Memory and Geometry in Bruno: Some Analogies

Found Sci (2014) 19:69–88 DOI 10.1007/s10699-012-9314-7

Memory and Geometry in Bruno: Some Analogies Manuel Mertens

Published online: 12 February 2013 © Springer Science+Business Media Dordrecht 2013

Abstract In 1588 the Italian philosopher Giordano Bruno wrote a treatise against the mathematicians and philosophers of his time (Articuli centum et sexaginta adversus huius tempestatis mathematicos atque philosophos), which he dedicated to the emperor Rudolph II. The ‘oddities’ thus presented to the emperor, as an alternative to sixteenth-century mathematics, have been studied from both a mathematical and a philosophical point of view. In addition to the philosophical approach, this article indicates analogies between the Nolan’s geometry and his art of memory. Bearing in mind that Bruno was a teacher in the ars memoriae, the manner in which mnemonic aspects are woven into his mathematical thinking is brought out. Keywords

Art of memory · History of mathematics · History of philosophy

1 Introduction More than once Bruno’s mathematics have been approached from a predominantly mathematical point of view, and this has resulted in a rather negative judgement. Tocco (1889, p. V) referred to his mathematical oddities. Olschki (1924, p. 54) characterized his mathematical speculations as primitive and Koyré (1973, p. 57) simply called him un mathématicien exécrable. This mathematical perspective leads to incomprehensible results, and to that extent seems insufficient. On the other hand, there is the more philosophical approach. De Bernart (1986, p. 52) saw in Bruno’s diagrams a search for a theory of perception, and since the end of the nineteenth century a tendency has explored the philosopher’s geometry in relation to his atomism.1 The most comprehensive study, by Bönker-Vallon (1995), linked the Nolan’s mathematics with his metaphysics. Without any doubt the philosophical perspective correctly reflects the Nolan’s true incentive for putting his mind to a science towards which he adopted 1 Lasswitz (1884), Michel (1960), Monti (1980), Lüthy (2003).

M. Mertens (B) Centre for History of Science, Ghent University, Ghent, Belgium e-mail: [email protected]

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a critical attitude from the outset. Here I propose a perspective—in agreement with, and as a complement to, the philosophical point of view—that grasps how his geometry is intertwined with mnemonics. First, I will focus on Bruno in Paris as a teacher of the ars memoriae (1582). By some practical inventions he introduced ‘mathematics’ into his art of memory. Secondly, I will proceed to the Articuli centum et sexaginta adversus huius tempestatis mathematicos atque philosophos (Prague, 1588, henceforth Articuli adversus mathematicos) where some passages evoke his Parisian writings. Here the Nolan inserted mnemonics into his mathematics. Finally, I will look closely at memory and geometry in one of Bruno’s most important geometric figures from De triplici minimo et mensura (Frankfurt, 1591): the Atrium Veneris. The correspondences between the construction of this figure and the Statua et membra Cupidinis from Lampas triginta statuarum (Wittenberg, 1587) elucidate how the geometric map of the goddess’ atrium overlaps the memory map, on which the statue of her son and its members are built.

2 Lettor Straordinario et Provisionato in Paris Bruno’s nova filosofia—proclaiming an infinite universe with an infinite number of worlds— took root in a mind which was deeply occupied with the ars memoriae. This art had its origin in ancient rhetoric and made use of places (loci) and images (imagines) to remember data. Several testimonies describe the Nolan as a teacher in the art of memory, and the recent edition of his mnemonic works has shown the didactic character of a great part of these writings.2 One of these testimonies, a famous excerpt of Bruno’s inquisitional trial in which he explains to the judges how he succeeded in opening the doors to the French court, suggests why Bruno was to introduce mnemonics into his later Articuli adversus mathematicos dedicated to the emperor Rudolph II: […] and I went to Paris and started to give extraordinary lessons to make myself known and become famous; […] I gained such a name that King Henri III summoned me one day and asked me whether the memory which I had and which I taught was a natural memory or obtained by magic art; I gave him satisfaction, and from what I told him and had him try on his own, he understood that it worked not by magic arts, but by science. After that I printed a book on memory entitled De umbris idearum which I dedicated to His Majesty, whereupon he made me an extraordinary and endowed reader.3 It was thanks to his art of memory that Bruno was capable of attracting the attention of the King who rewarded the Nolan by taking him into his band of personal teachers, among whom were numerous Italians.4 The title page of the above mentioned De umbris idearum (Paris, 1582), suggests that the philosopher did not intend to write a vulgarizing best-seller 2 See Bomne I, pp. II–XIII and Bomne II, especially in the comment on Explicatio triginta sigillorum

(London, 1583) and Sigillus sigillorum (London, 1583). 3 ‘[…] e andai a Paris, dove me messi a legger una lettion straordinaria per farmi conoscer et far saggio di

me; […] acquistai nome tale che il re Henrico terzo mi fece chiamare un giorno, ricercandomi se la memoria che havevo et che professava era naturale o pur per arte magica; al qual diedi sodisfattione; et con quello che li dissi e feci provare a lui medesmo, conobbe che non era per arte magica ma per scientia. Et doppo questo feci stampar un libro de memoria sotto titolo De umbris idearum, il qual dedicai a Sua Maestà; et con questa occasione mi fece lettor straordinario et provisionato’ (Firpo 2000, pp. 49–51). I have considered, but do not reproduce Yates’ translation (1966, p. 200). 4 For Bruno as ‘lettore reale’, see Ricci (2000, p. 147).

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on the art of memory, but rather that he had serious philosophical aspirations.5 In fact this book proposed a new kind of epistemic method that emphasized the collaboration between will, memory and intellect—following Augustine’s three-way partition of the soul. Within this framework, the images used during the mnemonic practice had a value related to the shadows of the Neoplatonic ideas. Practising the art of memory should lead the artist gradually to the higher levels of being.6 Playing with the shadows, the Nolan was interiorizing the Neoplatonic world structure, and reproducing the relations between the several levels of being—an important influence from the art of Ramon Lull—so that he could climb the ladder of being and reach a view of the whole picture, emulating the first mind. This was the main point of De umbris idearum.7 Bruno was not interested in studying the elements separately, but wanted to understand them within the web that linked all beings together. The hand, joined to the arm, the foot to the leg and the eye to the forehead, obtain a greater recognisability when they are joined than when presented separately; like this, when no parts and species of the universe are separate and out of the order—which in the first mind is the most simple and perfect order beyond any number—what could we not understand, remember and do, if we apprehended them by connecting the one with the other and by unifying them according to reason?8 In his paragraphs on the traditional mnemonic places and images (subiecta and adiecta in his terminology) the teacher emphasized the advantage of animating them.9 Lullian wheels were then presented as a practical side of the art. The segments of the wheels (subiecta) became fine settings in which animated images (adiecta) replaced what were mere letters in Lull’s art. The artist needed to breathe life into the two-dimensional combinatory wheels, transforming them into three-dimensional living scenes, turning Lull’s algebra into a fantastical logic. Bruno was aware that in Lullism these wheels did not serve proper geometric reasons, but had a symbolic meaning. The circularity of Lull’s first combinatory wheel, for example, expressed the convertibility of the divine predicates in each other and in God.10 5 The title promises more than just a good memory. From the outset, incompetents are warned that this book is

destined not for them, but for the learned. Bomne I, p. 2: ‘De umbris idearum. Implicantibus artem, Quaerendi, Inveniendi, Iudicandi, Ordinandi et Applicandi: Ad internam scripturam, et non vulgares per memoriam operationes explicatis. […] Umbra profunda sumus, ne nos vexetis inepti. Non vos, sed doctos tam grave quaerit opus.’ 6 Bomne I, p. 64: ‘Umbra igitur visum preparat ad lucem. Umbra lucem temperat. Per umbram divinitas oculo esurientis sitientisque animae caliganti nuncias rerum species temperat atque propinat.’; Ibid., p. 104: ‘Ita ab umbris ad ideas patebit aditus et accessus et introitus.’ On the epistemic importance of De umbris idearum, see especially Spruit (1988). 7 Bomne I, p. 88: ‘Quem ordinem cum suis gradibus qui mente conceperit, similitudinem magni mundi contrahet aliam ab ea, quam secundum naturam habet in se ipso. Unde quasi per naturam agens, sine difficultate peraget universa.’ 8 Bomne I, p. 100: ‘Sicut manus brachio iuncta pesque cruri et oculus fronti, cum sunt composita, maiorem subeunt cognoscibilitatem quam posita seorsum, ita, cum de partibus et universi speciebus nil sit seorsum positum et exemptum ab ordine—qui simplicissimus, perfectissimus et citra numerum est in prima mente—si alias aliis connectendo et pro ratione uniendo concipimus, quid est quod non possimus intelligere, memorari et agere?’ 9 For the animation of both subiecta and adiecta, see Bomne I, p. 178: ‘Agere quoque intelligantur adiecta in subiecta et in subiectis, vel pati a subiectis vel in subiectis. Aliqua, inquam, actione vel passione vivificata habeantur, quatenus aliquo motu internum visum quasi sopitum exagitatione quadam expergefaciant, errando, transeundo, subeundo, adeundo, […], quo aliquid admoveant, pellant, trudant, excludant, abalienent, […].’ See also Bomne I, pp. 152, 166, 172, 174. 10 Lull (1991, p. 110); De compendiosa architectura et complemento artis Lulli, Bol, II, 2, p. 15.

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After the paragraphs on his subiecta and adiecta, he proceeded with the organum or instrument used by the soul to practise his art.11 This tool by which the cogitative faculty inquires and discerns is called the scrutinium, and had, according to the philosopher, never been considered before his time.12 It is compared to a stick one can use to stir a pile of acorns in search of a chestnut. ‘It is a kind of number, by which the cogitatio in its manner touches the conserved species, divides and separates them according to its capacity, collects, associates, mutates, forms orders and refers them to an established unity.’13 Thus Bruno indicates a form of natural rationality—operating through number units14 —that is already present in the pre-linguistic sphere of the mind. This ‘kind of number’, responsible for the ordering activity of the mind, linked his art to nature, in which a similar ordering activity is present.15 The ‘mathematical’ aspect of the scrutinium refers above all to order; a Pythagorean stress which returned in the subiecta semimathematica of his second work on the art of memory. This second treatise, Cantus Circaeus (Paris, 1582), offered a more didactic approach to his ars memoriae. A more elaborate treatment of mnemonic techniques here provided better access to the practical side of his art. Cantus Circaeus contains two dialogues. In the first, the sorceress Circe conjures the planetary spirits.16 When the conjuration is finished she asks her disciple Moeris to put more incense on the fire and to look through the window of the palace. Moeris is very surprised because of all the men that were outside before, only three or four are left and take refuge in a safe place. All the rest, of whom some hide in caves nearby, some fly into the branches of the trees, others hastily run to the seashore, and the domestic ones approach the gates of the palace, have changed into animals.17 Her mistress replies that the spell has not caused the human beings to change into animals, but has made the creatures take their original form again. The people Moeris previously saw were in fact animals in human shape.18 Then the two leave the palace and each time they encounter an animal, Circe describes its special features. Of course this gives Bruno an opportunity to criticize, allegorically, the different types in society. But to interpret this first dialogue as a sixteenth-century Animal Farm would be all too limited an approach. Certainly if we call to mind that one of the heretical points raised during his inquisitional trial concerned the theory of metempsychosis (circa animas hominum et animalium, Firpo 2000, p. 341), we will be warned off from any superficial reading. Already in Antiquity, Circe’s myth was interpreted by the Pythagoreans as an allegory for the cycle of reincarnations (Buffière 1973, p. 506). In opposition to the moral-allegorical reading—which denied true metempsychosis—of some 11 Bomne I, p. 180: ‘Reliqum est ut de organo, quo in proposito utitur anima, nonnihil determinemus.’ 12 Bomne I, p. 182: ‘Inter haec omnia quod scrutinium appellamus sive discerniculum—utpote quo cogitatio

inquirit atque discernit—instrumenti rationem sortiri facile constat. Quod ita communi nomine insignimus, quippe cum ad nostra usque tempora eius nulla facta fuerit consideratio, proprio celebrique nomine caret.’ 13 Bomne I, p. 188: ‘Est igitur scrutinium numerus quidam, quo cogitatio tangit modo suo species conservatas, eas pro sua facultate disterminando, disgregando, colligendo, applicando, immutando, formando, ordinando, inque seligendam unitatem referendo.’ 14 Bomne I, p. 186: ‘Ecce igitur scrutinii munus est ut unitates—ita enim dixerim multa una, ut verborum censoribus aliquid concaedam—sigillatim capiendae, per ipsum in ordinem disponantur.’ 15 For the relation between his art and nature, see Bomne I, pp. 122–132 and Sturlese (2000). 16 For Bruno’s composition of the conjuration (with an enumeration of animals, epithets and compound terms) in relation with the prescriptions of Renaissance magic in Ficino and Agrippa, see Bomne I, p. 749. 17 Bomne I, p. 620: ‘Mirabile visu, Circe, mirabile: de tot, quos vidimus, hominibus tres quatuorve tantum, qui trepidi ad tuta confugiunt, remansere. Caeteros omnes, quorum alii in proximas se recipiunt cavernas, alii in arborum ramos advolant, alii se dedunt in proximum mare precipites, alii domestici magis ad nostras fores adproperant, in diversi generis animantia video transformatos.’ In Vergil’s eighth Eclogue Moeris appears as a werewolf. 18 Ibid.: ‘Ii, qui adhuc perstant, veri sunt homines: illos nec vult neque potest cantus noster attigisse.’

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of his forerunners, Bruno believed that souls really transmigrated into other bodies, both human and animal (Granada 1993).19 In his eyes, one of the marvellous powers of his art was to go beyond the world of appearances to penetrate and judge the real nature of beings, to unmask the true soul in a body. Thus the spell of Circe, after which the true nature of beings is revealed to her disciple, symbolizes this ‘magical’ effect of Bruno’s art.20 In the second dialogue of Cantus Circaeus a more advanced student in Bruno’s art explains to a novice how to learn the first dialogue by heart. Looking back to the first dialogue, we realize that Bruno must have been a demanding teacher. The text of Circe’s invocation is not structured by rhythm or rhyme, which might have assisted the memory of the students. On the contrary, at first sight it appears to be an endless enumeration of animals, epithets and terms that describe the powers and operations of each planetary spirit. The novice is told that before memorizing this text, he should prepare his subiectum. This ‘subject’ can be understood literally as ‘what lies beneath’, as the screen on which the memory construction is projected and to which the memory images are added. Then the advanced disciple refers to three types of subject: This subject […] can be naturally composed, semi-mathematical or verbally posed. The naturally composed can be extremely general, spread out according to the extension of the universe, or very general after geographic size, or simply general to the extension of a continent; or it can be particular following the political extension, or very particular as in a domestic or economic sense, or at its most particular according to the quantity and number of rooms in a house.21 The first type, the naturally composed, can have different sizes. From the greatness of the extension of the universe it can be gradually reduced to the dimensions of the surface of a room. Preparing his subiectum for Circe’s dialogue, the student needed a subject in the ‘domestic or economic sense’. He had to draw out the sorceress’s palace with seven halls, one for each planetary spirit. Then each hall had to count three rooms, because in order to invoke these spirits, Circe needed to name the animals associated with that specific planet, which could be placed in the first room. Next a list of epithets had to be drawn up. Mnemonic images translating these epithets could be put in the second room, while the enumeration of compound terms that described the powers and operations of the planet could be stored in the third room. Thus in the seven halls of Circe’s palace the whole conjuration could be stored. To memorize Circe’s comments on the different animals, the students might have used a ‘geographical subject’, divided into a field nearby the palace for the domestic animals, 19 Pico and Ficino tend to interpret Plato’s and Plotinus’ adhesion to metempsychosis allegorically, in opposition to the Nolan’s literal reading. See Granada (1993, pp. 42, 50–51). Next to many allusions to metempsychosis in the Nolan’s Cabala del cavallo pegaseo (London, 1585) and Spaccio della bestia trionfante (London, 1584), an amusing reference to the doctrine is found in a testimony against Bruno of a former fellow prisoner in Firpo (2000, p. 341): ‘Essendo egli in letto, andai a trovarlo e trovandoli vicino un ragnetto, l’ammazzai, e lui mi disse ch’havevo fatto male, e cominciò a discorrere, che in quelli animali poteva esser l’anima di qualche suo amico, perché l’anime, morto il corpo, andavano d’un corpo in un’altro, et affirmava, che lui era stato altre volte in questo mondo, e che molte altre volte saria tornato dopo che fosse morto, o in corpo humano, o di bestia; et io ridevo, e lui mi riprendeva, che io mi burlassi di queste cose.’ 20 A similar reading of this dialogue is offered by Matteoli and Sturlese (2007, pp. 480–481). For the kind of magic and the political background of Cantus Circaeus, see Perrone-Compagni (2000). 21 ‘Subiectum vero istud […] esse potest vel compositum naturale, vel semimathematicum, vel verbale positum. Ipsum vero naturale vel potest esse communissimum, extentum iuxta latitudinem ambitus universi, vel communius iuxta latitudinem geographiae, vel commune iuxta latitudinem alicuius continentis, vel proprium iuxta latitudinem politicam, vel proprius iuxta latitudinem domesticam seu oeconomicam, vel propriissimum iuxta multitudinem atque numerum partium domus et particularum eiusdem’ (Bomne I, p. 672). For an almost identical passage, see also De umbris idearum, Ibid., pp. 150–152.

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a wood with caves for the wild animals and birds, and finally the seashore to observe the fishes. An important prescription is that the ground plan of the subiectum should reflect the structure of the text that is to be memorized. Thus the subdivision of Circe’s palace into seven halls, each containing three rooms, responds clearly to the sequence of her conjuration of the seven planetary spirits, each counting three lists of terms to be pronounced. After having grasped this common rule, the novice is also interested in the techniques of the more advanced disciples and asks about the semi-mathematical subjects. These fulfil a specific function and operate as a kind of compass, capable of determining the student’s precise coordinates inside a memory building. After all, the virtuoso should always know where he finds himself within his memory palace. What image is he looking at: the third, the thirteenth or the twenty-seventh? For the sake of this orientation Bruno invents a system to mark certain places so that their order should become immediately visible. For our teacher, mere numbers were useless for this purpose, because incapable of striking our imagination. Pure mathematical subjects are not useful when they are abstract and therefore cannot strike or move our fancy [...] What mathematicalia can procure by themselves, is order alone; and this can be found in a twofold way—in one of them it succeeds very well—in figures namely, and in numbers.22 Thus marking the images with numbers, the artist will doubtless get confused and will not be able to determine which image he has in front of him. On the other hand, two systems devised by Bruno do succeed in fulfilling this orientating function; one using figures, the other (not mere) numbers. The semi-mathematical subject making use of figures is illustrated when Bruno designs a floor plan to memorize Aristotle’s Physics in his Figuratio Aristotelici Physici Auditus of 1586 (Paris, Fig. 1). The different rooms in the floor plan are formed after polygons so that the student is immediately informed on the order of the images he is looking at. Encountering images in the triangular room, he knows that he is treating an argument pertaining to the third article of a certain chapter. An image stored in the pentagonal room refers to the fifth article. Thus this semi-mathematical subject has the property of directly indicating order to the artist. But looking at the first paragraph ‘de quindecim imaginibus auditionis physicae figurativis’, where fifteen mythological figures denote the central themes of Aristotle’s Physics, we understand that this floor plan, as a real subiectum, was meant to lodge adiecta animata. Thus the eight books of Physics are re-organized by Bruno in agreement with the fifteen subjects treated in them.23 Of the figures that will animate the polygons, Minerva represents the principle, Thetis matter, Apollo the form, and so on.24 In this way is the method making use of polygons used in Figuratio Aristotelici Physici Auditus, and animated by mythological figures. It turns the whole enterprise into a rather crowded memory construction. As in the Lullian wheels of De umbris idearum, here too the abstract ‘geometry’ is accompanied by the animating concrete, a typical feature that will return in Bruno’s more serious geometry. 22 Bomne I, p. 684: ‘Subiecta pure mathematica usu venire non possunt, quandoquidem abstracta sunt et sua abstractione phantasiam pulsare vel movere non possunt, […] Illud ergo, quod valent praestare mathematicalia secundum se, est ordo solus; et hic in duobus inquiri potest—in quorum tamen uno feliciter succedit—in figuris videlicet et numeris.’ I do not consider here the third type of subject (subiectum verbale positum), it being irrelevant for this article. 23 Figuratio Aristotelici Physici Auditus, Bol I, 4, p. 141. 24 A similar procedure is found in Lampas triginta statuarum—although in this ars inventiva per triginta statuas, as the work is called by the author, invention is the main point, not memory—where thirty statues embody thirty philosophical concepts. This time Apollo stands for unity, Saturn for the principle, Prometheus for the efficient cause, and so on.

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Fig. 1 Figuratio Aristotelici Physici Auditus

Apart from figures, the order-indicating quality could also be found in the semi-mathematical numbers. This is the second system that Bruno devised to translate numbers into images which are capable of striking our imagination. The first ten numbers will be represented by certain objects. A column will stand for one, a gate for two, a tripod for three, and so on. Depending on the material of which these objects are made, the artist will know which decade is indicated. Thus a linen column will stand for one, a wooden column for eleven, an iron column for twenty-one and so forth.25 These figurate numbers are inserted inside memory buildings to indicate at any time the coordinates of the artist. Although Bruno adopts no posture towards mathematics, it is clear that pure numbers are not useful for his art of memory, as their abstraction cannot meet its needs. Therefore he invents two systems for translating order, normally indicated by numbers, into images. This then is how Bruno introduces ‘mathematics’ into his art of memory. After his two books on memory, the philosopher dedicated a treatise to the already mentioned art of Ramon Lull, De compendiosa architectura et complemento artis Lulli (Paris, 1582), at the end of which he stated that his addition of mnemonics was necessary to perfect this art.26 Referring to his former subiecta and adiecta, the Nolan offered a floor plan, in the corners of which a person performing an action with a marked object and an attribute (homo, actio, insigne and adstans) represented the four significations (indicated by the vowels in Fig. 2) of the nine elements of Lull’s alphabet (from B to K). Figure 2 presents the room in which the four significations of B (Deus, Bonitas, Differentia, Utrum) were remembered. Thus in nine rooms all the possible significations of Lull’s elements were stored. The combinatory activity itself then resulted in scenes, in which these vivid images (vivas habeant imagines) crossed a room to put themselves in relation with each other (Fig. 3).27 In her chapter ‘La geometria della memoria’ De Bernart interpreted these mnemonic additions to Lullism in a precise epistemological tradition. She explained how Bruno reformed the Aristotelian and Platonic relationship between substance and accident through Duns Scotus and Cusanus. The scotist logic of infinite possibility transcended Aristotelian logic. The intelligibility of 25 Bomne I, p. 687: ‘Veruntamen ipsi numeri non valent repraesentare, sed ordinem tantummodo insinuare. Applicentur igitur rebus aliquibus naturalibus et per easdem colorentur atque formentur. Destinentur ergo pro primo denario linea, pro secundo lignea, pro tertio ferrea, […]’ For a further elaboration of this system, see Bruno’s sixteenth seal ‘de numeratore’ in Explicatio triginta sigillorum, Bomne II, p. 60 and pp. 130–132. 26 De compendiosa architectura et complemento artis Lulli, Bol, II, 2, p. 62: ‘Et tu lector, postquam intellexeris ista, necessario Lullium intelliges, et videbis nos multum addidisse, quantum ad facilitatem, ordinis rationem, distinctionem et sufficientiam disciplinae spectat, et aliud quid de memoria, quod non est pars istius disciplinae, sed ad ipsius retentionem est maximum necessarium. Videbis etiam nos addidisse quod est de substantia artis usque ad complementum ipsius, quod nec Lullius fecit, nec alium fecisse vidimus’. 27 Ibid., pp. 17–21.

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Fig. 2 Ad alphabeti retentionem

Fig. 3 Atrium quadratum novem discriminatum conceptaculis

the accident—for which Bruno spoke in De umbris idearum against Plato28 —emphasized the partly empirical origin of knowledge and the ascent from the sensible to the intelligible through the different faculties of the soul. One of these faculties is memory. It is engraved with geometric figures by the cogitatio and filled up by empirical information offered by the sensus communis. For De Bernart (1986, p. 77), these geometrical figures individuate the mathematical constants of experience, that mediate the passage from the empirical to the rational sphere of knowledge. In her eyes the additions proposed to memorize the Lullian figures become meaningful in this epistemic context. On the other hand, Bruno’s mnemonic practice dates from his youth (adhuc puer). Originally, it was not designed as an alternative endeavour to overcome epistemic problems—as presented by De Bernart—but was his modus philosophandi and the key to his success from the start.29 Although Bruno must have been familiar with mathematical astronomy, having lectured on Sacrobosco’s Sphere before coming to Paris, the rudimentary ‘geometrical’ elements seen until now were limited to Lullian combinatory wheels, figures designed to facilitate the memorization of them, and polygons that functioned as semi-mathematical subjects—all animated by vivid images. Besides, the ordering activity of the scrutinium according to number units—also the basis for the subiecta semimathematica—already suggested a Pythagorean inspiration, and brought Bruno’s art closer to nature, where a similar ordering activity is present. As such, his ars memoriae is analogous to his later ars geometrica—equally based on number units—which the philosopher also brought into agreement with nature. After his dialogues on the infinite universe and his encounter with the compass of Mordente, the philosopher put his mind more seriously to geometry, which procured for him an intellectual 28 Bomne I, p. 112: ‘Accidentium ideas non posuit Plato, […] Et nos in proposito ideo omnium volumus esse ideas, quia ab omni conceptabili ad easdem conscendimus.’ 29 Explicatio triginta sigillorum (Bomne II, p. 114): ‘Hoc principium extitit, quo ad artis memoranda rationes adsequendas sum promotus. Ipsum adhuc puer ex monimentis Ravennatis expiscare potui.’ In 1585 Bruno told his Parisian confidant Cotin about an earlier success. As a friar he was ‘appellé de Naples à Rome par le pape Pius V et le cardinal Rebiba, amené en une coche, pour monstrer sa Mémoire artificielle, récita en hébreu à tout endroit le psolme Fundamenta, et enseigna quelque peu de ceste art audit Rebiba’ (Spampanato 1921, p. 654).

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instrument with which to explore such basic themes of his new cosmology as atoms and the vacuum. Towards the end of his career he even aspired to the vacant chair of mathematics in Padova. In the event the chair was obtained by Galileo in September 1592, a few months after Bruno was arrested (Ricci 2000, p. 471). But in the geometry of this later period, vivid images realized the construction of his most important figures. This fantastical imaginary assistance was already present in the Parisian treatises and suggests a profound link between his late metaphysical-geometric figures and the topology of his ars memoriae.

3 Prague: Philasophers, Geameters and Domini Circiterizantes We have noted how Bruno introduced ‘mathematics’ into his memory structures by his subiecta semimathematica, and that Lullian figures were vivified through mnemonic additions. Now we can look at how he introduced mnemonics into his Articuli adversus mathematicos. The dedicatory epistle to Rudolph II gives a beautiful account of Bruno’s intellectual experience in sixteenth-century Europe. Unfortunately this Europe was torn apart by religious wars, and counted more disparate sects than there had ever been or would be human generations on the earth.30 In this description of the regrettable political situation, we find a sudden exhortation to the emperor: Let us not be equal to brutes and barbarians, but be transformed into the image of him, who makes his sun come up over the good and the bad, and makes the rain of his graces come down over the right and the wrong.31 Although here the Nolan’s wish is formulated by recalling a Biblical verse, he is making exactly the same suggestion as in his De umbris idearum, dedicated to Henri III: let us emulate the first mind. He then draws a bead on the sages of his time, whose number ought to exceed the infinite quantity of fools.32 The worst cause of general ignorance is that it is forbidden by law to dispute, and prescribed that it is praiseworthy to remain firm about the received theories and to adhere to an opinion once conceived. For Bruno, ‘this ruling refers well to those beasts that are no longer human, but formed after the image and similitude of humans.’33 Again an idea is echoed that Bruno had already developed in his Parisian Cantus Circaeus, when after her spell Circe informed her disciple that the people she had seen were in fact not human, but animals in a human shape. The philosopher probably hoped to achieve a similar success as he had enjoyed at the French court. Besides, the political affinity between Prague and Paris, just like the sympathy of the emperor for Henri III, was certainly not unknown to Bruno and might even have encouraged him to present his art to Rudolph II (Cengiarotti 2004, p. 296). This suggests an almost provocative attitude, in which Bruno presents himself as a possible private tutor of the emperor: 30 Articuli adversus mathematicos, Bol I, 3, p. 3: ‘Itaque de tam variis et diversimode sectis opinantibus longe pluribus quam sint atque fuerint in mundo generationes, […]’. 31 Ibid., p. 4 ‘[…] ne brutis barbarisque similes consistamus, sed in illius transferamur imaginem, qui solem suum oriri facit super bonos et malos, et gratiarum pluviam super iustos instillat et iniustos.’ Cf. Mt. 5, 45. See also Lampas triginta statuarum (Bom, p. 1262). 32 ‘quasi sapientum numerus infinitum stultorum numerum superare vel exaequare vel ad ipsum propius accedere debeat, […]’ (Bol I, 3, p. 5). 33 ‘quod sane statutum ad bestiales illos bene refertur, qui iam non homines, sed ad imaginem et similitudinem hominum sunt effincti, […]’ (Bol I, 3, p. 6).

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As His Majesty already by himself understands many of these things, he would be able to see and judge everything easily by my indications, […]34 Although the emperor already knows much, he ‘would be able to see and judge everything easily’ by the personal indications of the Nolan (me indicante). The teacher reappears. Bearing in mind how Circe unmasked animal souls in human bodies to express the unveiling effect of Bruno’s art, we can imagine what the philosopher might have meant when he promised the emperor that he ‘would be able to see and judge everything easily’. Thus the dedicatory epistle to Rudolph II contains references to the Nolan art as it has been exposed in his Parisian treatises. Bruno then proceeds to the matter in hand. He sets up a list of axioms and theorems. From the first axiom it is clear what his true concern is: the universe and its minimum.35 The axioms are followed by the theorems of the minimum. I. The division of nature and art cannot continue to the infinite (although it is still undetermined), but ends necessarily with a certain term. II. What does not refer to a figure, is not a minimum; each part needs to be a part of some figure.36 These first two theorems of the minimum articulate very well the outline of Bruno’s attack on the mathematicians. The first dictates that in correspondence with nature, art also needs a minimum. It appears that not only his ars memoriae, but also his ars geometrica is consonant to nature. Bruno has already reproached geometry as being cut adrift from nature in Cena delle ceneri (London, 1584).37 The criticism resounds in this theorem aimed at the sixteenth-century geometry of the continuum. If atoms constitute the natural world, geometry should not deny but embrace the minimum.38 The second fixes the necessary link between the minimum and the figure, in which it has to take part. Without the figure, no minimum can exist. In Sigillus sigillorum the importance of the figure, embracing both Aristotelian categories of quantity and quality and located in the emanationist movement of descensus and ascensus had already been emphasized.39 In his Articuli adversus mathematicos the figure is anterior to measure.40 Thus sensible figuration becomes a necessary condition for the rational act of mensuration. All minima (the smallest triangle, circle, angle, and so on) are 34 ‘Iam hic C.M.T. cum per se plurima horum facillime intelligere, meque indicante facillime videre et iudicare omnia possit, […]’ (Bol I, 3, p. 7). 35 Bol I, 3, p. 10: ‘Universum est maximum. Totum est maius et perfectum. Pars est minus, imperfectum, et proximior mensura. Individuum est minimum, nec perfectum neque imperfectum, et communissima mensura […]’ See Heuser-Keßler (1991). 36 ‘I. Tum naturae, tum artis resolutio non est ad infinitum (licet interdum indeterminata), sed certum terminum definit necessario. II. Quod nullius est figurae, non est minimum; pars nempe omnis alicuius est figurae’ (Bol I, 3, p. 10). 37 Cena delle ceneri (BOeuC, II, p. 243): ‘Raggio reflesso e diretto, angolo acuto et ottuso, linea perpendicolare, incidente e piana, arco maggiore et minore, aspetto tale e quale, son circostanze matematiche e non cause naturali. Altro è giocare con la geometria, altro è verificare con la natura.’ 38 For an illustration of this geometry of the continuum, Ricci refers to a sixteenth-century preface of Euclid’s Elementa (Paris, 1557), where the removal of the indivisible residue is celebrated as an announcement of free and true geometry. Bruno instead restores this atomic residue (Ricci 2000, p. 436). 39 Bomne II, p. 272; ‘Figura vero quaedam est non sine qualitate quantitas, non sine quantitate qualitas, sed in quantitate qualitas, non lux, non color, non lucis colorisque vestigium—hanc etenim quandoque tactu iudicamus—non pura quantitas, non pura qualitas, sed ex utraque et in utraque unum.’ The emanationist movement is described successively through lux, color, figura and forma. 40 Bol I, 3, p. 19: ‘Prius igitur figuram esse oportet quam mensuram.’

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Fig. 4 Figura mentis

‘minima sensibilia’. The abstraction of pure ‘mathematica’, which he has already rejected in Cantus Circaeus, does not fit into this project of concrete geometry.41 After the enumeration of axioms and theorems we finally come to the articuli themselves. The reader is told that the arguments presented are directed against the vulgar theses and hypotheses of the geometers and astronomers, who have been operating without a justified criterion for measure, which has resulted necessarily in reprehensible calculations. And this tells especially against the astronomers, because the loss of measure can be seen clearly in their calculating methods, based on trigonometry and involving approximations, instead of a Pythagorean geometry being developed out of real number units.42 With heavy irony Bruno refers to the measurers that make use of tables filled with approximate calculations as the ‘domini circiterizantes’. For him this weakness is due to their modus mensurandi, losing itself in tables filled with irrational numbers and lacking a minimum. But how then does he propose to measure? Does the Nolan propose an alternative? Indeed he does. Ready to teach, I thus reveal three all-generating figures in order to open in them the universal terms of this art […] so that the demonstration and evidence itself of the case, beyond any number, will be submitted to the senses, and I do not—like others who contemplate a statue—unfold the foot first, then the eye, next the forehead and the rest separately, but the whole in all its parts and each part in the whole, so that both will be illuminated by a mutual collation.43 Here Bruno presents his three all-generating figures (figurae omniparentes) on which he will build his own geometry. Three years later they will return in his De triplici minimo et mensura and they are often referred to also in his Praelectiones geometricae (Padova, 1591).44 They form the basis out of which the Nolan develops his geometry. But again a bell should 41 Cf. note 22. For Bruno’s concrete geometry, see Otto (1991). 42 ‘Ut enim extra unitatem nullus est numerus, ita et extra nostrum minimum nulla est mensura. Mensura-

toribus ubi recursus est ad illam infelicem artem triangularum, et sinuum atque chordarum tabulas, amissio mensurae in aperto est […], ut tandem cum suo paulo plus vel minus opus absolutum exprimant, nobis iam non geometrae seu mensuratores, sed quasi mensuratores, quasi geometrae sunt appellandi, vel (si mavis) domini circiterizantes’ (Bol I, 3, p. 29). For Bruno in the context of sixteenth-century mathematics, involving trigonometry, see Aquilecchia (1993). On his geometrical constructions as based on real numbers, see Heipcke et al. (1991). 43 ‘Figuras ergo tres omniparentes […] docturus ante oculos obiicio, ut in ipsis universos artis huiusce terminos aperiam […] ut […] demonstratio et ipsa rei evidentia omnibus absoluta numeris ex omni parte sensibus subiiciatur, neque aliorum more contemplandae statuae nunc quidem pedem, nunc oculum, nunc frontem, nunc vero caetera seorsim, sed omnia in singulis et singula in omnibus, ut mutua collatione amplius illustrentur, explicamus’ (Bol I, 3, pp. 19–20). 44 To avoid confusion I remark that Gabriele (2001, p. 375) switches the second and third of these figurae omniparentes. Nor are the three figures repeated exactly in De triplici minimo et mensura, where the figura Amoris from the Articuli adversus mathematicos (Fig. 6) is called atrium Minervae and one similar (but not identical) to the figura intellectus (Fig. 5) is called the atrium Veneris (Fig. 8). The Praelectiones geometricae date from 1591 (Aquilecchia 1964, p. XIV).

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Fig. 5 Figura intellectus

Fig. 6 Figura amoris

ring. The way in which his methodology is presented here recalls the one exposed in Paris. A demonstration ‘beyond any number’ betrays his endeavour to operate according to the most simple and perfect order of the first mind, which is ‘beyond any number’.45 And more explicitly, the example of the statue, used to illustrate how Bruno’s geometry seeks to reach a view of the whole, repeats almost literally his words from De umbris idearum, where he asserted that the parts and species of the universe ‘obtain a greater recognisability when they are joined than when presented separately’.46 Thus the metaphor used to illustrate the core of his geometric method, developing an art of mensuration out of three figures in which all particular problems can be understood, describes the major aim of his ars memoriae as well.47 These figurae omniparentes serve the Brunian geometric practice, but their design seems rather to answer philosophical-symbolic motifs. The first is called the figure of the mind (Fig. 4) because it is constructed by four intersecting circles that penetrate each other through the centre, thus involving each other mutually. For Bruno the features of this figure symbolize the mind, containing the universals and bringing them into its unity.48 Next the figure of the intellect (Fig. 5) counts seven touching circles that do not intersect, just as the intellect 45 See note 8. 46 See note 8. 47 This focus on the mutual commitment of the part and the whole reflects Bruno’s adherence to Cusanus’ etymological link between mens and mensura. Understanding the whole and measuring the parts depend on each other. See De Cusa (1983, p. 108): ‘Amplius ad mentis tractatum descende et dicito: Esto, quod “mens” a “mensura” dicatur, ut ratio mensurationis sit causa nominis: quid mentem esse velis?’ 48 Bol I, 3, p. 20: ‘Nomina […] non citra rationem destinavimus, ut prima, quae quatuor circulis mutuo se per centra penetrantibus, implicantibus atque coinsitis perficitur, figura mentis universa continentis et in unitate quidam implicantis appelletur.’

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is capable of distinguishing everything and distributing all according to its own reasons.49 Finally the figure of love (Fig. 6) is expounded through intersecting as well as touching circles, to translate ‘the substance of the universe which is now contrary, then concordant’. And just like this substance, this figure ‘preserves perpetual concordance in contrariety and contrariety in concordance, in union distinction and in distinction union, in unity multitude and in multitude unity.’50 After the presentation of his three most fertile figures, the philosopher asserts that they relate not only to geometry, but to all grounds of knowing, contemplating and operating.51 Then he continues his discourse against the geometers of his time, ignorant of the minimum (whose calculations therefore imply approximations), who might be called geameters, just as the philosophers might be philasophers.52 After all, apart from one vocal (the alphaprivative), a philasopher is approximately the same as a philosopher. Thus Bruno ironically emphasizes again the importance of the minimum, without which no measure will ever be possible and consequently neither a geometer nor philosophy will exist.53 The geometric practice itself is not elaborated very well in the Articuli adversus mathematicos. Several of these practices are elucidated in his later Praelectiones geometricae. But of course his three figurae omniparentes fail to incorporate all geometry.54 The Nolan admits that in most cases his figures will answer, and will need no other explanation, but not always (ut si non semper, plerumque tamen absque alia explicatione demonstratio sensibus subiiciatur).55 So he has already announced that for some problems he will have to introduce new figures, and this results in the beautiful collection of diagrams accompanying this book. Some deal with the division of the continuum, others with the construction of polygons or parallels. But once again, Bruno’s geometry is not void of mnemonic implications. Indeed a purely mnemonic figure is inserted (Fig. 7). At first sight this figure has nothing to do with geometry. A similar figure can be found in Cantus Circaeus to illustrate a system invented to translate syllables into images. The well in the middle (puteus in Latin) is agreed to indicate that the first letter of the syllable is a ‘d’.56 The human figure next to the well will perform 49 Ibid.,‘Secunda, constans septem se attingentibus circulis, nempe in punctis quo mutuo non penetrent et intersecent, figura Intellectus omnia distinguentis propriisque rationibus distribuentis appellatur.’ 50 Ibid., p. 21, ‘Tertia tandem, quae tum attingentibus tum intersecantibus se circulis explicatur, Amoris figura nuncupatur, quandoquidem substantia universi tum contraria est, tum quoque concors, utpote in contrarietate concordiam et in concordia contrarietatem […] in unitate multitudinem in multitudine unitatem perpetuo reservans.’ 51 Ibid., ‘Foecundissimae sunt figurae, quae non solum geometriam, sed et omnem sciendi, contemplandi et operandi rationem apprime referunt, quibus certe sine defectu non possunt esse pauciores et supervacuitate plures.’ 52 Ibid., p. 21: ‘Ignorantia minimi facit geometras huius saeculi esse geametras, et philosophos esse philasophos.’ 53 Ibid., p. 26: ‘Ut ergo praeter monadem nihil est, praeter atomos et puncta nullum est quantum, ita et praeter minimi portionem et definitionem nulla est mensura, nullus est geometra et nulla consequenter philosophia.’ 54 Rather than incorporating geometry, these figures aim to reconstruct the mutual commitment of the part and the whole. See Bönker-Vallon (1995, pp. 211–223). 55 Bol I, 3, p. 20. 56 Surprisingly alphabetical correspondence between the first letter of the object and the initial consonant of the syllable—which would facilitate the mnemonic practice—is not necessary in Cantus Circaeus. Next to arbor (b) and columna (c), puteus stands for ‘d’ (Bomne I, p. 732). Nor was it important in De umbris idearum (Bomne I, p. 228): ‘In quibus non requiritur necessario primum nominis agentis vel actionis elementum idem esse cum illo cuius est expressivum: sufficit enim ambo haec determinato huic significando esse adscripta.’ Although this woodcut was possibly not made by the Nolan himself—unlike the other figures in Articuli adversus mathematicos—it illustrates a specific technique and displays characteristic subjects of the Nolan’s mnemonics.

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Fig. 7 Puteus

actions in different directions and will thus procure the vowel of the syllable. Depending on the direction in which he moves different vowels can be represented. So when the man’s action is completed in the corner where we see the ‘a’, the whole scene indicates ‘da’. Now why is this figure with a specific mnemonic function inserted into a book against mathematicians? Does this figura subalterna illustrate a specific geometric problem that cannot be solved by one of the figurae principales? This seems improbable. Maybe then our Nolan teacher wants to fascinate the emperor by his art, to encourage him to be taught and initiated into his mnemonics? This is more likely. Yet there might be a more valid reason for inserting this mnemonic figure into a book against mathematicians, which stresses the shared desire of both the mnemonic and geometric art to correspond to nature. If we take a closer look at the images in the four corners of the figure, another analogy with the Parisian works comes to the surface. We perceive two globes at the right. The one on top is a stellar globe that can be recognized by the belt of the zodiac. The one below represents the earth on which a continent is drawn out. In the corners on the left we see two different floor plans. At the top is presented an emanationist emblem or Lullian wheel, while below is repeated a mnemonic addition of the Nolan to memorize the elements of Lull’s art (Fig. 2) from De compendiosa architectura et complemento artis Lulli. In fact each corner contains a subiectum of another size, recalling the passage of Cantus Circaeus where the extensions of the naturally composed subject were described (see note 21). In a similar passage in De umbris idearum, after the particular (a house) and the more particular (a room), the most particular subject was called ‘the atom, not atom in the literal meaning, but of this sort.’57 These ‘mnemonic atoms’ are niches or receptacles in which images can be located, just like the place at the well. Looking back to this subordinate figure in the Articuli adversus mathematicos, we realize that it displays not four subjects, but five, as the centre of the picture—the well—represents Bruno’s ‘mnemonic atom’. In this way, the Nolan art of memory is faithful to nature in having a term, as was prescribed in the first theorem of the minimum in Articuli adversus mathematicos. Following this conjecture the presence of this figure in his attack against mathematicians—denying the minimum in disagreement with nature—does make sense.

4 Atrium Veneris and Statua et Membra Cupidinis Bruno’s rudimentary ‘geometry’ of his early period was always assisted by vivid images. In Prague, echoes of his Parisian works have reminded us that the author of the Articuli 57 On the subiectum the philosopher says the following in De umbris idearum, Bomne I, p. 152: ‘Aliud est proprium, nempe si placeat oeconomicum. Aliud est magis proprium, tetrathomum videlicet vel pentathomum. Aliud est propriissimum, quod est athomum, athomum, inquam, non simpliciter, sed in isto genere.’

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adversus mathematicos was a teacher in the ars memoriae. A purely mnemonic figure was inserted in this treatise. Once again abstract geometry was accompanied by the animating concrete. This old feature even characterizes Bruno’s figurae omniparentes, the core of the philosopher’s geometry. In De triplici minimo et mensura—where a minimum triplex of number, monad and atom is postulated—they are presented as archetypes and assigned to Apollo, Minerva and Venus.58 Their construction is explained in verses that stage mythological persons drawing the figures by their movements. How this trinity of archetypes (mens as unity, intellectus as distinction and amor as concordance in contrariety) depicts the unfolding of the unity in multiplicity, and how their geometric construction is based on Bruno’s theory of numbers as real entities, has been demonstrated by Bönker-Vallon (1995, pp. 205–223) and others.59 But why they are realized by the grace of vivid images remains a difficult question. Bönker-Vallon (1995, p. 220) speculates that the Nolan assigned mythological gods to these figures in order to clarify the triadic structure of his mathematics according to Renaissance tradition. Gabriele (2001, p. 496) links the mythological personages more explicitly to the art of memory, referring to the acting persons on the wheels of De umbris idearum. This geometry has clearly mnemonic aspects, but it remains problematic for scholars seeking to relate the Nolan’s geometric to his mnemonic maps.60 A comparison between the atrium Veneris and Bruno’s statua et membra Cupidinis as worked out in Lampas triginta statuarum enlarges our comprehension of the mnemonic aspects of this geometry. The construction of theatrium Veneris in De triplici minimo et mensura is described in the following way: Behold the sacred temple of Venus, honoured and venerable, In the centre of which stands Amor, father of geniuses. Four concentric circles surround the sanctuary. They are divided after equal intervals by one radius, Which from the centre drew the first circle. This circle is marked by the same radius with six points, Through which Amor, with six arrows, shoots wholes up to the outermost circle, And divides the penetrated circles in as many parts; Ordering that each of all the penetrated points Takes its proper deity. Thus the quadruple order counts Twelve, positioned there to be seen in this sequence: First comes the species of Bonum, the second place is taken by Saint Concordia, who is followed by Dilectio and Emphasis and Fascinium, burning Furor and sweet Gratia. The following order first presents harmonic Honor, Friendly Indulgentia with the image of the sacred fire, Gracious Chorea of a procession of nymphs and dryads, Then comes playful Lascivia, soft Musica, Novitas of nature. The third order first shows Ornatum and peace from the sources of Poros, Which are followed by Querela, together with deprived quiet, With Rigor comes ardent Spes and Trepidatio of mind. 58 Bol I, 3, p. 274: ‘Sunt tres principio archetypi, quarum in facie/ omnis momenti norma est mensuraeque atque figurae./ Do primam Phoebo, quadratque secunda Minervae,/ Tertiaque est Veneris, siquidem propriumque sigillum/ Agnoscunt harum in vultu et secreta profundo.’ 59 See also Heipcke et al. (1991, p. 159). 60 This objective has been attempted by Vianello (2003) who linked passages from Bruno’s Italian dialogues to the figurae omniparentes; a rather unconvincing enterprise.

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Fig. 8 Atrium Veneris

The fourth order presents twelve woods, Which all take the name of the nearby deity.61 In Venus’s temple Amor stands at the centre of four concentric circles. He penetrates the circles with six arrows, dividing them into as many parts. Each intersection of these radii and the circles is occupied by a personification that one way or another is related to Venus. Of course they are chosen to mark the intersections of the geometric figure by their initials (Fig. 8). But there is more. The crowd is not randomly composed merely for the use of their initials. The centrifugal motion of Amor’s arrows passes different orders with characteristics (emotions, effects, related concepts) of love. In the periphery some ‘negative’ features are found, as raptae coniuncta Querela quieti and Trepidatio mentis. A similar cluster of terms around the statue of Amor (now Cupid) returns in Lampas triginta statuarum, where thirty statues are described and accompanied by thirty attributes. Each statue embodies a concept. The thirty statues with their thirty attributes are all-embracing and serve the art of invention to procure each subject a collection of related features by which middle terms for argumentations are found.62 The field of Venus is used by the philosopher to describe concordance in re. Next the statue of Venus treats concordance in voluntate, after which the arrows or knots of Cupid symbolize concordance in actione. The order in which the figurative language is displayed in Lampas triginta statuarum recalls the one used for the construction of the atrium Veneris in De triplici minimo et mensura, where Amor was placed in the centre of Venus’s temple, from where he shot his arrows. Next in Cupid’s members are contained thirty concepts, among which we can recognize most of the personifications that constructed the atrium Veneris. This series ends with the rather ‘negative’ features of love (timor et zelotypia, suspicio), just as was the case in the periphery of Bruno’s geometric figure. The following list shows the correspondences between the two works. 61 Bol I, 3, pp. 281–282: ‘En Veneris cultum, sanctum et venerabile templum,/ In cuius medio pater est Amor ingeniorum./ Arcanum quadruplex cyclus concentricus ambit./ Hos aequis radius distinxerat intervallis/ Unus, qui a medio primum est resolutus in orbem,/ Quem senis radius partivit finibus idem/ Per quos fodit Amor spiculis ora extima senis,/ Et totidem in partes penetratos dividit orbes;/ Praecipiens proprium ut capiant penetralia numen/ Cuncta sigillatim. Bissena hinc ordo quaternus/ Connumerat, serie hac ibi conspicienda locantur./ Prima Boni est species, Concordia sancta secundum est/ Nacta locum, hanc sequitur Dilectio et Emphasis atque/ Fascinium, Furor ardescens et Gratia suavis./ Harmonicum ordo sequens primo praesentat Honorem/ Idoloque Ignis sacro Indulgentia mitis./ Nympharum Dryadumque choris pergrata Chorea,/ Succedit ludens Lascivia, Musica mollis,/ Naturae Novitas. At tertius ordo repostum/ Primo habet Ornatum et Pori de fontibu’pacem,/ Quos sequitur raptae coniuncta Querela quieti/ Cumque Rigore ardens Spes et Trepidatio mentis./ Quartus bis senos lucos deinc ordo reposcit,/ Queis singillatim est vicino a numine nomen;’ I reproduce the names of the deities in Latin so that the personifications can be linked to the letters in Fig. 8. 62 Bom, p. 938: ‘Existimamus nullam esse proponibilem quaestionem quae subterfugere possit saltem unam ex istis ideis, quae per omnia et singula ipsius membra atque circumstantias numerum mediorum suppeditare non valeat.’

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Atrium Veneris (De triplici minimo et mensura, Bol I, 3, pp. 281–282)

De statua et membris Cupidinis (Lampas triginta statuarum, Bom, pp. 1266–1276)

1. Amor 2. Spicula Amoris 3. Species Boni 4. Concordia sancta 5. Fascinium 6. Gratia suavis

1. Cupido 2. Tela Cupidinis 3. Benevolentia 4. Concordia in actione 5. Fascinatio 6. Gratia, per quam quod dulce est, consequenter est gratum. 7. Gaudium 8. Incantatio…sicut et musicae et harmoniae ex individua quadam consonantia oritur complacentia, seu dulcedo 9. Inebriato Poro…: ex superfluitate enim et luxu ingruente in Poeniam 10. Timor et zelotypia 11. Spes 12. Suspicio

7. Ludens Lascivia 8. Musica mollis

9. Pori de fontibu’pacem 10. Raptae Querela Quieti 11. Spes 12. Trepidatio mentis

The lists are not identical: nor is this the central point. What is important is that a similar cluster of characteristics surrounds Cupid’s statue in both of the philosopher’s works. The correspondences shed a mnemonic light on the personifications of the geometric map of atrium Veneris. They were not merely chosen to indicate the intersecting points with their initials, but partook of one of Bruno’s memory patterns that had already been exposed in Lampas triginta statuarum.63 In this case, the mnemonic and the geometric maps can be seen to overlap. The author of Lampas triginta statuarum did not present himself as inventing the method of the statues—the use and form of the ancient philosophy and prisci theologi—but as reviving it in his century.64 With sensible figures, fabricated by imagination and fantasy, those things which are removed from the senses, are signified.65 Thus the thirty statues and their features referred to higher concepts, removed from the senses. The same counts for the personifications in his figurae omniparentes, which were capable of incorporating many peculiar geometric problems. The personifications of attributes or effects of love animated the geometric atrium and as sensible images—just like the statues—referred to and signified what was removed from the senses: the archetype of love.

5 Conclusion By focussing first on the Parisian mnemonic works, I have tried to outline a certain attitude of Bruno’s towards mathematics before he ventured upon his Articuli adversus mathematicos. The rudimentary ‘geometric’ figures stemming from his teaching period (Lullian wheels, figures to remember Lull’s alphabet and polygons) were animated by vivid images. The scrutinium or tool by which the rational faculty inquires and discerns, operated through 63 The first version of Lampas triginta statuarum dates from 1587. 64 Bom, p. 940: ‘Itaque usum atque formam antiquae philosophiae et priscorum theologorum revocabimus,

qui nimirum arcane naturae eiusmodi typis et similitudinibus non tantum velare consueverunt, quantum declarare, explicare, in seriem digerere, et faciliori memoriae retentioni accomodare. […] Non ergo huius docendi rationis primi sumus inventores, sed forte in hoc tempore […] primi exsuscitatores, […]’. 65 Ibid.: ‘Sensibilia erunt figuratae species et opera phantasiae et imaginationis fabrefactae, per quas subinde volumus ea, quae a sensu remotiora, significari.’

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number units. Its ordering activity linked the Nolan art to nature and foretold the Pythagorean inspiration of his later ‘natural’ geometry. Yet, mere numbers were of no use in his art of memory, since they were incapable of striking our imagination. Therefore as an innovator he introduced his semi-mathematical subjects (polygons or images for numbers). These ‘visible translations of numbers’ foreshadowed the concrete geometry through figures as it was developed in Prague. Entering into Bruno’s epistle to Rudolph II we encountered several echoes of his Parisian treatises. I argued that in order fully to understand this treatise, we had to bear in mind that the Nolan was a teacher of the ars memoriae, in Prague as well as in Paris. He criticized the astronomers and geometers, the ‘domini circiterizantes’, who—lacking a minimum—made use of ‘the unfortunate art of trigonometry’, for a great part based on approximations. But the philosopher did not limit himself to attacking his contemporaries; he also proposed his own method, which sought to collect all the geometric practices in only three figurae omniparentes in order to understand totality at a glance, a capital feature of his art of memory. Still, from the start it was suggested that not all geometric problems could be treated within these three figures. This is why he introduced subordinate figures. I focused on one of these, which at first sight had nothing to do with geometry. I conjectured that this mnemonic figure appeared, not as a mere ogle to seduce the emperor to his art, but as a representation of the minimum—the major concern of the whole book—of Bruno’s art, which goes hand in hand with nature. Thus Bruno’s ars memoriae is not only present at the surface of the Articuli adversus mathematicos, but is even analogous to the central method exposed in it. Finally I confronted one of these all-generating figures with a statue from Lampas triginta statuarum. I argued that the correspondences between the atrium Veneris and the Statua et membra Cupidinis, although far from identical, lead us to presume that in the Nolan’s mind, both are constructed on a common memory pattern. While the geometric figures are presented as archetypes, the sensible images—according to the ancient philosophy of Lampas triginta statuarum—signify ideas removed from the senses. In this way the geometrical and mnemonic images collaborate to represent Bruno’s trinity. Abstract geometry is coloured by imaginary personifications translating emotions, effects and related concepts, that all point to the archetype itself, removed from the senses.

Bibliographical Note on Bruno’s Works For Bruno’s works on memory and magic, I was able to make use of the recent critical editions, accompanied by Italian translations and ample comments, published by Adelphi Edizioni (Opere mnemotecniche, Opere magiche). All the illustrations presented here are taken from Corpus Iconographicum, also published by Adelphi Edizioni. For his Italian dialogues I refer to the bilingual edition (Italian–French) of Les Belles Lettres (Oeuvres Complètes) that was initiated by the great philologist Aquilecchia, whose contribution to our understanding of the Nolan philosopher can hardly be overrated. Regretfully, for the Nolan’s mathematical writings, I still had to rely on the nineteenth-century edition of the Latin works (Opera latine conscripta), obviously without translation. The praelectiones geometricae—preserved in a manuscript at the University Library of Jena—only came to the surface in the twentieth century and, together with Ars deformationum, were published by Aquilecchia in 1964. Because Bruno’s Latin is rather complicated, I have chosen to paraphrase or translate all passages into English, accompanied by the original texts in footnotes. When a work of Bruno is mentioned for the first time, date and place of publication are given. The following list first shows the

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major editions with their seals as generally used in Bruno-studies, succeeded by the works cited in the present article. G. Bruno: 1879–91, Opera latine conscripta publicis sumptibus edita recensebat F. Fiorentino, Morano, Neapoli-Florentiae (indicated as Bol, followed by part and volume) G. Bruno: 1993, Oeuvres Complètes, Les Belles Lettres, Paris (BOeuC, followed by volume) G. Bruno: 2000, Opere magiche, Adelphi, Milano (Bom) G. Bruno: 2001, Corpus Iconographicum, Le incisioni nelle opere a stampa, a cura di M. Gabriele, Adelphi, Milano G. Bruno: 2004–2009, Opere mnemotecniche, Adelphi, Milano (Bomne, followed by volume) G. Bruno: 1964, Praelectiones geometricae e Ars deformationum, a cura di G. Aquilecchia, Edizioni di Storia e Letteratura, Roma De umbris idearum (Paris, 1582, in Bomne I) Cantus Circaeus (Paris, 1582, in Bomne I) De compendiosa architectura et complemento artis Lulli (Paris, 1582, in Bol II, 2) Explicatio triginta sigillorum (London, 1583, in Bomne II) Sigillus Sigillorum (London, 1583, in Bomne II) Cena delle ceneri (London, 1584, in BOeuC II) Spaccio della bestia trionfante (London, 1584, in BOeuC V) Cabala del cavallo pegaseo (London, 1585, in BOeuC VI) Figuratio Aristotelici Physici Auditus (Paris, 1586, in Bol I, 4) Lampas triginta statuarum (Wittenberg, 1587, in Bom) Articuli centum et sexaginta adversus huius tempestatis mathematicos atque philosophos (Prague, 1588, in Bol I, 3) De triplici minimo et mensura (Frankfurt, 1591, in Bol I, 3) Praelectiones geometricae (Padova, 1591, in Praelectiones geometricae e Ars deformationum) Acknowledgments The generous support of the Research Foundation Flanders (FWO) made the present article possible. I wholeheartedly thank Marco Matteoli, who inspired this additional result of my doctoral thesis during a stay in Florence during spring 2009. I am also much obliged to my promoter Wim Verbaal, who made many useful suggestions, with patience and confidence. And, although he passed away silently more than a year ago, I still owe many thanks to my former promoter Fernand Hallyn, whose academic personality will not be forgotten.

References Aquilecchia, G. (1964). G. Bruno Praelectiones geometricae e Ars deformationum, a cura di G. Aquilecchia. Roma: Edizioni di Storia e Letteratura. Aquilecchia, G. (1993). Bruno e la matematica a lui contemporanea. In: Schede Bruniane (1950–1991) (pp. 311–318). Roma: Vechiarelli Editore. Bönker-Vallon, A. (1995). Metaphysik und Mathematik bei Giordano Bruno. Berlin: Akademie Verlag. Buffière, F. (1973). Les mythes d’Homère et la pensée grecque (2nd ed.). Paris: Les Belles Lettres. Cengiarotti, G. (2004). Giordano Bruno a Praga; progetti di riforma. In: A. Ingegno, & A. Perfetti (Eds.), Giordano Bruno nella cultura del suo tempo, atti del convegno organizzato dall’università di Urbino e dall’Istituto italiano per gli studi filosofici (Urbino-San Leo, 23–24 settembre 2000) (pp. 285–305). Napoli: La Città del Sole. De Bernart, L. (1986). Immaginazione e scienza in Giordano Bruno. Pisa: Ets Editrice.

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De Cusa, N. (1983). Idiota de sapientia, de mente, de staticis experimentis. In Nicolai De Cusa Opera Omnia (Vol. 5). Hamburgi: In aedibus Felicis Meiner. Firpo, L. (2000). Le Procès. Paris: Les Belles Lettres. Gabriele, M. (2001). G. Bruno Corpus Iconographicum, Le incisioni nelle opere a stampa. a cura di M. Gabriele. Milano: Adelphi. Granada, M. A. (1993). Giordano Bruno et la “dignitas hominis”. Présence et modification d’un motif du platonisme de la Renaissance’. Nouvelles de la République des Letters, XII, 35–89. Heipcke, K., Neuser, W., & Wicke, E. (1991). Über die Dialektik der Natur und der Naturerkenntnis. Anmerkungen zur Giordano Brunos De Monade, Numero et Figura. In K. Heipcke, W. Neuser, & E. Wicke (Eds.), Die Frankfurter Schriften Giordano Brunos und ihre Voraussetzungen (pp. 145–162). Weinheim: VCH Verlagsgesellschaft. Heuser-Keßler, M.-L. (1991). Maximum und Minimum. Zu Brunos Grundlegung der Geometrie in den Articuli adversus mathematicos und ihrer weiterführenden Anwendung in Keplers Neujahrsgabe oder Vom sechseckigen Schnee. In K. Heipcke, W. Neuser, & E. Wicke (Eds.), Die Frankfurter Schriften Giordano Brunos und ihre Voraussetzungen (pp. 181–197). Weinheim: VCH Verlagsgesellschaft. Koyré, A. (1973). Études d’histoire de la pensée scientifique. Paris: Gallimard. Lasswitz, K. (1884) Giordano Bruno und die Atomistik. Vierteljahrsschrift für wissenschaftliche Philosophie, VIII, 18–55. Lull, R. (1991). L’art bref. Paris: Les Éditions du Cerf. Lüthy, C. (2003). Entia & Sphaerae: due aspetti dell’atomismo bruniano. In E. Canone (Ed.), La filosofia di Giordano Bruno, problemi ermeneutici e storiografici, Convegno internazionale, Roma, 23–24 ottobre, 1998 (pp. 165–198). Firenze: Olschki. Matteoli, M., & Sturlese, R. (2007). Il canto di Circe e la ‘magia’ della nuova arte della memoria del Bruno. In F. Meroi (Ed.), La magia nell’Europa moderna, Tra antica sapienza e filosofia naturale, Atti del convegno, Firenze, 2–4 ottobre, 2003 (pp. 480–81). Firenze: Olschki. Michel, P.-H. (1960). L’atomisme de Giordano Bruno. In La science au seizième siècle. Colloque international De Royaumont, 1–4 juillet, 1957 (pp. 251–264). Paris: Hermann. Monti, C. (1980). Introduction to G. Bruno. Opere latine. Torino: Unione Tipografico-Editrice Torinese. Olschki, L. (1924). Giordano Bruno. Deutsche Vierteljahresschrift für Literaturwissenschaft und Geistesgeschichte (vol. 2, pp. 1–79). Otto, S. (1991). Figur, Imagination, Intention. Zu Brunos Begründung seiner konkreten Geometrie. In K. Heipcke, W. Neuser, & E. Wicke (Eds.), Die Frankfurter Schriften Giordano Brunos und ihre Voraussetzungen (pp. 37–50). Weinheim: VCH Verlagsgesellschaft. Perrone-Compagni, V. (2000). Minime occultum chaos, la magia riordinatrice del Cantus Circaeus. Bruniana & Campanelliana, VI(2), 281–297. Ricci, S. (2000). Giordano Bruno nell’Europa del Cinquecento. Roma: Salerno Editrice. Spruit, L. (1988). Il problema della conoscenza in Giordano Bruno. Napoli: Bibliopolis. Spampanato, V. (1921). Vita di Giordano Bruno con documenti editi e inediti. Messina: Principato. Sturlese, R. (2000). Arte della natura e arte della memoria in Giordano Bruno. Rinascimento XL, 123–141. Tocco, F. (1889). Introduction to G. Bruno, Opera latine conscripta publicis sumptibus edita, recensebat F. Fiorentino, Morano, Neapoli-Florentiae I, 3. Vianello, F. (2003). Le mappe magiche di Giordano Bruno. Sapere Edizioni. Yates, F. A. (1966). The art of memory. Chicago, London: University of Chicago Press.

Author Biography Manuel Mertens is doctoral researcher funded by the Research Foundation Flanders (FWO). As member of the Centre for History of Science at Ghent University, his field of interest primarily concerns Renaissance philosophy as transition between Medieval and modern scientific methodology. Within this field, his research pays special attention to the analysis of discourses of knowledge. His doctoral thesis discusses the relationship between the mnemonic and magical writings of the important sixteenth-century philosopher Giordano Bruno by taking into account his ‘art of writing’. For now this research resulted in an article on Bruno’s version of the art of Ramon Lull and its fortunes, published in Bruniana & Campanelliana 2009 (2).

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Memory and Geometry in Bruno: Some Analogies

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