gnomon o Zhou [dynasty])
g
763–79h
Early rst century (?)i
Continued
Demonstration in Chinese and Vietnamese mathematics able 15.2 Continued reatises listed in Number the Xin ang shu 8
9
10
Identi ed as the ollowing extant treatises
Wu jing suan
Wu jing suan shu
Zhen Luan ( . c. 570 )
(Computations
(Computational
in the vebooks) classical
procedures [ound] in the ve classical books)
Zhui shu (Mending procedures)
Not extant
Qi gu (Continuation [o the work] o ancient [authors])
Qi gu suan jing
Wang Xiaotong C. 626
Zu Chongzhi
(429–500)j
12
C. 570
Second hal o the fh century
(Computational (b. ?– d. afer 626 )k treatise on the continuation [traditions] o ancient [mathematicians])
11
Date o compilation
Author
Ji yi (Records o [things] lef behind or posterity)
Shu shu ji yi
San deng shu (Numbers o three ranks)
Not extant
(Records o the procedures o numbering lef behind or posterity)
Xu Yue (b. beore 185 – d. afer 227)
Dong Quan
C. 220
Prior to 570
Notes: a A book entitled Sun zi by one Sun Chao o the Jin dynasty (265–420) is mentioned in the lists o proscribed books o the third through the tenth century, see An and Zhang 1992: 51; it is not impossible that this was the mathematical treatise
b
c
d
or prototype the amous waritswritten in c.and fhnot century . treatise Sun zi bing fa on the art o Qian Baocong suggested that the treatise was compiled in c. 400 ; he also believed that the extant version was altered during the Sui (581–618) and ang (618–907) dynasties, see SJSSa: 275; Guo 2001: 14. In some sources the treatise credited to the authorship o Zhen Luan , . c. 570, see SJSSa: 409. Te date suggested by Qian Baocong; he also suggested that the extant version o the text may have been modi ed in the seventh century , see SJSSa: 409, Guo
517
518 Notes: (continued) 2001: 18. Compare with the date ‘ fh century? Very approximately’ suggested by Martzloff 1997: 124. e Liu Hui in his ‘Preace’ o 263 suggested that the treatise was compiled on the basis o an ancient prototype by Zhang Cang (?–152 ) and Geng Shouchang ( . rst century ), see SJSSb: 83; or a discussion, see CG2004: 127. Te opinion o Liu Hui is one o the numerous theories concerning the date o compilation o the treatise; or an overview, see Li 1982. See also Cullen 1993a. f Compare with the date ‘200 –300 ’ suggested by Martzloff 1997: 124. g h
i
j
k
Qian Baocong suggested that the treatise was completed between 466 and 485 (SJSSa: 325), while Feng Lisheng argued or the interval 431–50 (Guo 2001: 16). Guo 2001: 25. Te text o the srcinal treatise written by Xiahou Yang most probably in the rst hal o the fh centurywas lost by the eleventh century and replaced by a compilation o Han Yan written in 763–79; see SJSSa: 551. Te dates suggested or this treatise vary considerably; I adopt here the viewpoint o Cullen 1993b and 1996, being well aware o other opinions concerning the date o compilation. Martzloff 1997: 124 provides a hardly acceptable period o time: ‘100 (?) – 600 ’. Wang Xiaotong in his ‘Preace’ to theQi gu suan jing mentions Zu Gengzhi (b. beore c. 480 – d. afer 525) and not his ather Zu Chongzhi as the author o the treatise (SJSSb: 415). Martzloff 1997: 125 suggests or Wang’s lietime the datesc‘. 650–750’ which are impossible given that his treatise was included in the collection o 656 .
below. Te list could not be chronological either, given that according to the conventional chronology the Zhou bi certainly was considered to antedate the treatise Hai dao and yet was listed afer it. Te only suggestion that seems plausible is that the list ollowed the order in which the treatises were actually studied.25 According to the ang liu dian (Six Codes o the ang [Dynasty]) and to the Jiu ang shu (Old History o the ang [Dynasty]), the students o the College were subdivided into two groups each comprising feen students. Te rst group studied treatises [1–8], and the second one treatises [9–10].26 In able 15.1 and below I reer to the textbooks o the groups [1–8] and [9–10] as constituting a ‘regular programme’ and an ‘advanced programme’, respectively, given that the extant version o the treatise [10] (in contains more solution diffi cult mathematical methodsand than those ound in [1–8] particular, o cubic equations), that the now lost treatise [9] was, according to Li Chuneng, a difficult book (and, 25
26
An almost identical list can be ound in the Jiu ang shu (Old History o the ang [dynasty]) (JS 44: 17b), yet the order o the treatises in the ‘regular programme’ is different: Jiu zhang, Hai dao, Sun zi, Wu cao, Zhang Qiujian, Xiahou Yang, and Zhou bi. Te San deng shu is mentioned as San deng. LD 21: 10b, JS 44: 17b. Te Xin ang shu only mentions that the number o students amounts to thirty, see XS 44: 1b, des Rotours 1932: 133.
Demonstration in Chinese and Vietnamese mathematics
as becomes clear rom an inspection o the number o years allotted to the study o the treatises, the most difficult book in either programme).27 Te study in each programme required seven years. Books [11–12] were studied simultaneously with the other treatises in both programmes; the time necessary or their study was not speci ed.28 Te conventional identi cation o the twelve treatises constituting the curriculum is ound in a number o modern works and is summarized in able 15.2. Te conventional identi cation o the ang dynasty textbooks with the extant treatises contains a number o points that have never been sufficiently clari ed. For instance, there are three treatises listed in the bibliographical section o the dynastic history Xin ang shu which, hypothetically, might be identi ed as the textbook Jiu zhang listed in able 15.1 and mentioned in the chapter on state examination o the same history: they are the Jiu zhang suan shu compiled by Xu Yue, the Jiu zhang suan jing compiled by Zhen Luan (XS 59: 13a), and the Jiu zhang suan shu commented on (zhu ) by Li Chuneng (XS 59: 13b), all three treatises in nine chapters (juan ). I the latter treatise is assumed to be the textbook used or instruction, it remains unclear whether it was identical with the only extant Song dynasty edition o the treatise commented (zhu ) by Liu Hui and accompanied with the explanations o the commentaries (zhu shi ) by Li Chuneng (see below). Te Zhang Qiujian rom the curriculum could be either the Zhang Qiujian suan jing in one juan commented on by Zhen Luan (XS 59: 13a), or a three-juan edition o the treatise commented on by Li Chuneng (XS 59: 13b); however, the earliest (and only extant) Song dynasty edition in three juan mentions Zhen Luan as the commentator while containing only commentaries signed by Li Chuneng (SJSSb: 343). As or the treatise listed in the curriculum as Xiahou Yang, the bibliographical chapter o the Xin ang shu mentions two books the titles o which bear reerence to this name: one is the Xiahou Yang suan jing commented on by Zhen Luan, and 27
28
Li Chuneng wrote about Zu Chongzhi and his book as ollows: ‘ [He] was the best o mathematicians. Te title the book [he] compiled Mending . Nothoroughly one o the the aculty [lit. ‘unctionaries’] o theo[Mathematical?] College iswas able to procedures comprehend proound [ideas it contained]. Tis is why [they] abandoned [the book] without [even trying] to understand [it].’ (SS 16: 4a). Martzloff ’s translation o the last part o this quotation reads ‘He [Zu Chongzhi – A.V.] was excluded (rom the textbooks used or teaching) because none o the students o the Imperial College could understand him’ (Martzloff 1997: 45, n. 22), and it is somewhat misleading, since Li Chuneng’s statement was clearly pointed against the personnel o the College (and not against its students), while the high esteem he expressed or the book o Zu Chongzhi was apparently related to his decision to introduce the Zhui shu into the curriculum as the cornerstone o the advanced programme. Siu and Volkov 1999.
519
520
the other is the Xiahou Yang suan jing authored by one Han Yan whose lietime has been a matter o controversy. Te hypothesis advanced by Qian Baocong and adopted by other modern authors states that the received book is dated o the eighth century (SJSSb: 25–7), yet the extant version contains three juan unlike the treatises listed in the Xin ang shu, both containing only one juan.
Te examination procedure Tere were two kinds o examinations held in the Mathematical College: (1) the regular tests conducted every ten days, and (2) the examinations at the end o the year. Te regular tests included three questions: two on memorization o a 2000-word excerpt and one on the ‘general meaning’ (da yi ) o the excerpt. Te examination at the end o each year was held orally; students were asked ten questions on the ‘general meaning’. It seems that there was no graduation examination at the end o the entire course.29 Tose who successully graduated rom the College were allowed to take the examination or the doctoral degree ming suan 30 together with some other categories o candidates.31 Te examination included two parts. Te task or the rst part was to write an essay answering ten questions related to one o the two programmes, ‘regular’ or ‘advanced’. Te second part o the examination in both cases consisted o a test on the memorization o the treatises San deng shu and Shu shu ji yi held in the orm o ‘examination by quotation’ (literally, ‘strip reading’tie du or ‘strip [reading] o classics’ tie jing ).32 Te Xin ang shu provides the ollowing description o the examination procedure o the rst part: 29
30
31 32
See XS 44: 2a; or translation see des Rotours 1932: 141–2, or a discussion o the procedure see Siu and Volkov 1999. Literally, ‘[He Who] Understood Computations’ (or ‘Learned in Mathematics’ , as Lee 2000: 138 suggests); the ‘he’ in the translation is imposed by the historical setting in which only men were admissible to the state examinations. Te appellation o the degree (and o the related examination) was thus similar to the other titles reerring to the degrees and examinations on the Conucian classics (ming jing , lit. ‘[He Who] Understood the Classics’), law (ming fa , lit. ‘[He Who] Understood the [Juridical] Norms’), calligraphy and writing (ming zi , lit. ‘[He Who] Understood the [Chinese] Characters’); see des Rotours 1932: 128. See des Rotours 1932: 128, n. 1 or a detailed description o the candidates. On the procedure o the ‘examination by quotation’ see des Rotours 1932: 30–31, 141, n. 2; Siu and Volkov 1999: 91, n. 41; see also Lee 2000: 142.
Demonstration in Chinese and Vietnamese mathematics (XS 44: 1b–2a)
All [the candidates examined in] the Mathematical College 33 [have to] produce records34 o ‘general meaning’ or ten35 tasks [represented with] mathematical problems (lit. ‘problems and answers’).36 [Tey have to] elucidate the numerical values [o the problems], [and to] design [computational] procedures [that would solve them]. [Tey] elucidate in detail the internal structure o the [computational] procedures [they designed].37 [I they do] so, then they pass. [When they are] tested 33
34
35
36
37
Des Rotours 1932 : 154 suggests ‘For mathematical studies . . .’ (‘Pour l’étude des mathématiques . . .’); his suggestion shows that he may have been perplexed by the heterogeneous headings o the paragraphs describing the examinations: in some cases the beginning o the description mentions the degree, as in the case o the law examination or the degrees jin shi and ming fa: . . . ‘All [the candidates or the degree] jin shi. . .’; . . . ‘All [the candidates or the degree] ming fa . . .’ (XS 44: 2b, ll. 11–12), while in the case o the examinations or the degrees ming zi and ming suan the names o the corresponding schools, shu xue and suan xue were mentioned instead (XS 44: 2b, ll. 13–14). Tis speci cation o the institution can mean that the candidates were examined in the respective college and/or the only candidates admitted to the examination were those who graduated rom it. Te word used here, lu , does not appear in the description o other examinations; des Rotours 1932: 154, n. 3 writes ‘I am not certain o my translation, because I don’t understand well the meaning o the word lu ’ (‘Je ne suis pas certain de ma traduction car je ne comprends pas bien le sens du mot lu .’). Indeed, the term lu looks somewhat inappropriate in the context o examination, since one o its principal meanings is ‘to copy, to record’. My interpretation o this term as ‘writing a protocol [o computations]’ is discussed below. Tis emendation o the srcinal text containing the word ben (‘srcinal’) is based on three premises. Firstly, the descriptions o the other examinations in the Xin ang shu containing the clause ‘VX’ with a verb V with the meaning ‘to examine’, ‘to ask’, etc., always have a numeral in the position o X, e.g., (‘ask [to complete] ten tasks on general meaning’), the examination or the degree ming jing (XS 44: 2b, ln.3); (‘ask [to complete] 50 tasks on general meaning’), the examination on the degree ming jing, option ‘Tree [Great] commentaries’ (XS 44: 2b, lns. 5–6); (‘ask [to complete] 100 tasks on general meaning’), the examination on the degree ming jing, option ‘[Dynastic] Histories’ (XS 44: 2b, l. 8); (‘to pass [examination consisting o] 100 tasks on general meaning’), the examination on the Rites of the Kai-Yuan era (XS 44: 2b, l. 4), and (‘ask [to complete] one task on general meaning’) in the description o the oral tests held every ten days in the Mathematical College ( XS 44: 2a, l. 5). Secondly, ten is indeed the number o the tasks the candidates were supposed to complete in this particular case. Tirdly, the word ben (as well as its modi cation ) ound in all the extant editions o the history is graphically relatively close to the word ‘ten’ , and the alteration o the text may have happened in an early edition and reproduced in later editions. Te interpretation o the term wen da as ‘[mathematical] problem’ was argued or in Siu and Volkov 1999. A slightly different translation o the two central excerpts o this paragraph was offered in Siu and Volkov 1999: 92. See also des Rotours 1932: 154–5.
521
522 with three tasks on the Jiu zhang, and with one task on each [o the treatises] Hai dao, Sun zi, Wu cao, Zhang Qiujian, Xiahou Yang, Zhou bi, Wu jing suan, [they] pass [i out o] ten [tasks they complete] six. [For the treatises] Ji yi and San deng shu, [they do] ‘strip reading’, and or ten [excerpts they] succeed [i they complete] nine. [When they are] tested with the Zhui shu and Qi gu, [they] produce records o ‘general meaning’ taking mathematical problems [as the examination tasks], [they have to] elucidate the numerical values [o the problems], [and to] design [computational] procedures [that would solve them]. [Tey] elucidate in detail the internal structure o the [computational] procedures [they designed]. As or those [trea38 [the candidates have to] make tises/examination papers] without commentaries, the numerical data coherent, to design [computational] procedures and [should] not make mistakes in the meaning and in the structure [o the procedures]. [I they do] so, then they pass. For the Zhui shu [there are] seven tasks; or Qi gu [there are] three tasks. [Tey] pass [i out o] ten [tasks they complete] six. [For the treatises] Ji yi and San deng shu, [they do] ‘strip reading’, and or ten [excerpts they] succeed [i they complete] nine. [Under the conditions listed above] they pass the degree examination, [but i they drop] one treatise [o the two], even i [they] completed six [tasks out o ten], [they] will not obtain the degree.39
Tis excerpt leaves several questions unanswered. In particular, it remains unclear whether the examination works o the candidates were written in 40
the ormatrelevant as taskstoontheother subjects, contents or whether theytreatises. had some specisame c ormat mathematical o the In Siu and Volkov (1999) the authors suggested the ollowing hypothesis: the candidates were given mathematical problems similar (but not identical) to those contained in the treatises o the chosen ‘programme’, that is, problems belonging to the categories or which the candidates knew the solutions yet with modi ed numerical parameters. Te change o parameters may have implied a modi cation, sometimes considerable, o the known algorithms 38
39
40
Te meaning o this phrase remains unclear; see a discussion o it in the concluding section o the present article. Te last remark apparently could reer to the case when the candidate ailed all the tasks related to the Qi gu . A discussion o the expression ‘general meaning’ is necessary here. Tis term occurs only in the descriptions o the examinations on the degrees ming jing (in the general description and in the description o two options; see above), ming suan , examination on the Rites of the Kai-Yuan era, as well as the description o the instruction in the Mathematical College (see above). One can suggest that the term ‘questions on meaning’ reers to a kind o task ocusing on the capacity o the examinee to provide a plausible interpretation o a given text or texts. Lee offers two examples o questions and answers on ‘general meaning’, da yi (interestingly, he renders this very term as ‘written elucidation’) in the context o examination on Conucian classics; he suggests that this kind o questions ‘tested mainly amiliarity, that is, memory, o the classics’ (Lee 2000: 142).
Demonstration in Chinese and Vietnamese mathematics
needed or the solution o the problems.41 In other words, the candidates were asked to design algorithms that were not mere replicas o the algorithms ound in the textbooks (otherwise the examination would have been reduced to a simple test o the students’ memory) but their generic versions designed according to the modi ed parameters. Tis hypothesis, however appealing it might have seemed, could not be provided by Siu and Volkov with any supporting evidence since the examination papers written by the candidates during the mathematics examinations o the ang and the Song dynasties do not now exist. However, rather unexpectedly, a supporting piece o evidence was ound in a Vietnamese mathematical treatise.
Mathematics examinations in traditional Vietnam: the case of a model examination paper Te available inormation concerning the traditional Vietnamese mathematics and the relevant reerences to the earlier works can be ound elsewhere;42 it can be very brie y summarized as ollows. Te number o extant mathematical treatises amounts to twenty-two; the earliest extant treatise is conventionally credited to an author o the feenth century while the other treatises were compiled in the eighteenth to early twentieth centuries. Teir style and contents are very close to those o Chinese mathematical treatises compiled prior to the introduction o Western mathematics into China.43 Te Vietnamese system o state education and civil examinations similar to the Chinese one dates back to the eleventh century , yet Chinese education and examinations were present in Vietnam well beore that time, since the country technically remained a province o China until the mid tenth century.44 Tere is no inormation about institutions speci cally ocused on mathematics education, yet historical records mention the examinations in ‘counting/computations’ (Viet.toán ) that took place in 1077, 1179, 1261, 1363, 1404, 1437, 1472, 1505, 1698, 1711, 1725, 1732, 1747, 1762, 1767, and 41
42 43
44
Tis statement was made in Siu and Volkov 1999 and amply illustrated in Siu 1999 and Siu 2004: 174–7. Volkov 2002; 2008; 2009. Te reader can nd more details on the extant treatises in Volkov 2009: 156–9; the descriptions in Volkov 2002 and Volkov 2008 do not take into account the most recent ndings. Te reader can nddescriptions o the traditional Vietnamese education in Richomme 1905 : 9–28; ran 1942; Vu 1959: 28–57; Nguyen 1961: 10–40; Woodside 1988: 169–233. Te short description o Ennis 1936: 162–4 draws upon the early yet still useul works o Luro (1878) and Schreiner (1900).
523
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1777.45 Te mentions are very short and do not provide any inormation concerning the contents and the procedure o the examinations. Since the state mathematics examinations were abolished in China by the end o the Song dynasty (960–1279), one can only guess what may have been the procedure and the contents o the Vietnamese state mathematics examinations and their relationship with the Chinese examinations o the ang and Song dynasties. o my knowledge, no srcinal Vietnamese mathematics examination papers have been ound so ar. Fortunately, there exists a ‘model’ mathematics examination paper published in 1820 by Phan Huy Khuông , apparently in order to provide the students with an idea o the best way to answer an examination question. Phan placed the mock examination essay that occupied almost six pages in the last, ourth chapter o his treatise entitled Chỉ minh lập thành toán pháp (Guidance or understanding the Ready-made Computational Methods) (CML 4: 30a–32b). Tis text sheds light on the examination procedure in Vietnam; moreover, it indirectly corroborates the hypothesis concerning the Chinese examination procedure mentioned in the section above. Te srcinal manuscript is preserved in the library o the Institute or Han-Nom Studies (Hanoi).46 In my work I used a micro lm copy o the manuscript preserved in the library o the Ecole rançaise d’Extrême Orient (Paris). Te catalogue ran and Gros (1993) provides only very sparse inormation about the author and the contents o the book. Te treatise opens with a picture o an abacus (p. 3a) which is an exact reproduction o the picture ound in the Chinese mathematical treatise Suan fa tong zong (Summarized undamentals o computational methods) by Cheng Dawei compiled in 1592 (SFZ: 113). Te picture is ollowed by a table o correspondences between powers o 10, monetary units, units o length, weight, and volume (p. 3b). wo ollowing pages present thirty-two diagrams o various plane gures (reerred to as ‘shapes o leds’, Chin. tian shi ) (pp. 4a–b) o which the areas are calculated in Chapter 2 o the treatise. Te model examination essay consists o a solution o a mathematical problem written by an imaginary examinee; or the ull translation o the examination paper see Appendix . Te problem reads as ollows: three categories o officials, A, B and C, are to be remunerated with 1000 cân o silver, yet out o this amount only the sum S = 5292 lượng was supposed to be distributed among the unctionaries.47 It is claimed in the 45 46 47
Volkov 2002 . It is listed under number 433 in ran and Gros 1993: 258. Cân and lượng , technically, are measures o weight (1 cân = 16 lượng), but were also used as monetary units in China and Vietnam, being applied to silver.
Demonstration in Chinese and Vietnamese mathematics
problem that the at-rate distribution method cannot be used to distribute this amount, and the method o weighted distribution is proposed instead. Te ratio o the amounts to be given to the unctionaries o the three ranks is 7 : 5 : 2, and the numbers o unctionaries o each rank are NA = 8, NB = 20 and NC = 300, respectively. Tere are two questions: (1) to nd the amount o silver to award each unctionary o the categories A, B and C, and (2) to nd the total amount o money allotted to each group o the unctionaries. In modern terms, this is a problem on weighted distribution: one has to nd the values x1, x2, . . ., xn given that x1+x2+. . .+xn = S and x1: x2: . . . : xn :: k1: k2: . . . : kn or given weighting coefficients k1, k2, . . ., kn. Problems o this type as well as the standard procedure or their solution equivalent to the ormula xj
Sk j n
ki i 1
are ound in a number o Chinese and Vietnamese mathematical treatises beginning with the Chinese mathematical treatises Suan shu shu (Writing on computations with counting rods)48 and Jiu zhang suan shu.49 However, the problem ound in the Vietnamese treatise contains a particularity: it is known that there are three different ranks o unctionaries, and or all unctionaries o the same rank the weighting coeffi cients are the same; in our notation, k1= k2 = . . . = k8 = kA = 7, k9= k10 = . . . = k28 = kB = 5, k29= k30 = . . . = k328 = kC = 2, and one is asked to nd the values xA, xB, xC (xA = x1 = . . . = x8 , xB = x9 = . . . = x28, and xC = x29 = . . . = x328) such that xA : xB : xC :: kA : kB : kC, and NA·xA + NB·xB + NC·xC = S. Te examinee is also asked to nd the total amount o money allotted to each group o unctionaries, that is, to calculate the values XA= x1 + . . . + x8, XB = x9 + . . . + x28 and XC = x29 + . . . + x328. In this chapter I use the term ‘aggregated weighted distribution’ to identiy the category o problems on weighted distribution in which the ‘sharers’ be subdivided B, C,.the. .weighting containingcoeffi NA, cients NB, NCare ,. . . sharers,can respectively, such into that groups in each A, group the same and equal to kA, kB, kC,. . . . Any problem on aggregated weighted distribution apparently can be solved with the classical algorithm cited above, yet in several sources a modi ed version o the method was used: the 48
49
Te earliest extant Chinese mathematical treatise Suan shu shu was completed no later than the early second century ; or English translations, see Cullen 2004 and Dauben 2008. Cullen 2004: 43–51, 54–6; Dauben 2008: 114–21, 126–7; CG2004: 282–99.
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addition o the weighting coefficients is done in two steps: rst, theweighting coefficients are multiplied by the numbers o ‘sharers’ in the respective groups, second, the results o the multiplications are summed up: K = NA· kA + NB·kB + NC·kC + . . . . Te earliest problem on aggregated distribution in China is also ound in the Jiu zhang suan shu (problem 7 o chapter 3):50 there are two groups containing three and two persons, respectively, k1= k2 = k3 = 3, k4= k5 = 2, S = 5 (SJSSb: 112). However, the solution offered in the Chinese treatise does not treat speci cally this particularity o the condition; the procedure simply suggests to set the weighting coefficients as 3, 3, 3, 2, 2 and to proceed according to the ‘classical’ method. Chronologically, the earliest extant Chinese treatise eaturing the multiplication o the numbers o sharers in each category by the respective weights NA·kA, NB·kB, NC·kC is the Sun zi suan jing; problem 24 o the second chapter (juan) o the treatise belongs to this type and contains a detailed description o the computational procedure (SJSSb: 274). Problems o this type are also ound in the Zhang Qiujian suan jing (problem 17 o chapter 1 and problem 13 o chapter 2, SJSSb: 303–4, 315–16), Suan xue qi meng (Introduction to the learning o computations, 1299) by Zhu Shijie (dates unknown) (problem 50 o chapter 2, SXQM: 1161), Jiu zhang suan fa bi lei da quan cat (Great compendium o the computational methods o nine egories [and their] generics, 1450) by Wu Jing (dates unknown)51 and Suan fa tong zong (Summarized undamentals o computational methods, 1592) by Cheng Dawei (1533–1606) (Problems 8, 15 and 31 o chapter 5, SFZ: 377, 383, 294, respectively).52 Te problems on weighted distribution can be ound in a number o Vietnamese mathematical treatises. Te most interesting case is the systematic introduction o the method ound in the Ý rai toán pháp nhất đắc lục (A Record of What Ý rai Got Right in Computational Methods, preace 1829) compiled by Nguyễn Hữu Tận .53 As or the treatise under investigation Chỉ minh lập thành toán pháp, chapter 4 contains thirty-eight problems o which twelve are devoted to weighted 50
51
52
53
Te Suan shu shu does n ot contain problems on aggregated sharing: in all six problems related to the weighted distribution (problems 11–16, 21 in Cullen 2004) the weights o the sharers are all different. Problems 5, 33, 36 and 44 o chapter 3 (DQ 3: 3a, 14b, 17b, 21b) belong to the category o ‘aggregated weighted distribution’, but only problem 5 (analogous to problem 7 o the Jiu zhang suan shu) is solved with the ‘classical’ algorithm used in the Jiu zhang suan shu. o numerate the problems, I countthe problems per se as well as generalized rules given without numerical data. Volkov orthcoming.
Demonstration in Chinese and Vietnamese mathematics
distribution (Problems 5–7, 10–11, 14–19, 38).54 Among them, only two problems deal with the ‘aggregated sharers’, namely, problem 6 and problem 38 (which is the problem solved in the ‘model examination paper’). Problem 6 represents a case o a ‘mixed’ weighted distribution combining ‘solitary’ and ‘aggregated’ sharers. In this problem one deals with the unds raised by a temple.55 Te setting is as ollows (CML 4: 6a–7b): Te total amount o 240 cân o gold was collected; 3 parts o the total amount were obtained rom selling incense, 6 parts rom a ‘senior donator’, 24 ordinary male donators contributed 4 parts each and 5 ordinary emale donators contributed 3 parts each. In modern notation one has to nd the values x1, x2, . . ., xn, n = 31, given that x1+x2+ . . . +xn = S and x1 : x2 : . . . : xn :: k1: k2: . . . : kn or the given weighting coefficients k1 = 3, k2 = 6, ki = 4 or i = 3, . . ., 26 and kj = 3 or j = 27, . . ., 31. Te procedure provided in the treatise can be written in modern terms as ollows: – – – –
one has to calculate the sum o the coefficientsk1+ k2 = 9; nd the value k3+k4+ +k26 as 24·k3 = 96; nd the value k27+k28+ +k31 as 5·k27 = 15; nd the sum K = k1+k2+ +kn = (k1+ k2)+(k3+k4+ +x26) +(k27+k28+ . . . +x31) = 9+96+15 = 120; – use the obtained total value K to divide the total amount o money and to obtain the ‘constant norm’ S/K; – now one obtains the amounts o money xi corresponding to the weights ki: the money or incense x1 = k1·(S/K), the money o the senior donator x2 = k2 · (S/K), the money o each ordinary male donator xi = ki · (S/K), i = 3, . . ., 26, and the money o each ordinary emale donator xi = ki · (S/K), i = 27, . . ., 31; – to obtain the money donated by each group, the reader is given the cases o the incense and the senior donator as examples: here the obtained value S/K is to be used again, and one is told to multiply this value by the ‘parts’ corresponding to the group. In the case o the incense and the senior donator it will correspond to x1 = k1 · (S/K) and x2 = k2 · (S/K), respectively. Te reader then is told that the 54
55
It still remains unclear how many problems there were in the srcinal version. In the micro lm o the manuscript preserved in the Ecole rançaise d’Extrême Orient (Paris) the text o problem 14 beginning on page 16a is incomplete. Moreover, Problem 17 (p. 18a) on ‘8:2 distribution’ is misplaced in the section on ‘6:4 distribution’. Tese two details suggest that at least one page o the srcinal treatise was not copied by the copyist and other pages may have been copied in a wrong order. Te wording o the problem makes it unclear whether the money is supposed to be obtained, or given by, the temple; I provided my translation in assuming that historically Vietnamese temples usually obtained rather than distributed money.
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remaining operations would be similar. Instead o computing the impact o the ordinary male and emale donators as NM·xM and NF·xF, where NM = 24, xM = x3, NF = 5, xF = x27, the reader is told to compute these values as (24·k3) · (S/K) and (5·k27) · (S/K), respectively. It appears plausible to suggest that the author o the Vietnamese treatise at this point reinterpreted the data, and considered each entire group o male and emale donators as ‘collective donators’ o the donated money, possessing K = N · k and K = N · k ‘shares’; M M 3 F F 27 – the problem is concluded with a check-up o the obtained answer; one has to check whether the sum o the amounts obtained rom each source is equal to the total amount o the raised money. It is not veri ed whether the portions o money coming rom the our sources indeed constitute the given ratio. Now we can return to the model examination paper. Te solution o the imaginary examinee contains six parts: (1) a ormal introduction (p. 30b, ll. 8–11); (2) an explanation why only a part o the awarded silver was actually given to the unctionaries (p. 31a, lls. 1–6); (3) an explanation o the act that the at-rate distribution could not work (p. 31a, l. 6 – p. 31b, l. 4); (4) a rewording and a solution o the weighted distribution problem (p. 31b, l. 4 – p. 32b, l. 5); (5) a veri cation o the answer (p. 32b, lls. 5–7); (6) a ormal ending o the examination paper (p. 32b, lls. 7–9). Te reader will notice that the examination paper contains more than a solution o just one problem. Te imaginary examinee is supposed to check the proposed data, nd an explanation or the seeming discrepancy ound in the condition (it is stated that 1000 cân = 16000 lượng is to be given to the unctionaries, yet the amount o money distributed among them was only 5292 lượng), and solve two problems, one on at-rate and the other on weighted distribution. Te suggested solution o the weighted distribution problem runs as ollows: in order to nd xA, xB and xC, at the rst step the sum K = k1+ k2 + + k328 is calculated; to do so, the imaginar y examinee calculatesNA·kA = 56, NB·kB = 100, NC·kC = 600 and adds them up to obtain K = 756. Te term used to reer these products is rather particular: while talking about the weights kA, kto B, kC the examinee uses the word ‘shares/parts’ (Chinese fen ), but when passing to the ‘aggregated shares/parts’ NA·kA, NB·kB, NC·kC he employs a combination o two characters (Chinese fenlü) ‘parts– coefficients’ or ‘multiples o shares/parts’; I shall return to this term later. At the second step, the total amount o money, S = 5292 lượng, is divided by K yielding 7 lượng, called the ‘constant norm’, as in problem 6. Te
Demonstration in Chinese and Vietnamese mathematics
amounts o money xA, xB and xC to be obtained by each unctionary o the group A, B, C are calculated as the ‘constant norm’ multiplied by kA, kB, kC, respectively.56 In the second part o the solution the imaginary examinee looks or X A, XB and XC which obviously could be ound as NA·xA, NB·xB, NC·xC once xA, xB and xC have been calculated. However, the suggested solution is different: or example, or group A, the author suggests the calculation o (NA·kA)· (S/K) instead o calculating N ·[(S·k )/K]; or groups B and C similar A A operations are perormed. Once again, it can be understood as i the author considered each entire group A, B and C as one ‘collective recipient’ o the awarded money, possessing KA = NA·kA, KB = NB·kB and KC = NC·kC ‘shares’, respectively, while the sum o the ‘shares’ KA + KA + KC remained equal to K.
Examinations and commentaries Te solution o the model problem provided in the treatise was based on the algorithm or the ‘aggregated sharers’ ound in a number o Chinese and Vietnamese mathematical treatises, yet it would be reasonable to suggest that the imaginary examinee was supposed to design his solution on the basis o the inormation ound in the same treatise. Indeed, the treatise provides two sources o such inormation: (1) a general description o the algorithm o weighted distribution (CML 4: 4b–5a), and (2) the aorementioned problem 6 o chapter 4 on distribution o donations. A cursory inspection o these two sources suggests that the solution in the model paper was designed by analogy with the solution o problem 6; in particular, the term ‘parts–multiples’ (or ‘multiples o parts’) ound in the solution o the model problem does appear in the solution o problem 6 but not in the algorithm introduced on p. 5a. It is especially interesting that in this case the Vietnamese author used the term lü , since the concept o lü was one o the key elements in the conceptual system presented in Liu Hui’s commentary on the Jiu zhang suan shu. In modern notation, a number A is a lü (a ‘proportional’, or ‘multiple’) o another number, A ′, i one can establish a proportion in which both numbers occupy the same positions in the ratios involved: A : B : . . . :: A′ : B′ : . . . .57 However, the term 56
57
In Volkov 2008 I suggested a mathematically correct yet ‘modernizing’ reconstruction o the rst part o the Vietnamese procedure. For a detailed discussion o the term, see CG2004: 135–6, 956–9. Martzloff 1997: 196–7 employs the term ‘model’ (i.e. one number can be used as a ‘model’, a representative, o another number).
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‘parts–multiples’ (or ‘multiples o parts’?) fenlü introduced by the Vietnamese author appears to be unparalleled in the Chinese mathematical texts o the rst millennium . Te solution o the imaginary examinee was supposed to be designed as a modi cation o the solution o a problem rom the treatise he, presumably, was supposed to be amiliar with. In other words, the examination paper was based on a problem already solved and discussed earlier, but with a modi ed structure (three groups o unctionaries instead o the combination o two individual and two collective donators) and altered numerical data. Te entire ormat o the examination paper was larger than just one problem: it was rather that o a ‘research project’ in which a given situation was approached with two mathematical ‘models’, one o at-rate distribution (rejected as neither tting into the numerical data nor corresponding to the hierarchical structure o the group o unctionaries) and one o weighted distribution. Te mathematical contents o the particular problem solved in the Vietnamese model examination paper are not as important or the present discussion as the very ormat o the essay suggested by the author o the treatise who apparently was well acquainted with the actual examination procedure. Most importantly or the present discussion, the Vietnamese model examination paper ts, to a large extent, into the ormat described in the ang dynasty Chinese source mentioned above, namely: (1) the core o the examination task consists o a mathematical problem; (2) the examinee ‘elucidates’ the ‘numerical values’ provided in the given problem (that is, checks the consistency o the given numerical data), and (3) he ‘designs a computational procedure’ o which (4) the ‘structure/rationale’ he discusses in detail, that is, he provides a detailed solution in which every step is commented upon. Te imaginary Vietnamese examinee styles his text as i he operates with a counting instrument to obtain his result while writing down the results o the operations he is perorming. It would be reasonable to assume that the Chinese candidates o the ang dynasty also employed their counting rods during the examination to solve the problems given to them. I this assumption is correct, their solutions must have contained the protocols o perormed computations that would have looked rather similar to that ound in the Vietnamese model examination paper. Tis observation makes it tempting to interpret the term lu (‘records, protocols’) employed in the description o the mathematics examinations in the Xin ang shu quoted above as reerring to this particular eature o the mathematics examination papers.
Demonstration in Chinese and Vietnamese mathematics
Back to China When constructing his solution, the imaginary Vietnamese examinee produced a text the structure o which to a large extent resembles the solution already provided in the treatise, namely, in problem 6 o the same chapter. One can conjecture that the Chinese examinees o the ang dynasty were also supposed to base their solutions on those provided in the respective mathematical textbooks. Here we come to the ocal point o the present chapter, namely, the role the commentaries ound in Chinese mathematical treatises played in mathematical instruction and examinations. able 15.3 provides the names o the commentators o the extant ten mathematical treatises used in the Mathematical College o the ang dynasty. able 15.3 shows that the treatises used or instruction all incorporated commentaries, unlike the extant treatises listed under numbers 1 and 2. Te history o transmission o the treatises is so obscure that even i the names o the commentators in the extant treatises coincide with those mentioned in the bibliographies listed in able 15.3, it remains unknown whether the extant commentaries are indeed identical with those used in the Mathematical College o the ang dynasty. An inspection o the extant commentaries listed in able 15.3 shows that they differ considerably as ar as their style and contents are considered. Te commentaries are mainly ocused on the computational procedures designed or solution o the problems, yet the ormats adopted by their authors were not the same. Liu Hui’s commentary on theJiu zhang suan shu contains parts written in different styles: the commentator interpreted the operations with ractions exempli ed in the treatise using especially coined mathematical terms; used diagrams o plane gures and descriptions o (probably imaginary) three-dimensional models or solution o geometrical and algebraic problems; provided detailed computations in case o the calculation o the value o π close in style to Liu Xiaosun’s cao or lef only obscure indications which, however, may have been reerring to some speci c mathematical contents.58 Te commentaries o another enigmatic gure, Zhao Shuang or Zhao Junqing (conventionally these two names are believed to be the aliases o the commentator Zhao Ying mentioned in 58
For the srcinal text, translation and discussion see CG2004, as well as the works o other authors quoted by Chemla and Guo; on the geometrical diagrams see Volkov 2007. Tis variety o styles can make one ponder over the authenticity o the received commentary conventionally credited to the authorship o the person known as Liu Hui whose biographical data remain unknown, yet the latter problem, certainly important, is not pertinent in the context o the present inquiry.
531
532 able 15.3. Te extant ang dynasty mathematical textbooks and their
commentatorsa
Number
Te extant treatises used in the Mathematical College
Commentators as speci ed in official histories
1
Sun zi suan jing
Zhen Luan (Jiu ang shu);b Li Chuneng
Commentator(s) o the extant treatises None.
(Computational treatise (Xin ang shu); Li o Master Sun) Chuneng (Song shi) 2
Wu cao suan jing
Li Chuneng et al. (Song shi)
None
(Computational treatise o ve departments) 3
Jiu zhang suan shu (Computational procedures o nine categories)
Li Chuneng (Xin ang shu); Liu Hui; Li Chuneng et al. (Song shi)c
Liu Hui; Li Chuneng et al.
4
Hai dao suan jing
Li Chuneng (Xin ang shu)
Li Chuneng et al.
(Computational treatise [beginning with a problem] about a sea island) 5
Zhang Qiujian suan jing Zhen Luan; Li Chuneng (Xin ang (Computational treatise shu) o Zhang Qiujian)
Liu Xiaosun ; Li Chuneng et al.d
6
Xiahou Yang suan jing
Te author (Han Yan , ang dynasty)
7
8
Zhen Luan (Jiu ang shu and Xin ang shu)
(Computational treatise o Xiahou Yang) Zhou bi suan jing Zhao Ying ;e Zhen Luan (Jiu ang (Computational shu);f Zhao Ying; treatise on the gnomon Zhen Luan; Li o Zhou [dynasty]) Chuneng (Xin ang shu)g Wu jing suan shu Li Chuneng (Xin ang shu); Li Chuneng (Computational (Song shi)h procedures [ound] in the ve classical books)
Zhao Junqing ; Zhen Luan; Li Chuneng et al.
Li Chuneng et al.
Continued
Demonstration in Chinese and Vietnamese mathematics able 15.3 Continued
Number
Te extant treatises used in the Mathematical College
Commentators as speci ed in official histories
9
Qi gu suan jing
Li Chuneng (?) (Jiu ang shu);i Li Chuneng (Xin ang
(Computational
Commentator(s) o the extant treatises Te author (Wang Xiaotong )
treatise on the shu) continuation o [traditions] o ancient [mathematicians]) 10
a
b c d
Shu shu ji yi Zhen Luan (Jiu ang (Records o the shu and Xin ang shu) procedures o numbering lef behind or posterity)
Zhen Luan
Li 1977: 269–271 quotes these and other sources mentioning the names o commentators. Zhen Luan is mentioned as the commentator and the authorJS ( 47: 6b). Te title is mentioned as Jiu zhang suan jing (SS 207: 3b). Liu Xiaosun o the Sui dynasty (581–618) authored the ‘computations’, cao .
e
f
g
h i
Conventionally identi ed as Zhao Junqing also known as Zhao Shuang , the author o the commentary ound in the extant edition o the treatise. Te Jiu ang shu mentions three different editions o the treatise, two commented upon by Zhao Ying and Zhen Luan, and onecompiled by Li Chuneng (JS 47: 5b). Te Xin ang shu mentions our different editions commented upon by the three commentators separately (Li Chuneng is credited with the authorship o two commentaries) (XS 59: 12b, 13b). In the Song shi the treatise is mentioned as authored by Wang XiaotongSS ( 207: 3a). In the Jiu ang shu both Wang Xiaotong and Li Chuneng are mentioned as the authors (JS 47: 6b); probably, the text o the history is corrupted and Li Chuneng was srcinally mentioned as a commentator.
bibliographical chapters o dynastic histories, as able 15.3 shows), whose lietime presumably was not too distant rom that o Liu Hui, offer a slightly narrower range o styles. Te best-known contribution o Zhao is his justi cation o a series o quadratic identities with the help o geometrical diagrams, to a certain extent similar to those used by Liu Hui in his commentaries on the ninth chapter o the Jiu zhang suan shu.59 Te actual intentions that Liu Hui and Zhao Shuang had when writing their commentaries on theJiu zhang suan shu and Zhou bi suan jing, respectively, do not seem related to any kind o educational activity. However, 59
Gillon 1977; Cullen 1996: 206–17; CG2004: 695–701.
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their commentaries on the oldest and presumably highly respected texts in the collection o the textbooks were edited in the seventh century to be used or instruction. Te commentaries arguably compiled by Li Chuneng and his team or educational purposes thus may have corresponded most closely to the style o work with ancient texts practised by the instructors o the Mathematical College.60 Yet the commentary on theHai dao suan jing by Li Chuneng et al. did not discuss the rationale o the methods; instead, the commentators explained the terms occurring in the conditions o the problems and reproduced the procedures provided by Liu Hui with plugged numerical parameters. Tat is, or Li Chuneng the relevant interpretation o a procedure consisted o a correct identi cation o the parameters involved and the operations with them. Te parts o Li Chuneng’s commentary devoted to calculations look similar to the ‘computations’cao ( ) added by Liu Xiaosun to the Zhang Qiujian suan jing, and both texts resemble closely the computations in the Vietnamese model examination paper. Tese observations suggest the ollowing conjecture. Even though the ormat o the ang dynasty examination papers remains unknown, the ormat adopted by the author o the model examination work in the Vietnamese treatise ts surprisingly well into the short description o the ang dynasty mathematics examinations quoted above. Te imaginary Vietnamese examinee used as his model the solution o a generic problem ound elsewhere in the same treatise and, in particular, provided detailed calculations close enough to those ound in the model problem. Now, what kind o explanations o the ‘meaning’ o the given problems were the actual Chinese examinees o the ang dynasty expected to provide? It is perhaps not too daring to conjecture that their writings were supposed to resemble those provided by the commentators o the treatises used as textbooks. In other words, it appears plausible to suggest that the commentaries o Liu Hui, Zhao Shuang, Li Chuneng and others ound in the treatises used or instruction in the Mathematical College were used as the models or the examination papers; not only did they provide the students with methods used to investigate the validity o the computational procedures presented in the treatises, but they also established the particular ormat to be imitated by the candidates when writing their examination essays. Te phrase wu zhu zhe ound in the description o the mathematics examinations in the ‘advanced programme’ and rendered above 60
It appears quite probable that the commentarial activity o Zhen Luan who produced a set o commented mathematical treatises in the second hal o the sixth century was directly related to a system o state mathematics education established, as some authors have suggested, at the Court o the Northern Zhou dynasty (see above).
Demonstration in Chinese and Vietnamese mathematics
as ‘As or those [texts/papers] without commentaries’ can be understood in at least three different ways: (1) it reers to a commentary expected to be written by the examinee in his examination paper but omitted or some reason; (2) it reers to a commentary missing in one o the two treatises o the ‘advanced programme’ which constituted the topic o the examination, and (3) the word ‘commentary’ zhu had here the technical meaning ‘to preappoint a candidate to a position’.61 Te third option hardly seems to be relevant in this particular context. Siu and Volkov (1999) have argued or the rst option mainly on the basis o the inspection o the only extant treatise o the ‘advanced programme’, theQi gu suan jing by Wang Xiaotong in which almost all the problems are provided with commentaries. However, a large part o the srcinal treatise is lost: according to the bibliographical sections o the Jiu ang shu and Xin ang shu, the book srcinally contained our juan (JS 47: 6b; XS 59: 14a) while the Song shi mentions only one juan (SS 207: 1a). Te extant version contains only twenty problems; the texts o problems 17–20 and o the respective commentaries are partly lost (SJSSb: 434–5). It is thereore impossible to know whether every single problem o the ang dynasty version o the treatise was commented upon by Li Chuneng, or whether a certain number o the problems were lef without commentaries.62 Moreover, nothing can be known about Li Chuneng’s commentaries on the second book o the ‘advanced programme’, theZhui shu by Zu Chongzhi, since the book had already been lost by the time o the Song dynasty; it is equally possible that only some problems contained commentaries. I this was the case, the phrase wu zhu zhe, ‘as or those without commentaries’, may have reerred to paradigmatic problems rom the treatises used as textbooks in the ‘advanced programme’ which did not contain commentaries on certain problems. Tis option leads to the ollowing hypothesis: in the ‘advanced programme’ examination tasks were compiled on the basis o problems rom the Qi gu suan jing and Zhui shu; i the srcinal problem contained a commentary, the examination criteria were the same as in the ‘regular programme’ examination: the examinee had to ‘elucidate numbers’ and to ‘elucidate in detail the internal structure o the [computational] procedure’, that is, to compile a text similar to the srcinal commentary. I the problem taken as the model or the examination task did not contain a commentary, the candidate was not asked to provide ‘elucidations’ but to ‘make the numerical data coherent’, and ‘not to make mistakes in the meaning and 61 62
Des Rotours 1934: 43, 49, 217, 244, 266, 268; Hucker 1985: 182, nos. 1407–8. Te interested reader will nd the annotated translation by Berezkina 1975 highly useul.
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in the structure’ o the procedure. Each o the terms employed here most probably had a precise technical meaning difficult or even impossible to restore, yet one can saely conjecture that in the latter case the examinee was supposed to provide a sequence o correct operations leading to the solution without their detailed justi cation. I the phrase about the ‘lack o the commentaries’ reerred to the compilations o the examinees, one can suggest that they were supposed to write their explanations in the ormat similar to that o the officially established commentaries and, most probably, used these commentaries as the best available models. I the second interpretation o the phrase is correct, the description o the examination procedure suggests an even larger role o the commentaries ound in the treatises used or instruction. Whichever interpretation o the phrase ‘as or those without commentaries’ is adopted, the role o the commentaries is apparent: they were not only providing explanations or justi cations o the algorithms ound in the treatises, but also became the models or the examination papers.
Conclusions Until recently the historians o Chinese mathematics tacitly assumed that the commentaries on mathematical texts, especially those authored by Liu Hui and Zhao Shuang, were ‘purely mathematical works’ written by proessional mathematicians or unidenti ed target groups, presumably small communities o experts and disciples. Tis assumption is most probably correct; my hypothesis is that the embedding o Liu Hui’s and Zhao Shuang’s commentaries into the context o state education radically changed the way in which they were interpreted and used. Afer having been edited by the team o Li Chuneng, the commentaries on the treatises constituting the curriculum set the guidelines or the instructors and students o the Mathematical College. More speci cally, in order to demonstrate their correct understanding o an algorithm ound in a mathematical treatise, the students and examinees had to perorm the operations the algorithm prescribed with the correctly inserted numerical values. Tis reconstruction is corroborated by at least three documents: (1) the commentaries o Li Chuneng’s team on the Hai dao suan jing written in the seventh century with the purpose o being used as didactical material in the Mathematical College and conspicuously eaturing computations perormed according to the algorithms devised by Liu Hui; (2) the aorementioned description o ang examinations, and (3) the Vietnamese model examination paper. Te
Demonstration in Chinese and Vietnamese mathematics
commentaries o Li Chuneng on the Hai dao suan jing may have naturally become paradigmatic texts imitated by the authors o examination essays devoted to this particular text, and one can conjecture that the commentaries o Liu Hui and Zhao Shuang, containing justi cations o the algorithms, in turn also may have been employed by the students and examinees as models in their oral presentations and written examinations. Te commentaries thus provided the standards o persuasiveness and consistency and shaped the style and structure o the mathematical discourse in the branch o the traditional Chinese mathematics perpetuated within the network o official educational institutions o the rst mille nnium .
Acknowledgements I would like to express my gratitude to two anonymous reerees or their valuable suggestions, to Karine Chemla or her personal and proessional support throughout the preparation o the chapter, and to the Institute or Advanced Study, Princeton where the rst draf o the paper was completed in 2007. Te nancial support or my work in France and Vietnam in 2006–7 was provided by the National Science Council, aiwan (grant no. 95–2411-H-007–037), by the Leading Edge Research Foundation o the National sing Hua University, aiwan (grant no. 95N2521E21) and by the Institut National de Recherche Pédagogique, France; I would like to express my gratitude to all these institutions.
Appendix Te rst part o the Appendix contains the srcinal text o the ‘model examination paper’ rom the Chỉ minh lập thành toán pháp (Guidance or understanding o the Ready-Made Computational Methods) by Phan Huy Khuông’s (CML 4: 30a–32b). When reproducing the text, I preserved the srcinal layout, that is, one line o the srcinal corresponds to one line o the transcription below. Te srcinal text does not contain punctuation, and I introduce my own. Te emendations o the text are indicated with the brackets 〈〉 and : ‘ 〈A〉B’ means that the sequence o characters A is suggested to be replaced by the sequence B
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(either A or B can be an empty sequence, that is, 〈A〉 alone means that the sequence A is to be suppressed and Bstands or the sequence B which is to be added). Te second part o the Appendix contains its English translation with the reerences to the page and line numbers o the srcinal. /p. 30a/ 63
/p. 30b/
/p. 31a/
/p. 31b/ 63
Tis is the title o the section separated rom the main body o the text with an indent.
Demonstration in Chinese and Vietnamese mathematics
/p. 32a/
/p. 32b/
ranslation
/p. 30a/ [1] Imitation o a composition o a mathematical problem [written according to] the ormat o an examination paper. [2] Question: [Let us suppose that] now there is money to award [unctionaries], the total amount is 1000 cân (). As or this amount o money, the award assigned to a given [group o] unctionaries [3] had the value o 5292 lượng (). Te award was promised to 328 people affiliated with the given establishment.
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[4] [I one] intends to distribute equally according to the number o the people, [then] [it will be] uneven and there will be a remainder o 4 phan 8 li. [5] [I or those] ranging rom the high to the low [positions] the pattern o ‘equal rank [distribution]’ () [is applied], [then] there is [something] incorrect. So, the method o ‘ at-rate distribution’ cannot be universally applied, [it is] already [6] clear. Now, [one] wishes to use this money to be applied equally [within one rank] according to unequal ranks o the aorementioned corpus [o unctionaries] in separating them into three ranks [as ollows]. Rank A: /p. 30b/ [1] 8 persons, each person obtains 7 parts; rank B: 20 persons, each person obtains 5 parts; rank C, 300 [2] persons, each person obtains 2 parts. [I we proceed in this way], then what will be the [amounts o money corresponding to] the parts obtained by all the people and the due amount [o money] or each [o the three groups o unctionaries]? [We] examine all [3] the experts in computations [who] s‘ tudy and exercise’,64 [those who] penetrate into the subtleness o weights and measures, [who can] dis-
[4] [5] [6] [7] [8]
64 65
66
67
tinguish and differentiate, analyse adequately, [those who know how to] arrange and dispose [the counting rods], in order to inspect the simple as well as the proound [matters].65 Answer:66 each person o rank A obtains 49 lượng o silver; the due amount is 392 lượng; each person o rank B obtains 35 lượng o silver; the due amount is 700 lượng; each person o rank C obtains 14 lượng o silver; the due amount is 4200 lượng. Response [o the examinee]: [I,] so-and-so,67 say: [this] computational method involves [the operations o] multiplication and division and does not go beyond the [method o] ‘distribution according to grades’
A quotation rom the rst chapter o theConucian classic Lun yu (Te Analects). Probably, this paragraph is a ormal ending appended to every problem proposed to candidates at the examination. Te answer is written in smaller characters; it is possible that the answer was supposed to be written by the examinee in the blank space lef afer the word ‘answer’. A sel-depreciatory (Chinese reading yu) indicates the position in which the actual name is to be inserted.
Demonstration in Chinese and Vietnamese mathematics
().68 Firstly,69 there is an amount [to be distributed]; [secondly],70 there are grades. [9] Here is the computational problem [proposed] by those in charge,71 and [what is below is] how I answered it. Now it is clear that what is asked in the problem [10] is solely concerned with the awarded money kindly dispatched to the given groups [o unctionaries]. [One] brie y discussed the ‘ atrate distribution’, [and afer that] used the ‘distribution according to grades’ as the principal [11] method. I know that [this] computational method has unlimited miraculous applications! I, so-and-so, ask or counting rods72 to ‘arrange and dispose’ them.73 /p. 31a/ [1] As or the very [phrase] ‘[Let us suppose that] now there is money to award [unctionaries], the total is 1000 cân [o silver]’, [I] make it [= this amount] uniorm [with other units] using [the actor] 16, [which is] the ‘norm’ o cân.74 [2] Te total amount [thus] obtained is 16000 lượng. ‘[I] benevolence is maniested by the superiors, [then] necessarily the subjects are kindly 75
awarded.’ As ar as this money [3] is concerned, the said unctionaries cared about their bene t and could not themselves accept to keep [the money] privately. Tereore [4] what the granting authorities kept out o the amount o awarded money was a deposited amount o 10708 lượng.76 68
69 70 71 72
73
74
75
76
Tis is the term or weighted distribution ound in chapter 3 o the Jiu zhang suan shu; see SJSSb: 109ff. Or: ‘in the upper [position]’. Or: ‘in the lower [position]’. Here the term may be a ormal title o an official; see Hucker 1985: 162. It is worth noting that counting rods and not the abacus are mentioned here. According to the report o Giovanni Filippo de Marini (1608–82), counting rods were still in use in Vietnam as late as the mid seventeenth century; see Volkov 2009: 160–4. However, one cannot rule out the possibility that the term toán may have been used here as a metaphorical reerence to a counting instrument in general. Probably, a quotation rom the ending o the problem ‘. . . [those who] arrange and dispose [the counting rods], in order to inspect . . .’ Tat is, 1 cân = 16 lượng , thereore to convert an amount o money rom cân to lượng one has to multiply it by 16. Tis phrase does not have any particular mathematical meaning and appears to be a quotation rom a text that I have been unable to identiy. Tat is, the authorities retained some amount o money or the good o the unctionaries. Tis is but a tentative rendering o a rather obscure paragraph explaining why not the entire
541
542
[5] As or the [remaining] 5292 lượng, [one] measures this amount o money or the awarded [6] aoresaid corpus o 328 unctionaries. Ten [i] this money is given to this establishment [= unctionaries] and [they are] awarded in the same way, [7] [it is as] i [one] uses the method o ‘ at-rate distribution’. [One] sets above77 the number o the persons, [one] sets below 78 this [amount o] money.79 Using the divisor [one] divides [the amount o money] by the ‘evaluation division’ and by ‘returning [division]’;80 [8] [one] immediately establishes that every person obtains 16 lượng 1 tien 3 phan 4 li.81 Tus this money [could] not [9] be entirely [paid and] there would be a remainder o 4 phan 8 li.82 o get the actual [value], one can use the methods o reduction o ractions to common denominator and o injection [o integer parts o mixed ractions] into numerators.83 [I] these men’s categories /p. 31b/ [1] are classi ed as high and low, being at the same level [only within] one rank, [then] this action is not to be called ‘analysing [correctly] the inner structure [o it]’84 and there is [something] [2] unreliable. [I] this is so, then the ‘method o at-rate distribution’
77 78 79
80
amount o 16000 lượng was distributed among the unctionaries and why the 10708 lượng should have been deducted rom the srcinal amount o 1000 cân. Or: ‘ rstly’. Or: ‘secondly’. I the counting instrument supposed to be used is the counting rods, then the positions o the operands (divisor in the upper position and the dividend in the lower position) differs rom the classical Chinese disposition o the operands represented with the counting rods (divisor below and the dividend above) described in the Sun zi suan jing(see SJSSb: 262). Te standard methods o division perormed with the abacus I am aware o all assume that the dividend is to be set in the lef (= upper) part o the abacus, and the divisor in its right part. I am thankul to K. Chemla who drew my attention to this particularity o the Vietnamese method (private communication, 2008). For a very short discussion o the methods o division shang chu and gui (in Mandarin transcription o the characters) mentioned here see LD1987: 181–3.
81 82 83
84
Indeed, 5292 ÷ 328 = 16.134(14634). Tat is, 5292 − 328·16.134 = 5292 − 5291.952 = 0.048. Tis phrase can be understood as saying that one can obtain an exact value i a common raction is used instead o decimal one. Tis rather rough translation o the expression (qing qi li in Mandarin transcription) would require a long discussion o the term qing which cannot be offered here; the interested reader is reerred to CG2004: 970 or an interpretation o the term as employed by Liu Hui.
Demonstration in Chinese and Vietnamese mathematics
[3]
[4] [5] [6]
[7] [8] [9]
cannot be applied to all [the unctionaries]. Tis is why [one] does not need to ‘dispose and arrange’ [the counting rods in order to solve the problem in this way] and [can] trust [what was stated] in the problem [viz., that the at-rat e distribution method cannot be used]. It is already clear that this is so! Also, ranging the people according to their unequal capacities, [one has to give them] larger or smaller awards. So, those who are superior will obtain more, those who are inerior will obtain less. One distributes it [according to] unequal ranks. Te [distribution] pattern certainly [should be] like this. Tereore [one will] use this amount o 5292 lượng to distribute this [money] among the aorementioned corpus o 328 persons [while applying] the ‘weighted distribution’ [method] and having the number o the people subdivided into three ranks. Rank A: 8 persons, each person obtains 7 parts. Rank B: 20 persons, each person obtains 5 parts. Rank C: 300 persons, each person obtains 2 parts. Tis is the method o ‘weighted distribution’ or [this] problem. Tis method should be applied [as ollows]: rst o all, [I]85 set [on the counting device] 8 persons o rank A, multiply them by 7 parts, obtain
the product, 56 /p. 32a/ [1] parts–multiples.86 Again [I] set 20 persons o rank B, multiply them by ve parts, obtain the product, 100 [2] parts–multiples. Also [I] set 300 persons o rank C, multiply them by two parts, obtain the product, 600 parts–multiples. [3] Ten in an auxiliary [position o the counting instrument I] add the three [amounts] o parts–multiples, and obtain in total 756 parts– multiples. [4] [I] take it as the ‘norm’ [= divisor]. And at this moment [I] set 5292 lượng o this money to be the dividend. Ten [5] [I] divide [this dividend] by the norm, set it [= the result, on the counting instrument], and [thus] obtain [that] one part–multiple equals seven lượng. [I] keep it [on the counting instrument] as the ‘constant norm’ and multiply by it 85
86
I translate this part o the examination paper in rst person, since its imaginary author is assumed to perorm operations with a counting device (hence ‘set’) and to comment on them. On the term ‘part–multiple’ (Chinese fenlü ) see the discussion above, pp. 529–30.
543
544
[6] the parts-multiples o each rank. Tat is, rst o all, I shall take the seven parts o each [unctionary] o rank A, multiply it [by seven lượng], and establish that each man o rank A [7] obtains 49 lượng o silver. Ten again, [I] take ve parts o each [unctionary] o rank B, also multiply it [by seven lượng], establish that each person o rank B [8] obtains 35 lượng o silver. Again, [I] take two parts o each [unctionary] o rank C, also multiply it [by seven lượng], [9] establish that each person o rank C obtains 14 lượng o silver. Here [the computation] o the [amount o] silver allotted to each person o each rank is already /p. 32b/ [1] completed. As or the due [amount o money] or each [rank], [I take] the aggregated parts o each rank, and [I] shall similarly multiply [it] by the ‘constant norm’, and thus [2] will know the due amounts. Tat is, [I] multiply the aggregated 56 parts–multiples or the rank A [by the ‘constant norm’] and establish the [amount o] silver due to [all the unctionaries o] the rank A, [3] 392 lượng. Again, [I] multiply the aggregated 100 parts–multiples or
[4]
[5] [6] [7]
[8]
the rank B [by the ‘constant norm’] and establish the [amount o] silver due to [all the unctionaries o] the rank B, 700 lượng. Also, [I] multiply the aggregated 600 parts–multiples or the rank C [by the ‘constant norm’] and establish the [amount o] silver due to [all the unctionaries o] the rank C, 4200 lượng. Te silver due to each rank is thereby already established! As or the ‘return to the srcin’,87 [I] add together the amounts o silver due to the three ranks A, B and C, uniting them together, and establish the srcinal [amount o] silver, 5292 lượng. [I,] so-and-so, am not clever as ar as the ‘learning’ [is concerned]; [I only] roughly know the ‘arrangement and disposition’ [o the counting rods] or the [computational] methods and schemes ( );88 [I] am bad at what [I] do, and still do not know how to ‘distinguish and differentiate’89 between ‘excessive and insufficient’. Now, in answering the question [I] came up with a shallow and approximate answer
87
88
89
Tat is, the check-up conducted in order to veriy whether the answer obtained corresponds to the conditions o the problem. Te imaginary examinee apparently makes an allusion to the nal part o theproblem mentioning ‘. . . [those who] arrange and dispose [the counting rods] . . .’ Once again, this is a quote rom the nal part o the pro blem ‘. . . [those who] . . . distinguish and differentiate’.
Demonstration in Chinese and Vietnamese mathematics
[9] to it. Was it correct or wrong? Hope that those in charge will make [a right] decision. With best regards.90
Appendix Tis Appendix contains a list o the titles o Chinese mathematical treatises mentioned in the paper in Chinese characters, pinyin transliteration, Wade-Giles transliteration used in Anglo-Saxon countries and in aiwan, my translation o the title, and the translation adopted in Martzloff 1997.91 Te treatises are listed alphabetically according to the pinyin transliteration o their titles.
90 91
A ormal ending. Martzloff 1997: 17, 20, 56, 124–5, 129.
545
ed pto da no it las na r
de tp od a no it al sn ar
eisl G -e da W
in
anl ito at pu m o C dn al Is ae S
erp pa s hti in
es tia er tl a on tia tu p om C
tion rea tli sn ar t
gn ih c usan oa iat H
ip fa an gnsu hca ihu C
ngij na su oa ida H
ie ibl af na su nag zh uiJ
79 91 olff zt ra M
n iot ar eti ls na rt iny ni P
es en hi C ni e til
on an C ] m el bo rp a hti w gn i nin ge [b
dn a sli a se at uo ab
e isv ne he rp om C yll u F eh to m ui dn pe ocm ta er G
anl iot tua p om C ]o no it ecll o C [ o sd oh te m anl oi ta upt ocm
sr tep ha C en i N ni sd oh te M ]r i het dn a[ se rio ge act en ni
dn a s em l obr P w e [N ht i w
s icr en ge
yg ol an A by d sei ev D ]s el u R
sn ito p rci se rP aln iot tua ]s p el m u o w R C se urd ec rpo la n iot at up m o C dn a s m leb or Pt ne cin [A ht i
sr tpe ah C en i N in
se rio ge ta c en in o
se reg o e s D n ee iiot r d T ar eh e to ht tr no A s e eto
eh t o onn a C aln iot tua p m o
nte ic n A o oni ta un it no C C no e its ae o trl n a iot on a tia uin tu t p noc om e C th
tn ei nca o ] nso iitd ar [t
s]n iac it a m het a [m
alc goi orl e m u N -o hm itr TN A s re d ksn ud inh ra oec be ee r f p e hrt he lg o t ni s o re ebr sd bm or u m u ce n N R o
aun ’h c a ite l
u sh an gnsu hca ihu C
g inh nc usa uk ’hi C
na uq da
u sh na su nag hz iuJ
ngij na su u ig Q
hus gn ed anS
uhs gn te na S
o ec uro Sl ar en e
s sse ec or P G sl at ne am dn u yit de re zir ts a po m ro m Su
ih c u sh uh S
g uns t ngu’ t af aun S
iy ji hus uh S
gn oz g otn fa n uaS
s ohd te M no tai tu p m o C sd oh te m l noa it at up m oc o
] edt se gg us no i atl nsa rt o N [ sn o tia utp m oc on gn irit W
shu hsu n uSa
dos r gn tni ocu h it w
eh t ot n ito ucd or tn I gn nri ela eh t ot oni ctu do tr In
gn e im ’ eühc hs n uSa
ec ne ic Sl a oni ta utp om C
onn a C la no tia tu p m o C ’is zn u S
het o no na C la oni ta utp om C
s oni tc eS veit ar t ins i m d A vei F
s oin ta tu p ocm o
sei ta er tl a oin ta tu p m o C
sei ta er tl a oin ta tu p om C
s net m rat p de ev o
gn i ch aun s uz t unS
uSn re ts a M o
gn hic uans oa’ st u W
sn ito irp cs er Pl a oni ta utp om C se urd ec rpo la n iot at up m o C
sc i sas l C ev iF het o la c ssi lac ve e th ni ]d n ou [
la oni t tau p m o C ’s nga Y u hoa i
s’ n iaj iu Q gn ha
es ti ear tl an ito at up om C
nga Y ouh ai X o
es ti ear tl an ito at up om C
g inh c
anu s eni h ’iu-ch C gn ha C
no an X C Z
kso bo
onn a C anl iot at pu m o C
sn ito at up om C icn o onm Z G
de] ste g usg oni ta ls na tr o N [
ouh Z o no m on g eh t on
se udr ceo pr gn dni e M
o no na C yt asn y D uo h
ajni ui Q gn ha Z o
es tia er tl a on tia tu p om C
g inh c
gn i ch anu s pei)( ip uo h C
u sih uh C
y]t ans y [d
hus n sua g inh c u W
na su g anY uo hias H
ngij na su n jiau i Q nag hZ
gn ji n uas ib uo hZ
u shi huZ
hsu uh s na Su
g en m qi e xu n uSa
g inj anu izs nu S
ngij na su o ca u W
hus na su gn ji u W
gn ji n uas gn a Y ouh iaX
548 Bibliography Primary sources
CML – Phan Huy Khuông. Chỉ minh lập thành toán pháp (Guidance or understanding o theReady-made Computational Methods). Manuscript A1240 preserved in the library o the Institute or Han-Nom Studies, Hanoi, Vietnam.
DQ –(Great Wu Jing (1450)oJiuthe zhang suan fa bi leimethods da quano nine categories [and compendium computational their] generics). In Zhongguo kexue jishu dianji tonghui (Comprehensive collection o written sources on Chinese science and technology), (gen. ed.) Ren Jiyu, Shuxue juan (Mathematical section), (section ed.) Guo Shuchun vol. : 5–333. Zhengzhou (1993). JS – Liu Xu (gen. ed). Jiu angshu (Old history o the ang [Dynasty]). aibei (1956). SFZ – Mei Rongzhao and Li Zhaohua (eds.). Cheng Dawei . Suanfa tongzong jiaoshi (An annotated edition o the Summarized Fundamentals of Computational Methods). Heei (1990). SJSSa – Qian Baocong (ed.). Suanjing shishu (en canonical books on computation). Beijing (1963). SJSSb – Guo Shuchun , Liu Dun (eds.). Suanjing shishu (en Canonical Books on Computation). aibei 2001). ( SS – uo uo et al. (eds.). Song shi (History o the Song [dynasty]). aibei (1956). SXQM – Zhu Shijie (1299) Suan xue qi meng (Introduction to the learning o computations). InZhongguo kexue jishu dianji tonghui (C omprehensive collection o written sources on Chinese science and technology), (gen. ed.) Ren Jiyu , Shuxue juan (Mathematical section), (section ed.) Guo Shuchun , vol. : 1123–200. Zhengzhou (1993). LD – Zhang Jiuling et al. ang liu dian (Te six codes o the ang [dynasty]). In Wenyuange Siku quanshu, vol. 595: 3–293. aibei (1983). XS – Ouyang Xiu and Song Qi . Xin ang shu (New history o the ang [Dynasty]). aibei 1( 956).
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An Pingqiu , Zhang Peiheng (eds.) (1992) Zhongguo lidai jinshu mulu (Catalogues o the Forbidden Books in China Trough Generations).Zhongguo jinshu daguan (General History o Forbidden Books in China), vol. . aibei. Bailly, C. (1857) Réforme de la géométrie. Paris.
Demonstration in Chinese and Vietnamese mathematics Berezkina, E. (1975) ‘Matematicheskii traktat o prodolzhenii drevnikh [metodov]. Vang Syao-tun’ (Mathematical treatise on the continuation o ancient [methods], [by] Wang Xiaotong), Istoriko-matematicheskie issledovaniya20: 329–71. Biot, É. C. (1847) Essai sur l’histoire de l’instruction publique en Chine, et de la corporation des lettrés, depuis les anciens temps jusqu’à nos jours: ouvrage entièrement rédigé d’après les documents chinois . Paris. Cullen, C. (1993a) ‘Chiu chang suan shu [= Jiu zhang suan shu]’, inEarly Chinese exts, ed. M. Loewe. Berkeley, CA: 16–23. (1993b) ‘Chou pi suan ching [= Zhou bi suan jing]’, inEarly Chinese exts, ed. M. Loewe. Berkeley, CA: 33–8. (1996) Astronomy and Mathematics in Ancient China: Te Zhou bi suan jing. Cambridge. (2004) Te Suan shu shu ‘Writing on reckoning’: A ranslation of a Chinese Mathematical Collection of the Second Century , With Explanatory Commentary. Needham Research Institute Working Papers vol. . Cambridge. Dauben, J. W. (2008) ‘ Suan shu shu: a book on numbers and computations’, Archive for the History of Exact Sciences 62: 91–178. Des Rotours, R. (1932) Le traité des examens, traduit de la Nouvelle histoire des ’ang (chap. 44–45). Paris. Ennis, . E. (1936) French Policy and Developments in Indochina. Chicago, IL. Gillon, B. S. (1977) ‘Introduction, translation, and discussion o Chao ChunCh’ing’s “Notes to the Diagrams o Short Legs and Long Legs and o Squares and Circles”’,Historia Mathematica4: 253–93. Guo Shuchun (2001). ‘Guanyu “Suan jing shi shu”’ (On the Suan jing shi shu [en classical mathematical treatises]). InSJSSb: 1–28. Hucker, C. O. (1985) A Dictionary of Official itles in Imperial China. Stanord, CA. Lee, . H. C. (2000). Education in raditional China: A History. Leiden. Li Di (1982) ‘Jiu zhang suan shu’ zhengming wenti de gaishu “ ” (An overview o the controversial questions [related to] the Jiu zhang suan shu). In Jiu zhang suan shuyu Liu Hui (Te Jiu zhang suan shuand Liu Hui), ed. Wu Wenjun. Beijing: 28–50. Li Yan (1977). ‘ang Song Yuan Ming shuxue jiaoyu zhidu’
(Te system o [Chinese] mathematics education the ang, Song, , Zhong Yuan and Ming dynasties). In Li Yan suan shi o luncong (Collected Papers on the History o Chinese Mathematics). aibei : vol. .1: 253–85. Libbrecht, U. (1973) Chinese Mathematics in the Tirteenth Century:Te Shu-shu chiu-chang of Ch’in Chiu-shao. Cambridge, MA. Luro, J.-B.-E. (1878) Le pays d’Annam:Étude sur l’organisation politique et sociale des annamites. Paris.
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550 Martzloff, J.-C. (1997) A History of Chinese Mathematics. Berlin. Mikami Yoshio (1913) Te Development of Mathematics in China and Japan. Leipzig. (Reprinted New York 1961.) Needham, J. (with the collaboration o Wang Ling) 1( 959). Science and Civilisation in China, vol. . Cambridge. Nguyễn Danh Sành (1961) ‘Contribution à l’étude des concours littéraires et militaires au Viet-Nam’, PhD thesis, University o Rennes. Richomme, M. (1905) ‘De l’instruction publique en Indo-Chine’, PhD thesis, Faculté de droit et des sciences économiques de l’Université de Paris. Robson, E. (2001) ‘Neither Sherlock Holmes nor Babylon: a reassessment o Plimpton 322’, Historia Mathematica28: 167–206. Schreiner, A. (1900) Les institutions annamites en Basse Cochinchine. Saigon. Siu Man-Keung (1995) ‘Mathematics education in ancient China: what lessons do we learn rom it?’ Historia Scientiarum4: 223–32. (1999) ‘How did candidates pass the state examinations in mathematics in the ang dynasty (618–917)? – Myth o the “Conucian-Heritage-Culture” classroom’, inActes du Colloque ‘Histoire et épistemologie dans l’éducation mathématique’. Louvain: 321–34. (2004) ‘Official curriculum in mathematics in ancient China: how did candidates study or the examination?’ In How Chinese Learn Mathematics: Perspectives from Insiders, ed. Fan Lianghuo, Wong Ngai-Ying, Cai Jina and Li Shiqi. Singapore: 157–85. Siu Man-Keung and Volkov A., (1999) ‘Official curriculum in traditional Chinese mathematics: How did candidates pass the examinations?’, Historia Scientiarum 9: 87–99. Sun Peiqing (ed.) (2000) Zhongguo jiaoyu shi (History o education in China). Shanghai. ran Van rai 1942) ( ‘L’enseignement traditionnel en An-Nam’, PhD theis, University o Paris. Volkov, A. (2002) ‘On the srcins o the oan phap dai thanh(Great Compendium o Mathematical Methods)’, inFrom China to Paris: 2000 Years’ ransmission of Mathematical Ideas, ed. Y. Dold-Samplonius, J. W. Dauben, M. Folkerts and B. van Dalen. Stuttgart: 369–410. (2007) ‘Geometrical diagrams in Liu Hui’s commentary on theJiuzhang suanshu’, inGraphics and ext in the Production of echnical Knowledge in China, 425–59.ed. F. Bray, V. Doroeeva-Lichtmann and G. Métailié.Leiden: (2008) ‘Mathematics in Vietnam’, inEncyclopaedia of the History of Non-Western Science: Natural Science, echnology and Medicine , ed. H. Selin. Heidelberg: 1425–32. (2009) ‘Mathematics and mathematics education in traditional Vietnam ’, in Oxford Handbook of the History of Mathematics, ed. E. Robson and J. Stedall. Oxord: 153–76.
Demonstration in Chinese and Vietnamese mathematics (orthcoming). ‘Didactical dimensions o mathematical problems: “weighted distribution” in a Vietnamese mathematical treatise’. Vu am Ich (1959) A Historical Survey of Educational Developments in Vietnam. A special issue o the Bulletin of the Bureau of School Service, by College o Education, University o Kentucky, Lexington, 32(2) (December). Wong Ngai-Ying (2004) ‘Te CHC learner’s phenomenon: its implications on mathematics education’, inHow Chinese Learn Mathematics: Perspectives from Insiders, ed. Fan Lianghuo, Wong Ngai-Ying, Cai Jina and Li Shiqi. Singapore: 503–34. Woodside, A. B. (1988) Vietnam and the Chinese Model: A Comparative Study of Vietnamese and Chinese Government in the First Half of the Nineteenth Century. Cambridge, MA. Yan Dunjie (2000) Zu Chongzhi kexue zhuzuo jiaoshi (An annotated edition o the Scienti c Works o Zu Chongzhi). Shenyang. Yushkevich, A. P. (1955) ‘O dostizheniyakh kitaiskikh uchenykh v oblasti matematiki’ (On the achievements o Chinese scholars in the eld o mathematics). Istoriko-matematicheskie issledovaniya (Studies in the history of mathematics) 8: 539–72. (In Russian.)
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16
A ormal system o the Gougu method: a study on Li Rui’sDetailed Outline of Mathematical Procedures for the Right-Angled riangle
In contrast to the deductive structure developed in Euclid’s Elements, which is always taken as the model or ancient Greek mathematical reasoning, the structure o most ancient Chinese mathematical books could be described as that o a collection o problems and procedures. Moreover, these procedures were mostly described within the context o numerical problems. As historians have argued, in some ancient Chinese mathematical texts there are proos establishing the correctness o the algorithms included. However, these proos were mostly written by subsequent mathematicians, and were contained in commentaries attached to related procedures. 1 Tereore, as these proos were speci cally brought to bear on procedures that were taken rom texts that already existed, they could seldom orm a system by themselves, and hence the reasoning model in them looks exible. Tis raises two related questions: when did Chinese mathematicians think o developing a ormal system o mathematics in their books? Moreover, could the mathematical results developed in ancient China be presented systematically and ormally? In this chapter, I shall rely on a Chinese mathematical book, the Gougu Suanshu Xicao (hereafer abbreviated as GGSX, Detailed Outline of Mathematical Procedures for the Right-Angled riangle, 1806), to investigate these questions. Furthermore, I hope that the discussion will shed some light upon questions such as why and in which context a ormal system o mathematics emerged in China.
1
552
Te best-known examples o proos in ancient Chinese mathematical texts are those Liu Hui provided in his commentary to Jiuzhang suanshu. For greater detail, see Guo Shuchun 1987; Chemla 1992; CG2004: 3–70; Wu Wenjun 1978.
A formal system of the Gougu method
1. Detailed Outline of Mathematical Procedures for the Right-Angled Triangle: a formal system for the ‘Procedure of the right-angled triangle (Gougu)’ Te table of contents ofDetailed Outline of Mathematical Procedures for the Right-Angled Triangle (GGSX) In the mathematics developed in ancient China, the study o the numerical relations between the sides o a right-angled triangle (the ‘gougu’ shape) and the side or diameter o its inscribed and circumscribed square or circle ormed a sel-contained system, which was entitled the ‘gougu Procedure’ (, Gougu shu). In contrast to the mathematics developed in Europe, in which the sides o a right-angled triangle were generally named as sides around the right angle and hypotenuse, in ancient China, the two sides o the right angle had different names, the longer one being named gu, and the shorter one gou (Figure 16.1).2 Te GGSX was completed in 1806 by the Chinese mathematician Li Rui (1769–1817). 3 Te whole book was devoted to methods or solving a rightangled triangle when two o the ollowing thirteen items attached to it are known:4 Te gou, the shortest one o the two sides around the right angle Te gu, the longest one o the two sides around the right angle Te hypotenuse Te sum o gou and gu Te difference between gou and gu Te sum o gou and the hypotenuse 2
3 4
In his commentary on Te Nine Chapters of Mathematical Procedures , Liu Hui gave the ollowing de nition: ‘Te shorter side is named gou, (and) the longer side is named gu’ (Liu Hui, Commentary, inJiuzhang suanshu, chapter 9, 1a). When different names are used, it is easy to describe the calculation between them and to name the quantities they yield. In what ollows, we will come back to these quantities. During the sixteenth and seventeenth centuries, some Chinese mathematicians named gu the vertical side o the right-angled triangle, and gou the horizontal side (see Gu Yingxiang, Discussion on Gougu, in Gu Yingxiang, 2a). InGGSX, the author Li Rui named the sides in the ancient way, which he described in his text. Liu Dun 1993. In Yang Hui’sXiangjie Jiuzhang Suanfa (A Detailed Explanation of Te Nine Chapters o Mathematical Procedures, completed in 1261), there is a table containing all o these thirteen items. Guo Shuchun 1988 argues that the main part o this book was written by Jia Xian, and that Yang Hui only provided commentaries on it. I this is so, these thirteen terms were already sorted out in the eleventh century. Note that the names o the last our terms included in Yang Hui Suanfa(Mathematical Methods by Yang Hui) are not the same as those Li Rui uses in his book. For example, Yang Hui (45a) expressed the sum o gou and the sum o gu and hypotenuse as ‘sum o hypotenuse and the sum (ogou and gu)’.
553
554
gu
gou
Figure 16.1 Te gougu shape (right-angled triangle).
Te difference between gou and the hypotenuse Te sum o gu and the hypotenuse Te difference between gu and the hypotenuse. Te sum o gou and the sum (o gu and the hypotenuse) Te sum o gou and the difference (between the hypotenuse and gu) Te difference between gou and the sum (o gu and the hypotenuse) Te difference between gou and the difference (between the hypotenuse and gu).5 I we denote gou, gu and the hypotenuse by a, b and c respectively, able 16.1 below contains the ollowing items: Li Rui’s text is composed o two parts – the table o contents and the main text. Both are presented in the orm o a ormal system. First, let us have a look at the table o contents. Te table o contents o the GGSX is a list o seventy-eight problems. We know rom a basic theorem in combinatorics that i we choose two items out o thirteen, we can have seventy-eight combinations. Tereore, the table o contents o the GGSX in act includes all the problems that can be raised in relation to the topic o the book. Tis means that Li Rui’s solutions to the whole set o problems concerning the right-angled triangle are included in the book. In the table o contents, the problems are laid out according to two different models. We shall come back to them below.
5
One may think that there could be other terms, such as the sum o hypotenuse and the difference between gou and gu. Tat could be denoted as hypotenuse + ( gu – gou). However, it is equal to (hypotenuse + gu)−gou. In act, this table includes the three sides o a right-angled triangle and the positive differences and sums that can be derived rom them.
A formal system of the Gougu method
able 16.1 Te thirteen items o the ‘Gougu Procedure’ a (gou) b (gu) c (xian) b + a (gougu he) b − a (gougu jiao) c + a (gouxian he) c − a (gouxian jiao) b + c (guxian he) c – b (guxian jiao) a + b + c (gouhe he) b + c – a (gouhe jiao) a + c – b (goujiao he) a – c + b (goujiao jiao)
Here is a translation o the rst parto the table o contents, 6 in which gou is rendered as a, gu as b and the hypotenuse as c. I designate the difference in layout by two marks that I place at the beginning o each item. • • • ә ә ә ә • • ә
a, b (being given), nd c a, c (being given), nd b b, c (being given), nd a a, a + b (being given), subtract a rom the sum, the remainder is b, enter into this problem by the procedure o a and b a, b − a (being given), add a to the difference, the sum is b, enter into this by the procedure o a and b a, a + c (being given), subtract a rom the sum, the remainder is c, enter into this by the procedure o a and c a, c − a (being given), add a to the difference, enter into this by the procedure o a and c a, b + c (being given), nd b and c a, c − b (being given), nd b and c b, a + b (being given), subtract the difference rom b, the remainder is a, enter into this by the procedure o a and b
• b, a + c (being given), nd a and c • b, c − a (being given), nd a and c • b, c + b (being given), subtract b rom the sum, the remainder is c, enter into this by the procedure o b and c 6
For the complete table ocontents, see Appendix 1. In the srcinal text, there is no mark at the beginning o each problem in the table o contents. In order to clariy the structure o the table and the book, I attach a mark, a circle or a square, to each problem (see below, p. 570).
555
556
ә b, c − b (being given), add b to the difference, the sum is c, enter into this by the procedure o b and c ә c, a + b (being given), nd a and b • c, b − a (being given), nd a and b ә c, a + c (being given), subtract c rom the sum, the remainder is a, enter into this by the method o a and c ә c, c − a (being given), subtract the difference rom c, the remainder is a, enter into this by the procedure o a and c. ә c, b + c (being given), subtract c rom the sum, the remainder is b, enter into this by the procedure o b and c. • ... • a + b, (b + c)−a (being given), nd a, b, and c (two problems). • ... • b – a, a – (c – b) (being given), nd a, b, and c (our problems). • a + c, (b + c)-a (being given), nd a, b, and c (two problems). • ... • a + c, a – (c – b) (being given), nd a, b, and c (two problems). • ... • c – a, a+ (c – b) (being given), nd a, b, and c (our problems). • . . .7 Te table o contents maintains this ormal order. For every problem in it, two items are given. Te rst item is chosen ollowing the order o able 16.1, while the second item is the one coming afer the rst item given in able 16.1 and is also chosen according to the order o able 16.1. For example, in the rst orty-two problems, the ollowing pairs o items are given: a, b; a, c; a, b + a; a, b − a; a, c + a; a, c − a; a, b + c; a, c − b; a, a + b + c; a, b + c − a; a, a+c−b; a, a-c+b; b, c; b, b+a; b, b−a; b, c+a; b, c − a; b, b + c; b, c − b; b, a + b + c; b, b + c − a; b, a + c − b; b, a−c+b; c, b + a; c, b − a; c, c + a; c, c−a; c, b + c; c, c−b; c, a + b + c; c, b + c−a; c, a+c−b; c, a − c + b; b + a, b − a; b + a, c + a; b + a, c − a; b + a, b + c; b + a, c−b; b + a, a + b + c; b + a, b + c − a; b + a, a + c − b; b+a, a − c + b; ...
Trough this arrangement, the author o the GGSX, Li Rui, gave every problem in the book a de nite position in the table o contents and i we 7
Li Rui 1806, able o contents (Mu,), 1a–6b.
A formal system of the Gougu method
want, we can gure out the position o a problem by the items given in the problem.8 From the above discussion, we see that the list in the table o contents displays a ormal system. Let us analyse the structure o the outlines o problems included in the table o contents. I have translated the beginning o the list o problems into English, and I attached a symbol to each problem at the beginning o the translation o its outline. Te layout o all the problems marked with a black circle is generally the same. All contain two sentences. Te rst sentence is composed o the names o two items, without any conjunction between them. Te second begins with a verb, qiu (, ‘ nd’), and ends with the names o the sides o a right-angled triangle which are sought or in that particular problem. Te problems marked with a square are composed o three parts. Te rst sentence also consists o the names o two items, without any conjunction. Te second part contains one or two procedures. Trough the procedure, the items given in the rst sentence are transormed into items mentioned in a previous problem. Te third sentence is a statement, which begins with yi (, ‘according to, relying on’). A procedure named by the two items that are the result o the transormation in the second sentence is then mentioned, and the sentence ends with ruzhi (, ‘enter into it’).9 Consequently, it is clear that both the order in which the list o problems is given in the table o contents and the way in which the outlines o the procedures are given are all arranged in a systematic way. However, this is 8
9
Trough this arrangement, Li Rui also ensured that he would not leave any problem out. Another Chinese mathematician, Wang Lai, one o Li Rui’s riends, gave the general solution to the problem o computing the number o combinations on things taken two or more at a time. See Wang Lai (1799?). Wang Lai does not provide the exact date o the completion o this book; however, he mentions that he attained the results contained in it in 1799. For details o the compilation o Wang Lai’s book, see Li Zhaohua 1993 . Te whole item means that one solves the problem according to the procedure o the problem in which the resultant items are given. Only the problems marked with a black circle are contained in the main text o the book, the ones marked with a square appearing only in the table o contents. In act, through the sentences just described, these problems are transormed into one o the other problems. Tese sentences not only give the way o transorming one problem into another, but also give the reasons why this problem could be solved by the procedure mentioned in the third sentence. For example, the ourth problem reads ‘a, a + b (being given), subtract a rom the sum, the remainder is b, enter into this by the procedure o a, b.’ Te rstsentence makes precise the data given in the problem, and the last one indicates that the procedure or the rst problem solves this new problem, while the middle one yields the reason or this, that is, (b + a ) − a = a. In other words,a is given, and it is shown how b can be ound. Te problem can hence be solved with the procedure o the problem, the data o which are a and b. In this way, even though only the problems marked with black circles are solved in the main part o the book, the book indicates how to solve the entire set o seventy-eight problems. We will come back to this point later.
557
558
not the only argument on which we rely to reach the conclusion that the GGSX has a ormal struc ture. An examination o the main text o the GGSX also proves revealing and is particularly signi cant or our argument.
Te main text of the GGSX Te main text contains the twenty- ve category items hose (t marked with black circles above and in the complete list o problems in the GGSX as given in the Appendix below). All the seventy-eight problems in the GGSX are solved in terms o these twenty- ve problems.10 Now, let us analyse how Li Rui presents and solves these problems in the GGSX. Te translation o the ourth problem in the GGSX is given here as an example, and reads as ollows. Suppose gou is (equal to) 12, (and) the sum o gu and the hypotenuse is (equal to) 72. One asks how much gu and the hypotenuse are. Answer: gu, 35; the hypotenuse, 37. Procedure: subtract the two squares one rom the other, halve the remainder and take it as the dividend, take the sum o gu and hypotenuse as the divisor, divide the dividend by the divisor, (one) gets the gu; subtract the gu rom the sum, the remainder is the hypotenuse. 0
Outline: set up gu as the celestial unknown; multiplying it by itsel, one gets
0 ,11 1
which makes the square o gu. Further, one places (on the computing surace) gou, 12; multiplying it by itsel, one gets 144, which makes the square o gou.
10
11
See n. 10 . In act, the main text o GGSX contains thirty-three problems. For some problems, a note is attached to the outline, which says ‘two problems’ or ‘our problems’ (see the table o contents). Tis is not simply because Li Rui wants to give more examples to special problems. He has better reasons or this. Te rst kindo problem that is represented by two examples is the one or which ‘a+b and (b+c)−a (being given), [it is asked to] nd a, b, and c.’ For this problem Li Rui gives two examples. One relates to the condition (b+c)−a > a+b, whereas the other illustrates the condition (b+c)–a < a+b. For these two examples, even though the procedure used is the same, in relation to the difference in the conditions, Li Rui has to provide two cases. He gives two different demonstrations and constructs different diagrams or each o them. In the thirteenth century, Li Ye had already encountered this kind o difficulty. Li Rui edited Li Ye’sCeyuan Haijing in 1797, so it is likely that he may have been in uenced by his research on Li Ye. In one problem, Li Rui provided two different groups o answers or a second-degree equation. Tis is due to his study on the theory o equations. For Li Rui’s study on equations, see Liu Dun 1989. In ancient China, the degree o the unknown was indicated by the position o its coefficient. In the GGSX, the degree o an unknown attached to a given coefficient increases rom top to bottom. Tis polynomial is equivalent to 0 + x0+ 1x2. For an explanation o the tianyuan method, see LD1987.
A formal system of the Gougu method
144
Adding the two squares yields
0
, which makes the square o the hypotenuse.
1
(Put it aside on the lef.) Further, one places the sum o gu and the hypotenuse, 72; subtracting rom this the celestial unknown, the gu, one gets 5184
72
−1
, which makes the
hypotenuse. Multiplying it by itsel, one gets −144 , which makes a quantity equal 1
to (the number put aside on the lef). Eliminating with the lef (number), one gets 5040
2520
; halving both o them, one gets , the upper one is the dividend, the −144 − 72 lower one is the divisor, (dividing), one gets 35, hence the gu. Subtracting the gu rom the sum o gu and the hypotenuse, 72, there remains 37, hence the hypotenuse. Tis conorms to what was asked (see Figure 16.2). Explanation: in the square o the sum, there is one piece o the square o gu, one piece o the square o hypotenuse, and twice the product o gu and the hypotenuse. [Subtracting the square o gou rom within it, the remainder is twice the square o gu, subtracting the square o gou rom the square o the hypotenuse, the remainder is the square o gu]12 and twice the product o gu and hypotenuse. Halving them makes the square o gu and the product o gu and hypotenuse. Join the two areas together, hence this is the multiplication by one another o gu and the sum o gu and hypotenuse, so, dividing it by the sum, one gets the gu.13
Except or the rst three problems, the layout o every problem is exactly the same as in the above example. In other words, the text or each problem is composed o the same components: a numerical problem, an answer to the problem, a general procedure without speci c numbers, an outline that sets out the computations using the tianyuan algebraic method, and an explanation, which may be regarded as a general and rigorous proo with a diagram.14 Furthermore, the order o the different parts remains the same throughout the whole book.15 Consequently, not only do most o the 12
13 14
15
In the srcinal text, characters contained in square brackets were printed in smaller size than the main text. Tis arrangement indicates that Li Rui did not think that this part belonged to the main text. In act, this part provides the reasoning o the previous statement. Li Rui generally provides reasons or his argument and statements in this way throughout the whole work. Li Rui 1806: 8b–9a. Te explanation does not discuss the meaning o the problem or the procedure, but it highlights the reasons why the procedure given is correct. Tis is why it can essentially be considered as a proo o the procedure ollowing the problem. In the rst problem, Li Rui triesto reconstruct the demonstration o the ‘Pythagoras theorem’ (which in present-day Chinese is called the ‘Gougu theorem’, whereas in the past
559
560
gou
subtract
gou
e s u n e t o p y h
square of gu
gu
gu
hypotenuse
Figure 16.2 Li Rui’s diagram or his explanation or the ourth problem inDetailed Outline of Mathematical Procedures for the Right-Angled riangle .
problems in the GGSX have the same layout in general, but also the parts o every problem are similarly arranged in a ormal way. Concerning the ve parts o each problem in the main text o GGSX, there is not much that can be said about the rst three. Te structure o the presentation o each problem and its solution remains the same or the whole book. Te only changes concern the numerical values in the problem and the answers as well as the concrete procedures. We shall ocus our analysis on the last two parts o the presentation o each problem: the outline and the explanation. Let us begin our analysis o the structure o these parts o the problems in the GGSX with an inspection o the outline o the calculations. For each o the thirty-three problems contained in the book, Li Rui gives an outline o the calculations. And except or the rst three problems, they all bring into play the tianyuan method.16 Te rst step is to set up thecelestial unknown. In addition, Li Rui ollows a strict rule in choosing the unknown. Te rule
16
it was called ‘Gougu procedure’), as given by Liu Hui around the year 263. Strictly speaking, the demonstration is not a rigorous one, and it is unknown whether it re ects Liu Hui’s srcinal proo or not. For Li Rui’s demonstration o the Pythagorean theorem, see ian Miao, orthcoming. For Liu Hui and his proo o the Pythagoras theorem, see Wu Wenjun 1978 ; Guo Shuchun 1992; Chemla 1992; CG2004. ianyuan algebra is a method or solving problems. It makes use o polynomials with one indeterminate, expressed according to a place-value system, in order to nd out analgebraic equation that solves the problem. Te equation was also written down according to a place-
A formal system of the Gougu method
is: i a, gou, is not known in the problem, he sets a as the unknown. I a is known, and b is not known, he sets b as the unknown.17 Te second step o the outline consists o establishing the tianyuan equation. o analyse this step, we shall give two examples to show the ormal way in which Li Rui does this. Problem 9 reads as ollows: Suppose there is the gou (which is equal to) 33, the difference between the hypotenuse and gu (which is equal to) 11. Ask or the same items as the previous problem (gu and hypotenuse).
0
Outline: set up gu as the celestial unknown, multiplying it by itsel, [one] gets
0, 1
which makes the square o gu. Further, one places gou 33; multiplying it by itsel, one gets 1088, which makes the square o gou. Adding the two squares 1088
together yields
0
, which is the square o the hypotenuse. (Put it aside on the
1
lef.) Further, one places the difference between the hypotenuse and gu, 11, adding 121
it to the celestial unknown, gu, one gets
11 1
, multiplying it by itsel, one gets
22
,
1
which makes a quantity equal to (the number put aside on the lef). Eliminating with the lef (number), one gets −868 , halving both the upper and the lower, one gets 22 −484 , the upper one is the dividend, the lower one is the divisor, (dividing), one 11
gets 44, which is the gu.18
All the thirty problems ollow the same pattern. First, Li Rui tries to nd the expression o gou and gu on the basis o the items that are known. He then multiplies each by itsel respectively, adds the squares to each other, and puts the result on the lef. In a second part, he looks or an expression
17
18
value notation. Te expression o polynomials and equations makes use o the representation o numbers with counting rods in a place-value number system. Moreover, the notation uses the tianyuan, which is supposed to be the unknown and which is represented by a position. Tis method ourished inthirteenth- to ourteenth-century China. However, it seems that Chinese scholars and mathematicians could no longer understand this algebraic method by the sixteenth century. In the eighteenth century, Chinese mathematicians rediscovered this ancient method, and Li Rui, author o GGSX, made the most outstanding contribution to restoring it. For tianyuan algebra, see Qian B aocong 1982. On the revival o the tianyuan method in eighteenth-century China, see ian Miao 1999. Only in the third problem, in which a and b are known, is c chosen as unknown. Tis problem is solved by a direct application o the Pythagorean theorem, and thus the tianyuan method is not used. Li Rui 1806: 9b.
561
562
or the hypotenuse, and squares it. Finally, by eliminating the square o the hypotenuse and the expression put on the lef side, he gets the equation. Te second example (problem 58) shows that Li Rui deliberately ollowed the same pattern in the whole book. Suppose there is the sum o gou and the hypotenuse (equal to) 676, the difference between the sum (o gu and the hypotenuse) and gou is 560. One asks how much the same items as in the previous problem (gou, gu and the hypotenuse) are. 0
Draf: set up gou as the celestial unknown, multiplying it by itsel, one gets
0 , which 1
makes the square o gou. Further one places the sum o the hypotenuse and gou, 676, and subtracting gou rom it, one gets the ollowing:
676
−1
, which is the
hypotenuse. Further, one places the difference between the sum and gou, 560; 560
adding the celestial unknown gou to it, one gets the ollowing ormula,
1
, which
676
makes the sum o gu and the hypotenuse. Subtracting the hypotenuse, −1 rom it, one gets
−116 2
, which makes gu; multiplying it by itsel, one gets the ollowing
13456
ormula, − 464 , which is the square o gu. Adding the two squares together, one 4
13456
gets the ollowing ormula: − 464 , which makes the square o the hypotenuse. (Put it 5
676
456976
on the lef.) Further, multiplying the −1 hypotenuse, −1, by itsel, one gets −1352 , 1
which makes a quantity equal to (the number put aside on the lef). Eliminating the −443520
lef (number), one gets 110880
one gets
222
888
; dividing all the numbers rom top to bottom by 4,
4
. Solve the equation o the second degree.
1
One gets 240, which is the gou. Get the gu and hypotenuse according to procedure. Tis answers the problem.19
In modern algebra, the above outline could be reormulated into the ollowing procedure: ake a, the gou, as x, then, a2 = x2 as c + a = 676 19
Li Rui 1806: 28b–29a.
A formal system of the Gougu method
so c = c + a – a = 676 − x as c + b – a = 560 so, c + b = 560 + x and, b = c + b – c = 560 + x – (676 – x) = – 116 + 2 x then, b2 = 13456 –x + 4x2 and, c2 = a2 + b2 = 13456 – 464x + 5x2 while, c = 676 – x so, c2 = 456976 – 1352x + x2 thus, 13456 – 464x + 5x2 = 456976 – 1352x + x2 so, – 443520 + 888x2 = 0 x = 240.
In this problem, relying on the items given in the outline, Li Rui rst nds the hypotenuse. However, he does not multiply the hypotenuse by itsel, to put the result on the lef side. Instead, he seeks to nd the gu, and, only then, he adds the square o gou and gu and puts the result to the lef. It is only in the second step that he computes the square o the hypotenuse and eliminates the result with the number placed on the lef side. It is clear that the nal equation could not be affected by which number was rst put on the lef side, and there are reasons to believe that Liu Rui certainly understood this point. Only one reason can account or why Li Rui insisted on determining the gu rst, namely, that he wanted to ollow the same ormat in presenting each o the outlines. From the evidence analysed above, we can conclude that throughout the whole book Li Rui ollows a ormal pattern or the outline o calculation. Let us now consider how Li Rui presents his explanations in his book. What kinds o rules does Li Rui ollow to ormulate his proos? Te eighth problem o the book reads: Suppose the hypotenuse is (equal to) 75, and the sum o gou and gu (equal to) 93. One asks how much the gou and gu are.
Te procedure given is as ollows: Subtract the two squares one rom the other, halve the remainder and take it as the negative constant. ake the sum (ogou and gu ) asasthe coeffi o the rst degree o the unknown, and the negative one thepositive coefficient ocient the highest degree o the unknown. Extracting the second degree equation, one gets the gou. Subtracting gou rom the sum, the remainder is gu.20
20
Li Rui 1806: 11b.
563
564
gu
gou
e nus ote hyp
g o u
subtract
h y
difference
d i f f e r e n c e
subtract
p o te
n
u s
e
g o u
s u b t r a c t
g u
h
y p o te
n u s
subtract
e
g o u
gou
gu
Figure 16.3 Li Rui’s diagram or hisfor explanation or the riangle eighth .problem inDetailed Outline of Mathematical Procedures the Right-Angled
Tis procedure may be represented in modern algebraic terms by the ollowing equation: x2 + (a + b)x − [(a + b)2 − c2]/2 = 0 whose solution is x = a. Li Rui’s explanation may be translated as ollows: Explanation: in the square o the sum, there are our pieces o the product o gou (a) and gu (b), one piece o the square o the difference between gou and gu. In the square o the hypotenuse, there are twice the product o gou and gu, and one piece o the square o the difference (between gou and gu). Subtracting one rom the other, theproduct, remainder is twice thethe product o gou andand gu. Halving piece o the which is also product o gou the sumit,oone andone gougets gu minus the square o gou. Tereore, take the sum as the negative coefficient o the rst degree o the unknown.21
Now, let us inquire into the process o explanation (seeFigure 16.3). In the rst step, Li Rui decomposes the two ‘squares’ mentioned at the beginning o 21
Li Rui 1806: 12a.
A formal system of the Gougu method
the procedure and gets the geometrical expression o the difference between them, then he transorms hal o the difference between the two squares, which is the negative constant term o the equation described in the procedure, into an expression involving the unknown,gou, and gu. Ten, he urther changes the product ogou and gu into an expression depending on the unknown, gou, and the given item,gou+gu. Tis yields the same expression as the equation o the procedure. Tereore, the explanation corresponds exactly to the procedure. With the diagram, the explanation is in act a geometrical proo to account or the correctness o the general procedure. Except or the rst three, all the proos in the book are obtained by exactly the same process. Tereore, we may conclude that the proos are also produced in a uniormly ormal way. o recapitulate, in the whole work Li Rui ollows a ormal way or the outline o the calculation, through which a tianyuan algebraic equation – the procedure – is ound, as well as or his proos. With this ormal structure o the book, he produces a ormal system or the gougu procedure strictly based on traditional methods developed in ancient China. From this, we see that the ancient Chinese methods could be used to present mathematical knowledge in the shape o a ormal system.
2. Li Rui’s intention in developing a formal system of the Gougu methods From the above discussion, we see that the GGSX is shaped as a ormal and complete system or solving right-angled triangles (gougu shape in Chinese). In this section o the chapter, I will tackle two problems. First, did Li Rui deliberately plan his GGSX as a ormal work? I the answer is yes, we shall then seek to understand why he was interested in creating such a ormal system o gougu procedures, and what he wanted to show to his readers through such a system. First, we must establish that Li Rui consciously developed his system. Let us start by summing up the characteristics o the ormal expression o the system in the GGSX. (1) Te organization o the table o contents o the GGSX ollows a consistent pattern. (2) Te layout o the problems in the main text ollows a consistent pattern too. (3) ianyuan algebra is used or all the outlines o calculation in the text except the rst three.
565
566
(4) Li Rui ollows a ormal and systematic way o choosing the unknown, and seeking the equation in the outline. (5) Te proos are derived rom the corresponding procedure strictly using the same process and methods. Now, let us see whether it was necessary or Li Rui to ollow all the steps listed above. It is clear that there should be no need or the layout o the table o contents and all problems in the main text to ollow a consistent pattern. Moreover, most ancient Chinese mathematical books do not share this eature. We may thus saely assume that i Li Rui took the trouble to design his book in this ormal way, he did so intentionally. Let us now come to the third and ourth eatures. In ancient China, the study o gougu procedures has a history that precedes the invention o tianyuan algebra. In the Nine Chapters of Mathematical Procedures, an entire chapter is devoted to gougu problems, or the solution o which procedures are given.22 And we have evidence showing that up to the third century, Liu Hui and Zhao Shuang gave proos to some ormulas.23 Although their diagrams are lost, other books survive that include proos o some o the ormulas, and Li Rui was amiliar with most o them.24 Tereore, he could easily have studied these results and proos. In act, in some cases, the proos could have been more easily and clearly presented without using the tianyuan methods. Tereore, it was not necessary or Li Rui to use tianyuan algebra or all the problems and proos in which he used it. So, it is not ar-etched to conclude that Li Rui chose to use tianyuan algebra deliberately. Furthermore, there was no need or him to ollow exactly the same order to obtain his equations. As we have already showed above, it was not necessary to obtain systematically rst the gou and gu, and only then the xian or hypotenuse. Nor was it necessary to systematically look or the equation on the basis o the Pythagorean theorem, as Li Rui did. A number o ormulas existed in ancient Chinese mathematical books, such as Te Nine Chapters, Yang Hui’sXiangjie jiuzhang suanfa and Li Ye’sCeyuan haijing. Xu Guangqi and Mei Wending also provide several 22
23 24
According to Guo Shuchun, the main part o the Nine Chapters of Mathematical Procedures , including the ‘Gougu’ chapter, was already ormed beore the rst century . See Guo Shuchun 1992. On the ‘Gougu procedure’ in the Nine Chapters, see Guo Shuchun 1992: 83. On Liu Hui’s proo oGougu procedures, see CG2004: 704–7. In 1797, the year he compiled the Chouren zhuan, a collection o biographies o mathematicians and astronomers, Li Rui made a serious study o all the mathematical texts that existed in his time, including Yang Hui’sXiangjie Jiuzhang Suanfa, Xu Guangqi’sGougu yi and Mei Wending’sGougu juyu. On Xu Guangqi’sGougu yi and Mei Wending’s Gougu juyu, see ian Miao, orthcoming.
A formal system of the Gougu method
ormulas in their books. Li Rui studied all o these books beore he compiled the GGSX. Had he wanted to do so, he could have used these ormulas to nd hisequations more easily. Clearly, he insisted on ollowing a uniorm pattern throughout his book. Te same remark applies to the proos. It was also not necessary to ollow exactly the same approach throughout. But again, clearly Li Rui obstinately chooses to stick to a rule he has set or himsel. From the above analysis, one can reasonably conclude that Li Rui deliberately shaped a ormal system o gougu problems in his book. Tis conclusion leads us to our last problem: what did he intend to show his readers in orming such a system? Li Rui’s preace to the GGSX gives us some hints. He writes: [As or] the Dao o mathematics, the important thing is that one must thoroughly understand the great principles (Yi ). [I one] seeks [methods] by minor parts, even i his [method] is in accordance (with the problem) in number, it can not be looked upon as a method. In the year o Bingyin, Xu Yunan (Naian) and Wan Xiaolian (Qiyun) studied with me, [the knowledge they learned] also came down to gougu mathematics. In the ree time between our discussions, [I] compiled this book and showed it to them. In order to (let them) know that even i procedures are produced according to [speci c] problems, they still have a consistent [reason 25
behind] them.
Tis passage rom the preace shows clearly that Li Rui did not aim at achieving new discoveries when he composed the GGSX. His aim was to show that there was a consistent reason or theory in mathematics. His essential motivation or writing the book was without doubt didactic.26 However, there may have been another reason why Li Rui wrote such a book. Possibly he hoped to show that the mathematical results developed in ancient China had consistent reasons and had their own system. His intention in doing so might have been to reject the opinion that Chinese mathematical books only provided procedures or concrete problems. I do not have hard evidence to support my argument, but considering the context within which the GGSX was compiled sheds some light on Li Rui’s intention and provides additional support to my argument. In 1607, the rst Chinese translation o Euclid’s Elements (the rst six books) was published under the title Jihe yuanben.27 Te two translators, Matteo Ricci and Xu Guangqi, claimed that giving reasons or mathematical 25 26 27
Li Rui 1806: preace. See Liu Dun 1993. On the transmission o the Elements in China, see Engelriet 1998, Engelriet 1993.
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methods and leaving the reader with no doubt about mathematical knowledge were the essence o Western mathematics. wo years afer the publication o the Jihe yuanben, in 1609, Xu Guangqi composed Gougu yi (Te Principle of Gougu).28 o interpret the word yi, we have to brie y mention what Xu Guangqi said in the preace o another book, Celiang fayi (1607). Peter Engelriet offers the ollowing analysis: He [Xu Guangqi] makes a distinction that proves very important in his conception o Western mathematics: a distinction between methods and yi. Te word yi can take on a wide range o meanings, but it is obvious that in this context it must reer to the proos and explanations given in Western mathematics. For Xu Guangqi states explicitly that only afer the Jihe yuanben had been translated was it possible to transmit the yi o the metho ds. Moreover, the Western methods o surveying are not essentially different rom the methods transmitted in the Zhoubi suanjing and the Jiuzhang suanshu. What makes Western mathematics more valuable is that it supplies explanations which show why the methods are correct.29
In his Gougu yi, Xu Guangqi sums up the main topics Chinese mathematicians addressed with respect to gougu problems. He stresses that these problems could only be solved on the basis o the Pythagorean theorem,30 to be ound as Proposition 47 o Book in the Elements. He argues: In the old Nine Chapters, there are also [methods] o nding the gou and gu rom each other, [ nding] the inscribed square and circle, and [ nding] the sums and the differences rom each other.31 But it is only capable o stating the methods, and it is not capable o discussing its principles (yi). Te methods established [in it] are in disorder and shallow, and do not bear reading.32
What is signi cant or us here is that both Xu Guangqi and Li Rui use the word yi. While Xu argues that Te Nine Chapters did not talk about 28 29
30
31
32
See Engelriet 1998: 297–8. Engelriet 1998: 297. Engelriet discusses the meaning o yi and the srcin o this term in Xu Guangqi ’s book in more detail in Engelriet 1993. Te example Xu Guangqi quotes earlier in his text is rom the Zhoubi suanjing(Te Mathematical Canon of the Zhoubi, dating rom the beginning o the rst century ). Tis text contains a general statement o the Pythagorean theorem, including a paragraph which could be regarded as a general proo o it (see Ch’en Liang-ts’so 1982 , Li Jimin 1993). In the third century, Zhao Shuang and Liu Hui present clearer proos in their commentaries to the Zhoubi Suanjing and Te Nine Chapters, respectively. See Qian Baocong 1982; Guo Shuchun 1985; CG2004: 704–45. Te Nine Chapters of Mathematical Procedures(dated rom the rst century to therst century ). Tis is one o the most important mathematical classics o ancient China. Te ninth chapter o this book is devoted to gougu methods. See Guo Shuchun 1985. Xu Guangqi, preace.
A formal system of the Gougu method
yi, Li Rui argues that one has to understand the yi.33 However, although Xu believed that traditional mathematical learning could not provide any ‘principle’ or the gougu procedure, Li Rui developed a ormal system based on traditional methods and mathematical terms.34 It thereore seems reasonable to assume that one o the reasons why Li Rui wrote the GGSX was that he wanted to demonstrate that the traditional methods could be developed into systems and, in doing so, one could also orm a system o consistent reasoning.35 Let me sum up brie y my conclusions. In 1806, the Chinese mathematician Li Rui shaped a ormal system based on the gougu procedure. In his work, in seeking procedures and the proo o their correctness, Li Rui strictly ollows traditional methods and terms. Tis provides evidence or whether there could have been a ormal system in mathematical research in ancient China. Further analysis shows that Li Rui deliberately constructed such a ormal system. Even i he may have had only a didactical aim in mind, it appears that the context o tension between Western mathematical methods and Chinese traditional methods may well lie at the bottom o Li Rui’s motivation or compiling theGGSX.
Acknowledgements Tis paper was primarily nished during the academic year October 2001– September 2002, when I worked in the University o Paris 7, cooperating with Proessor Karine Chemla and being nancially supported by the Ministryo Research, France. It was presented at the seminar (March 2002 – June 2002) organized by Karine Chemla, Geoffrey Lloyd, Ian Mueller and Reviel Netz. I bene ted a great deal rom the attentive discussion and valuable suggestions o all the participants, including Karine Chemla, Catherine Jami, Geoffrey Lloyd, Ian Mueller, Reviel Netz and Alexei Volkov. Joseph W. Dauben and John Moffett read the outline o this chapter, and provided 33
34
35
In Liu Hui’s commentary (263 oTe Nine Chapters of Mathematical Procedures), he uses the character yi to indicate the reason behind the procedures provided in the Nine Chapters. On the meaning o yi in ancient Chinese mathematical texts, see ChemlaYi ‘ ’ ()’, in CG2004: 1022–3. From 1797 onwards, Li Rui began helping Ruan Yuan tocompile theChouren zhuan (Biographies o mathematicians and astronomers). Te main sources or this book were rom the Siku quanshu. Xu Guangqi’sGougu yi was included in this e ncyclopaedia. Tereore, one can saely assume that Li Rui studied Xu Guangqi’s book and knew Xu’s opinions concerning traditional gougu procedures. For a detailed analysis o Li Rui’ s attitude towards Western and traditional mathematics and detailed arguments concerning the compilation o the GGSX, see ian Miao 1999, ian Miao 2005.
569
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valuable suggestions. I am especially in debt to Karine Chemla and Catherine Jami. As we were discussing and working together requently during the year, it is not easy to identiy all the inspirations I have gained rom them. Tereore, here, I would like to take this chance to express my gratitude to them.
Appendix Te content o the Detailed Outline of Mathematical Procedures for the Right-Angled riangle Problema
Given
Find
•1 •2 •3 4 5 6 7
a, b a, c b, c a, b + a a, b − a a, c + a a, c − a
c b a b, c b, c b, c b, c
•8 •9 10 11 •12 •13 14 15 •16 •17 18 19 20 21 22
a, c + b a, c − b b, b + a b, b − a b, c + a b, c − a b, c + b b, c − b c, a + b c, b − a c, c + a c, c − a c, b + c c, c − b a + b, b − a
b, c b, c a, c a, c a, c a, c a, c a, c a, b a, b a, b a, b a, b a, b a, b, c
•23 •24 25 26 •27 •28 29 30
a + b, a + c a + b, c − a a + b, c + b a + b, c − b b − a, a + c b − a, c − a b − a, b + c b − a, c − b
a, b , c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c
Other
Problem 1 Problem 1 Problem 2 Problem 2
b
Problem 1 Problem 1
Problem 3 Problem 3
Problem 2 Problem 2 Problem 3 Problem 3 Problem 1
Problem 24 Problem 23
Problem 27 Problem 28
Continued
A formal system of the Gougu method
Appendix Continued Problema
Given
Find
31
a + c, c − a a + c, c + b a + c, c − b c − a, b + c c − a, c − b
a, b, c a, b, c a, b, c a, b, c a, b, c
•50a
c + b, c – b a, a + b + c a, c + b − a a, a + c − b a, a − c + b b, a + b + c b, c + b − a b, a + c − b b, a − c + b c, a + b + c c, b + c − a c, a + c − b c, a − c + b a + b, a + b + c a + b, c + b – a
a, b, c b, c b, c b, c b, c a, c a, c a, c a, c a, b a, b a, b a, b a, b, c a, b, c
Problem 3 Problem 8 Problem 8 Problem 9 Problem 9 Problem 12 Problem 13 Problem 12 Problem 13 Problem 16 Problem 17 Problem 17 Problem 16 Problem 16 a+b>c+b–a
•50b •51 52 •53 54 55 •56a
a + b, c + b − a a + b, a + c − b a + b, a − c + b b – a, a + b + c b − a, b + c − a b − a, a + c − b b − a, a − c + b
a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c
a+b
•56b
b − a, a − c + b
a, b, c
•56c
b − a, a − c + b
a, b, c
•56d 57 •58a
b − a, a − c + b a + c, a + b + c a + c, b + c − a
a, b, c a, b, c a, b, c
•58b 59 •60a •60b
a + cb + c − a a + c, a + c − b a + c, a − c + b a + c, a − c + b
a, b, c a, b, c a, b, c a, b, c
•32 33 34 •35
36
37 38 39 40 41 42 43 44 45 46 47 48 49
Other Problem 2 Problem 27 Problem 23 Problem 24 Problem 28
Problem 16 Problem 17 Problem 17 b − a > a − c + b; (b − a) − (a − c + b) > a – c + b b − a > a − c + b; (b − a) − (a − c + b) < a − c + b b − a < a − c + b; (a – c + b) − (b − a) > b − a b − a < a − c + b; (a − c + b) − (b − a)< b − a Problem 12 a+c> b+c−a a+c< b+c−a Problem 12 In this Problem, two answers are given. Tis means there are two different right-angled triangles with the same data a + c and a−c + b Continued
571
572
Appendix Continued Problema
Given
Find
•61 62 •63a
c−a, a+b+c c−a, c+b−a c − a, a + c − b
a, b, c a, b, c a, b, c
•63b
c − a, a + c − b
a, b, c
•63c •63d 64 65 66 •67 •68 •69 •70 71 72 73 74 75
c − a, a + c − b c − a, a + c − b c − a, a − c + b c + b, a + b + c c + b, b + c − a c + b, a + c − b c + b, a − c + b c − b, a + b + c c − b, b + c − a cv− b, a + c − b c − b, a − c + b a + b + c, b + c − a a + b + c, a + c − b a + b + c, a − c + b
a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c a, b, c
76 77
b + c − a, a + c − b a, b, c b + c − a, a − c + b a, b, c a + c − b, a − c + b a, b, c
78
Other Problem 13 c − a > a + c − b, (c − a) − (a + c − b)> a + c − b c − a > a + c − b, (c − a) − (a + c − b) < a + c − b c − a < a + c − b, (a + c − b) − (c − a) > c − a c − a < a + c − b, (a + c − b) − (c − a) < c − a Problem 13 Problem 8 Problem 8
Problem 9 Problem 9 Problem 8 Problem 12 Problem 16 Problem 17 Problem 13 Problem 9
Notes: a Te sign ‘•’ indicates problems that were also discussed by Xu Guangqi in the Meaning of Gougu. b Tis means ‘Solve this problem using the method o problem 1’. Similarly hereafer.
Bibliography Chemla, K. (1992) ‘Résonances entre démonstration et procédure: remarques sur le commentaire de Liu Hui aux Neuf Chapitres sur les Procédures Mathématiques’, in Regards Obliques sur l’Argumentation en Chine, ed. K. Chemla. Extrême-Orient, Extrême-Occident14: 91–129. Ch’en Liang-ts’so (Chen Liangzuo) (1982) ‘Zhao Shuang Gougu yuanang tuzhu zhi yanjiu ’(Research on the commentary by Zhao Shuang on the gures o thebase and the height, the square and the circle). Dalu zazhi 64: 33–52. Engelriet, P. (1993) ‘Te Chinese Euclid and its European context’, in L’Europe en Chine, ed. C. Jami and H. Delahaye. Paris: 111–35. (1998) Euclid in China: Te Genesis of the First ranslation of Euclid’s Elements in 1607 and its Reception up to 1723. Leiden.
A formal system of the Gougu method
Gu Yingxiang (1553) Gougu Suanshu (Mathematical Procedures on Gougu), 1553 edn. Guo Shuchun (1985) ‘Jiuzhang suanshu “Gougu” zhang de jiaokan he Liu Hui gougu lilun xitong chutan ’ (Editorial research on the Gougu chapter in the Nine Chapters and the system o Liu Hui’sgougu theory). Ziran kexue shi yanjiu4: 295–304. (1987) ‘Shilun Liu Hui de shuxue lilun tixi ’ (A study on Liu Hui’s System o Mathematical Teory). Ziran bianzheng fa
tongxu 9: 42–48. (1988) ‘Jia Xian Huangdi Jiuzhang Suanjing Xicaochutan ’, Ziran kexue shi yanjiu7: 328–334. (1992) Gudai shijie shuxue taidou Liu Hui (Liu Hui: A Leading Figure of Ancient World Mathematics). Jinan. Li Jimin (1993) ‘“Shang Gao dingli” bianzheng “”’ (extual research on “Shang Gao” theorem). Ziran kexueshi yanjiu12: 29–41. Li Rui (1806) Gougu Suanshu Xicao (Detailed Outline of Mathematical Procedures of Gougu), in Li shi Yishu (Posthumous writings of Mr. Li), ed. Ding Quzhong (1875). Li Zhaohua (1993) ‘Wang Lai Dijian Shuli Sanliang SuanjingLüelun ’ (A short discussion on Wang Lai’sDijian Shuli and Sanliang Suanjing), in antian Sanyou, ed. Horng WannSheng. aibei: 227–37. Liu Dun (1993) ‘Gougu suanshu xicao tiyao ’ (Introduction to the Detailed Outline of Mathematical Procedures for the , Right-Angled riangle), in Zhongguo kexue jishu dianji tonghui, shuxue juan ed. Guo Shuchun , vol. . Zhengzhou: 67–9. Qian Baocong (1982) Zhongguo shuxue shi (History of Mathematics in China). Beijing. ian Miao (1999) ‘Jiegenfang, ianyuan, and Daishu: algebra in Qing China’, Historia Scientiarum9 (1): 101–119. (2005) Zhongguo Shuxue de Xihua Licheng (Te Westernization of Mathematics in China). Jinan. (orthcoming) ‘Rejection and adoption o Western mathematics: Chinese mathematician’s study on theGougu Procedure rom the beginning o 17th to the beginning o 18th century’. WangDijian Lai ), in (1799?) (Te 4, Dijian Shuli, book Mathematical Teory of Jiashu tang edn. Hengzhai Suanxue Wu Wenjun (1978) ‘Churu xiangbu yuanli ’ (Te Out–in Principle), in Zhongguo gudai keji chengjiu (Scienti c and echnological Achievementsin Ancient China). Beijing: 80–100. Xu Guangqi (1609) Gougu Yi (Te Principle of Gougu), ianxue chuhan edn. Yang Hui (1261) Xiangjie Jiuzhang Suanfa (A Detailed ), Yijia tang edn. Explanation of the Nine Chapters of Mathematical Procedures
573
Index
abacus, see tool for calculation Abbenes, J. G.J., 191 abbreviation, 36–7, 328–40, 349, 355–6 absence of demonstrations,1, 9, 56, 276, 277, 280, 289 abstraction, 38, 354, 360, 452, 469, 471, 484, 485; see also generality, paradigm Abū al-Wafā , 286, 289 Academy of Science (Paris), 275 ācārya, 262, 266, 488 accuracy (indifference to visual),see diagram Acerbi, F.,205 Adelard of Bath,86, 87, 89, 90, 91, 93, 95, 105, 106, 109, 115, 117, 126–7, 130 aggregated shares/parts, see enlü Akkadian, 384–6, 391, 398, 411, 414 Akkadian method,373 Alexandria, 361 algebra, 6, 9, 11, 43–4, 57, 317–19, 450 in India,6, 235 Old Babylonian “algebra”,364–80 numerical interpretation,369 symbolic, 57, 327–8, 330 syncopated,327–8, 330 Algebra with Arithmetic and Mensuration, 242, 246, 247 algebraic analysis, see analysis algebraic formula,46, 391, 393, 395 algebraic proof,6, 7, 46, 47–51, 57–60 Diophantus’ impact on the text of algebraic proof, 39, 318–25 in an algorithmiccontext, 47–51, 59, 423–86 history of, 7, 9, 39, 50, 60, 426, 450, 480–4 validity of linked to the set of numbers with which one operates, 50, 311–26 algorithm, 9–10, 18, 38, 45, 51–3, 56, 59–60, 260, 270, 271–2, 359, 423–86, 487–8, 497–8, 500, 503, 506–8 algorithms reverse ofone another,47, 50; see also reverse algorithm as list of operations,38, 40, 44, 59, 425, 428–9, 432, 436, 438–54, 500 as statements proved to be correct,9–10, 18, ʾ
574
31, 38, 39–51, 53, 55, 57, 59–60, 428, 436 computing reciprocals (or ‘reciprocal algorithm’),44–7 cancelling theeffect of one another,45–7, 50, 447, 455 meaning of, 41–2, 44–6, 52–3, 62, 500, 503, 507 procedure for the eld with the greatest generality,431–6, 439–40, 442–4, 459, 475–6, 483 transparent,38, 40, 42, 48–9, 63; see also transparency use of, in proofs, 49, 426–81, 487–8, 497–8, 503, 508 see also factorization, procedure, square root, text of an algorithm Allard, A.,359 Allman, G., 279 allograph, 36, 37, 330, 338, 340–1, 359 Almagest, see Ptolemy altitude, 494–6, 498 An Pingqiu , 517 an-Nayrîzî (Abû l’ Abbâs al-Fadl ibn Hâtim an-Nayrîzî, also Nayzīrī (al-)) Analects (Te), see Lun yu analysis, 6, 12, 44, 280 algebraic, 6, 9, 242, 245, 246 indeterminate, 287 in Diophantus’Arithmetics, 36, 44, 350–3 see also demonstration Andersen, K.,150 annotation,86, 95, 103, 120 Anschauungsgeometrie, 276, 280, 285, 289 anti-Arab ideology,275 Antiphon, 296–7 Apollonius of Perga,1, 69, 70, 135, 140 Conics Book I, 149 manuscript Vatican206, see manuscripts (Greek) Proposition I13, 149 Proposition I16, 145–6 applications, 13, 274, 287
Index
approximations,40, 301, 452–8 Arabic science, 2, 5, 7, 21–3, 37, 43–4, 48, 274–5, 286–9 ‘Arabs’,5, 274, 287 handling of ‘Greek’ mathematics,286–9, 339–40 imitators of the Greeks,289–90 Archimedes,1, 2, 5, 20, 24–6, 28, 42, 62, 66, 69, 135, 140, 163–204, 276, 299–300, 305–6, 308, 351, 362 Heiberg’s edition of Archimedes’ writings,20, 24–6, 86 Heiberg forcing divisions between types of propositions and components of propositions onto texts,26 mechanical way of discovery,28, 42 Palimpsest, 86, 147–8, 164–5, 179–80, 187, 189, 192, 195 Archimedes, works by Arenarius, 178, 180–1, 186, 188–9, 190, 195, 203 Cattle Problem, 178, 186, 188, 203 Centres o Weights o Solids, 171 Conoids and Spheroids, 171, 178, 186, 188–9, 192–3, 199–201, 203 Floating Bodies, 164, 171, 178, 186–9, 194, 203 Measurement o Circle, 164–5, 171, 178–9, 186–8, 203 Method, 28, 42, 171, 178–9, 186–9, 195–203, 299–300, 308 proposition14, 148, 158, 196–8 Planes in Equilibria, 164, 171, 177–9, 186–9, 193–5, 203 Polyhedra, 171, 188 Quadrature o Parabola, 171, 178, 186, 188–9, 193–4, 203 Sphere and Cylinder, 140, 146, 164–76, 178–9, 181–4, 186–9, 193–5, 203 manuscript Vatican Ottob. 1850, see manuscripts (Latin) Spiral Lines, 164–5, 171, 176, 178, 186–9, 190, 195, 203 Stomachion, 171, 178, 186–9, 203 Archytas,190, 295, 298–9, 304 Aristarchus,305 On the Sizes and Distances o the Sun and Moon, 305 scholia, 156 Aristophanes, 297 Aristotle, 1, 17, 26–7, 66, 295–8, 300, 302–8, 325, 362–3, 377, 381 Prior Analytics and Euclid’s Elements, 377
Posterior Analyticsand Euclid’s Elements, 1, 26–7 theory of demonstration in thePosterior Analytics, 1, 17, 26–7, 66, 303–4 arithmetic,9, 33, 267, 294, 297–8, 300, 302, 440, 451, 459, 473, 482–3, 507 with fractions, 50, 426, 431–6, 441–3, 447, 451–80, 483 Arithmetica, see Arithmetics, Diophantus of Alexandria arithmetical reasoning,8, 34, 263, 269, 311–26, 504, 507 Arithmetics, 35–9, 44, 46, 283–4 critical analysis of annery’s edition,36 see also Diophantus of Alexandria Arnauld, A.,18 Arneth, A., 278–9 Arnzen, R., 87, 131 arti cial languages,45–6, 65 Āryabhat.a, 244, 282, 487–90, 494, 500–1, 504–8 Āryabhat. īya, 51–3, 66, 487, 489–92, 494, 498–501, 504–8 Āryabhat. īyabhās.ya, 487–508 Asiatic Society,230, 232, 239, 242, 257, 273 Assayag, J.,228, 256, 259 astronomy,265, 274–5, 294, 297–8, 300, 304, 494–8, 508 history of Indian astronomy, 237, 239, 241, 258, 261, 262, 264, 272–3, 276, 494–8 inequality of the moon, 275 practical, 274 spherical, 274 Athenian publicaccounts, 10 Atiyah, M.,16, 17, 64 August, E. F.,137 authenticity,79, 95, 97–9, 100–5, 110 Autolycus,139 autonomous practical knowledge,381 auxiliary construction, 209, 220, 221, 224 Averroes, 207, 223 axiom, 14, 304–5, 308; see also starting points axiomatization,15, 62 axiomatic–deductive structure,14, 15, 23, 57–8, 62 in the nineteenth century, early twentieth century, 12, 20, 26 outside mathematicsin ancient Greece,15, 29 ba gu wen ‘eight-leggedessays’,513 Babylonian mathematics,1, 5, 12, 14, 18, 20, 31, 37, 39–49, 50, 55, 59, 62, 65, 370, 377, 379–81 seen as empirical,363
575
576
Index
Babylonian mathematics (cont.) see also mathematical education Bachet, C., 325 backtracking huan , 437–8, 447, 457–8, 482–3, 539, 544 Bacon, Francis, 381 Bailly, C.,512 Bailly, J.-S.,234, 236–9 Bakhshali manuscript, 260, 501, 503, 506 Banerji, H. C.,247–8, 257 Barker, A.D., 302 Barozzi, 206, 207 Bashmakova, I. G., 328 Ben Miled, M., 87, 131 Berezkina, E., 535 Berggren, J. L.,149 Bernays, P., 312 Bertier, J.,311 Besthorn, R. O., 130, 138 Bhāskara I, 51–2, 66, 67, 270–2, 487–508 Bhāskara II (also Bhascara, Bhaskaracarya), 66, 235, 248, 249–50, 254, 264, 276, 290 Biancani, 206 Biernatzki, K. L., 275 Bija-Ganita (also Vija-Ganita), 235, 240, 241, 243–6, 249, 252, 254, 276, 487 Biot, E., 1, 5, 56, 66, 275, 511–13 Biot, J.-B.,5, 9, 245, 274–5 Birch, A. H.,285 Bīrūnī (al-), 290 Blanchard, A., 72, 132, 134, 187 Blue, G., 3, 66 Bombelli, R., 284 Book o Mathematical Procedures, see Suan shu shu Bos, H., 290 Bourbaki, N., 69, 74, 120, 132, 328 Bouvet, J.,3 Boyer, C. B., 328 Brahmagupta, 228, 240, 243, 246, 255, 264, 273, 508 Brahmasphutasiddhanta, 243–5, 508 Brentjes, S.,78, 85, 87, 89, 116, 118, 119, 132–3 Bretschneider,E., 279 Brianchon, C. J.,5 Britton, J. P., 385 broad lines,379 Brownson, C. D., 304 Bruins, E., 370–1, 385–6, 400, 410, 509 Bryson, 296 Buckland, C. E., 257 Burgess, J.,237, 257 Burkert, W.,295 Burning Mirrors(On), see Diocles
Burnyeat, M.F., 303 Burrow, R.,231–6, 239, 242, 247, 257 Busard, H. L.L., 86, 89, 130, 152 Calcidius Plato’simaeus, 156 calculation, 10, 40, 263, 266, 283, 498, 507 as opposed to ‘proof’, 10, 40, blind, 37, 507 devaluation of, calculation as a mathematical activity, 10, 12, 40, 50, 60, 263 see also Babylonian mathematics, computation Campanus of Novara,69, 78, 79, 81, 86, 127, 130, 134 cân (Vietnamese, Chinese:jin, measure of weight), 524, 527–8, 539, 541–2 Canaan, ., 340 canon, 53, 64; see also classic Cantor, M., 277–83, 285–9 cao ‘computations’,531, 533–4 Cardano, G.,289 Carnot, L., 5 Carra de Vaux, Bernard, 292 case of gure,23, 58, 90, 111, 115, 124, 128, 152–3 Catena, 206 Cavigneaux, A., 384, 405, 410 certainty, 2, 11, 12–17, 30, 62, 265, 281 actors’ perception of what yields certainty,14 certainty as possibly entailing losses for mathematics and history of mathematics,17, 18, 32, 41, 62 Ceyuan haijing, Sea Mirror o the Circle Measurements, 58, 450, 558, 566 Ch’en Liang-ts’so (Chen Liangzuo) , 572 Chang’an, 514 chang a ‘constantnorm’,527–8, 539 Charette, F.,5, 6–9, 10, 53, 228, 257 Charpin, D., 387 Chasles, M., 5, 278 Chattopadhyaya,D., 273 checking, 10, 44–5, 505, 507 Chemla, K., 1–68, 354, 423–86, 531, 537, 542, 560 Cheng Dawei, 512, 524, 526 Chỉ minh lập thành toán pháp, 524, 526, 537 China, 1, 2–3, 5, 7, 10, 12–13, 14, 16, 18, 24, 31, 41–2, 47–9, 51–9, 62, 63, 64, 65, 66, 67, 68, 423–86, 510–14, 523–4, 526, 531, 552–73 bureaucracy, 10, 54
Index
Cicero, 362 circle (ratio between circumference and diameter, between area and square of diameter),427–31, 438, 444–5, 447–8, 453, 468 circulation of practices of proof,3, 19, 30, 43, 48, 52, 53–9 citizens, 294 claims of identity,13, 64 classic Chinese (jing ), 47–51, 423–86 commentaries justifying, 51–3 meaning of,48–9, 51–3 proof in the commentaries on a classic or treatise,47–53, 56, 263, 423–86, 498–507 Sanskrit treatise or classic,51–3, 56, 260–7, 487–8, 501 see also canon Clavius, C., 2–3 codex, 71–3, 84, 88 co-latitude, 495–7 Colebrooke, H. ., 6–7, 9, 11, 42, 63, 228–9, 235–56, 257, 260, 275, 276, 279, 285 Collection, see Pappus College of calligraphy, see Shu xue Collegio Romano,2 colonialism,228, 229 commentaries,8, 76, 85–6, 87, 260, 262–3, 267, 270, 272–3, 487–9, 490, 493, 498–508, 511, 519, 521–2, 529, 531, 533–7 justifying their interpretation of a classic, 487–9 proof in the commentaries on a classic,8, 47–50, 56, 263, 423–86, 490, 498–507 see also Li Chunfeng, Liu Hui,zhu ‘commentary communication (‘make’ … ‘communicatetong ( )’), 436–8, 440, 450, 463–8, 470–3, 476, 538, 542 complexi cation (of fractions)an , 461, 466–7 computation,9–10, 17, 37–8, 40–1, 45–7, 50, 58, 271, 355–6, 358–9, 446, 448, 456, 465, 470, 474, 489, 491, 493–4, 497–501, 505 computationalreasoning, 260 execution, 45, 46, 426, 432–7, 442–3, 459, 462–4, 470–1, 477 layout,45, 46, 432–5, 437–8, 443, 472, 477–8, 483 see also calculation, cao, historiography
of mathematics (historiography of computation) Computational procedures [ound] in the Five Classical books, see Wu jing suan shu Computational procedures o Nine categories, see Jiu zhang suan shu Computational treatise [beginning with a problem] about a sea island, see Hai dao suan jing Computational treatise o Five Departments, see Wu cao suan jing Computational treatise o Master Sun, see Sun zi suan jing Computational treatise oXiahou Yang, see Xiahou Yang suan jing Computational treatise on the continuation o [traditions] o ancient [mathematicians], see Qi gu suan jing Computational treatise on the Gnomon o Zhou [dynasty], see Zhou bi suan jing Computations in the Five Classical Books, see Wu jing suan Comte, A.,5, 11, 292 Conica, see Apollonius of Perga (Conics) conjecture,300–1 consensus theory of proof,229 consequence (logically deduced), 274 constant norm,see chang a Continuation [o traditions] o ancient [mathematicians], see Qi gu continuity,24, 43–4, 54–9 historical continuity between modes of argumentation in Babylonian scribal milieus and proofs in Arabic algebraic texts, 43–4, 48 historical continuity between thirteenthcentury algebraic texts in China and Li Rui’s approach to the right-angled triangle, 58–9 correctness of algorithms (or procedures),9, 10, 18, 31, 33, 36, 38–60, 269, 271–2, 423–86 inscription designed to note down an algorithm while pointing out the reasons for its correctness,38, 46, 63 simultaneously prescribing operations and giving rationale of a procedure,40, 42, 43, 48–9, 63; see also transparency counting rods, see suan creation of speci c languages,65 Diophantus’Arithmetics, 37–8 formulaic expressions,37, 63
577
578
Index
critical editions, 138, 261 historical analysis of critical editions,20–6, 35–9, 136, 163–204, 336–9 history of proof and history of critical editions, 21, 22, 23, 25–6, 36 tacitly conveyingnineteenth-century or early-twentieth-century representations,22, 24, 26, 36 critique, in Old Babylonian mathematics,377, 380 Crombie, A., 291 cross-contamination,see diagram Crossley, J.N., 423, 427, 433, 486 cube root, 410–11, 413–14 Cullen, C., 423, 485, 518, 525–6, 533 cultures of proof,42, 229 Cuomo, S.,301 curriculum, 389, 410–11, 413–14, 512–15, 519, 536 Curtze, M., 279 Czinczenheim, C., 139–40, 156 Czwalina-Allenstein, A., 163 D’Alembert, J.,6, 12 D’Ooge, Martin Luther, 311, 313, 324 da yi ‘general meaning’,520–2 Dain, A., 70, 133 Dalmia, 264, 272 Datta, B., 271, 272 Dauben, J.W., 5, 65, 423, 485, 525 Davis, M., 17 Davis, S., 233, 236–9, 242, 249, 256, 257 De Groot, J., 213 De Haas, F.A. J., 210 De Rijk, L. M.,217 De Young, G.,78, 87, 89, 91, 104, 105, 110, 117, 130, 133 Decorps-Foulquier, M.,145 deductive structure, 2, 14–15, 20, 22–3, 26, 29–30, 52, 57–8, 62, 103–4, 109–10, 111–13, 119, 320–5, 440 de nition, 14, 26, 92, 93, 107, 115, 124, 126, 303–5, 308, 313–16, 318 and explanation,209, 216 arithmetical de nition,318 formal and material de nition,211, 214–15, 217 of polygonal numbers,313–16 physical and mathematical de nition, 213–14 see also number, starting points Delambre, J.,5, 245, 247, 256, 258 Delire, J. M.,262, 271–2 Democritus, 295
demonstration,1, 3, 11, 26–7, 64, 66, 229, 231, 242, 245–52, 254, 256, 260, 271–2, 274, 276, 280, 300–4, 306–8, 325 algebraic demonstration, 6, 280–1; see also algebraic proof and Euclid’s Elements I.32, 208–9, 219–21, 288; see also Euclid and syllogistic form, 27, 206, 209, 222, 224, 225 demonstratio potissima, 223 demonstratio propter quid, 223 demonstratio quid est, 223 illustrative [Anschauungsbeweis], 6, 276, 280, 285, 289 inductive, 3, 11, 285 in early-modernEurope, 2, 3, 7, 53, 289–90 logical, 5, 22, 274 physical versus mathematical, 214, 217–18, 221 primitive, 285, 290, 291 rigorous,275, 289, 290 status anduse of, in Arabic treatises,5, 288–9 visual, 6, 251–2, 272 see also analysis, geometry, proof, synthesis denominator mu , 423, 431–4, 436, 438, 446, 448, 459–61, 463–71, 475–80, 538, 542 Densmore, D., 145 Des Rotours, R., 513, 515, 518, 520–1, 535 Detailed Explanation o the Nine Chapters o Mathematical Procedures, see Xiangjie Jiuzhang suana Detailed Outline o Mathematical Procedures or the Right-Angled riangle, see Gougu Suanshu Xicao deuteronomy,329, 353, 359 Dharampal, 257, 258 diagram, 22, 24–5, 32–4, 41–2, 52–3, 58, 270, 272, 303, 320, 323, 500–3, 506–8, 524, 531, 533 Archimedes’,24, 140, 164–76 Chinese commentaries,24, 52, 58 critical edition of,22, 24–5, 32, 136, 138–40, 156–7 cross-contamination of manuscript diagrams, 153–6 Euclid, 24, 136–9, 140–5, 147, 149, 152 generality of, 143, 157; see also generality generic, 24 having particular dimensions,6, 24, 32, 41–2 Heiberg turns Archimedes’ diagrams into mere ‘aids’, dispensable elements, 25 iconic, 172–5
Index
iconically representing the numbers as con gurations of units,33, 313–16, 320, 323 indifference to geometric shape,143, 145–8, 154, 160 indifference to metrical accuracy,143–4, 153–5, 160 indifference to visual accuracy, 143–8 interpreting a diagram in terms of weight, 28, 42 lettered, 135, 157–8 Li Ye ’s Yigu yanduan (1259),58 manuscript,24–5, 32, 135–60 mathematical reconstruction,24, 135–6 overspeci cation,25, 140–5, 154–5, 157, 160 parallel between geometrical gures and problems,32, 41–2, 44, 48, 63 redrawing,136–7, 140, 153–9 representation of geometric object,143, 157 representationsof numbers as lines, 34 Sanskrit,24, 42, 270, 272, 500–3, 506–8 schematic representation,148–5, 157–60 speci c, 24, 25, 33 three-dimensional representation,169–72, 531 traditional, from China,58 Yang Hui ’s 1261 commentary on Te Nine Chapters, 58 dialect, 179–80, 188–91 didactical explanation,373, 376–7, 380 creating conceptual connections,376–7 in Old Babylonian mathematics,370 Dijian Shuli (Te Mathematical Teory o Dijian), 573 Dijksterhuis, E. J.,186 Dimashqî (Abû ‘Uthmân ad-),85 Dinostratus,296 Diocles, 76, On Burning Mirrors, 158, 162 Diogenes of Apollonia,306 Diophantus of Alexandria (also Diophantos, Diophantus),9, 26, 33–9, 43, 44, 57, 63, 242, 244–5, 283–5, 287, 311, 318–25, 327–62 Directorate of Education,see Guo zi jian Directorate of National Youth, see Guo zi jian disaggregation (of parts)san , 461, 465–6, 469–70 discussion, 262–3, 270, 488–9, 507 on the relative value of proofs,4, 11, 13, 14, 15–17, 27, 28, 29–30 on the standards of proof as asocial phenomenon,29–30
dispositif, 41–3, 59, 62, 69 distortion of sources,1–2, 289 distribution,527, 529, 540 aggregated [weighted], 525–6 at-rate,525, 528, 530 weighted, 525–30 dividend, see shi division, 426–84, 491, 493, 498–9, 540, 542 as multiplication by the reciprocal,390 at a stroke lianchu , 448–9, 451, 471–5, 480, 482–3 division together bingchu , 474, 482 evaluation, see shang chu execution of,432–51, 460, 471–2, 476–8 in return baochu , 453, 457–9, 478–9, 482–3 of fractions, 426, 427, 430–1, 463–71 of integers plus fractions (procedure for ‘directly sharing’ jingenshu ), 436, 442, 461–75, 477, 483 returning [division], see gui [chu] square root extractions as division,47, 454–8 divisor, see a Djebbar, A.,78, 81, 84, 87, 89, 107, 117, 118, 119, 120, 132, 133, 134 Donahue, W. H., 145 Dong Quan , 517 Dorandi, ., 70, 71, 133 Du Shiran , 427, 433 Du Zhigeng , 67 Duhem, P.,292 Dunhuang, 510 Durand-Richard, M.-J.,12 early-modern Europe,2, 6, 7, 11, 18, 30, 53, 289, 290 Ecole Polytechnique,5, 66 Egypt, 5, 7, 285 Egyptians, 285–6 eight-legged essays, see ba gu wen Eisenlohr, A.,285 Eleatic philosophers,295 Elements (Euclid), 2–3, 4–5, 7, 9, 11, 13, 14, 18, 20–4, 26–7, 28, 33, 36, 37, 39, 69–134, 135–6, 140, 152, 168, 184, 267, 295, 298, 304, 306, 308, 552, 567–8 Book , 149 Book , 149 Chinese version of Euclid’sElements, 2–3, 13, 56, 65 Clavius’ edition of theElements, 2–3, 56, 67 direct tradition,21, 74–7, 78–9, 113–14, 124–5
579
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Elements (Euclid) (cont.) divergences between proofs in the various manuscripts, 22, 90 Edition August,82, 137 Edition Gregory,82, 138 Edition Grynaeus, 82, 138 Edition Heiberg,70, 74, 77, 82–3, 100, 117, 130, 136–7, 141–5, 153–4 Edition Peyrard,82, 130 indirect tradition,21–2, 75–7, 78–9, 85–8, 93, 100, 104, 105, 113–19, 122, 124–5 logical gaps,21–2, 100–4 manuscripts Bodleian 301, 73, 78, 83, 111 Bologna 18–19, 78, 81–2, 86, 101, 115, 121–2 Escorial 221,78 Florence 28.3, 78 Rabat 53, 152, 160 Uppsala 20, 152, 160 Vatican190, 5, 73, 78, 80, 82–5, 91, 92, 94–5, 97–8, 100, 101, 111, 115, 116, 128–9 Vienna 31, 78 see also manuscripts (Greek) problem, 39, 146–7 Proposition .1, 97–8 Proposition .7, 141 Proposition .13, 138 Proposition .22, 155–6 Proposition .35, 141–2 Proposition .44, 143–4, 155 Proposition .45, 91, 121 Proposition .7, 144 Proposition .14, 91, 102, 121 Proposition .21, 154–6 Proposition .24, 82 Proposition .25, 115, 152 Proposition .31, 112, 152 Proposition .33, 115 Proposition .35, 115 Proposition .36, 115, 152 Proposition .37, 103, 115, 121 Proposition .5, 115, 152 Proposition .10, 103 Proposition .16, 146–7, 153 Proposition .20, 112, 142–3 Proposition .22, 112 Proposition .31, 112 Proposition .33, 80, 85 Proposition .2, 104, 117 Proposition .19, 82, 115 Proposition .9, 112
Proposition .12, 102 Proposition .13, 102 Proposition .13 vulgo, 82 Proposition .16, 121 Proposition .72, 91, 116 Proposition .111, 116 Proposition .1, 94–8 Proposition .12, 137 Proposition .31, 91 Proposition .33, 149 Proposition .34, 91, 101 Proposition .37, 112 Proposition .38, 82, 137 Proposition .38 vulgo, 80–1, 103 Proposition .39, 136 Proposition .5, 102, 103, 121 Proposition .6, 92, 102, 103 Proposition .7, 102, 121 Proposition .8, 102, 121 Proposition .9, 101 Proposition .10, 103 Proposition .11, 101, 103 Proposition .13, 101–2 Proposition .14, 101–2 Proposition .15, 100–2 Proposition .17, 103, 114–15, 137 Proposition .6, 80 Proposition .11, 118 Proposition .15, 136 Propositions Book ,141 Propositions Books – ,141 Propositions Book .2–10, 112 Propositions Book .9–13, 105–6 Propositions Book .11–12, 121 Propositions Book .14–17, 104–5 Propositions Book .22–3, 115, 117, 126, 127 Propositions Book .26–7, 104 Propositions Book .30–1, 104, 117 Propositions Book .5–8, 104 Propositions Book .27–8, 121 Propositions Book .31–2, 91 Propositions Book .66–70, 107–9, 115 Propositions Book 1. 03–7, 107–9, 115 Propositions Book 3. 6- .17, 81, 82, 119, 121–2 Propositions Book .1–5, 112, 115 ‘wrong text’ of Euclid’s Elements, 20, 66, 84, 87–8, 93, 114, 122, 134 see also Euclid Eneström, G. H.,279 Engelfriet, P., 3, 13, 65, 567–8 engineering, 299
Index
Engroff, J. W., 87, 106, 107, 116–18, 130 enlightenment,228, 232, 258, 272 post-enlightenment,228 Ennis, . E., 523 epistemological values attached to proof,4, 13–14, 18, 28–35, 52, 62, conviction, 15, 17–18, 29 fruitfulness, 17, 28, 32, 61 increased clarity,17, 18, 61 understanding,17, 18, 41–4, 49, 59 see also certainty,generality, incontrovertibility, rigour, value epitomization,116, 117, 120–1 equalities, 37, 49, 62 transformed qua equalities, 38, 44, 49, 449–50 equalization tong , 436, 463–6, 468, 470–1 equation,287, 328, 348, 360 general, 57–8 Pell, 290 simultaneous linear,509 symbolism forequations in China,56–9 third-degree,289, 518 ethnic characteristic, 286 Euclid, 1, 43, 69, 70–1, 136, 267, 276, 282, 288, 304, 308, 317–19, 362–3, 376 geometric algebra of Book ofElements, 319–20 Optics, 141, 150 Phenomena, 139, 141 see also Elements Eudoxus, 295–6, 299, 304–6 Euler, L.,15, 290 Eurocentrism (history of), 292 Europe,261, 264 European imperialism, 10 missionaries,2–3, 10 vernacular languages,2, 37 Eutocius, 178–9, 299 commentary on Apollonius’Conics, 145 commentary on Archimedes’Sphere and Cylinder, 158 evaluation division,see shang chu; see also division evangelism,2–3 exactness of results,40, 50, 268, 437, 440, 447, 453–8, 482 examination by quotation, see tie jing; see also tie du excision (practice of), see philology exhaustion (method of ), 30, 296, 299, 308 explanation,273, 487–8, 490, 497–501, 507, 559–60, 563–5, 567
by a formula,392, 395 gures introduced for types of ‘explanation’, 41–3, 499–501 in mathematics,41–2, 51–3, 214, 224 in natural sciences,214 pratipadita, pradarsa to refer to algorithms solving a problem,53, 497–501 through middle terms,207–9, 214–24 a ‘divisor, norm’,431–4, 459–62, 467–72,
478, 527–8, 538–9, 541, 543–4 factorization, 401, 408, 410–16 algorithm,405, 411, 416 Fârâbî (al-), Abû Nasr Muhammad ibn Muhammad ibn arhân,87, 131, 133 Fecht, R.,136 en ‘parts, shares’,528, 538–40, 543–4; see also fractions, number (fractions) Feng Lisheng, 518 enlü ‘aggregated shares/parts’, ‘multiples of shares/parts’, parts-coefficients’, ‘parts-multiples’,528–30, 539, 543–4 Filliozat, J.,230, 257 Five Departments, see Wu cao Folkerts, M.,86, 130, 133, 279, 550 Follieri, E., 73, 133 formal system,552–3, 557, 565, 569 formulaic expression,37, 357–8 Fowler, D.H., 73, 133, 318, 327 fractions, 332, 356, 390, 402, 431–80 arithmetical operations on fractions, 458–80, 483; see also complexi cation, division, multiplication, simpli cation as parts, 434, 436, 458, 460–1, 463, 464–70, 473, 475–6, 478 as a pair of numbers,435, 459–61, 467, 471–2, 478, 483 see also number Freudenthal, G.,87, 131, 133 Freudenthal, H., 143 Friberg, J.,385–6, 404–5, 410–11, 416 Gadd, C. J., 386, 410–11 Galen, 304–6 Ganesa, 241, 248–9, 276 Gardies, J.-L.,18, 65 Gauss, K. F., 279, 381 general meaning,see da yi generality,8, 24, 31–2, 33–8, 52, 57–8, 60–3, 425, 432, 434–6, 441, 450, 452, 456, 462, 467–70, 476–9, 498, 507 in Diophantus’Arithmetics, 38, 347–9 in Diophantus’On Polygonal numbers, 33–5
581
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generality (cont.) in Nicomachus’Introduction to Arithmetic, 33–5 of diagrams,see diagram of objects, 33, 427, 460–1, 474, 481 of problems,57, 424, 463 of proofs, 429, 441, 468, 471, 474, 481 see also paradigm, rule generalization,264, 286, 313, 322, 324 Geng Shouchang, 518 Geography, see Ptolemy geometrical demonstration,3, 6, 8, 29, 41–2, 58, 65, 245–6, 249, 252, 271 geometrical problem,41, 146–7, 151, 158 geometrical progression, 385, 391, 399, 405, 408–9, 416 geometry, 2–3, 41, 48–9, 232–4, 236, 239, 242–3, 245–6, 248–9, 256, 260, 267, 271–3, 278, 294–5, 297–301, 304, 491–503, 507, 512 cut-and-paste, 312 Greek, 1, 9, 12, 14, 20, 22, 24, 26, 28–30, 43, 52, 66, 135–60, 267, 275, 277, 425 Indian, 8–9, 260, 267, 271–3, 280–1, 289, 491–503, 507 practical, 286, 289 Gerard of Cremona,86, 87, 89, 90, 91, 93, 95, 96, 98, 103–5, 106, 109, 117, 119, 126–7, 130, 133 German logicians, 288 Gillon, B. S.,533 Gnomon o the Zhou, see Zhou bi Gnomon o the Zhou [dynasty](Te), see Zhou bi gnomons,492–5, 498 Goldin, O., 223 Gougu procedure (‘Pythagorean procedure’),553, 566, 569 Gougu Suanshu (Mathematical Procedures on Gougu), 553 Gougu Suanshu Xicao (abbreviated asGGSX), Detailed Outline o Mathematical Procedures or the Right-Angled riangle), 552–4, 556, 558, 560, 564–5, 567, 569–70 Gougu yi , Te Principle o Gougu, 566, 568–9 Gow, J.,324 Grabiner,J., 13, 65 grammar,31, 51, 261, 281 empirical, 281 etymological, 281 logical, 281 syntactical, 281, 357–9
Great Compendium o the Computational Methods o Nine Categories [and their] Generics, see Jiu zhang suan a bi lei da quan greater common divisor (in Chinese, ‘equal number’,dengshu ), 470 Greek philosophy and mathematical proof, 1, 362 Greek science,2, 3, 5, 7–8, 12, 14, 20, 21–4, 26, 28–30, 32–4, 37–8, 52, 56, 60, 62, 66, 67 Greeks, 1, 2, 5, 8, 9, 264, 267, 274, 280–4, 286–91 ‘apodictic rationality’,280 ‘Greek miracle’, 1 ‘logical Greek’,274–92 Gregory (Gregorius), D.,82, 138, 275–6, 279, 280, 284, 285, 290 Groningen, B. A. van., 340, 361 Grundlagen der Mathematik, 312 Grynée (Grynaeus), S., 82, 138 Gu Yingxiang , 553 gui [chu] ‘returning [division]’, 538–9, 542 Guidance or Understanding theReady-made Computational Methods,see Chỉ minh lập thành toán pháp Günther, S.,279 Guo Shuchun , 12, 42, 49, 52, 55, 423–85, 531, 560 Guo zi jian ‘Directorate of Education’, 514
Hacking, I.,3, 10, 11, 13, 15, 40, 65 Hai dao , 515–16, 518, 522; see also Hai dao suan jing Hai dao suan jing, 516, 532, 534, 536–7, 546; see also Hai dao Hajjâj (al-) (al-Hajjâj ibn Yûsuf ibn Matar),78, 86, 89, 91, 93, 105, 106, 109–10, 117–21, 130, 133 Han Yan, 516, 518, 520, 532 Hankel, H., 277–90 Hankinson, R. J., 306 Harari, O., 14, 26–7, 39, 65 harmonics, 294–5, 297–8, 304 Hart, R., 291 Hayashi, ., 12, 260, 488, 501, 503, 506 Haytham (‘Ibn al-), Abû ‘Alî al-Hasan ibn al-Hasan Ibn al-Haytham,85, 87, 132 Heath, .,83, 163, 186, 311, 317–18, 323, 330, 345, 349
Index heavenly unknown (tianyuan ‘celestial srcin’), 57 Hebrew science,2 Heiberg, Johann,20–8, 32, 85, 86, 89, 113, 130, 133, 148, 279 edition of Archimedes’ writings,24–6, 86, 163–204 edition of theConics, 145 edition of Teodosius’Spherics, 139 edition of theElements,20–4, 70, 74, 77–8, 130, 136, 138–9, 141–5, 153–5 philological choices and their impact on the editing of the proofs,21, 22, 23, 25–6, 82–3, 99 Heinz, B., 229, 257 Helbing, M.O.,207 Hellenocentrism, 282–3 Hermann of Carinthia,86, 117, 127, 130, 133 Hero of Alexandria,71, 76, 85, 104, 111, 113, 115, 117, 282, 283, 285, 290, 298, 300, 306, 345, 362 Dioptra, 139 Herodotus, 294, 302 heuristics, 308 heuristic patterns of mathematical creation, 277 Hilbert, D., 17, 65, 100, 312 Hiller, E.,311 Hilprecht, H. V., 405, 421 Hindus, 5, 265, 274, 279 ‘apt to numerical computations’,284 Hippias,296–7 Hippocrates of Chios, 295–6, 304 Hippocratic treatises,301–2 historiography,274, 291–2, 302 ‘antiquarian’ style ofscholarship,279 evolution of European historiography of science with respect to ‘non-Western’ proofs, 4–14, 20, 50, 53 computation,9–10, 12, 40, 60 in uences in mathematics,53–4, 282, 284–8 mathematical education,56 mathematical proof, 1, 4–14, 19–28, 53, 277, 280, 289 ‘non-Western’mathematics,7–9, 260, 266, 274–92 of Islamic science,287, 292 of mathematics,4–14, 19–28, 53, 56, 135–40, 248, 252, 256, 258, 260, 266 of science, 5, 10, 11, 13, 26 presentist historiography and Platonic approach,279
history of French Orientalism,230, 275, 292 history of mathematics,261–3, 277–9, 289 ‘cultural’,279 history of the philosophy of science,5, 11, 13, 27 Hoche, R.,311 Hoffmann, J. E.,279 Holwell, J. Z.,257 homogenizationqi , 464, 467–8, 473 Horng Wann-sheng,57, 65 Høyrup, J.,1, 9, 12, 37, 39–44, 48, 49, 53, 60, 65, 343–4, 362–83, 398 Huangdi Jiuzhang Suanjing Xicao , 573 Hultsch, F., 279, 339 hydrostatics, 305 Hypatia,345, 350 hypotheses, 301, 303, 305, 503 Hypsicles,324–5 Iamblichus,299 icon, see diagram Ideler, L., 278 igi, 390–1 Ikeyama, S., 487, 508 imagination, 282, 290 imperialism,10, 291–2 incontrovertibility,2, 4, 14, 15, 18, 300, 304, 306–8 Inden, R., 230, 257 India, 5, 6–9, 11–12, 14, 16, 18, 37, 51–3, 59, 67, 260–3, 267, 272–3, 482, 487, 508 Europe’s knowledge of Indian mathematics, 11–12, 275–7 history of Indian astronomy, 237, 241, 258, 260–2, 267, 272–3, 494–8 history of Indian mathematics,228, 241, 246, 256, 260–3, 267, 272–3, 487–508 Indian algebra,6, 228, 231, 234, 238, 242, 244, 245, 246 srcins of Indian mathematics,231, 233–4 representationsof, 229 Indians,260, 264, 269, 280 ‘computing skills’,282 ‘Indians’ mode of thought’, 280 ‘less sensitive (einühlig) logic’, 283 ‘more intuitive rationality’, 280 Indology,257 British, 230 French,231 German, 6, 261, 273 induction, 11, 282, 312–13, 316–18 Institute forHan-Nom Studies, 524
583
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Index
instrumentation,274 intellectual inferiority,274, 292 interpretation,6, 9, 11, 22–3, 28, 30, 38–9, 43–8, 51–5, 60, 260, 271, 423–84, 487–90, 492–4, 498, 500, 503, 506–7 of operations,38, 41, 42, 44, 48, 431, 439, 443–4, 446, 450, 458, 464–6, 473, 476, 480, 483, 498; see also meaning problems in the interpretation of the sources, 9, 26–7, 29–32, 38–40, 44–6, 49, 63, 468 proof based on an interpretation involving a plate, beads and pebbles,43 skills required for,38–43 Introduction to Arithmetic, 33, 311–18 Introduction to the learning o computations , see Suan xue qi meng intuition ‘Indian’style of mathematical practice, 280 modern mathematics,282, 290 inversion an , 440, 442, 451, 459, 472, 475–80, 482–3 inverse operations,437, 450, 452–8, 460, 472, 474, 482 of an algorithm,455 see also algorithm, reverse algorithm Irigoin, J.,70, 72, 73, 83, 133 irrationals, see number Ishâq ibn Hunayn (Abû Ya’qûb Ishâq ibn Hunayn),78, 86, 89, 91, 93, 96, 103, 104, 106, 107, 109–10, 116–19, 126–7 Isocrates, 298 iteration, 385, 393, 396–7, 402, 413, 416–17 Jaffe, A., 15–17, 64, 66 Jain, P. K., 12, 66, 487, 489 Jami, C., 3, 20, 65, 66, 68, 230, 257 Japan,54, 424 Jardine, N.,223 Jayyânî (al-), Abû ‘Abdallâh Muhammad ibn Mu’âd al-Jayyânî al-Qâsî, 87, 132 Jesuits, 2–3, 67, 238, 257, 258, 275, 511 astronomers,230, 241 historiography,238 Ji yi , 515, 517, 522; see also Shu shu ji yi (Records o the procedures o numbering lef behind or posterity) Jia Xian , 553, 573 Jiegenang (to borrow a root),573 Jihe yuanben , Elements o geometry, 567–8 Jin dynasty, 517 Jin dynasty, 514 jin shi (degree), 521
Jiu ang shu , 518, 532–3, 535 Jiu zhang , 515–16, 518–19, 522; see also Jiu zhang suan jing, Jiu zhang suan shu Jiu zhang suan a bi lei da quan , 526, 546 Jiu zhang suan jing , 519, 533; see also Jiu zhang suan shu Jiu zhang suan shu , 47–50, 54–6, 58, 67, 376, 384, 387, 389–90, 399, 403, 406–9, 411, 420, 423–86, 511–12, 516, 519, 525–6, 529, 531–3, 541, 546, 553, 566, 568–9; see also Jiu zhang suan jing Joannès, F., 387 Johannes de inemue,86, 127, 130 Johnson, W.,192 Jones, A, 71, 139, 149–50, 304 Jones, W., 237–8, 257, 258 juan , 519–20, 526, 535 Junge, G., 86, 130, 131, 161 justi cation, 260, 263–4, 269–70, 388, 417, 488–90, 498 implicit in equation algebra,367 in Old Babylonian mathematics,366–7, 369–80 mathematical and didactical explanation, 370, 377 mathematical justi cation, 260, 263–4, 269–70, 363, 377, 488–90 Kai-Yuan li (Rites o the Kai-Yuan era), 521–2 Kant, I., 377 Karaji (al-),287 Kejariwal, 230, 257 Keller, A., 8, 12, 20, 51–3, 60, 66, 260–72 , 487–508 Khayyâm (al-), ‘Umar,87, 132 Khwarizmi (al-),43, 65, 67, 286, 288, 350 Klamroth, M.,21–3, 77–9, 81, 85, 89, 99, 100, 113–14, 116–19, 133 Klein, J., 328 Kline, M., 1, 9, 40, 363–4, 370, 383 Knorr, W., 20, 21, 66, 70, 79, 81, 84–5, 87–8, 95, 99, 110, 111, 114–15, 116–19, 122, 134, 149–50, 185–6, 198, 295, 345, 351 Korea, 54, 424 Kramer, S. N., 386, 410–11 Krob, D.,43, 66 Kuhi (al-), 289
Lakatos, I., 15, 52, 66 Langins, J.,5, 66
Index
Lardinois, R., 256, 259 latin science, 2, 21–3, 37 latitude,495–8 Laudan, L., 5, 66 layout of text,388, 412–16, 418 columns as related to the statement of rules which ground the correctness of the algorithm,45–6 created for a kindof texts, 44–7 spatial elements of,45 Le Gentil, G. H. J. B.,234, 236, 238, 239, 258 Lear, J.,304 Lee Hong-Chi, Tomas,513–14, 520, 522 Leibniz, G. W.,3, 15, 65, 67 Leiden, 399, see manuscripts (Arabic) Lejeune, A., 139 lemma, 23, 92, 100, 103, 125, 126, 129 Lenstra, H. W., 255, 258 Lévy, .,89, 105, 106, 134 li (Vietnamese, Chinese:li, measure of weight),538, 540, 542 Li Chunfeng, 47, 54, 55, 424, 429, 435, 442,467,473,480,483, 518–19, 532–7 Li Di , 518 Li Jimin , 423, 427, 429, 452, 459, 460, 468, 485, 568 Li Rui , 56–9, 63, 553–4, 556, 560–1, 563, 565–9 Li Yan, 427, 433, 514, 533 Li Ye , 57–8, 450, 558, 566 Li Zhaohua , 548, 557 Libbrecht, U.,513 Libri, G., 278 Lilavati, 235–6, 240, 243, 245–9, 257, 262, 276 linear perspective,137, 139, 148–51 literal signs,6, 245, 328 literate mathematics,344–7, 351–2, 356–7 Liu Dun , 548, 553, 567 Liu Hui , 47, 54–5, 64, 67, 423–86, 516, 518–19, 529, 531–4, 536–7, 542, 552–3, 560, 566 Liu Xiaosun , 531–4 Lloyd, A. C., 210 Lloyd, G. E. R., 1–2, 10, 14, 28–30, 34, 38, 42, 60, 66, 294–308 logic, 281–3, 288 wanting amongst the Indians,282 see also syllogism Loria, G., 279 lü , 427, 440, 445, 448, 468–72, 474–5, 479, 485, 529
‘lüs put in relation with each other’xiangyu lü , 460–1, 470 lu ‘Protocol [of computations]’,521, 530 Lun, A. W.-C., 423, 427, 433, 486 Lun yu , 540 lượng (Vietnamese, Chinese:liang, measure of weight),524, 528, 539–44 Luro, J.-B.-E.,523 MacKenzie, D.,10, 66, 229, 258 Mâhânî (al-), Abû ‘Abdallâh Muhammad ibn ‘Isâ, 87, 131, 134 Mancosu, P.,14, 66 Mandelbrot, B.,16, 17 manuscripts (Arabic),78, 86, 109, 117–19 Codex Leidensis 399, 85, 87, 106, 117, 119, 120, 130, 139 Rabat 53, 152 Uppsala 20, 152 manuscripts (Greek),72, 73, 77–8, 80, 83, 100, 106, 107, 110–11, 329, 331, 333, 335–40 Bodleian 301, 73, 78, 83, 111, 142–5, 147, 152–4, 156 Bologna 18–19, 78, 81–2, 86, 101, 115, 121–2, 144, 146, 156 Florence 28.3, 78, 147, 156 Paris 2344, 78 Paris 2466, 78, 147 Teonine mss.,80–1, 82–5, 91, 92, 93, 94–5, 97–8, 101, 113, 115, 128–9 Vatican190, 73, 78, 80, 82–5, 91, 92, 93, 94–5, 97–8, 100, 101, 111, 113, 115, 116, 128–9, 137–8, 141–2, 144, 147, 149, 154–6 Vatican204, 140, 150–1 Vatican206, 145–6, 149 Vienna 31, 78, 144, 147, 153–5, 156 see also Archimedes (Palimpsest) manuscripts (Latin), Vatican Ottob. 1850, 146, 165 Marini, Giovanni Filippode, 541 Martija-Ochoa, I., 1 Martzloff, J.-C.,2–3, 66–7, 293, 510, 513, 518–19, 529, 545–6 Massoutié, G.,323 Master Sun, see Sun zi masters of truth,1 Mathematical College,see Suan xue mathematical education,2, 3, 44, 54–6, 389, 511, 513–14, 523, 534 administrative sources on state education system, 54–5
585
586
Index
mathematical education (cont.) in scribal schools, 41, 43–4, 62, 387, 389–90, 403, 405, 408–10, 412–16; see also scribal school mathematical education and commentaries, 54–5 mathematical examinations in China, 54–5, 514, 520, 523–4, 530–1, 534, 536–7 in East Asia, 54, 55 in Vietnam, 523, 524 textbooks used in educational system, 2–3, 54, 509–10, 513–14, 516, 518–19, 523, 531–2, 534–5 mathematical proofs, motley practices of,6, 15, 16, 19, 28–30, 34, 50, 61–4 mathematics, 260–73, 487–8, 503, 508 ‘Chinese’,3, 275, 509, 511, 513, 537 division betweenpure and applied, 12 ‘Egyptian’,285–6 ‘foreign in uences’,282, 284–8 ‘Greek’,6, 282–5 ‘Hindu’,264 ‘Indian’,7, 260–1, 267, 275–7, 280–2, 503 ‘Islamic’,286–9 ‘Muslim’,279, 289 ‘Oriental’,7, 277ff orientalized form of,283–5 pre-scienti c, 291 scienti c, 291 Maximus Planudes,284 McKeon, R. M.,382, 383 McKirahan, R., 223 McNamee, K.,340 meaning, 32, 45–6, 57, 263, 265, 269, 488, 490–1, 500–1, 503 ‘higher meaning’ of an algorithm,52, 481 formal meaning,466, 468 material meaning,42, 466–7 meaning (yi ), ‘meaning, intention’ of a computation, of a procedure,42, 48, 426, 431, 436–40, 446–8, 453, 455, 464–7, 480–1 meaning (yi’ ), ‘meaning, signi cation’ of a procedure, 52, 55, 58, 481, 521–2, 534–5 meaning as opposed to value,40, 46, 438–9, 446–7, 453, 455 meaning in Sanskrit commentaries,51–2, 263, 265, 269, 488, 490–1, 500–1, 503 meaning of operations,37–8, 42, 48, 59, 62–3, 444, 480–1 use of the context of a problem to formulate
the ‘meaning’ of procedures,42, 44, 48, 444–6 use of the geometrical analysis of a body to formulate the ‘meaning’ of procedures, 44, 48 see also da yi ‘general meaning’ mechanical considerations,42, 277 mechanics, 298–9 medicine, 301–2, 304–5 Mei Juecheng , 57 Mei Rongzhao, 548 Mei Wending , 66, 566 Mendell, H.,222 Mending Procedures, see Zhui shu Menge, H., 70, 130, 139 Meno, 8, 15 Mesopotamia, 1, 5, 12, 14, 18, 20, 31, 37, 39–49, 50, 55, 59, 62, 65, 282, 384, 387, 389, 400, 410 meteorology,295, 297, 301 Method, see Archimedes, works by method, 11, 30–2, 34, 36, 37, 42–3, 50, 57, 66, 260, 264, 266–7, 268, 270–1, 273, 497, 505–7 debates about proper methods in numerous domains of inquiry in ancient Greece, 29, 36, 294–308 see also exhaustion (method of) methodology of science, 4, 8, 11 Michaels, A., 293 Mikami Yoshio, 510 Mill, J. S.,11 Ming dynasty, 510, 513 ming a (lit. ‘[He Who] Understood the [Juridical] Norms’),520–1 ming jing (lit. ‘[He Who] Understood the Classics’), 520–2 ming suan (lit. ‘[He Who] Understood the Computations’),520–2 ming zi (lit. ‘[He Who] Understood the [Chinese] Characters’),520–1 Minkowski, C., 261, 273 mithartum, 366–7 modes of analysis,243 Moerbeke, see William of Moerbeke Mogenet, J.,73, 131, 139 Molland, G.,288 Montucla, E.,234, 237–8, 258 Morrison D.,217 Moses ibn ibbon,87, 131 motivations for writing down proofs,15, 17, 19, 31–2, 39–41, 52–3, 481; see also epistemological values attached to proof
Index
Mueller, I.,33–5, 37, 62, 100, 296, 306, 311–26 Mugler, C.,164, 186 multiple, see lü multiples of shares/parts,see enlü multiplication,426–84, 491, 493, 499, 506, 526, 540 execution of, 432–5, 443, 460 of fractions (procedure for multiplying fractions ), 426, 427, 467, 472, 475–80 of integers plus fractions (procedure for the eld with the greatest generality ), 433–6, 439–40, 442–4, 459, 475–6, 483 with sexagesimal place value notation, 388–91, 394–7, 400, 402–4, 415–17 see also tables Murdoch, J.,83, 86, 89, 120, 134 Muroi, K., 411 Murr, S.,258 music theory,297, 302 Nâsir ad-Dîn at-ûsî,78, 89, 105, 109–10, 117, 118 nation,284 ‘non-geometricalnation’,282 ‘oriental nations’,5, 274 Nayrīzī (al-) (also an-Nayrîsî),138–9 commentaries on theElements, 76, 85, 87, 113, 117, 130, 131, 132, 133, 138–9 Needham, J., 57, 67, 510, 513 neopythagoreanism,362 Neo-Sumerian,390 Nesselmann, G. H.,278, 323–4, 327–8, 330, 336 Netz, R., 24–6, 30, 35–9, 46, 64, 135, 140, 145, 148, 158, 163–205, 306–7, 329, 341, 351 Neugebauer, O.,37, 71, 136, 139, 345, 363, 369–70, 376, 386, 389, 392, 410, 412 New History o the ang [dynasty](Te), see Xin ang shu Nguyễn Danh Sành,523 Nguyễn Hữu Tận, 526 Nicomachus of Gerasa,33–4, 311–18 Nine categories, see Jiu zhang Nine Chapters (Te), abbreviation ofNine Chapters on Mathematical Procedures (Te), see Jiu zhang suan shu Nine Chapters on Mathematical Procedures (Te), abbreviated asTe Nine Chapters, see Jiu zhang suan shu Nippur,384, 387, 389, 390, 399, 403, 406–9, 411, 420, 421, 422 Noack, B., 156
Noel, W., 148 ‘non-Western’,10, 20, 50 astronomy,228 mathematics, 229 norm, see a norm of concreteness,379 Northern Zhou dynasty, 514, 534 numbers,9, 10, 33–9, 44–7, 60, 268, 263, 283, 311–26, 441, 452–5, 460, 489, 504 abstract, 389, 402, 460, 462, 467, 469 actual rst number, 311–12 and algebraic proof,50, 59, 423–86 as con guration,33 as multiplicities of units,33, 311–12 classi cation of,317 de nition ofpolygonal, 33–5, 313–16 fractions, 50, 423, 426, 431–8, 441, 447, 454–7, 459–80, 482–4 geometric or con gurational representation of, 34, 313–16, 470 Greek way of writing, 37, 311–12 integers plus fractions,431–8, 440, 447, 453–7, 460–77 integers,35, 50, 431–8, 441, 452–3, 456–7, 460, 463–77 irrational,283 quadratic irrationals,50, 452–7, 472, 483–5 natural representationof, 311–12 negative,244, 283, 388, 563 polygonal,33–5, 62, 311–26 positive, 244, 283, 388, 563 potential rst number,312 rational, 283 regular,44–5, 390–1, 394–5, 397, 400, 402–3, 413, 416–17 representation of,9, 34, 37 results of divisions,50, 431–3, 437, 440–1, 447, 453–60, 472, 478–80, 482–3 results of root extraction,50, 452–8, 482 sequence of, 33, 34, 312–19 table of,34, 317 yielded by procedure of generation,33 see also fractions, sexagesimal place value notation Numbers o three ranks, see San deng shu numerator zi , 423, 431–4, 436, 459–61, 464–71, 475–80 numerical methods, 44, 290 Nuñez, P.,380, 383 objects composite versus incomposite,211–12, 214–15, 218
587
588
Index
objects (cont.) material versus immaterial, 207, 212, 219, 225 mathematical, 207–8, 212–14, 218–19, 221–2, 225 ontological status of,208, 212, 219, 221, 225–6 Oelsner, J.,390 Old Babylonian ‘algebra’,see algebra mathematical terminology,364–7, 374, 377 mathematical texts,40, 364, 367, 371, 374, 377, 379, 509 period, 364, 384, 387–9, 399, 410 Old History o the ang [dynasty], see Jiu ang shu operations, 38, 40–6, 48–50, 58, 59, 61, 62, 460, 490–1, 494, 498, 500 arithmetical,509; see also fractions, procedure on statements of equality,38, 49, 449–50 symbols to carry out operations in the Arithmetics, 37, 38 optics, 298, 304 Optics, see Euclid Ptolemy oral teaching in Old Babylonian mathematics, 370, 376 oral versus written,16, 19, 53, 503, 507–8 order (change of) in Euclid’s Elements, 23, 90, 92, 93, 99, 105–7, 114–15, 125, 127, 129 Oriental, 7, 9, 286, 288, 291 ‘Oriental nations’,5, 274 ‘Oriental science’,5, 7, 275, 291 Orientalism, 228, 230, 256–7, 258, 259, 278, 279, 288, 291 Orientals, 9, 282, 290 computationaland algebraical operations, 282 ‘imaginative Orientals’, 274 Otte, M., 351 outline, 552–3, 558, 560–3 Ouyang Xiu , 548 overspeci cation,see diagram overvaluation of some features attached to proof, 4 incontrovertibility of itsconclusion, 4, 14, 18 rigour of its conduct,4, 15 Palimpsest, see Archimedes Pañcasiddhānta, 262–5, 273 pandit, 264, 270, 272 Pān.ini, 281 Paninian grammar,51, 281 Panza, M., 351
Pappus of Alexandria,76, 85–6, 111, 113, 131, 171, 298–9, 301, 331 Collection, 139 comments on Euclid’s Optics 35, 150 Commentary to Ptolemy’s Almagest, 139 papyri, 71, 73–4, 340, 344–5 paradigm, 31, 32, 38, 41–2, 58, 63, 424, 457, 484 Parry, M., 190 parts, see en, fractions parts of the productjien , 434, 436, 439–40, 443, 455, 465 parts-coefficients, see enlü parts-multiples, see enlü Pascal, B., 18, 65 Pascal triangle,512 Pasquali, G., 70, 134 Patte, F.,12, 56, 67 Peacock, G., 11 Peng Hao , 423, 430, 456–7, 473, 485 Pereyra, 206, 223–4 perspective, see linear perspective persuasion, 302,306 Peyrard, F.,5, 66, 80–1, 82, 130 phan (Vietnamese, Chinese:en, measure of weight), 538, 540, 542 Phan Huy Khuông, 524, 537 Phenomena, see Euclid Philolaus, 295, 298 philology, 74–7, 261–6, 278 format, 72, 84, 164, 191–5, 203, 345, 347, 353, 359, 405–7, 530, 534, 536, 539 practice of excision,176–85 Philoponus, J., 27, 207–22 philosophy,3–4, 10, 13, 15, 294–5, 303–4, 306–7 history of, 15 Indian, 280–1 see also history of the philosophy of science physics, 295–6, 298, 300 Piccolomini, A., 206, 223, 225 Pingree, D.,262, 273 Plato, 8, 15, 66, 179, 294, 297–300, 302–4, 306 Playfair, J.,231–4, 238–9, 242–3, 247, 249, 258, 276–7, 279, 280, 293 Plimpton 322 (cuneiform tablet), 509 Plooij, E. B.,87, 132, 134 Plutarch, 298–9 Pococke, E., 288 Poincaré, H.,15, 168–9, 175 politics of knowledge,4, 10, 67, 228, 258 of the historiography of mathematical proof, 5, 10, 59
Index
Polygonal numbers(On), 33–4, 311–26 Bachet’seditio princepsof the Greek text, 325 Polygons (On), 313–16 polynomial algebra,57–8 Poncelet, J. V., 5, 15 porism, 23, 91, 92, 93, 103–4, 105, 115–16, 124, 126, 128 porismata, 284 Poselger, F. ., 323 positions, 435, 459, 478, 483 array of,434–5, 478 positivism, 292 Posterior Analytics, 206, 207, 208, 209–10, 211, 215, 216, 217, 223, 325 postulates, 26, 301, 305 of parallels,288 practical orientation,6, 8 practical as opposed to speculative orientation,5 ‘practical orientation’ of the mathematics of the Arabs,5 ‘practical orientation’of the mathematics in the Sulbasutras, 8, 12, 260, 266, 268, 270, 272 practices of computation,40, 45; see also tool for calculation practices of proof,1, 2, 4, 11–12, 15, 17, 21–3, 28–30, 31–2, 35, 38, 41, 47–51, 54–9, 61–3, 425, 426, 448–9, 462, 471, 483 history of, 19, 23, 30, 38, 43, 53, 60, 480–4 shaping of, 15, 18, 20, 32, 35, 38, 59, 62–3 pratyayakaran.a, 498, 503, 505 predication,210, 212, 222 essential, 208, 209–12, 218 prediction,16, 300 principle (arkhê), 112, 312 principle , 567–9 Principle o Gougu, see Gougu yi Prior Analytics, 377, 383 problem (mathematical),17, 31, 35–44, 47–8, 55–9, 65, 260, 295, 300, 387, 413, 427–9, 449, 452, 462–4, 467, 480, 491, 493, 498, 505, 507, 509–10, 512, 516, 522–32, 534–5, 539–41, 543–4, 546, 570–2 as general statements, 38, 57, 424, 441 as paradigms, 31, 63, 522, 529, 534 category of problems,38, 424, 463, 510, 525 da ‘answer’,55, 520–1 Diophantus’ problems relating to integers, 35–8 explanation pratipadita (Sanskrit) of an algorithm by means of problems,53
introduced by the term ‘to look for’ qiu ( ), 444–6 parallel between geometrical gures and problems,41–2, 44, 48 particular problems,41, 58, 423, 424, 441 problems with which the understanding of the effect of operations can be grasped, 41–2, 44, 48–9, 481 use of problems in proofs,41–2, 44, 48–9, 53, 63, 65, 425, 445–6, 462–4 wen ‘problem’ (Chinese), 55, 520–1, 538, 541 problem-solving,35–5, 57, 285 procedure,263, 269, 271–2, 313, 487, 489–90, 492–4, 498–501, 503, 505, 507 arithmetical,33, 313, 507 fundamental, 52, 61, 425, 451, 476, 480–1 see also algorithm Proclus of Lycia,27, 76, 121, 131, 206, 207, 208, 219–22, 224, 298, 304–6, 362 professionalizationof science, 4–5, 11 programme for a history of mathematical proof, 18–19, 59–64 programme of study in Mathematical College, 519, 522 advanced, 518–20, 534–5 regular,518, 520, 535 Prony, G.,382 proof, 89–94, 99, 260, 263, 269–71, 265, 312, 317–25, 444–9, 498–507, 512, 559–60, 563, 565–9 activity of proving as tied to other dimensions of mathematical activity,16, 19, 43, 51, 53, 55, 60 actors’ perception of proof,4, 263, 270, 498–507 alternative proof, 89–90, 107–10, 112, 114 analogical proof,91–2, 120 double proofs,23, 83, 89–90, 93, 99, 107–10, 114, 124, 126, 129 elementary techniques of proof,30, 33, 44, 59–60, 62 functions ascribed to proof in mathematical work, 15–19, 41, 263, 270 general proofs,see generality goals of proof,13, 14–15, 18–19, 28–35, 38, 41, 51–2, 58, 61–2 key operationsin proof, 425–52, 480–1 pattern of argument,2, 25–6, 30, 35 potential proof,91–2, 120 proof and algorithm,39–51, 423–84 proof as bringing clarity,17, 18, 61 proof as bringing reliability,17
589
590
Index
proof (cont.) proof as establishing mathematical attributes that belong to their subjects essentially, 27 proof as providing corrections,17 proof as providing feedback,17 proof as support of a vision for the structure of a mathematical object,17, 33–4 proof as yielding clues to new and unexpected phenomena,17, 31, 52 proof as yielding ideas,17 proof as yielding mathematical concepts,17, 31, 52 proof as yielding new insights,17 proof as yielding techniques,17, 30–2, 38, 41, 52, 61 proof as yielding understanding,see epistemological values attached to proof proof as yielding unexpected new data,17 proof by example,316–18 proof by mathematicalinduction, 320–5 proof for statements related to numbers and computations,9 proof in the wording,40, 48, 468; see also transparency proof of the correctness of algorithms,9–10, 18, 31, 38, 39–51, 53, 55, 57, 59–60, 423–84, 498–507 proofs as a source of knowledge,17, 52, 429, 448, 471 proofs as opposed to arguments,15, 16, 28, 29 proofs as opposed to insights,16 proofs highlight relationships between algorithms, 52 relations between proofs,23, 445 rewriting a proof for already well-established statements, 17 rigorous proof withdiorismos, 289 role of proof in the process of shaping ‘European civilization’ as superior to the others, 2–3, 4–5, 10 substitution of proof,23, 90, 99, 107–10, 111, 125, 127, 129 technical terms for proof,41, 42, 48, 52, 55, 425, 431, 448–9, 451, 456–8, 464–8, 473, 481–3, 498 there is more to proof than mere deduction, 52 tool-box, 30, uses of proof,2, 4 see also meaning upapatti
proportional, see lü proposition, 3, 5, 8, 23, 26, 31, 274, 314–19 arithmetical and general propositions,33–4 purely arithmetical propositions,34, 319 Protagoras, 297 protocol of computations,see lu Proust, C., 20, 44–7, 50, 389–90, 402, 405, 420 Ptolemy,300, 306 Almagest, 140 Geography, 149 Optics, 139 Pyenson, L., 292 pyramid, circumscribed to a truncated pyramid, 430, 432, 436, 438–9, 444, 447 ‘truncated pyramid with square or rectangular base’angting , 427, 429–32, 436, 438–9, 441, 443–6, 455 ‘truncated pyramid with a circular base’ yuanting , 426–52, 468, 476 see also volume (cone) Pythagoras, 295 ‘procedure of the right-angled triangle (gougushu )’, 56 Pythagorean theorem,3, 8, 58, 252, 490–2, 494, 497–8, 501–2, 507 Pythagoreans, 311 Qi gu , 515, 517, 522; see also Qi gu suan jing Qi gu suan jing, 511, 517–18, 533, 535, 546 Qian Baocong, 517–18, 520, 561, 568 Qin Jiushao (alsoCh’in Chiu-shao), , 549 qing , 538, 542 quadratures, 295–7, 304 quadrilateral, 277 Quaestio de certitudine mathematicarum, 206, 223 quantity, 431–7, 442, 453–7, 459–79, 482–4 as con guration of numbers,432, 435, 460, 472, 477–8, 483 Quinn, F., 15–17, 64, 66 quotation, 75, 76, 77, 85, 163, 179, 184, 366, 367, 430, 520 Qusta Ibn Lūqā, 361
race ‘race apt to numericalcomputations’,284 Indo-Aryan races, 292 Semitic races, 292 Rackham, H., 362
Index
Raeder, J., 161 Ragep, J.,292, 293 Raina, D.,6–8, 9, 12, 228, 230, 238, 245, 247, 258 Raj, K., 237, 258 Rashed, R., 43, 67, 87, 132, 330 Rav, Y.,15, 17, 67 re-interpretation,500 recension,89, 109–10, 120 al-Maghribî (Muhyî al-Dîn al-Maghribî) recension,120 the so-called Pseudo-ûsîrecension,89, 106, 109–10, 117, 120, 288 reciprocals, 44–7; see also algorithm Records o [things] lef behind or posterity , see Ji yi; see also Shu shu ji yi Records o the procedures o numbering lef behind or posterity, see Shu shu ji yi Record o What Ý rai [=Nguyễn Hữu Tận] Got Right in Computational Methods (A), see Ý rai toán pháp nhất đắc lục redrawing, see diagram regular number,see number Renaissance, 27, 291 Renan, E.,292 restoring u , 437, 447, 453–8, 460, 473–4, 481–3 results, 5, 28, 40–3, 44–7, 50, 59, 427, 429, 431, 432–40, 444, 446, 448, 455, 460, 465–6, 479 emphasis on,277 reverse algorithm,384, 397, 404, 415 revival of past practices of proof,56 in China, 56–9 rewriting of lists of operations,44, 49, 52, 438–52; see also transformations Reynolds, L. G., 70, 71, 72, 134 Rhind Papyrus,285, 289 Ricci, M., 2–3, 56, 67, 567 Richomme, M., 523 right-angled triangle,8, 56–8, 265, 268, 270, 491–2, 494, 497–8, 507 rigour,4, 6, 12, 14, 15, 290 as a burden, verging on rigidity,7 lack of,6, 7, 290 of the Greek geometry,12, 14, 277 Rites o the Kai-Yuan era, see Kai-Yuan li Robert of Chester,86, 117, 127, 130 Robson, E., 384–6, 389, 396, 404–5, 410–11, 417, 509 Rocher, R.,237, 258 romanticism, 280, 291 Rome, A.,139, 162
Rommevaux, S.,78, 81, 84, 89, 118, 119, 120, 134 Ross, W. D., 325 Rota, G.-C., 17, 67 Rotours, R. des, 513, 515, 518, 520–1, 535 Rsine, 494–8 rule, 5, 6–7, 46, 265, 267–8, 270–1, 274, 278, 280, 281, 283, 285, 286, 288, 489–90, 498–9, 501, 503–7 as opposed to proof, 5, 9 general,402–3 rule of ve,504–5 rule of three,490–1, 493–8, 500–3, 505, 508 in Chinese, ‘procedure of suppose’ jinyou ( shu ), 451, 468, 472, 474–5, 479 trigonometrical,233 Russell, B.,225 Rutten, M., 370–1 Sabra, A. I., 120, 132, 134 Sachs, A. J.,45, 384–6, 388, 391–3, 399, 404–5, 410, 416–17 Said, E. W., 228, 258 Saito, K.,23–5, 30–2, 52, 67, 138, 141–2, 144, 146, 148, 150, 152–5, 158 San deng , 518; see also San deng shu San deng shu , 515, 517–18, 520–2, 546 Sanskrit,6, 7, 51, 56, 260–1, 264–5, 269, 272, 487, 491, 501, 506–8 texts, 6–8, 24, 42, 51–2, 63, 260–1, 264–5, 269, 272, 487, 507–8 Sarma, S.R., 490, 508 Sato, .,198 schemes, see shi scholium, 86, 95, 97–8, 103 Schreiner, A.,523 Schulz, O.,323 Schuster, J. A., 4, 8, 67 science, 10, 11, 13, 265–7, 269, 508 free inquiry versus lack of science, 8 idea of the unity ofscience, 11 sciences of India, 228, 265, 508 value of science in the eyes of the public, 2–4, 11 publications devoted to the ‘scienti c method’,11 values attached to science, 8, 265–6 scienti c management,381; see also taylorism scienti c writing,274 obstacle to logical proofs, 281 obstacle to the formulation of theorems,281 Scriba, C.J., 5, 65, 279, 293 scribal practice, 327, 329, 331, 333, 339–40
591
592
Index
scribal school, 384, 386–7, 390, 405; see also Babylonian mathematics scroll, 71–2, 84 Sea island, see Hai dao Sea mirror o the circle measurements, see Ceyuan haijing Sédillot, J.-J.,274 Sédillot, L.-A., 274 self-evidence, 305, 378, 380 separation of ‘Western’ from ‘non-Western’ science, 10, 53, 56, 59, 291 sequence, direct sequence and reverse sequence, 398–403, 405, 406, 409–10, 412, 415, 417 sequence of operations or calculations, 397–8, 404 see also algorithm, geometric progression Sesiano, J., 347, 350 sexagesimal place value notation, 384, 388–9 shang chu ‘evaluation division’,538, 542; see also division shapes of elds,see tian shi shaping of a scienti ccommunity,4–5, 11 shares, see en Shen Kangshen, 423, 485, 486 shi ‘dividend’, 431–4, 459–62, 467–72, 478 shi ‘schemes’,539, 544 Shu shu ji yi , 517, 520, 533, 546 Shu xue ‘College of calligraphy’,521 Shukla, K. S., 487–508 Shuo wen jie zi (dictionary), 511 Sidoli, N., 23–5, 139, 150, 158 silk, 469 simplicity,435, 442 Simplicius, 76, 85, 121, 131, 134, 296 simpli cation,431 of an algorithm,471 of fractions (‘procedure for simplifying parts’ yueen shu ), 431, 437, 460–1, 467, 469–70 sinology, 275 Siu Man-Keung, 513, 515, 519–23, 535 Six Codes o the ang [Dynasty] (Te), see ang liu dian Smith, Adam, 382 Smith, A. M.,304 social context for proof,18–19, 43, 53, 60–1 development and promotion of one tradition as opposed to another,60 interpreting a classic,47–53, 60, 423–84 professionalization of scientists,4, 11 rivalry between competing schools of thought, 1, 29–30, 59
teaching, 11, 44, 53–5, 60 see also mathematical education Socrates, 298, 300, 303 solids, 427–8 Song dynasty, 510, 513–14, 519, 523–4, 535 Song Qi , 548 Song shi (History of the Song (dynasty)), 532–3, 535 Sonnerat, P., 236, 237, 258 sophists, 296–7 speculative trend,286, 302, 510 sphere,453 as real globe, 150, 158–9 Sphere and Cylinder, see Archimedes, works by spherical geometry,139, 150–1, 159 Spherics, see Teodosius square root,386, 388, 410–16 Srinivas, M. D., 7, 12, 51, 67, 260, 273, 487, 508 Staal, F.,267, 273 Stache-Rosen, 261, 273 Stamatis, E.,70, 86, 130, 136, 138, 164 standard formulation,494 starting points,2, 14, 300, 303–5 debates about starting points in ancient Greece, 2, 14, 29 see also axiom, de nition State University, see Guo zi jian statics, 298, 304–5 Stedall, J.,288, 550 Stern, M., 279 Strachey, E.,276, 285 strip [reading] of classics [examination],see tie jing strip reading [examination],see tie du style of proof,25–6 style ‘characterizing distinct civilizations’, 9, 61, 61 styles of practising mathematics,9, 34–5, 278 distinctiveness of the Western scienti c style, 1, 9, 10, 56, 291 ‘Greek style’, 8–9, 278 ‘Indian style’,278, 281–2, 290 ‘intuitive, illustrative and unre ected style’, 291 ‘Oriental style’,9, 285 ‘systematic andaxiomatic–deductivestyle’,291 suan ‘counting rods’, 432, 511, 530, 538, 540–4 suan ‘operations with counting rods’, see toán Suan a tong zong, 512, 524, 526, 546 Suana tongzong jiaoshi (An annotated edition of theSummarized
Index
undamentals o computational methods), 548 Suan jing shi shu (en classical mathematical treatises), 548–9 Suan shu shu, 47, 423, 430, 434–5, 456–7, 469, 473–5, 482, 485, 510, 525–6, 547 Suan xue ‘MathematicalCollege’,511, 514, 518–22, 531–4, 536 Suan xue qi meng, 526, 547 Sude, B.,87, 132 Sui dynasty , 514, 517, 533 Suidas, 345 Sulbasutras, 8, 12, 260–73, 282, 506 Sumerian, 384–5, 390–1, 398 Summarized undamentals o computational methods, see Suan a tong zong Sun Chao , 517 Sun Peiqing , 550 Sun zi , 515–18, 520, 522; see also Sun zi suan jing, Sun zi bing a Sun zi bing a, 517 Sun zi suan jing, 511, 516, 526, 532, 542, 547 Supervisorate of National Youth, see Guo zi jian supporting argument,15, 16 surface for computing,see tool for calculation surveyability,37, 46 Surya Siddhanta, 233, 236, 239, 240, 241, 249 Suryadasa, 508 Susa, 41–3, 371, 376 Suter, H., 279 sutra, 51, 260, 262, 265–71, 487–508 syllogism, 27, 280 symbol, 6, 35–7, 38, 327–8, 330–6, 341, 349 symbolism, 35, 37, 39, 63, 284 and the development of the reasonings to establish the solutions to problems, 35–7, 57–8 Diophantus, 35–7, 63, 284, 330–41 for equationsand polynomials in China, 57–8 Vieta,36 synthesis, 44, 280; see also demonstration systematization,336, 342, 345, 347 Szabó, Á.,295
tables cubes and cube roots, 414 metrological,389 multiplication,400, 402, 403 numerical,385, 387, 389, 402, 406 reciprocals,385, 390–1, 393–4, 397, 400, 402, 403, 405, 417, 421
squares and square roots,412, 414 see also trigonometry tablets, Mesopotamian mathematical, AO 8862,379 BM 13901, 367–8, 373–4 CBS 1215, 384–6 IM 43993,376 IM 54472, 386, 410–11 Ni 10241, 395, 405–6, 420 MS , 371 MS , 371–2 MS , 371, 374–5, 379 UE 6/2 222, 386, 410–11 VA 6505,385–6, 391 VA 8390, 44, 364 YBC 8633, 376 YBC 6967, 377–8 ak, J. G., van der, 156 aliaferro, R. C., 145 ang dynasty , 54, 513–14, 516–17, 519, 523–4, 530–2, 534–5 ang liu dian , 514, 518 annery, P., 36, 279, 282, 284, 290–2, 318 critical edition of Diophantus’Arithmetics, 36, 318–25, 330–2, 336–9, 345 artaglia,N., 289 task, see tiao assora, R., 30, 67 aylor, J.,276 taylorism, 380–2 chernetska,N., 148, 162, 205 technical text,37, 40, 42–3, 60 shaping of technical texts for proof,35, 37, 40–1, 43, 62–3 technique of reasoning,33, 38, 53, 50–2, 58, 61, 320–5 eltscher,K., 229, 230, 259 en classical mathematical treatises, see Suan jing shi shu terminology,265, 311–18, 425, 431, 436–7, 440, 446–9, 451, 456, 464, 470 geometrical,318, 427 text of an algorithm,9, 39–42, 45–6, 48–9, 51, 59, 63, 487–90, 492–4, 498, 500, 503, 506–7 operations on the text of algorithms,438–52 see also text of proof, transformations text of a proof,16, 22, 26, 32, 35–9, 42–3, 62–3 arti cial text, 39, 45–6, 65 design of text as an indicator of the context,63 reading the text of a proof,30–2, 38, 40, 52 text of algorithm pointing out the reasons for its correctness,39–42, 48
593
594
Index
textual techniques,43, 44, 60 textual tradition,78 direct/indirect, 74–7, 78–9, 88–93, 101, 104–5, 107, 113–19, 124–5 Tâbit ibn Qurra,78, 86, 88, 89, 91, 93, 96, 103–5, 106, 107, 109–10, 116–19, 126–7 Tales, 295 Te Nine Chapters o Mathematical Procedures , see Jiu zhang suan shu Teaetetus, 304 Teocritus, 190, 191 Teodorus, 304 Teodosius, 139 On Days and Nights, 136 Spherics, 139–40, 150, 158 scholia, 156 Spherics 6, 150 Spherics 15, 151, 159 Spherics manuscript Vatican204, 140, 150–1 theology, 2–3, 13, 300, 304 Teon (of Alexandria),79–81, 83–5, 88, 131 Commentary to Ptolemy’s Almagest, 139 Teon of Smyrna, 311 theorem, 10, 16, 17, 39, 43, 56, 67 as statementsproved to be true, 39, 425 see also Pythagoras (Pythagoreantheorem) theoretical value,16, 425; see also epistemological values attached to proof theory, 17, 290 Egyptian (inductive), 286 Greek (deductive),1, 5, 26–7, 286 Tibaut, G.F. W., 8–9, 12, 260–73 Tom, R., 16, 17 Tomaidis, Y., 348 Tomson, W., 86, 130, 131, 161 Tucydides, 302, 306 Tureau-Dangin,F., 364, 369 Turston, W. P., 15, 16, 17, 20, 67 Tymaridas,282 ian Miao , 56–9, 552–73 tian shi , 524 ianyuan algebra , celestial/heavenly unknown’ algebra,57–8, 559, 561, 565–6 tiao ‘task’, 520–2, 530, 535 tie du ‘strip reading [examination]’, 520–2 tie jing ‘examination by quotation’,520 tien (Vietnamese, Chinese:qian, measure of weight), 538, 542
ihon, A., 73, 131 toán (Vietnamese, Chinese:suan) ‘operations with counting rods’, 523 okyo Metro,159 omitano,206 tool for calculation abacus, 394, 404 surface in ancient China,426, 432–5, 437–42, 459–60, 467, 472, 477–8, 483 oomer,G. J., 158, 169, 311 oth, I., 12 ran Vanrai, 523 transformations,44, 49, 426, 430–83 accomplished in the algorithm as list of operations, 43–4, 49–50, 59, 429, 438, 440, 442, 446–50 cancelling opposed operations (eliminating inverse operations that follow each other), 50, 439–40, 442, 446–8, 450, 452–9 dividing at a stroke (lianchu ), 448–9, 471–5, 480, 489 inserting an algorithm, 44, 432–3, 435–43, 459, 461, 464, 476 inverting the order of operations,442, 448, 451, 459, 475–80, 482 post xing operations to the text of an algorithm, 443–6 pre xing operations to the text of an algorithm, 430 validity of algebraic transformations of algorithms, 49–51, 59–60, 426, 438, 440, 447, 450–2, 454–6, 458–80, 482 translation, 2–3, 5, 6, 21–2, 36, 56, 75–7, 79, 81, 87–8, 109–10, 122 transliteration,71–3, 111 transparency,63, 354 as an actor’s category,48, 457 ideal of, 38–40 in Babylonian tablets,39–40, 42, 46 in Diophantus’Arithmetics, 37–8 in writings from ancient China,42, 48, 436, 440, 457, 479 why the texts of algorithms are not all transparent,49–51, 440–1 trapezoid, 498–500 redennick, H., 377 trigonometry,277 trigonometrical rules, 233 trigonometrical tables,233 ummers, P. M. J. E., 87, 132 uo uo, 548 usi [pseudo-] (also Nas.īr al-Dīn al-. ūsī, Nâsir
Index
ad-Dîn at-ûsî),89, 106, 109, 117, 120, 288 ybjerg, K.,300 ycho Brahe,275 uniformity,13, 15, 26, 57–8, 435, 442 unit(s), 311–26, 436, 452, 460–6, 468–70, 472, 475 spatial con gurations of units, 33, 313–16 unknown, 35, 57, 328, 331, 334–6, 349–50 upapatti (Sanskrit),249, 487, 489, 498, 501–3 Ur,384, 386–7, 406, 411 Uruk, 387, 406, 410 utility, 298–9 Vahabzadeh,B., 87, 131, 132, 134 value, 4, 10, 19, 28, 62, attached to rigour, 17 Van der Waerden,B. L., 363 Van Haelst, J.,84, 134 variant, 84, 88–92, 92–4, 98–9 global/local, 89–91, 93 philological variants/deliberate alterations, 84, 88 post-actum explanations,95–8, 120 Veldhuis, N.,408 Ver Eecke, P., 163, 186, 311, 323 veri cation, 5, 44, 51, 260, 271, 274, 399, 410–12, 414–15, 417, 488, 498, 503–7 Vesalius, 381 Vidal, D., 241, 256, 259 Vieta, F.,328 Vietnam, 54–5, 511, 523–4, 537, 541 Vietnamese system of state education,523 Vija-Ganita, see Bija-Ganita Vitrac, B.,22–3, 69–134, 136–7, 202 Volkov, A.,1, 44, 53–6, 60, 423, 452, 486, 509, 513, 515, 519–24, 526, 529, 531, 535, 541 volume of the cone,426–52, 468 of the cylinder,445, 455 of the regular tetrahedron,501–2 see also pyramid von Braunmühl, A., 279 Vu am Ich, 523 Wagner, D. B.,12, 17 Wallis, J., 6, 12, 254, 288 Wang Lai, 557 Wang Ling, 57, 67, 550 Wang Xiaotong, 517–18, 533, 535
ways of thinking Greek, 275 Indian, 275 weighting coefficients, 525–7 Weissenborn, H., 79, 134 Weitzmann, K., 170 Wertheim, P., 323 Wessel, C., 381 Whewell, W.,5, 11, 67 Whish, C. M., 7 Whittaker, J., Widmaier, R.,3, 67 William of Moerbeke,146, 164–5, 176 manuscript Vatican Ottob. 1850, see manuscripts (Latin) Wilson, N.,70, 71, 72, 134 Wittgenstein,L., 327 Woepcke,F., 85–6, 279, 286 Wolff, C., 382 Wong Ngai-Ying,513 Woodside, A. B.,523 Writing on computations with counting rods, see Suan shu shu Writing on reckoning, see Suan shu shu written versus oral, see oral versus written Wu cao , 515–16, 518, 522; see also Wu cao suan jing Wu cao suan jing, 516, 532, 547 Wu Jing, 526 Wu jingsuan , 515, 517, 522; see also Wu jingsuan shu Wu jingsuan shu , 517, 532, 547 Wu Wenjun, 12, 67, 549, 560 Wylie, A., 275 Xenophon,298 Xi’an , 514 Xiahou Yang , 515–16, 518–19, 522; see also Xiahou Yangsuan jing Xiahou Yang, 518 Xiahou Yang[Master], see Xiahou Yang Xiahou Yangsuan jing , 516, 519–20, 532, 547 Xiangjie Jiuzhang suana (A Detailed Explanation o the Nine Chapters o Mathematical Procedures), 56–9, 553, 566 Xin ang shu , 514–21, 530, 532–3, 535 Xu Guangqi 2–3, 13, 56, 58, 66, 567 Xu Shen , 511 Xu Yue 517, 519
595
596
Index Ý rai toán pháp nhất đắc lục , 526 Yabuuti, K., 57, 68 Yan Dunjie, 515 Yang Hui , 566 Yeo, R. R., 4, 5, 8, 11, 67, 68 Yushkevich,A. P., 510
Zamberti, B.,79–80, 134 zenith distance,494–5 Zeuthen, H. G., 277–84, 287, 289, 290 Zhang Cang , 518 Zhang Heng , 453 Zhang Jiuling, 548 Zhang Peiheng, 517 Zhang Qiujian, 515–16, 518–19, 522; see also Zhang Qiujiansuan jing Zhang Qiujian [Master], see Zhang Qiujian
Zhang Qiujiansuan jing , 516, 519, 526, 532, 534, 547 Zhao Junqing, see Zhao Shuang Zhao Shuang (also Zhao Junqing, Zhao Ying ), 531–4, 536–7, 566 Zhao Ying , see Zhao Shuang Zhen Luan , 517, 519, 532–4 Zhou bi , 512, 515–16, 518, 522; see also Zhou bi suan jing Zhou bi suan jing, 511–12, 516,
532–3, 547 zhu ‘commentary’, 519, 534–5; see aslo commentaries Zhu Shijie 526 Zhui shu , 515, 517, 519, 522, 535, 547 Zu Chongzhi, 517–19, 535 Zu Gengzhi , 518 Županov, I. G.,230, 259
