LEONARDIAN FLUID MECHANICS
I -
HISTO RY OF OF KINEMAT ICS
II - INCEPTION OF MODERN KINEMATICS By ENZOMACAGNO
IIHR Monograph No. 112 Iowa Institute of o f Hydraulic Hydraulic Research Research The University of Iowa Iowa City, Iowa 52242-1585 August 1991
LEONARDIAN FLUID MECHANICS
I - HISTOR HIS TORY Y OF KINEMATICS KINEMAT ICS II - INCEPTION OF MODERN KINEMATICS By ENZO MACAGNO
Sponsored by National National Science Foundation Foundation and National National Endowment Endowment for f or the Hum Humanities anities
IIHR Monograph No. 112 Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242-1585 August 1991
LEONARDIAN FLUID MECHANICS
I - HISTOR HIS TORY Y OF KINEMATICS KINEMAT ICS II - INCEPTION OF MODERN KINEMATICS By ENZO MACAGNO
Sponsored by National National Science Foundation Foundation and National National Endowment Endowment for f or the Hum Humanities anities
IIHR Monograph No. 112 Iowa Institute of Hydraulic Research The University of Iowa Iowa City, Iowa 52242-1585 August 1991
TAKE TO KINEMATICS, IT WILL REWARD YOU; IT IS MORE FECUND THAN GEOMETRY; IT ADDS A FOURTH DIMENSION TO GEOMETRY
Advice of Chevyshev to Sylvester.
ADDENDUM IIHR MONOGRAPH 112 It takes a few weeks for monographs like this to be printed. In this case, those weeks were spent in a lecture trip. I took the opportunity of visiting several libraries to expand, if possible, the view of what kinematics is nowadays. Everywhere I found very co-operative librarians and thus, in a short time, I was able to collect more information useful for my project on the History of Kinematics. I am thankful to all of them. This addendum would have not been possible without their help and understanding. I am also indebted to persons who interacted with me during my lectures and periods of contact in different institutions; through their questions and critical comments they helped me very much. 1. FURTHER READING During this trip, I read the Eye and the Brain by Richard Gregory [1990]. I found the chapter on vision of movement really fascinating. There is a hint of such an application of kinematics in Hartenberg 1964 [see comment on page 28 of this monograph]. Surely, this is a topic that must be included in a comprehensive history of kinematics. Therefore, the studies like those of Helmholtz in his Physiological Optics must be taken into account. It seems amazing to me that velocity may be sensed by humans without involving a simultaneous measurement of a time interval. 2. COMPUTATIONAL KINEMATICS I have referred to the importance of computational kinematics in this monograph. This is an old branch of kinematics, but it has been enhanced by the use of modem computers. During a seminar at the Massachusetts Institute of Technology, I had the opportunity of learning about computational experimental kinematics aimed at gaining an understanding of arrays of vortices in general and also in connection with fish-like propulsion. I thought that Ampère would have enjoyed watching at such an elegant realization of the study of motion in the way defined in the famous essay in which he introduced the notion of modem kinematics. 3. APPLICATIONS OF KINEMATICS On page 62 of this monograph, I have included lists of different areas of human endeavor in which kinematics is the object of study and/or finds applications. In these lectures, due in part to having left at home the corresponding slides, I took a different approach. Similarly, to Keynes notions of positive and negative analogies [Macagno 1986, in this Addendum] , I decided to define positive and negative lists, and ask the persons in the audience to make their own list of those fields in which kinematics has no application. The reader may want to try it also.
4.-ADDmONAL BIBLIOGRAPHY ABRAHAM, Ralph 1982. Dynamics: the Geometry of Behavior. Aerial Press, Santa Cruz, CA. BARBOUR, Julian B. 1989. Absolute or Relative Motion? : A Study from Machian Point of View of the Discovery and the Structure of Dynamic Theories. Cambridge University Press, New York. BARUT, Asim O. 1989. Geometry and Physics: non-Newtonian Forms of Dynami cs. Bibliopolis, Napoli. COSTABEL, Pierre 1960. Leibniz et la dvnamique: les textes de 1692. Hermann, Paris. Contains comments on the general rule for the composition of movements. EMANUEL, Nikolai M. 1982. Kinetics of Experimental Tumor Processes. Pergamon Press, Oxford. (Translation of Kinetika eksperimentalnykh opukholevykh protsessov.) FORTI, Simone 1974. Handbook in Moti on. Press of the Nova Scotia College of Art and Design, Halifax, N.S. GREGORY, Richard L. 1990. Eve and Brain. Princeton University Press, Princeton, NJ. (Fourth Edition with new introduction). See especially Chapter Seven, which deals with vision of movement. The entire book makes for a very enjoyable reading. Anyway, unless you read the first six chapters, it is difficult to profit from the reading of the seventh chapter. HO, Chung You and Jen SRIWATTANATHAMMA 1990. Robot Kinematics: Symbolic Automation and Numerical Synthesis. Alex Pubi. Corp., Norwood, NJ. JENKINS,Jim and Dave QUICK, 1989. Motion. Kinetic Art. Peregrine Smith Books, Salt Lake City. MACAGNO, E. 1986. Analogies in Leonardo' s Studies of Flow Phenomena. Vinciani. Centro Ricerche Leonardiane, Brescia, Italy.
Studi
MOORE, Carol-Lynne and Kaoro YAMAMOTO, 1988. Bevond Words: Movement Observation and Analysis. Gordon and Breach, New York. This book deals with aesthetics of movement, dancing and choreography. [See also Souriau 1983] MANA, Demetrios 1979. A Kinematic Model of the Swing Leg for Clinical Applications. MS Thesis, MIT, Dept. Aeronautics and Astronautics, Cambridge, MA. MARTIN, W.N., Ed. 1988. Motion Understanding: Robot and Human Vision. Kluwer Academic Publishers, Boston. MUYBRIDGE, Eadweard 1955. The Human Figure in Motion. Dover Pubi., New York (Introduction by Robert Taft.) Contains a selection of 195 plates from the 11-volume work, Animal Locomotion, first published in 1887.
MUYBRIDGE, Eadweard 1957. Animals in Motion. Dover Pubi., New York (Edited by Lewis S. Brown.) Contains a selection of 183 plates from the 11-volume work, Animal Locomotion, first published in 1887. MUYBRIDGE, Eadweard 1973. Animal Locomotion: the Muybridge work at the University of Pennsyl vania. Amo Press, New York. Contains contributions by W.D. Marks, H. Allen and F.X. Dercum. MUYBRIDGE, Eadweard 1979 Muybridge’s Complete Human and Animal Locomotion. Dover Pubi., New York (Edited by Anita Ventura Mozley.) MUYBRIDGE, Eadwear d 1984. The Male and Female Figure in Motion: 60 Classic Photographic Sequences. Dover Pubi., New York. SARNOWSKY, Jurgen 1989. Die aristotelisch-scholastische Theorie der Bewegung: Studien Kommentar Albert von Sachsen zur Phvsik des Aristoteles. Aschendorf, Munster. SOURIAU Paul, 1983. The Estheti cs of Movement . University of Massachusets Press, Amherst (English version by Manon Souriau of Esthétique du mouvement.)
TABLE OF CONTENTS PART
I
1.
PREFACE................................................................................................................... iv
2.
GENERAL IDEAS....................................................................................................... 1
3.
S C O P E ...................................................................................................................... 16
4.
APPROACH AND METHODOLOGY........................................................................ 18
5.
EMPIRICAL KINEMATICS...................................................................................... 25
6.
THEORETICAL KINEMATICS.................................................................................36
7.
EXPERIMENTAL KINEMATICS ............................................................................ 53
8.
APPLIED KINEMATICS .........................................................................................59
9.
C O N C LU SIO N ....................................................................................................... 67
10.
BIB LIO GR AP HIC AL
RE FE RE N CE S.............................................................. 69
GEN ERAL...................................................................................
69
ANCIENT............................................................................................
72
MEDIEVAL - RENAISSANCE................................................................................
73
MODERN.................................................................................................................. 77 MACHINES - MECHANISMS.................................................................................85 EXPERIMENTAL KINEMATICS........................................................................... 87 REVIEW ARTICLES AND BIBLIOGRAPHIES ............................................ 91
i
TABLE OF CONTENTS PART
1.
II
NTRODUCTION....................................................................................................95
2. KINEMATICS BEFORE AMPERE ........................................................................... 98 3.
AMPÈRE's CONTRIBUTION...............................................................................104
4.
AFTER AMPÈRE.................................................................................................... 109
5.
CONCLUSIO N..................................................................................................... 112
6. ACKNOWLEDGEMENTS.......................................................................................115 7.
BIBLIOGRAPHY.................................................................................................... 116
APPENDIX
I
by
Matilde Macagno............................................................. 123
LEONARDO DA VINCI's SCIENTIFIC AND TECHNICAL KINEMATICS
ii
PREFACE
This monograph contains work for what I expect to constitute two separate volumes.
On the one hand, I believe that a comp rehe nsiv e hi story of kin emat -
ics has never been written; on the other hand, it seems necessary to write separately the history of the movement that led to the establishment of mathematical kinematics as a discipl ine of a standi ng similar to that of geometr y.
The
idea of a geometry in motion is already in Leonardo da Vinci and in embryonic form in older writings, but it flourishes only much later in the nineteenth century and it acquires full realization in our century.
I have chosen to integrate my initial efforts on the history of kinematics into the series
Le on ar dia n Flu id Mec hani cs
for several reasons, of which the
most compelling to me is that it was the study of Leonardo's manuscripts that led me to examine the entire history of kinematics because so much of Leonardo's writings and drawings is in fact about the study of motion in general and also in fields other than mechanics of fluids.
In second place is the
immediacy one feels when the time left is run nin g sho rter and shorte r, and one needs to put on recor d what seems a valuabl e effort.
If enough time is given
to me, then the two volumes will be ready in the near future; if not, at least the basi c pl an will be av ai lab le to ot he r schol ar s wh o may wa nt to br ing it to fruition.
iv
GENERAL IDEAS
Kinematics is the name for the study of motion that was introduced by Ampère in his famous essay on the classif icati on of sciences.
The intent of
Ampère was to create a new discipline in which motion would be studied without regard to the forces involved, that should be taken into account in dynamics.
He was certainly res pond ing to an existing t rend, and he knew very well
that the role of kinematics would be like that of a bridge connecting geometry and physic s.
I will disc uss in detail this histor ical moment in the second part
of this monogra ph. have
existed
What we need to establi sh now is that kinematical studies
much
before
Ampère,
and
even
before
Greek
science.
Kinematics, unlike its sister Geometry, is not reducible to a branch
of
Mathematics, although the movement initiated by Ampère has made modern kinematics very much into an exact science.
Apparently, a comprehensive history of kinematics has never been written. Following an investigation of this question for some years, I want to take some initial action by summarizing my findings and offering a preview of such a history.
If we take into account the classical tri logy: geometry , kinematics,
and dynamics, it is quite remarkable that both geometry and dynamics have been the ob jec t of imp or tan t hi st or ica l wr itings, wh il e ki ne ma tics has being considered only in some technical publications or in isolated chapters, always too specific, dealing with the history of mechanics.
1
The bias one finds in men who live a life of dedication to a given profession, or to a branch of a profession, is pervasive and is found in many works on kinematics.
In an extensive bibliography of kinematics published two
decades ago [de Groot 1970], we find almost nothing that does not relate directly or indirectly to the kinematics of machines and mechan isms.
By
chance (or mistake ?) there seem to be there only one entry which is on kinematics of continua .
There is absolut ely nothing on the mathematical kinemat -
ics that one can find in many other fields, in spite of the bibliography having started as an effo rt to cover "pure" kinematic s.
I tested the bibli ograp hy by
trying to find authors like Lagrange, Euler, Muybridge, Truesdell, but they were not included.
I think,
the rea son is simply that the kinematics of
deformable bodies was never contemplated by de Groot; of course, his was the rig ht to ch oos e the areas to be cove red .
I say all this not as a hars h criti cism
of such a formidable accomplishment, but only to make the point that it is difficult to agree on what is the thematic of kinematics, be it "pure" or "applied".
One can place kinematics somewhere between geometry and dynamics, but it shoul d be much close r to the latter than to the former.
Fro m the point of
view of dimensi ons, geometry implies only length, while kinematics requires length and time, and dynamics needs to cons ider length, mass and time.
One
must be careful, however, not to take a simplistic view, suggested by either dimensional analysis, or purely theoretical studies of displacements of geometric figu res , be they rigid or deformab le.
It may sound paradoxical , but
kinematics cannot be comprehended unless one knows dynamics, and even
2
more than that, unless one know s physics.
This is not sur pri sing , because
even the full understanding of geometry requires a knowledge of physics.
Kinematics is a science that deals with motion in the most general ways po ss ible.
We can see a wa ter wa ve prop agating while the wa ter itsel f st ays
very much where it was before the wave was produced, but there are other waves which travel as well, heat waves, pr essu re waves, etc.
There can obv i
ously be motion of matter, but there is also motion of form and other proper ties.
It is not clear if the notion of what constit utes the science of kinematics
enjoys a consensus even among those who are specialists in some of its aspects.
For instance, in the case of mechanical enginee rs,
one finds the
notion that kinematics is only about particles and rigid bodies, or chains of rigid bodies [see, e.g., 11.1 in Engineering Mechanics by I.H. Shames I960]. Kinematics is much less than mechanics in some ways, and much more in other ways.
It is not easy to desc rib e with a few se ntenc es what ki nematics
is. I will try to give an overal l pict ure of my vie ws in this cha pter, prec eding the statement of the scope of this project, and the methodology which I believe is appropriate.
Study and application of motions in technology is a very old human activ ity.
The historical developments are documented to a certain extent, but we
should harbor no doubt that there existed heuristic and intuitive notions of kinematics developed and abstracted during those technological developments about which we know only the prod uct, the machin e or the device, but do not know the thi nking associat ed with it.
There is also much complex kinemati cs
in dancing and in ballet, but what is scientific or even proto-scientific knowl 3
edge in such activities?.
Let us remembe r that also animals can perf orm very
complex kinematic feats but obviously know no kinematics as a science, either empirical or rationa l, a lthough s omethi ng like geometry and kinematic s must be encode d in the ir br ains .
We kn ow, from photographic and from cinema to-
graphic studies, how complex can be just the normal walking of animals, not to speak of other much more complex motions like those of predator and prey, and court ing evolu tions .
Why can one empirically devise and execute very
complex motions, buy it takes a very long time to come up with a theory for even very simple motions.
Nowada ys we are us ed to the no tion that techn ol ogy is he lpl ess wi tho ut science, but we know that historically technology came first and prepared the ground for the emergence of the first scientific notions.
Therefore much
kinematics and dynamics must have existed before any theory was available. And in fact, there in technology was the source of the science that was going to systematize, under the name of rational mechanics, much empirical knowledge.
But from the point of view of planning the study of the history of
kinematics, I think it preferable to study first the development of theoretical kinematics, then of its applications, and only after that investigate the roots of all that kno wl edg e.
I tend to agree with Tru esd el l [1977 ] in sayi ng that the
study of the history of a science without a theoretical system is, if not impossible, at least diffi cult and full of pitfalls.
Becaus e in the case of kinematics,
we have such a system, it seems preferable to begin with theoretical developments together, or followed by the study of the applicative developments.
4
The sources of kinematics, as those of geometr y, are in the long historical proce ss of dev elopme nt of empiri ca l knowl edge.
It is bel iev ed th at Thal es
visited Egypt and, having had the opportunity of learning there empirical geometry, developed such knowledge further and began to give it a rational ba si s, a the or et ica l system of th ink ing .
We kno w th at in Gr eec e there were
philosophers wh o ch allen ged the sc ien tific app roa ch to geo metry; ap pa ren tly they did not inhibit progress, and they may have had a role in making the geometers much more careful in their postu lati ons and proo fs.
Howe ver , the
critical attitude may have worked much better in the case of the study of motion, i.e. of kinema tics .
It is also true that, in hin dsi ght , we may see that
kinematics needed much more empirical and experimental knowledge before it could be distilled into the science that Ampère recognized as ready to be systematized.
In the long proce ss of devel oping the science of motion, one source lies in the observation of natural motion and flow phenomena, another in the constant invention of motion for our own body, for objects we handle, for devices we operate.
I believe that one of the fir st to make such a conn ect ion between
machines and mechanisms of early times and theoretical kinematics was Franz Reuleaux [See Weihe 1942] . There is also a pass age in the writing s of Henri Poincaré which reveals the tremendous importance of perceiving keenly the world around us, who live at the interfaces earth-air and air-water, to make possible the developme nt of a sc ience li ke ge om et ry ( and I wo uld add, kine matics ).
I will use at this point a section of a rece nt paper [Mac agno 1991]
5
commenting the ideas of Poincaré about the understanding of geometry and kinematics.
The difficult process of the human mind in understanding and properly describi ng motion was described by Henri Poincaré [1895].
To appreciate
Poincaré's argument, just remember the tremendous mental efforts spent to cope with Zeno's paradoxes which are about extremely simple kinds o f motion.
It is a sober ing experience, f or anybo dy who believes to know his or
her geometry, to try to describe in the simplest possible way the general motion of a rigid body from any one positio n to another.
This, as it could be
expected, was achieved sooner than the description of a motion in which the bo dy expe rienc es cha nge s in sh ap e.
At the ti me Poinc aré wr ote his comme nt s,
the study of the geometry of deformation was still a rather young discipline [see, e.g ., Hel mholt z 1858].
Poin caré pointed at the great difficul ty in fol-
lowing the motion of a body with important changes in shape; he noted that the great discovery in this field was the ability to follow the motion of small elements, that such an idea is very complex and could only have appeared relatively late in the histo ry of science.
It could, moreove r, not have aris en, if the
observation of solid bodies had not already taught us to distinguish and describe changes of position amounting to rigid motion.
Poincaré concluded:
"If, th erefore, there were no solid bodies in nature, there wo uld be no geo m e try . " And with out geo metr y, I wou ld add, no kinemat ics.
Even an initial survey of the historical developments of the studies of motion reveals immediately how little of permanent value and how much
6
wasted effort was contributed by those who chose to approach motion in a pu rel y spe cu lat ive way, without resorting to observations li ke the astronomers did from the beginning or to experimentation and invention as many early engineers also did.
Prob ably , ge ometry could emerge only after sur veyor s and
engineers for long centuries had made the perception and understanding of fundamental common traits almost obvi ous to any reflect ive mind.
Then it
be came possible to formu lat e pos tul at es and construct a logico-de ductive s y s tem of thinking.
But the great adventure of adding the variabl e time to geome
try prove d to be a formi dable task not easily accompl ished.
It is often said
that we have an advanced technology because of science, but perhaps it is the other way around, that science emerges after great technological progress. Once a body of science is available, more technology takes place and again a new chapt er in science is added, and so on and so forth.
At least, this seems
to be the pattern until recent times in which the acceleration of all processes may have fundamentally changed the interactions.
In the development of geometry, even today, there is a persistent trend aimed at excluding motion from the subject.
This appears to match the req ui
site of not considering force in kinematics, but this does not constitute a per fect analogy.
Although not usually mentioned explicitly, there is a basic role
for motion in geometry.
It has been unavoi dable from the very beginning.
There is an interesting experience that anybody with some knowledge of geometry can live, and it is to try to avoid any reference to motion in dis cussin g geometric quest ions.
For instance, try to explain how to construct, in
elementary geometry, a regular polyhedron without any reference to motion.
7
Another example: when we define bodies of revolution we customarily use the kinematics of rota tory motion.
The point is that motion is indeed unavoidable
in geometry, and it should be recognized also as useful, and at the same time troublesome, because it begs the question of preservation of rigidity in all geometrical operat ions.
On the other hand, anybody who has attempted the
study of motion has found unavoid able the clever use of geometry, or oth erwise has failed in contributing anything of permanent value.
In kinematics there has been very important activities of an experimental nature concerning not only the study of models and the visualization of flows, but al so that of ob jec ts li ke the hu man figur e and the bo di es of an imals.
Ther e
are many other phenomena that require the use of the methods of experimental phy sics,
an d I will dev ot e a cha pter to disc uss
ex per ime nt al
ki ne matics .
Although Greek science was never able to make much progress in kinematics, we must take into account some inroads that were undoubtedly of historical significance.
For instance, on composition of motions, Aristotle was right
when he stated that a body, actuated by two motions that are such that the distances traveled in the same time are in constant proportion, will move along the line of a parallelogram the length of whose sides are in constant ratio to each other.
[Dugas 1955].
descr ibe a curve.
If the ratio of the veloci ties vari es, the body will
I find this a very interesti ng passag e on kinematics.
We must take into account that Greek science was approached in different ways and that the loss of most of the documents resul ts in a biased view.
But,
according to Clagett [1955], three fundamental views were important and had
8
lasting effects.
Motion was impor tant in each of them, altho ugh its s tudy may
have been ill-conceived oftentimes, or abandoned to tackle instead of easier static phenom ena.
The first, the so-cal led material view of natur e lead to the
atomic theory of Leuci ppus and Democri tus,
and thus motion was given a
prominent pl ac e through the conception of a ceas el es s ag itat ion of the at oms. The second approach was mathematical (Plato and Pyth agor as).
The third
approach was the Aristotelian which can be classified as one of compromise in which matter and form are considered as bound in an inextricable way. Perhaps, the wide scientific interests of Aristotle led him inevitably in this direct ion.
Matter is in con sta nt change , and so is form.
In Aristotelian physics, motion ought exist because there is a center toward which all bodies tend to move, then because there is already a density stratifi cation in the world, i.e. a series of concentric spheres of different density, we may observe centrifugal trends when air is placed in water or centripetal effects when water is placed.
But if there woul d exist vacuum, no natural
motion could exist; i.e. no tendency tow ards a natural place.
Aristo tle also
said that if a body would be set in motion in vacuum it would not possibly come to rest.
The reaso n being that why woul d it stop at one place rathe r than
at another? [Dugas 1955].
René Dugas, when detecting some incipient notions of a more general view of time dependent phenomena in Aristotle's Physics and his
Treatise
on
H eave ns, expressed that the great philosopher did not differentiate, regarding
change, mechanical concepts from others with more general significance.
9
Indeed, why should the study of motion be confined to mechanics? ; in a broad sense, motion and change are ever ywher e in the universe.
It is not only use of
analogical language to speak of velocity (or rate of change) in chemical or bio logical pheno mena, or really in any kind of pheno mena.
Mathematically, it is
the time rate of change of any variable property that provides the common thread.
In the preface to his history of mechanics, René Dugas explains that Hellenistic dynamics failed because it did not know how to account for the resistances to motion and it lacked a precise kinematics of the accelerated motion.
Havi ng treated and develo ped so well geometry, there was no ability
to think in terms of time depend ent geometry.
Thus they were quite good in
statics and sterile in dynamics, not having been able to invent kinematics. During a long time, only the astronomers did what amounts to observational work on celestial kinematics, and they did it quite well, although the relative motion they were describing was not the simplest.
If hist ory has shown someth ing is that we in fact got from res earch ers and thinkers closer to technology most of the original and fruitful ideas, clear notions and useful resul ts.
Clagett [1955] has called attention to the notions
of Strato the Phy sic ist ( -287 ?).
Some of his work was attributed to
Aristotle, but their views were in fact quite differ ent.
I will include a sum
mary of Strato's concepts where I think it belongs, i.e. in the section of sci entific and engineering kinematics.
10
Except for empirical advance s, after Greek science came to a halt, there was little activit y in kine mati cs [See Pe der se n et al.
1990].
But late in
Medieval times, in the schools of Oxford and Paris, the theoretical study of kinematics of very simple motions saw a brilli ant period, but this movem ent in turn had little proj ectio ns for a long time.
[See the impor tant studies
of
Clagett 1948, 1950, 1959].
Kinematics was revived by Leonardo when, using scanty notions and con cepts received from medieval science, he developed his own body of knowl edge and built a sui generis kinematics with striking resemblance in some areas with the kinematics of centuries after him rather than to that before him. Unfortunately, Leonardo only left unpublished material, and a new interrup tion was unavoidable.
Leonardo developed his own rudimentary t rans form a
tion geometry, an aspect of his work that did not escape the keen eye of Hermann Weyl [1952 ] although he did not resea rch the matter fully.
The vast
work of Leonardo in proto-kinematics has been studied by M. Macagno, and much of what is summarized in this monograph about Leonardo's geometria che si prova col moto has already been exposed in her contributions [Macagno
M. 1987, 1990, 1991 ].
She has loo ked at Leo nar do with the eyes of an
applied mathematician, and she has seen as geometry in motion some aspects that here are considered as incipient kinematics in some cases, or as true kinematics in others.
There are studies in Leonardo's manuscripts, that show him as a student of motion in the roles of artisan and artist, e nginee r and scientist.
11
He studied the
shaping of different materials by the means available in art workshops of his time [See, e. g. the Codex Atlanticus].
This appears to have been at the root of
his excellent understa nding of the geometry of deformation.
Of course, there
is a great distance between Leonardo and Euler and Lagrange, but in several ways he is the first to dwell on what we call now the Lagrangian and Eulerian views of the motion of defor mable bodies [Macagno 1987 ].
As an engineer
and scientist, he studied in a large variety of contexts, pathlines of one parti cle, of syste ms of particl es, of points of rigid and deformabl e bodies.
In
some cases we see him examining how an artist can change manually the shape of the material with which he works .
In others he examines deformati on of
figures and elements thereof much in the way that was done by scientists of the nineteenth century, as Helmho ltz and Bertrand.
In others he determines
path and ve loc ities for el emen ts of me chanisms as a me ch an ica l eng ine er .
His
studies of flow of water, air and many other sub stanc es (like, sap, blood, mud, etc.) is remarkable and takes hundreds if not thousands of folios in his manus crip ts.
His kinematics of fluids is partially synt hesize d already in a
number of papers and mono grap hs [See Macagno 1989c].
There is still a great
need for modern-app roach studies of other areas of kinematics in Leona rdo' s manuscripts (e.g., elastic bodies).
The period dominated by Galileo and Newton saw the beginning of mechan ics as a science which, for macroscale phenomena, is still used
today.
Whatever kinematics was developed then, until Euler and Lagrange included, was embodi ed in the studies of statics and dynamics.
The histor ical study of
kinematics for this period must tackle the problem of extracting from the avail
12
able documentat ion what belongs to kinematics.
This task exists also for
many more recent writings in which the authors did not care to separate find ings in kinematics from their treatment of wider or deeper problems.
As one examines books on mechanics there is a period in which their title is Rationa l M echanic s .
The book s I used as a stu den t for my firs t cour se in
mechanics were Levi -Civi ta [1918] and Cisotti [1925] both with that title.
A
few words are needed to explain what is meant by rational mechanics. Truesdell defined what he considered as rational mechanics [Truesdell 1958]; paraph rasing his de fi ni ti on , one cou ld sa y as well rational kinematics is a part of mathematics .
He added that mechan ics is a mathemat ical science ; I would
rather say that it is a physico-mathematical science, which I believe is more in line with the first users of the terminology ( mécanique rationnelle , meccanica razionale ). In what (for once) I agree whol ehe art edl y with Tr ues del l is in his
dictum that mathematics is also a science of experience, of that kind of experi ence that happens in our brains, because as he put it, who would think that an oscilloscope is amore precise instrument than the brain?.
The notions of analysis and synthesis in kinematics must also be studied. Although apparently originated by mechanical engineers, the idea of moving from analysis to synthesis in the kinematics of mechanisms and machines [Reuleaux 1875 , Hartenberg 1964] is nowadays present in all fields of appli cation, and in a crude way must have been present also in previous empirical developments as made evident by the ever increasing efficiency of kinematic
13
devices through history.
In tracing the historical developments of kinematics,
and of any of its branches, synthesis deserves as much attention as analysis.
In kinematics, synthesis is defined as follows: systematic approach to the design of a mechanism to perfor m a given function.
According to Fergus on
this was at first explored by Reuleaux 1876 [see Introduction to Dover Edition and Ch. XIII].
Reulea ux made the point that one must master analysis before
engagi ng in synt hesi s.
There is also in Reulea ux an approach whi ch departs
from the approach of kinematics in Paris, but this seems to be essentially in aspects that pertain to the consider ati on of efforts and force s, which are by definition beyond kinematics.
The role of some kind of knowledge of kinematics in representational art is perha ps le ss obviou s than in techn ol og y, but, on ce we kn ow ho w to lo ok at the work of artists, the evidence is also overwhelming. ception
of flow
"rheograms"
are
among five
artists
millennia
begins old,
very while
early the
For instance, the perin
first
"rhe ograms " are five centuries old [Macagno 1984/5].
his tory,
and
important
some
scientific
Of course ,
artistic
kinematics (meaning mobiles, animati ons, devices of kinetic art) is much more recent, and is probably inspired by scientific and technical kinematics.
Kinematics
on
its
own
seems
to
begin
in
late
Medieval
and
early
Renaissance times, it flourishes briefly and secretly in Leonardo da Vinci, it be co me s an exact scien ce wi th Ga lileo and fol lowers, it on ly gets its mode rn
14
name and power ful mathematical apparatus about two centuri es ago.
It con tin -
ues to be researched and developed very actively in our times.
In this chapter, I have tried to show that in kinematics there are a number of facets that are worth examining and that can be taken as the basis for a scheme for a historical study.
In this fir st attempt, I shall use those activities
that appear to me as the four main walls of the great building of kinematics: empirical, theoretical, experimental, and applied.
15
SCOPE
Now that an ov era ll pi ct ur e of ki ne ma tics has been deli ne ated, warr anted to advance some ideas about a plan of action.
it se ems
The history o f kin e
matics I envi sion shou ld encomp ass art, science and techno logy.
But the main
initial thrust should be aimed at establishing the history of kinematics as a science, f rom its origins to rathe r recent times.
The main reas on for this
choice is that I believe that the history of physico-mathematical developments can be least subjec ted to subject ive influe nces.
Other aspects, that are more
susceptible of biases, should be investigated afterward.
Some developments in geometry are important in the history of kinematics, and this is evident in the terminology which we find in some areas of geome try; terms like "bodies of revolution", "tra nsla tion s", "sc rew ", "glides", etc. The only term that seems to elicit optical phenomena rather than a common type of motion phenomena is "reflection".
But as mathematics progresse d,
there were other areas of this science that became useful in the study of motion.
One shoul d recogn ize that the bounda ries be tween mathematics and
kinematics were not always clearly traced.
Howe ver , we should not recognize
any study as belonging really to kinematics unless a clear association of space and time is present.
16
Men surely have always had a visual and muscula r perceptio n of motion, of velocity, and acceleration, but it is fascinating that it took such a long time to examine those percept ions and generate notions in a useful abstract manner.
I
beli eve that, in du e ti me, we shou ld wr ite a hi st or y that it is no t on ly ba sed on written documents about the analysis of motion.
The generati on of ideas, con -
cepts, ability to produce motions physically, etc., should also be investigated and traced through all kinds of evidence and the archaeological material available. For me, the deep roots of kinematics are in the long period of empirical kinematics, and in the somewhat briefer period of experimental kinematics that prec ed ed the ma the matica l app roa ch to motion .
Th e lea rni ng exper ien ce of
Leonardo, who was an autodidact par excell ence , can serve as a relatively recent model to theorize on how humanity went from very empirical activity to a profoundly theoretical knowledge as we can witness in kinematics.
In what follows, I will discuss first the appropriate methodology for this pr oj ect, and then su mmar ize the vi ew s of the differ ent fa cets of ki ne ma tics that I have developed in the several years of work in this endeavor.
17
APPROACH AND METHODOLOGY
The quest ion of the adequate approach to, and methodo logy of, a history of kinematics is not obvious and surely should not be that of traditional history of science.
The adequate way to proceed must be consi dered carefully.
One of the
important parameters of the historical study of any science is the knowledge of the pre sen t status of that science in its most advanced forms.
If such a para me
ter is replaced by an elementary or popularized form, at a level similar to that of the peri od studied, we may seri ousl y fail to use the adequate approach.
I would
propose the criter ion th at it is no t advi sab le to eng ag e into the hist or y of a dis cipline for which one is not able to provide a good critical description of its pres ent st at us , and a reasonable pr ed iction of wh ere it is go ing .
Several concrete examples to illustrate the above argument can be given. One could think that since geometry was Euclidean for millennia, it is not nec essary to know non-Euclidean geometry to study the geometry of say the Medieval times, or the Rena iss anc e times.
But the fact is that for a long time
already we have had a much better perspective of the fifth postulate thanks to the wor ks of Loba chev sky , Gauss , Boliay and foll owers .
But we do not need
to cons ide r non- Euclidean geometry to find striking examples.
R. Marcolongo
[1932], a distinguished classic mathematician, despite careful studies, was not able to see that Leonardo da Vinci developed the basics of transformation geometry, while Hermann Weyl [1952], a contributor to modern geometrical
18
developments, only needed browsing the manuscripts to find material enough to formulate a theorem in group theory which he named after Leonardo.
Within my own experience, I can cite other striking examples, in the case of hydrostatics [Macagno 1985, Dijksterhuis 1957], the analysis of Archimedes work that is done by fluid-mechanicists is certainly much more penetrating than that of men who know only elementary physics as taught in high-school or in a general course of university physics, even if they are distinguished in other fields.
If one kno ws about tensori al calculus one can do an incis ive an alysis of
writings on hydrostatics, and one can see where Archimedes failed, and also where Leonard o da Vinci (with an entirely di ffer ent appr oach) failed also. interesting
to
realize
that
hydrostatics
was
decoupled
from
dynamics
It is by
Archimedes but not by Leonardo, who rarely failed to associate statics and dynamics.
Another example is the histo ry of the classic study of Newton ian
fluids, which is much better understood by those who are familiar with non Ne wt onian flui ds.
So me invoke the danger of anachronism to ob jec t to the use
of the best present knowledge to examine that of the past, and one can grant that there are dangers and extreme care is needed.
But the infor med pe rso n may try
to avoid some pitfalls, the un-informed person is simply blind to part of the pi ct ur e he want s to ex am in e, and he has no idea of wh at he can do to av oi d mistakes.
An adequate methodology in the study of the history of kinematics requires that one be familiar not only with theoretical studies, but also with experimental techniques, because contrary to geometry, kinematics has important components
19
which are obser vati onal , experiment al, and technical.
Therefo re, if one is not
also an experi mental ist one is really handi capped for this project.
Of course,
the knowledge of the science in its present status is a necessary but not suffi cient condition; one must discover how to analyze and synthetize its entire his torical development.
In the case of the history of kinematics there is a particular problem that must be considered carefully, because, as an independent discipline, is much you nger than what can be consi dered its sister, geometry.
Theref ore one must
trace kinematical developments that are usually buried between two historical layers, one is usually geometrical, the other is mechanical, or physical, or chemical or biological.
It is a pity that a very early book on history o f geometry
has been lost, because, perhaps that early historian faced a problem of this type, when trying to extract geometry from writings on different subjects in which it was integrated.
As one examines docu ments, one must be aware that not always
the authors have clearly delineated what is geometry, kinematics and dynamics in their writ ings, an approach rarely used [Macagno -Land weber 1958].
The
easies t case, from this point of view, seems to be in the field of mechanisms and machines, and to a certain extent in mechanics of the continuum, because for both there exist already some specific historical studies which can serve as models. [See the corresponding section in B ib liogra phic al Refe rences ].
To further illustrate the dire need for scientists to take an interest in the history of kinematics, I can cite many examples, but perhaps the consideration of
the
Hadamard
conditions
for
20
surfaces
and
layers
of
discontinuity
[Predvotilev 1962] is most fit for that purpose.
Studying such a work one can
gain a view that seems reserved to scientists because of a relatively high mathematical language
barrier.
Howev er,
it seems
that
if contr ibution
like
Hadamard's were left out from a history of kinematics much would be lost just be ca us e of be in g di ff icu lt to ana ly ze and ev al ua te by gene ral ists.
To na me jus t
a few fields in which such conditions are applied, I can mention acoustics, optics, mechanics of particles, explosions, combustion, aerodynamics, hydr odynamics, shock loading and unloading in solids.
A field in which some models for the methodology to be used exist is that of the kinematics associated to early astronomical observations and studies [Schi appar elli 1874 , Neu geb aue r 1962].
This is not an area wit hou t amazing
features, because if one compares the observational means of those times with modern instrumentation in this discipline, it seems almost unbelievable that so much was accomplished in the description of the motion of stars and planets. One interesting problem is why, if the motions of celestial bodies were (even with old geocentric theories) not more complex than that of some mechanisms, the degree of sophistication in the description of the motion was so different. One reason may be that in the case of mechanisms and machines there was no interest in describing motions accurately, but in physically producing them. Be as it may, we find quite a different approach needed in these two areas, be ca us e alm os t from its ve ry inc ep ti on , astron om y of fe rs more adv anc ed theoretical kinematics than that of mechanis ms.
Perha ps only the study of the
level and floods of the Nile constituted a match to the astronomical studies in the neighboring lands of Mesopotamia and Greece.
21
It seems necessary to point out that the history of kinematics should be the history of motion rather than that of the natural or man-made systems in which such motions are found, or that of the points of view adopted ( motion relative to the Earth, or relative to the Sun, f or instanc e).
Becau se of this, it makes
little difference that an elliptical motion is accomplished by a planet or by a material point in a mechanism.
Only if the study of motion has peculiariti es in
one discipline or another there is an interest in establishing a differentiation. This circumstances may be found in many cases; waves in water and in elastic media may have a unity in the fundamental equations, but then the mathemati cal and experimental techniques used in the actual study of the corresponding kinematics may be quite different, and had had a different historical develop ment [Tokaty 1971, Timos hen ko 1953 shoul d be consul ted] .
A similar situa
tion occurs in kinetic theories as discussed below.
Another point that certainly requires knowledge of science rather than any other else, is that the introduction of mass does not make necessarily some notions strictly dynamical, and outside the field of kinematics, they may remain strongly and fundamentally kinematical.
For instance, when one dis
cusses the formulation of the equations for conservation of volume in fluid mechanics one is still very much in the realm of kinematics, but such domain is not abandoned even when the equations for conservation of mass are formu lated; the two treatments are very similar, the mass distribution becoming the less si gnifi cant part of the statement. [ See Apppend ix I of Macagno 1989a]. Even the discussion of momentum flux and kinetic energy flux are fundamen tally kinemat ica l.
One coul d say, in general, that there are terms in the equ a
22
tions of dynami cs which are essent iall y kinematical.
When is it that one is
dealing essentially with kinematics, in spite of being in the field of dynamics, and when not, is a matter that requires careful examination, and of course, it may depend on the defin itio ns adopted.
In turb ulenc e theory , for instance, the
so called Reynolds stresses present this kind of problem.
In the study of kinetic theories of the different states ( solid, liquid, gas), and in the studies of mass transport processes, we find a role for kinematics which has to be traced careful ly by scientific hist oria ns.
If one examines the
basi c de fini tions and formulations in the theo ry of ma ss transport by fl ui d flows, for instance, the notion of velocity for a mixture of different species cannot be introduced without taking into account the different densities involve d [Bird 1963].
There is an analogy with a simpl er prob lem, that of
centers of gravity as they are defined and mathematically determined in some cours es of calculus.
The matter is treated in a purely
geometri c manner,
although it is really a problem in physics, especially if the mass-density dis tributi on is not unifor m.
When the density is not consta nt it cann ot be easily
ignored.
In the defini tion of velocity in a mixtur e of dif fere nt specie s, either the common mass density or the molar concent rati on are used. vect or at a poi nt is def in ed as v = S (jj- v- j / indicate mass density.
£
Thus the velocity
p- , whe re p is used to
The notion of mass flux is invol ved, but it is more of a
kinematical notion than a dynamica l one.
23
Or we could say that an incu rsi on in
a third dimension, that of mass, becomes necessary to define velocity ( a typical kinematical quantity if there is one) in this context.
Finally, there is the still more subtle question of when kinematics can be decoupl ed from dynamic s, and when not. This is import ant because, if one is aware of this possibility, kinematics in cases of coupling becomes a much more delicate subject to study, not only p e r opment.
se.
but also in its historical devel-
In the Chapter on Theoretical Kinemati cs, more will be said about
this question.
I hope that this discussion is helpful in any future study, my own or ot he rs '.
If a histor y of kinematics is to be written, it seems that some pitfalls
should be avoided that can be detected in the history already written for other discip lines .
I hope also that there is a challenge here to engineers and scien-
tists not to relinquish to less prepared scholars the responsibility for writing the history of kinematics or that of any other engineering or scientific disci pl in e.
If necessary, we can learn ab ou t history and its me thods, a task that
seems easier than that of historians learning what we know about engineering and science.
24
EMPIRICAL KINEMATICS
Reuleaux has influenced modern thinking about what are mechanisms, and his wide concept ions show in my incl udin g devices with solid, flexibl e, fluid pa rts as mec hanisms.
Th at do es no t mean , of course, that anything relating to
fluid flow is cons ider ed to be a mecha nism in this mono gra ph.
But the
boome rang is regarded as a mechanism in spite of consisting of a si ngl e so lid bo dy ; the othe r el em en t is the air.
Th e las so is a mechanism beca use in spit e
of being basically a single piece it has originally two elements one of which is bu il t at on e extreme by a properly done kn ot .
I ev en go as far as envisioning
as part of kinematics the operati ons invol ved in kneadin g, and in knitt ing and weavi ng.
I believ e that for a long time all this wa s in a purel y , but no t at all
crudely, empirical basis.
Nowa days, for example, we have engineers de sig n
ing mechanisms to knit in three dimensions which require a sophisticated knowledge of theoretical kinematics.
Each successful mechanism or machine of pre-historic or early historic times is in itself a study in motion of an empirical nature, but unavoidably related to notions and thoughts about displacements, velocities and accelera tions.
I am not saying that the inve ntor s of the boomer ang or the sha do of had
ideas similar to ours about velocity or acceleration, but their senso ria l pe rce p tions related to our modernly defined properties of motion were surely as keen, or more, than ours.
It seems inter esting to attempt to reach a co mp re
hension of the kinematics of early projectiles, machines, tools and instruments
25
with an eye to the kinemat ics involved.
The stude nt of the history of kinemat
ics would do well to construct models of different devices and mechanisms and operate them with his hands; it may be very revealing to discover how keenly can one sense acceleration in a muscular rather than a visual way [Macagno M. 1987].
No matter how long was the way from skil ful early pro
duction of certain motions to our scientific notions about them, it seems a more reliable and correct source than the persistent obscure and confusive lucubrati ons about motion by a number of phil osoph ers, although, there are no doubt some important philosophical contributions, like Zeno's paradoxes, who made permanently valid and disturbingly challenging contributions and stimu lated an ever more rigorous codification of our knowledge; however paradoxes are not sources of invention and creativity in the acquisition of new knowledge of the physical world and its phenomena and in the processes of invention.
Reuleaux [1875] believed that motion is an easier and more accessible idea to early cultur es than force [ See p. 224 of Reuleaux 1876 ].
He consi dered
the fire-drill as the first machine contrary to the then generalized belief that it was the lever.
He thou ght that we believe the lever to be the first machine
beca use we thi nk th at the first goal was to over co me gr ea t resistances.
He
be liev ed that what first at tra ct s the op en consciousness, is re al ly the acc omp a nying motion.
He noted that childr en are much more attracted by motions than
by the for ce s imp lied in those sa me mo tions.
Re ul ea ux concluded that separ a
tion of the idea of force from that of motion is a very difficult mental opera tion and because of that we find it occur ring late and gradual ly.
Reuleaux
be liev ed that the un-erad icable fasci na tion with perpe tua l mo tion is roo ted in
26
the initial fasci nation with all kinds of motion.
Reuleau x [1876.
p. 222]
thought that rectilinear motion ( blow-tube, arc and arrow ) is not common among very old people s, how eve r primi tive such motion may appear to us.
I
would like to see these questions investigated more thoroughly.
Reul eaux 's section on ancient machines is extremely interesting.
He listed
the following as machines invented in antiquity: rollers to move great stones, carriages for war and transport, water wheels, toothed wheels, pulleys, certain kinds of levers.
He also made a list of "basic mechanical compo nent s:
1. the eye-b ar type of link called crank in kinematic s (vague con necti on with what is today called this way).
Sometimes called lever.
2.
the wheel, includi ng toothe d wheels.
3.
the cam
4. the scr ew 5.
the ratchet (intermit tent motion devices).
6.
the tension -comp ressio n organs ( chains, strings, belts, hydraulic lines ).
Hartenberg [1964] agrees that all the six Reuleaux's "basic mechanical compo nents" were already invented in antiquity and put to use before the Christian era.
We can trace the above kind of empirical kinematics even further back in time if we examine the work of archaeologists who document the existence of different mechanisms in differ ent cultures all over the world.
27
The popular
idea that civilizations can be measured by having known or not the wheel is an indication of this possibili ty.
Howeve r, in my opinion, there are inventions
that are perhaps cleverer than the wheel, like the boomerang, the picota, the sling, the shadoof, the blow-tube, the arc and arrow, the "boleadoras" (a lariat with stone balls) which are very ingenious kinematical inventions and require much more kinemati cal skills to use them prop erly .
Of cours e, the wheel is
probably the mo re ve rsatile ki ne ma tic el em en t we po ssess.
Th er e is kinemat
ics not only in utilitarian inventions; we find it in toys, games, dance there is also much of kinematics analyzed and synthesized by our first computer, the br ai n, wi th mainl y the aids of eye and ha nd .
I would go as fa r as to say that
there is much kinematics to be studied in the motions performed and created by animals, but this would overextend an already too vast program.
An
interesting
aspect
of
inborn
kinematics
has
been
pointed
out
by
Hartenberg [1964, p. 117-8]: we are experts at integrating velocities and making predictions of where a body with which we can collide will be in the next few seconds.
This is vital in cro ssi ng busy, or not so busy,
There is no dynami cs invo lved, p ure kinematics.
streets.
This ability to avoid colli
sion, or to escape danger by estimating velocities must be as old as mankind, must in fact be inherited from even older than human ancestors.
For the history of kinematics one should study the mechanisms found in early cultures from the point of view of their motion, leaving to others a sys tematic study of the origins, development and skill ful use of mechanisms in diff erent particular cultures.
For instance, in the case of the Egyptian shadoof
28
[as represented in Fig. 171 Reuleaux 1875] one does not immediately have the elements to evalua te it. The basic idea seems relati vely simple, but it does not appear as a device to which it is enough to provi de only muscle s.
It would
surely be difficult to design a robot that would operate this system with many degrees of freedom as efficiently as the many unknown men who have used it for such a long time.
I believe this is a way of jud gin g s ome mech ani sms.
The mechanical clock is a rather recent invention but the fact that it is enough to provide a weight or a winding spring to have it running by itself tells a story of effective but less complex kinematics, free from instabilitie s.
I have
constructed a model of a shadoof and tried my hand at it and I recommend the read er to do the same.
Or try the "bol eado ras " whic h seem simpler to make.
Again, I would like to see a robot throwing successfully the boleadoras as I saw it done as a child in the Argentinean pampas Mechanisms which still need an alert brain to function well are of a category different from those for which more or less brute force is enough.
During the long period of empirical kinematics, it is highly plausible to assume that motion was many times conceived regardless of considerations of force or power.
I supp ose that some mechani sms were aba ndoned becaus e not
enough power could be made available or because there were no materials appropriate for the construction.
This happened to a number of Leo nard o's
"invent ions", but some of them were good in the long term .
The great diffi-
culties to develop some mechanisms and machines should not be underestimated; Friedrich Klemm [1954] has discussed this question and has quoted the writings of some inventors which narrated such difficulties.
29
The idea of disregarding the forces involved recurred more than once in the hist ory of kinematics.
Mechanisms were consi dered in some periods as the
devices which essentially were designed to change one force into another, rat her than one motio n into another .
That it is really
better to begin by
studying motion disregarding the forces both for teaching and research had to be rediscovered more than once.
Th en the re is the question of wh at one
shoul d expect of a good kinematicist, his power s of analysis or his powe rs of synt hesis ?. Design and inventi on require always synthe sis rat her than analysis.
Thus, o f the remote past, we know the resul ts of synthe tical wor k and
very little, if anything, of the analytical work.
Should we conclude that our
ancestor s did not have a way of analyzing motion ?
•
There are obviously a number of mechanisms in our own bodies, in addition to much more complex physiol ogica l systems.
Mechanisms in which there
are chains of link ed bars are too analog ous to arms and legs [ Fis che r 1909] not to have resul ted from such inspir ation.
The oar could have had as ancestor
the arm and hand which can also be used for propulsion not only in swimming bu t al so in rowing.
If on e accept s Reuleaux's cl as si fica tion of me ch an ism s
and includes not only those made of solid pieces but consider the blowers for instance as mechanisms with fluid components, then the possibility of producing air flow with our mouth may very well be behind the invention of blowers. Probably the chain of links of so many mechanisms invented by different cultures started by the experience gained in producing motions with our own bo di es or its part s.
30
Some rotary motions were produced with human hands, like the alternative rotation of a rod by putting it in between the hands and moving them alternatively back and forth.
This is a very import ant kinematic feat becaus e it was
usefu l to drill holes and to prod uce fire [Tylor 1870 ]. Gei ger [1871] believed that rotary motion was the first that mankind produced by means of what could be called a mach ine.
I am not su re that this is true, alth ou gh Ge ige r ad du ce d
very good rea son s in sup por t of his idea.
In one case the rod is rotated a lter -
natively with the hands directly, and in another by means of a string or a cord that makes a loop around the rod.
Probably it was easier to conceive and develop alternating rotary motion than continuous rotat ion in one direction.
The wheel is less universal among
different peoples than some device to make fire.
Natural water current s pr o-
duce circulation or vortices at some places and one can presumably see a piece of wood rotating endlessly when caught in whirlpool.
But it seems a long way
to arrive at the point of designing and mounting a water wheel or a wind wheel.
We also know that the potte r wheel is a rathe r old inventi on.
At some
time circular, more or less uniform motion, must have been conceived and devices invente d that pro duc ed it.
The water wheel could be kept in motion
for as long as there was a steady flow of wate r prop ell ing it. The potte r wheel was probably kept in more less steady motion by either the feet of the artist or by the ha nds of an assi st ant . motion is extremel y inte rest ing.
The wa ys by which a mechanism wa s ke pt in In this study there is a theme, that I visual ize
as hovering over all the great variety of devices and gadgets, namely, what
31
kinds of motions were conceived and put into existence one way or the other?. If circul ar motion was already known by some people, did they co nvert it into rectil inear motion of some kind? given motion into another?.
Did they develop other conv ersi ons of a
Look at the motions rather than to the mechanical
devices.
When carriages were developed, their wheels would accomplish rotation and translation; the carriage itself mainly translation, but occasionally it shoul d change direction and rotate.
For the two carriage wheel, the change in
direction presented little difficulty, but a more complex kinematics character izes a four -whe el carriage which can easily turn.
Two- wheel chariots were
used in Western Asia, Egypt, Greec e for war at rather early times. they came from farther east, I ndia, China. Bible.
Perhaps
They are also mentioned in the
Reuleaux believes that a pr edece ssor of the wheel may have been the
rol ler under a heavy stone block.
To me this look s , kinema tica lly, very much
like the rod rotating between two hands.
The alternative rotation was used in
a later devel opment which seems related to the pot ter 's wheel, the lathe.
Both
in the East and the West this happened with different arrangements, but kine matically with the same type of motion.
To clarify a basic point, I would like to look at the piston and cylinder pu mp as a de vi ce that con ver ts an al ter na tive rec til ine ar motion into a uni direct ional motion (not unifo rm of course) of a water column.
Howev er, when
a centrifugal pump is used, a rotatory motion is converted into a rectilinear nearly uniform motion of a water column. This is an application of Reuleaux's
32
criterion of not restricting kinematics to rigid bodies, or linked rigid bodies; it was, in fact, fact, adopted in in ancient empirical kinematics.
The use of flexible or
fluid elements as integral parts of motion converte rs is very old. old.
Note that that
swimming is based on creating a certain fluid motion in order to propel our body bo dy in a gi ve n di r ec t io n.
We mu s t s t ud y me c ha n i s ms and an d c ha i ns or a s s e m
blie bl iess o f me c h a ni s ms f r o m the th e p oi nt o f vi ew of a ch ai n of mo ti on s c on c e i ve d already by our ancestors as part of their technology, science and art.
Modern kinematics has taught us that a solid body, which can be consid ered as rigid, can only perf orm a combination of translat ion and rotation.
For
pl an a r mo t i on s , th ere er e is a s i mp l if i c at i on , b e c a u s e any an y d i s p l a c e m e nt can ca n a lway lw ay s be ob t ai ne d by mean me an s o f j u s t a r ot at io n.
No t e that th at t r a ns l a t i on may ma y be c o n s i d
ered as a rotation around a center at at infinity. infinity.
In space motions , any displa ce
ment is equivalent to one by a screw screw or helical motion, i.e ., generally c om po se d o f a ro t a t io n and an d a t r an s l at i on .
This Th is ge ne r al t he or e m do e s not no t ex cl ude, ud e,
of course, descriptions of a specific kind for particular types of motion.
When we consider bodies like strings, belts, springs, or fluids like water and air, to the roto-translatory components an additional, much more complex componen t of motion, must be conside red.
Even solid bodies may have have to be
studied as deformable bodies, as in the case of studying the vibrations of a bu il di ng or a mac ma c hi ne , f or e xamp xa mple le .
Th e re is a well we ll de v e l o p e d t he or y of
deformable bodies which will be considered in the chapter on theoretical kine matics, but from the the point of view of empirical kinematics, an unde rst andi ng
33
of defor mati on surely existed f or some primitive devices and mechanisms to have been developed and perfected.
Some mechanisms are challenging if one tries to reconstruct a plausible path pa th t o w a r d th ei r i nv e nt i on .
A go od e xa mpl mp l e is the th e s c r e w - n u t me c ha ni s m
when used as a device to convert rotary motion into linear motion, for exam ple. pl e.
R e u l e a u x wa s pu z z l e d by wh e n the th e s c r e w - n u t c o mb i na t i o n made ma de its
appearance and why is predominantly right-hand rather than left-hand screw, although in antiquity there there are both [Reulea ux, p. p. 222 -223 ]. combinat ion is another puzzli ng motion convert er. gory.
The screw-ge ar
Turbine s are in this cate
And also centr ifugal pumps.
Kinematics is a field in which the engineer came in the wake of ingenious men and women, and then in the wake of engineers came mathematicians who finally erected a new dynamic geometry as the old static geometry had been erected more than than twenty centuries before from measuring, architecture, map pi ng and an d s u r ve yi ng .
In the following table, I have attempted to catalog the most important motion conversions in the long empirical period, which I believe continues to this day, because inventors still proceed in the same way; at least those many without a formal scientific or engineering training. training. tion is included for each item in the table.
34
Just one typical typical illustra
Rigid elements
Potter wheel
Flexible only
Lasso
Ri gi d- f l exi b l e
P u l l e y s a nd c o r d
Rigid-liquid
Oar, rudder
Rigid-gas
Boomerang
Flexible-fluid
Sails
Liquid-gas
Air pump with liquid piston
35
THEORETICAL KINEMATICS
Using just a single symbolic mathematical expression, we can say that the oretical kinematics is about Xj
=
Xj
(X, t).
What is meant by this is that, for each case of motion, we have a procedure to calculate the coordinates Xj of a set of material points, given the vector X that identifies each point ( X can be the initial-position vector) and the time t.
For the case of the continuum, we can find in Truesdell [1954a] more pre cise specifications, but the expression is applicable to any set of any number of particles starting with one and ending at infinity.
The above expr essi on is
of great generality, because it is applicable to either discrete systems or con tinua, and to all kind of models, det erminist ic or stochastic.
For example, we
can consider it as the representation of a particle that follows Newton's first law or of one that perf orms a ran dom walk, or anything in between.
It should
be und erstood al so th at our sy mb ol ic express ion may represent any disc onti nuities that we may want or need to include in our kinematic model; for instance quantic jumps in coordinates at any given time.
In kinematics, we are concerned not only with the positions of a point at certain times, but also with the displacement accomplished and the distance tr ave rse d in a certain time interval .
Hist ori call y, it took a long time to gr ad
ually introduce other functions of time like those for velocities and accelera
36
tions. For some probl ems, even higher derivative s have been intro duced, like the rate of change of the acceleration called jerk, and even other higher order derivatives.
[Harte nberg 1964],
Resal [1862] called this quantity the surac-
célération, but he noted that it was Transon, in 1845, who introduced the
notion.
It shoul d be realized that displa cemen t is not the same as dist ance tra-
versed; if a point describes a circle 20 cm in length to return to the same position, the displa cement is zero, the distance is 20 cm.
The cor re spon din g ang u-
lar displacement and distance are zero and 360°, respectively.
In this chapter, I am not attempting a systematic summary of the history of theoretical kinematics, not even a survey; rather, I intend to visit briefly a few highlights here and there along a period of about twenty five centuries.
I have
selected such material with an eye to those particular aspects that elicit some comments I believe useful for future work in this project.
The first steps in mathematical kinematics were accomplished for the motion of a single point representing either a small particle, or that of a rigid body undergoing a pu re tr an sla ti on .
Late r, the po int cou ld be in cur vi lin ear
motion either free or subject to some constraints as exemplified in one case by a project ile and in the othe r by a pen dulu m.
In genera l terms and for a planar
motion we can replace our above general expression by
xj = F(t) ,
37
x2 = G(t ).
In theory, one can always eliminate the time t between the two equations and obtain the equation x2
=
f( x x )
which is the pathline of the point in ques tion.
It is inte rest ing to realize that
this elimination of t entails a significant loss of information about the motion of the point.
If we study, e. g., Eudoxus model for planetary motion, we dis-
cover that there was a successful attempt to determine the corresponding pathline for a simple versi on of the model. inters ecti on of a cyli nder and a sphere.
This led to a curve defi ned by the This was certa inly inte rest ing because
one could visualize the path, but by itself it suppressed any information about how the particle moved along it. To better unders tand this point let us ass ume that we have
Xj = Rc os pt ,
x 2 = Rsin pt .
Then, elimination of t leads to
x l^
+
x 2^ =
^
,
which is the equation of a circu mfere nce of radius R.
But this last equation by
itsel f carries no informat ion with whic h we could, for example, determine the accelera tion of the given motion.
The last equation does not even tell us
where the particle was at the initial time t = 0.
38
Once differential calculus was developed, it became rather easy to calculate velocities and acceleratio ns when the parametr ic equat ions were known .
In the
last example, we only need to determine the vectors dx j/ dt and d Xj / dr .
We
can quickly find out that, in the given motion, the speed is constant but not the velocity (vector); the acceleration (vector) is centripetal.
Once the kinematics
of a point is developed, the kinematical theory of a rigid body, and that of chains of rigid bodies or mechanis ms may seem to become readily a ccessible. The history of this aspect is interesting because it is full of fascinating developments.
In the planar motion of mechan isms, we have come to con sid er a
pl ane at tac he d to each li nk.
As we usu ally refer the mo tion to a link that
remains fixed, its plane is usually chosen to refer to it the motions of the other pl an es .
Any figur e (inc lud ing its pl an e) in plan ar mo tion rot at es instan ta-
neously around a center.
If we think of two planes, one rotates relative to the
other around the inst antan eous center.
The names of Descarte s and Bernoul li
[1742] are related to the discovery of this center.
Although translation is thought to be understood by everybody, it is perhaps warrant ed to say a word of clarifi cation.
In transf orma tio n geome try and
in kinematics it does not have precisely the same meaning than in common language.
As we know, this happens with many words ( e.g. moment,
vortex, circ ulation, etc.).
work,
A set of points (those of a rigid b ody or those of a
fluid!) is performing a translation if the velocity vectors for all the points are identical.
Be mindful that this does not exclude the existence of acceleration,
and higher derivative s of the velocity.
The pathli nes of a body perf ormi ng a
translation do not necessarily have to be straight lines.
39
To give jus t one example of a more rec ent fi nding from mechanical engineering kinematics, Aronhold [1872] and Kennedy [1886] discovered independently a still more interesting theorem: The instantaneous centers of anv three links havi ng planar motion lie on the same straight line [Ha rt eng er g 1963]. Other notions are those of poles and cent rods.
In planar motion, if a figure
suffers a change of position (a given displacement ), in which it does not remain parallel to itself, it can only be reduced to a finite rotation around a po int whi ch is ca ll ed the po le.
If we ma ke a succession of displ aceme nt s,
there will be a succession of poles, they can be marked on the fixed plane and on the moving plane, thus yielding a fixed and a moving pol ygon.
If the
motion is continuous and not by steps, we generate a fixed and a moving curve, they are called centrods.In the case of wheel rolling on a pavement, or two gears, the cent rods are simple curves.
This knowle dge may be useful in
studying rather old discussions of circles rolling on a straight line [see, e.g. Heath 1921 and Clagett 1959 on one of Heron's problems].
The
first
mathematical
descriptions
of
fluid
flows
are
rather
re cent.
Relatively speaking, not much exists before Euler and Lagrange, although one should not neglect the many efforts ( many successful) of Leonardo da Vinci to describe the motions of air and water using almost purely geometrical procedures.
[Macagno,
Series of IIHR Monographs] .
The students of flow
be twe en the two Leonardos (d a Vinci and Eu le r) us ua lly tr ea te d the pr ob lems they considered as specific mechanical problems, and the historian faces the task of extracting from their writings what was actually purely kinematical.
40
This is probably easier after Euler and Lagrange, although some scholars ignore d the didactical facet of the appeal of Amp ère, , and did not separa te kinematics from dynamics.
Almost always, a discipline can be approached in many different ways; this is especially true of fluid mechanics, and I believe of continua mechanics also. Already Euler and Lagrange realized that there were two fundamental ways of descri bing flow, and more generally any motion with deformat ion.
In fact one
can use the two points of view also for rigid motions, but is generally not nec essary.
We refer nowadays to the Lagra ngian and the Eulerian d escr ipt ions ,
although, as already indicated, they can be found in geometric representations in Leonardo da Vinci' s manuscripts [Macagno 1991].
The history of the use
of this two representations of flow and transport phenomena is very interest ing and also very much influenced by the type of problems our Western civi lization has cons ider ed importan t at dif fere nt times.
But ret urni ng to the dif
ferent ways of describi ng motions, we find the emergence o f families o f lines like the pathlines, the streamlines, the streaklines, the fluid lines, to mention the most import ant ones.
Water in motion, like other flu ids, which seem so
amorphous when at rest, becomes highly structured to the eyes of any hydrodynamicist.
Artists have captur ed the many faces of water, but only scienti sts
have been able to depict the infinite variety of lines and surfaces that can be seen
in the midst of water bodies in motion.
In fluid mechanics, the notion of vortici ty has its own histor y.
It is rather
trivial in two-dimensional flows, because this is a property that seems to have
41
its full meaning only in thre e-di mensi onal space.
Once the essenti al vectorial
character of vorticity was recognized, the notion of vortex lines and vortex tubes was boun d to emerge, as it did.
But not until Truesdell publ ished his
masterful study of the kinematics of vorticity [Truesdell 1954a], was it possi ble to re al ize the fu ll signi ficance of this property of rotational flow s.
The kinematics of stochastic movements has a shorter history, but it is not as rece nt as some who spe ak about chaos are incl ined to believe. shoul d suffice.
Two samples
One, the studies of turbu lent fluctu ations [see,
to begin,
Reynolds 1894 ], and two, the work on random vibrations of machines and bu ildi ngs [Cr an da ll 1963].
I hope these few examples emphasize the importance for the historian of kinematics of being knowledgeable about modern kinematic theory.
Not until
we are aware of such theory, can we have the ability to put in the right per spective the great difficulties experienced historically to develop a rational appr oach to the stu dy of motion.
We can also realize t hat not many decades
ago there were still important gaps and pitfalls; and who knows if some do not exist even today in our knowled ge of motion.
This awarene ss is essential in
order to recognize that the science of motion progressed slowly but cannot be divided in a period that is scientific and one that is not scientific at any point in its histor y.
Let us not define as true science our level of scientific kn ow l
edge, or that of one or two centuries ago.
42
It is interesting to consider what must have been looked upon as the advanced theory at some remote times in the past.
The example I have chosen
for this combines
The author
kinematics
and dynamics.
of Mecha ni ca l
Problems (Aristotle ? ) [Dugas 1955] regards the law of the lever as a conse-
quence of the notable virtues of the circumference; the reasoning seems to be that something remarkable can be expected from something still more remarkable.
Conti nuin g the argument: In a lever with a fixed point,
the others
describe different arcs of circumference; the idea of opposites is then thrown in, and we have a philosophical piece handed to us to justify the theory of the lever.
We read that The properties of the balance are related to those of the
circle and the propertie s of the leve r to those o f the balance.
Ultimately mos t
of the motions in mechanics are related to the properties of the lever. [Dugas
1955, p. 19].
It appears that Aristotle tried to present motion in such a way that would circumvent the criticism of the Eleatic school which denied the possibility of the existe nce of motion and change. ing of the potential.
So, he cons ider ed moti on as the actuali z-
Change is a pro ces s going fro m potenti al existence to
actual existence (from one mode of existence to another ).
Aristotle co nsi d-
ered that there was one physics for sublunar regions, and another for the heavens.
The circular motion was cons ider ed to be the natura l motion for
celestial bodies.
Although Aristotle discusses in detail when a motion is faster than another by consi de ring the sp ac e traversed and the corr espon ding ti me,
43
he ne ve r
arri ved at what is so elementary for us: V = S/T. for instance, in the case S <
Tj
,
^
He consid ers several cases;
velocity
larger than
Vj
if
T?
. To divi de a dis tanc e by a time was not an accept able opera tio n, if it
was considered at all, while for us it is a common practice to refer to physical quantities of dimensions LT‘ *
or
LT"^ , or T'^ .
[Macagno 1971].
We
should be cautious about the limitations of a science that does not define velocity as we do it; such limitations may be less important than we may tend to believe.
Experi ence with computa tiona l fluid mechanics using finite diffe r
ences may show strikingly that one can calculate derivatives (e.g., velocities and accelerations) without the usual operation one learns in calculus courses.
When we examine the basic facets of mathematical kinematics, knowing what the ancient Greek scientist knew about geometry, it seems amazing that they did not create a similar science for the study of motion instead of embark ing in seemin gly- profo und dead-end alleys.
However, those interested in the
observation and study of the motion of the stars and planets surely accom pl ished, pr ac tica lly, po int s.
a st ud y of ki ne matics
of ob jec ts reduced to mat erial
In ef fect, perhaps the ear lies t sc ien tif ic st ud ies of ki nem at ics are th ose
perfor med by astronomers [ see, e.g ., Sch iap arel li 1874]. the
astronomical
studies
of
other
cultures,
We must no t for ge t
especially
those
of
the
Mesopotamia, where a numerical rather than a geometrical approach was taken [Neugebauer 1962].
44
Schiapparelli, Neugebauer, and others have published analyses of early astronomical studies, which are at the same time early examples of kinematics, early examples of Xj
=
Xj ( X , t). Neug ebauer has suggested that the con-
ics may have been discovered thanks to observations of the shadow of the sun dial. We must determine what else was done in ancient civilizations con cer ning studies of motion in other areas than astronomy.
The variati ons of level of
seas and rivers, although less regular than the motions of planets are an interesting possibility, as we know that there were important hydraulic works in ancient times, as vital, or perhaps more, than astronomic al knowle dge.
Be as
it may, I doubt very much that over periods of millennia we will find much more than tables, or perhaps, graphs, of positions of a particle as a function of time.
The notion of velocity remains very primitive when it is found, and
that of acceler ation; is surely in an even more primi tive state, but we shoul d not assume that they did not exist at all [see, e.g., Pedersen 1990].
The history of kinematics, as it can be found in astronomical observations of the remote past, should not be concerned with a number of otherwise very interesting aspects of astronomy, it should focus on the level of theoretical and empirical kinematics used in astronomy through the long history of that science.
From the strict point of view of kinematics, we may find more i nter-
esting the efforts to describe motion in a geocentric system than in an heliocentric syste m or in a galactic system.
Relative motion may become simp ler in
some cases, and the skills of the kinematicists less subject to a great challenge.
I would say that ellipses are less interesting than hippopedes .
45
Incidentally, at this point, it should be remembered that Aristarchus of Samos opted for what amounts to a Coperni can system.
To the rotation of the
Earth around its axis he added a revolution around the Sun, and he assumed the Sun and the stars to be fixed. plan et s rev olve arou nd the Sun.
He is sup po sed to have as sum ed that all the He was even ac cus ed of he resy fo r dist urbing
the center of universe with his ideas. Clagett [1955, p.
116] thou ght that
Copernicus should be called the Aristarchus of modern times.
In Greek ast ronom y we find a great use of circular paths, on a plane or on a sphere, or combinations thereof, and our interest resides on the kinematical probl em of describing pl an etar y pa th s inc lud ing retrogradation, but we must leave to others the study of the dynamical concepts behind such system.
Our
interest is in how advanced was the study of the motion used in a model, not whet her the model was poor or accurate dynamically speaking.
We would
like, for instance, to know if the notion of velocity was introduced or ignored, and how crude or ref ine d was the way in which it was defi ned.
Or if lack of
uniformity in a motion was studied or not, i.e., whether there was some protonotion of acceleration.
Eudox us and Ptolemy used dif feren t models,
and
rather than comparing them from the point of view of astronomical phenomena, we should consider how well the motion of the model was studied in each case.
Another inter esting probl em is how well did Aristotl e
Eudoxu s model in his comments about it.
understand
We know that other bodies, those
moving as projectiles on the Earth were also studied in Greece, and then there is an interest in comparing the level of kinematics in both fields, rather than conceptual error s and miscon cepti ons in one field or the other.
46
There was
some study of mechanis ms or devices to trace some curves .
In such a case, it
would be interesting to trace the understanding of kinematics if there was any; I mean whet her was there a point of view rep res ent ed by G(t)
?
or by
There
were
X
2
two
Xj
=
F(t) ,
X
2
=
= f( Xj ) ?
important
centers
of
astronomy
in
the
pre-Christian
Eurasian world, those two astrono mies proceeded in quite diffe rent ways.
We
owe to Neugebauer a good historical analysis of those astronomical sciences, and from the point of view of kinematics perhaps the salient features are the geometric
approach
Mesopotamia ns.
of
the
Greeks
and
the
numerical
approach
of
the
Kinematics is certainly much related to geometry, but it can
also be studied without any geometry, by means of numerical descriptions of functions as the Babylonians seem to have done it.
I believe that the idea that one may opt for a computational kinematics with little refer ence to geometry needs some lucubra tions.
Sup pos e that I have a
bad br ui se in my leg and it deve lop s into an el lip ti ca ll y shape d ul ce r of my skin, and that I am lucky and find a good dermatologist who treats it success fully.
Supp ose I measure regularly the two axes, Xj,X , of the ellipse and 2
make a table and feed it to a comput er.
Then I write a pr ogr am to find velocity
and acceleration for the point of the vector of compone nts
There
n0
material point moving in the physical space; but is it not this an example of computational kinematics ?. Who knows the f orc es acting on my ulcer? Surely not my der mat olog ist and all his coll eague s in the entire world .
But, maybe,
they can use my view of the kinematics of cicatrization to evaluate different
47
treatments.
Who know s
!.
May be a few average numbers
represe nting
velocity, or acceleration, of cicatrization is all they need, without ever drawing any geometrical diagram.
Was this the way the Babyloni an astr onomers
loo ked at their tabl es? Coul d they look at a table and see, with the same ease as we do when looking at a graph, that they had before their eyes a linear funct ion, a polyn omial of second or third degree? I tend to believe that all this is plausible.
Of the two important schemes of Eudoxus and Ptolemy, I will choose to call attention to that of Eudoxus because in passing I can refer to the excellent study due to Schiapparelli, a Milanese professor who took a very scholarly approach to this piece of history of science.
His was an approach which I
have considered as a model for some of my own studies, and that seems worth fol lowin g by others also.
Clagett [955] has praised this brilli ant rec ons tru c-
tion from scanty documentation.
The Babylonian approach can be viewed as a precursor of our curve fitting methods which now with the use of electronic computers can be applied with a high degree of refinement.
Eudoxus procedure is a precu rsor of the Fourier
analysis of complex motions.
In later Medieval times, Nicole Oresme used a representation in which a segment is taken perpendicular to a line to represent the intensity of a quantity at each point of a subject that affects the intensity. diagra ms in use today.
This soun ds much like
Thus longitudo (extension) is represented by a hori-
48
zontal line drawn in the direction of the subject.
The altitudo or latitudo
(height) of a segment at a given point is proportional to the intensio (intensity) of the prope rty at the point.
If the figure is a triangle it repr esen ts unifo rmly
difform quality ( unifformiter difformis).
A trap ezoi d is the same only that it
does not begin at zero, but at a certain value.
A recta ngle repre sent s a unifor m
quality , or we would say, of constant value.
Difformiter difformis is a qual-
ity which is repr esen ted by a curve. 62 different laws of variation.
Oresme made a classi ficat ion contai ning
This is superfluous within our methodology
governed by the use of advanced mathematics, but it may have been the right choice in Oresme's times.
Oresme extended the study of variation with time to what we would call now two and three variables.
When descr ibi ng velocit ies, Ores me noted that
they should be represented with a double extension, either in time or with respect to the subject.
He fou nd the meaning of the area of his diag rams.
For
instance in uniform notion, it becomes that the area gives the distance traveled. He called this integral, the total quality.
He cons ider ed a case requiri ng
the sum of a series: he made a diagram consisting of rectangles with base t/2, t/4, t/8 . . . . and hei ght i, 2i, 3 i ............. He stated that the distance traversed was four times the total quality of the fir st rectangle.
It seems that Oresme did
not reach the point of ex pre ssin g the law so famil iar to us: d = (l /2 )g t . According to Dugas [1955 p. 62] some scholastics discussed accelerated motion in theory, others the free fall motion , but they did not connect the two topics.
Was this a gap between theoreti cians and empiri cists ?
49
A
very
interesting
theoretical
medieval
kinematical
study
of
what
is
assumed as very simple motion is that of Oresme [Grant 1971] concerning two or more points ( sup pos ed to repr esent planets) in circula r motion.
Oresme
investigated the conjunction of two or more mobiles performing concentric circular motion s.
He was also concern ed with the repetit ion, or the predi ctabi l-
ity, of conj uncti ons, but he assume d that completely accurate posi tions can be verified to the point that one can in fact distinguish between coordinates that are given by ration al or irrati onal numbers .
Nonet heles s, this work of Oresme
deserves a place in the history of theoretical kinematics.
According to Dugas and Clagett, the developments in Oxford were more advanced, but with no geometric repre sentat ion. referred to certain
v e t e r e s ;
Oresme appears to have
and we can presume that they were the logicians of
the Oxford School ( who else? ).
Heyt esbu ry [1494] stated the rule for the
distance traversed in uniformly difform motion, that was also enunciated by Oresme.
He, howeve r, thoug ht that the effective velocity of a rotating body
was the maximum velocity.
What Heyt esbu ry studie d,
missed, was the notion of acceleration.
that in Paris was
Heyt esbu ry used an obsc ure language
but he su rel y int rod uce d toge ther with th at of ve loc it y the no tion of ac ce ler ation. The descri ption by Heyt esbu ry goes as follows, using modern termi nology:
If a body starts movi ng fro m rest, we can imagin e a case in which its
velocity increas es indefinitely.
In the same way we can imagine a more com-
pl ex ca se in wh ich the ve lo ci ty cha nge or ac ce ler at ion occ urs with in fi ni te variat ions, quick or slow.
The relation of acceleration to velocity is analogous
to that of velocity to distanc e traver sed. [Dugas 1955, p .67 ).
50
Clagett [1959]
says that about one century before, Gerard of Brussels had already considered the notions of Oxford.
He considers that Gerard wrote the first treatise on
kinematics.
In the study of mathematical kinematics, we must take into account that the history of analysis is surely going to be much easier than that of kinematic synthesis simply because it is a field that has been usually avoided by purely theoretical kinematicists.
The methodolog y for synthesi s, which includes nec
essaril y requi rement s of optimizati on in one way or another,
is much more
difficul t to put into practice and therefore to study histori cally.
As an example
we can consider the use of Chebyshev [1853 ] polynomials in the optimization of accuracy points in kinematics.
In some cases, of course, dynamics was uncoupled from dynamics because of lack of knowle dge.
Now, with more know led ge, does it make sense to
investigat e when kinematics can be uncou pled ? All fluid- mechani cists know that irrotational motion of an incompressible fluid can be studied in two steps, first the kinematics and then the dynamics. . This is a very interesting modern pr ob lem: wh en is it possible to deco uple ki nem at ics from dynami cs ?.
Fo r
instance, I am thinking of the ballerina who perhaps incidentally discovered that her speed of rotation could be varied by stretching out her arms or by br ingi ng the m clos e to he r bo dy . which the causes remained
Sh e lea rne d this as an empi rical law for
unkno wn.
The quest ion,
decouple kinematics from dynamics in this case?
51
for us,
is:
can we
The definition of kinematics seems to make everything simpler, but we must be concerned with the difficult question of the possibility of a meaningful deco upli ng of kinemati cs from dynamic s.
I am aware at least of one
notable exception: that of gas dynamics where even thermodynamics must interact with kinematics [True sdell 1954a].
For many who speak about chaos,
I have a ques tion :
can geometry and kinematics be decou pled from the dynam-
ics of turbu lence ?
Student s of fluid mechanics are sometimes puzzled by the
above mentioned analysis of frictionless fluid flow that is carried out without any discussion of the forces involved; forces which are then computed a posteriori.
They sometimes ask if there is not an invalidat ing mista ke in ignoring
the forces in the first step.
52
EXPERIMENTAL KINEMATICS
One may become so thrilled with the rational beauty of theoretical kinemat ics that it seems that the subject does not need observations and experiments. The temptation may be the same that lures many into considering geometry as unrelat ed to time and to phys ics .
I think there is an anal ogy be twee n k ine mat
ics and its relation with physics and geometry and its relation with kinematics. Geometry cannot be completely separated from experiments, for the simple reason that motion enters in one way or another, especially during periods of creation or
revision.
In
addition,
the
techniques
of
the
Mathematical
Laboratory have been, and still are, necessary to generate new themes in g-eometry. They, in fact, are more import ant in that role than in that of tea ch ing.
There is however, a difference, because most of kinematics has been,
and still is, developed by scientists and engineers who have understood the necessity of observation and experimentation in kinematics, not only as a tem po ra ry aid to be di scarde d eve nt ua lly , but as the ve ry essence of the subj ec t.
Historically, the great contributions to kinematics of ancient times are the observa tions of motion of astro nomers in Mesopota mia and Greece.
It is
inconceivable that such kinematics could have been purely mathematical, although the knowledge gained by using with great skill very crude instrumen tation was matched only by devising very clever mathematical models of motion and surp risi ngly accurate calculations.
As ast ronomy prog res sed,
more and more complex phenomena were studied, and the mathematics and
53
instrumentation required became of higher and higher sophistication, but observation of motion remained for a long time the central object of investiga tion.
At no moment in histor y can one say that astr on omy 's kinematics is a
pur ely the or eti ca l or mathe matica l science.
According to Clagett [1959], Strato the Physicist, a name given to him by Polybius, Cicero and Simplicius (fl. -278), wrote two treatises at least, which may be Mec han ics and On Au di bl es . We kno w of Strat o and his i deas be cause of the many attributions (which refer to about forty writings) by other authors. Only frag ments surv ive unfortuna tely.
Simplicius quotes from a treatise called
On motion [see repr oduct ion in Cohen & Drabkin 1948].
After explaining that
it is universally accepted that bodies in natural motion experience acceleration, he says that Strato asserts in his treatise On Motion that a body in natural motion completes the last part of its trajectory in the shor test time.
Then he
quotes Strato:
In the case o f bo dies moving thr ou gh the air this is wh at ha pp ens . observes water pouring down from
Fo r if one
a roof and falli ng fr om a considerable
height, the f l o w at the top is seen to be con ti nu ou s, but the water at the bottom falls to the ground in discontinuous part s.
Th is woul d never ha pp en unles s
the water traversed each successive space more swiftly . .
Stra to also used an argu ment which is cleare r than the one above.
He
called the attention of the reader to the mark left by the same stone when falling from a small height as compared to the mark when falling from a great
54
height.
He em phasize d that the stone was the same, the weigh t and size the
same, and that it is not impe lled by any dif fer en t force in each case. merely a case of acc eleration . dynam ic ana lysis.
It is
Clage tt sees in this a kinem atic rathe r than a
But I wo uld say that, for that, Stra to had to try some ways
of asse ssing spac es and times. He infers a differen t velocity by the use o f the complicated phenomenon of forming a dent on the ground. far form H eytesb ury, Oresme and Leona rdo.
I see Strato very
In fact, even Aristotle appears
more of a kinem aticist, if we con sider some pass ages only.
But the fact
remains that Strato was in these two cases an experimental kinematicist.
The use of experiments in kinematics is as natural and essential as it is in any ph ysic al science.
The mere fact that both spac e and time canno t be
defined, measured, or investigated, without instrumentation entering the picture in one way or another, makes the investigation of motion an essentially phy sical sc ienc e; it is tru e that it can co mpe te with geom etry in m athem atical structure and accuracy, but in no way can it be reduced to a theoretical mathematical subject , rega rdless of the field we consid er.
At all times, kinematics
has received great con tributions from engineers and applied scien tists.
I do
not forget that this is a classifica tion of profe ssio ns of our times, but there were people who under different terminology were playing the role of engineers and applied scie ntists at all times. of many other activities in the past.
What I say of eng inee ring can be said With the ever increasing technolog ical
pro gre ss since the Renais sance, in st ru m ents to observ e motion from the stars to the very small particles, and from all kinds of phenomena either natural or
55
man made, or created in laboratories, experimental kinematics has not ceased to grow during the same period.
The historical development of flow visualization is surely going to be one of the most interesting chapters in any comprehensive history of kinematics. Instances of flow visualization must have happened from the very beginning of an awareness of motion with some intent to understand it and describe it, beca use the m ost common flu id s, air and w ate r, are in many cases carr ie rs of naturally inco rpo rated tracers.
It is enough to think of dust, v apor, and smoke
in the case of air, of silt, leaves, and other floating or suspended materials in the case of water. The first eng ineer-sc ientist who did a great amount of work on experimental kinematics was Leonardo da Vinci, who studied the motion of parti cle s, of ri gid and defo rm able bodie s and of all kin ds o f fl uid s [M acagno M. 1987 to 1991].
The meth od has been extremely use ful to all stud ents o f
motions from very simple to extremely complex ones.
Leonardo has been credited with the invention of flow visualization tech niques in water; they may have very well been used before, but what is undoubtedly his merit is an impressive amount of observations of water flows, and he surely can be honored as the pioneer of the modern use of flow visual ization [Macagno 1989c, IIHR Mon ograp hs].
He made also notable studies of
the motion of granular materials induced in different ways ( gravity, vibra tions, centrifugation, etc. ), which in some cases were intended as models for the flow of liquids [Macagno 1982, 1991].
His stud ies of flow in flames
surely are remarkable although we do not know exactly what technique did he
56
use.
It seems that he disco vered a prim itive optical proc edu re by projecting
the shadow of the flames on a wall in some way.
The re has been grea t activity
in the field of visua lization of movem ents for more than a cen tury, since the invention of the pho tograp hic and the cinem atographic cam eras.
We have very
interesting aspects of kinematics from the study of Brownian motion to the flows in nebular stellar dynamics, from traffic flow to forest fires propaga tion, from laminar to turbulent flow, from sediment transport to blood flow. Every year, there are
sem inars, mee tings and co ng ress es entirely devoted to
visualization methods in differe nt fields [See, e.g ., Page ndarm et al. 1991 for a seminar with emphasis on computational kinematics. See Voller 1991 for modern video applications]
There are motions of such apparent complexity that, in the first phase of their study, the only po ssib le proc edu re is the experim ental.
I have already
mentioned the Brownian motion, for which some theory could be developed, but oth er m otions are st ill more dif fi cu lt to handle.
The stu die s of anim al
locomotion by Muybridge [1887] are already classical, but the experimental study of such motions continu es to this day [ e.g . A lexa nde r R.M. ed.
1977}.
Muybridge dem onstrated by taking photogra phs with a battery of cam eras, at a time in which the cinematograp hic camera was not available, that the pe rcep tion of a running horse by some artists was wrong, and that the horse is never airborne.
[Gom brich 1966, Raw lence - Cow ard 1990].
It is fasc inating that,
in order to have "living pictures" as Coward says, it was necessary before that there existed enough mechanical engineering kinematics to obtain both the
57
origin al and to realize its projec tion.
The "cinem atograph" needed a lot of
kinematics in addition to optics and chemistry to become efficient and useful.
Another two important lines of study for a history of kinematics are the instrumentation created or adapted for the study of motion and the models that are, and have been used, to inves tigate m otio ns,e xistin g or to created.
We
know that models of mechanisms and machines have been used for a long time (Vitruvius already challenged their value, and Leonardo discussed critically his asse rtion s five hund red years ago). Models were also used to dem onstrate different kinds of motion, like those which served to illustrate planetary motion.
Reuleaux [1876] created a large collection of models in Berlin [see
also articles by Kenn edy 1876 and Webb 1883].
In rather recent books on
engineering kinematics we find authors devoting a chapter or a section to physi cal kin em atic mode ls [see , e.g . H art enberg 1964 ].
Althou gh the fo ll o w -
ing is very recent history, nowadays we have the possibility of using computational models [See, e.g,. Voller 1991].
58
APPLIED KINEMATICS
Before speaking of applied kinematics, one must explain first what is the fundam ental kinematics that is being put into application.
I believe that we
should not identify fundam ental with theoretical mathem atical kinem atics, but include also the synthesis resulting from observations of and experimentation with motions of all kinds.
Fo r instance, if from flow visualiza tions of turb u
lent flow we infer that turbulence appears to consist of a whole spectrum of eddies of all kinds, we arrive at a non-theoretical knowledge which can be applied in the efficient generation of turbulence in a given flow, which is being designed to ac hi ev e a ce rtain ef fi ci ency of mixin g of two or more su b stance s.
Or, if we have obser ved v ery small length and time scale motions in
the midst of a liquid, we may apply this knowledge in several areas of physi co-ch em is tr y. Or we can use such know le dge fo r fo rm ula tion of th eo ri es.
In this chapter, as in preceding ones, I will not attempt a systematic survey nor a brief synopsis of the long history of applications of kinematics to many disciplines and activities.
I am trying instead to offe r a set of repr esen tative
examples beyond some already mentioned, like astronomy and mechanical enginee ring, which are classical fields of application.
By refe rring to a certain
number of diverse applications, I hope to convey the notion that they are in fact very numerous, and that a historical study of applied kinematics should be mainly the work of a team of scientists and engineers with an interest and also the necessary prepa ration in history of science.
59
It wou ld be better if some
coordination can be established, but it is important that enough people develop an interest and the will to accomplish such a task, otherwise the work may be done by persons without enough knowledge of the subject.
The fields of applied kinematics are so numerous that one hesitates to attempt the making of a list which everybody will surely find incomplete. Ho w ever , I believe that I have to offe r one. I am includin g among the applied fields those of science because we have now a body of knowledge which we can cla ssif y as an exact scienc e very close to geom etry in ancient times.
And
although one could say that kinematics is part of physics, one can also take the approach
of
placing
geometry
and
kinematics
in
the
realm
of
physico-
mathematical sciences. At least for the purp oses of listing d ifferent fields, of application of kinematics, I will take such a position.
For the applications to the sciences, we need only to take into account that it is dif fic ul t to find a scienc e in whic h motion has a neg ligib le role.
Even the
oldest branch of mechanics, statics, in some of its methods uses motion, and there fore kinem atics finds app lications.
Fo r the app lications to engineerin g,
we must take into account that different branches of engineering depend heav ily on kinematics: mechanical, aeronautical, hydraulic, environmental, naval, bio -m echanic al engin eerin g are goo d ex am ple s.
In some bra nches, the co ncern
is mainly with rigid motion, while in others it is the motion of fluids that takes the fundamental role.
60
In some cases, kinematics has been recognized as essential from the very begin nin g , whi le in oth ers th er e ha ve been dela ys. country to anothe r.
Th e ev en ts var y fr om one
Fo r instanc e, Dou glas P. Adams [Hain 1967] was of the
opinion that in the early 1950's no kinematics beyond the very elementary was taught in the Un ited States to mechanical engine ers.
I mu st say that at that
time, advanced kinematics was already taught to hydraulic engineers, and surely to aeronautical engineers also, because of the simple reason that the kinematics of fluids is much more complex than that of rigid, or even, elastic bodie s.
Th e acc om plish m ents of Euler, Cayle y, C hebysc hev, and oth er s in the
kinematics of rigid bodies were rarely, if ever, mentioned to m echanical engi neers [ Hain 1967 ], while hydraulic engineers were not unaware of the rather sophisticated studies of kinematics of vorticity by Truesdell [1954a].
Adams describes the changes and progress that occurred during and after the Second World War in the approach to kinematics of mechanisms and machines in the Unites States, which appears as having had great German influence.
In the field of fluid flow , we see a mixed influ enc e of German and
British origins [the schools of Prandtl, von Karman and Taylor, Rouse 1957] together with a flourishing of an already existing native tradition in the United States.
I think there is an intere sting line of resea rch , and not only for en gi
neering: that of the history of the reluctance to use fundamental principles and theories in all fields of app lications of kinematics.
One of the reaso ns for lack
of progress in science and technology one can observe historically in some place s is the narr ow -m in dedness of scie nti sts or engin eers in th at peri od.
We
seem to live in an age in which that does not e xis t, but it may v ery well be an
61
illusion .
Any way, in our times fundam ental kinematics is certainly applied to
the following fields:
SCIENCES
ENGINEERING
ASTRONOMY
MATERIALS
STELLAR DYNAMICS
TRANSPORTATION
PHYSICS
MECHANISMS AND MACHINES
GEOPHYSICS
HYDRAULICS
GEOLOGY
HYDRONAUTICS
OCEANOGRAPHY
AERO and ASTRONAUTICS
METEOROLOGY
TRANSPORT PHENOMENA
CHEMISTRY
BIO-ENGINEERING
BIOLOGICAL MOTIONS
ROBOTICS
BIOLOGICAL FLOWS
TRAFFIC FLOW
LOCOMOTION OF ANIMALS
COMPUTATIONAL KINEMATICS
AR T
EXE RCI SE & SPORTS
COMPOSITIONAL FLOW
INDIVIDUAL EXERCISE
KINETIC ART
GAMES
ANIMATION IN MOVIES
EXERCISE MACHINES
MOBILES
HUMAN BODY KINEMATICS
62
There is an application that was more important in the past than in our times: the transm ission of motion over relatively great distance s.
The advent
of electricity changed this field radically, but the history seems to have never been w ri tten.
One ex am ple of su ch a tr ansm is sio n exis te d in the fa m ous Marly
pum pin g in st all ati o ns in F rance [D ecr oss e 199 1] .
Ther e wer e many als o in
mining engineering when there was a chance of using the water power of a nearby water stream [Hartenberg 1964].
The general idea of drawing curves by means of mechanisms is old and still used with ever more soph isticated devices.
Since mathematicians usually are
reluctant to include time as part of mathematical considerations, and do not seem to favor any mathematics done with instruments, I think that we must consider the devices to draw all kinds of curves as mechanisms of interest to engineers and architects in first place, and also to physical scientists and artists.
Of cou rse, curve s can be drawn also by optical means, no t only
mechanical devices, and I would say that the corresponding kinematics is just one section of general kinem atics.
Desig ning and co nstr uc ting a device or a
machine that performs a certain desired motion is of course a vast endeavor and robotics is part of it.
To mechanical engineers we seem to owe the concepts of analysis and synthesis in kinematics; the analysis is a general study of given motions, while the second is the methodology by which one arrives at certain practical results. I think that what they call syn thesis is very much like design .
If so, this two
notions are applicable to all disciplines in which the production of certain
63
mo tions is an objective , and this includes science, eng ineering and art.
In the
eighteenth century there are two figures that may be considered as embodi ments of analy sis and synthes is at that time: Euler and Watt. Euler used math ematics with great skill in the his studies of motion in solid and fluid mechan ics; I am not awa re that he did any wo rk on syn the sis as defined above.
In
contrast, Watt excelled in kinematical synthesis, which is one way of applying the prin ciples of kinem atics. Among the intere sting c on tributio ns o f Watt, we have his mechanism for producing (with great approximation) straight-line motion; he considered this as one of his great engineering accomplishments. A warning seems warranted, however: We must guard ourselves from believ ing that necessarily, the best inventors should be either those who are well armed in the basic science or those who are not.
Many kinematical notions have been transferred to other fields via analogy, even to serv e in areas in which motion is not the object of study .
Thus we
find that the notions of pathlines are introduced in the study of stability of non -linear diffe ren tial equ ations [see, e.g ., Ross 1964].
The point for which
path lines are th ought to exis t is a poin t in the phase plan e; it is no t a particle or a ma terial poin t.
The "flow" patte rns that we can ob serv e in such an app li
cation are very similar to those that one sees in books of hy drody nam ics.
I
consider this a very interesting development because in the heart of mathemat ics, we find that "motion" plays a role, if not for logico-deductive proofs, for an elegant interpretation of behavior of solutions of differential equations.
64
Kinematics has found and it is finding application in chemistry; perhaps only elementary notions are being used, but the development is surely of interest.
Once a reaction starts it con tinues thro ugh some time and there fore one
can define a rate of production of a new compound, or a velocity of reaction. Reactions may reach equilibrium after some time.
If we wan t to prod uce a
certain substance, we must remove it so that the reaction continues; but this is another matter.
Our inte rest is here only in that the notion of veloc ity, and
eventually of acceleration , is as pre sen t in this field as in kinem atics o f pa rticles and bodies in mech anics.
There are othe r other proc esse s in which su b-
stances move at different speeds under different processes, like electrophoresis and chromatography, for which kinematics appears to have a role.
The motions of the human body have been studied from many different poin ts of view .
Rehab ilitati on p ro fessio n als need to kno w much ab out kin e-
matics of our body, and also seems to be needed by those who want to increase the prod uctiv ity of all kinds of worke rs [R abinbach 1991].
It seems
that many who begin working for the latter group end up in the hands of the form er group .
But there are many other aspects which are related to orga ns in
which motion at different scales is important, as in the case of contractions all along the dige stive system [Macagno 1980].
Other app lications in biology are
the studies of the motions of the heart and its valves, those of the eye, the flows in the circulatory, respiratory, and lymphatic systems.
With the advent of kinetic art, we began to see in Art Museums not only static pieces but many that were in motion or could be set into motion, or
65
could create the illusion of mo tion.
In many of them there was an application
of kinem atics involvin g some me chanism , [see , e.g. Rotative Demisphe re by Marcel Duchamp in Kozloff 1969] and thus we saw mechanisms could be used for obtaining not an utilitarian res ult but a esthetical effect. tains did that pre cise ly for millen nia. notion
of com pos itional
flow
In add ition to all this,
introduc ed recen tly
[Macagno
counterpoint to the old notion of compositional geometry.
66
Of course, founthere is the
1989 d] ,
as
a
CONCLUSION
When Ampère, in 1834, proposed the development of an independent science under the name of kinematics, there existed already many remarkable studies of motion in a literature pool extending over more than two millennia. The kinematics studies, however, were most of the time immersed in writings about ph iloso phy and ph ysics (in both the old and the new acce ption s). we have many more documen ts.
Now
The bibliograp hies by Haine and de Groot
alone, have nearly ten thousand entries, and we must take into account that they are practically restricted to one branch of engineering kinematics, in which almost exclusively rigid motion is considered; an immense variety of motions is thus overlook ed.
There are many thousan d more literature sou rces
in the other branches of engineering, in addition to those in several sciences. Moreover, kinematics is not limited to motions in science and engineering, it finds applications in many branches of art and technology, as illustrated in my list in the chapter on applied kinema tics.
Th erefore a com prehe nsive history
of kinematics appears as a monumental task.
For several years I have been working in surveying the entire field of kinem atics with a view to form ulate an outline for a histo ry o f the subjec t.
In
this contribu tion I have given a summ ary of my co nclu sion s, and of my views on how to proce ed.
I have also a ttempted to make a case fo r the writing of a
comprehensive history of kinematics but I did not mean to offer a preview of the work I have und ertake n.
Fo r the syste m atic work,
67
I believe that one
should start with the history of theoretical and experimental kinematics, and then proceed to work on the history of empirical kinematics on one side and on that of applied kinematics on the other .
I do not believe that one man alone can do more than draw the general lines of the history of kinematics, and I hope that my initial effort be followed by other histo rians of science.
I would like to see a numb er of engine ers and sci-
entists taking interest in this endeavor, because only they can tackle certain aspects of such a history .
As for my self, I will ende avo r to present in a few
years a synoptic systematic treatment of the four stems that I have decided to introduce and describe in this contribution.
68
BIBLIOGRAPHICAL REFERENCES
The following references are representative and do not constitute even an attempt at being comprehensive. They are divided in sections that have been convenient in my survey of documents considered useful for my work. From my files, I made this bibliographical selection that I considered useful for further work of my own, and possibly also for the work of others. The sections are under the following head ings: General, Ancient, Medieval-Renaissance, Modern, Machines and Mechanisms, Experimental Kinematics, Applied Kinematics, Review Articles and Bibliographies. These sections should be kept in mind when looking for any reference. GENERAL
This section is not directly related to the history of kinematics; it was useful when preparing myself for work in such a history. Analytic and transformation geometry are included because I found them useful to trace studies of motion of a protokinematic type. AGNESI, Maria Gaetana. 1748. Instituzioni analitiche. Milano. An analytical geometry, typical of her time, well known in the Continent, and published in English in 1801. ARCHIBALD, R.C. 1949. Outline of the history of mathem atics. 6th. ed. Math. Ass. of America. Too brief, but good bibliographic references (Boyer). BACHELARD, Suzanne 1970. The specificity of mathematical physics In Phenomenology and the Natural Sciences. Ed. Kockelmans J. J. and J.T._Kisiel, Northwestern UP, Evanston, 1970. Transl. from 1958-book. See discussion of Decartes's physics, and comparison with Newtonian physics. Bachelard says that most of Decartes' physics is based on geometry and not on algebraic thinking. Bachelard says that there is a tight coupling of mathematics and physics in our times; that one can no longer think physically without thinking mathematically. I believe such coupling to be much older. BOYER, C.B. 1956 History of Analytic Geometry. New York It contains a number of discussions of the role o f motion in geometry.
69
CHASLES, CHAS LES, Michel. Mich el. 1875 1875.. Aper
70
Important in connection with many mechanisms and machines which are examples of empirical kinematics. It contains also interesting comments on motion, as the ones regarding the inability of ancient Greece to develop a mathematical theory of motion. [See Dingier 1952]. 1952]. Important was also the reluctance (I suppose of an elite) to engage in practical applications of the theory ( See quotations from Plato). Plato). See also also quotations from Plutarch and Seneca. Seneca. Plutarch refers to Archimedes shying from writing on technical subjects { true ? }. }. In a last paragraph of this this chapter, Klemm quotes Guiraud [1900] who stated that he was not all that biased and that many others did manual work, and that there were workers in the assembly in Athens. See also quotations from a letter of Pascal explaining the great difficulties in finding skilled mechanics to construct his calculating calculating machine. machine. Klemm refers also to the difficulties found by Guerricke and his assistants. assistants. This is important as a hindrance in the the accomplishment of experiments. LANCZOS, LANCZOS , C. C. 1970. 1970. Space through the Ages. Ages. New York, Academic Academi c Press. A reference to Eudemus, a pupil of Aristotle, who wrote a History of Geometry now lost, is given on p. 10. 10. See also comments commen ts about motion in chapters on Einstein's Einst ein's theories. t heories. MALER ALER,, A. 1952. An An der Grenze von Scholasti Sch olastik k und Naturwis Natu rwisssens ssenschaft chaft.. Roma, 2nd 2 nd ed. MOREAU, J. 1965. 1965. L'espace et le temps selon Aristote. Aristo te. Editrice Editric e Antenore, Antenore , Padova. (No. 4 of Saggi e Testi.) NEUGEBAUER, O. 1948 1948 The Astronomical Astronom ical Origin of the Conic Sections. Proc. of the Am. Philos. Soc. S oc. v. XCII, p. p. 136-138. ( The shadows shad ows cast by sundials). NEUGEBAUER, O. 1962. 1962. The Exact Sciences Scienc es in Antiqu An tiquity. ity. Harper Har per Torchbooks Torch books,, New York. TIMOSHENKO, S. S. P. 1953. 1953. History History of Strength of Materials. McGraw-Hill, New York. See the account of the history of elasticity (Chapters Vili, Vi li, X, XIII.) Compare with the history of fluid mechanics [Tokaty 1971]. TOKATY, G. G. A. 1971 1971.. A History and Philosophy Philosoph y of Fluid Mechan M echanics. ics. G.T. G.T. Foulis, Henleyon-Thames, England. TRUDEAU, R J. 1987 1987.. The non-Euclidean Revolution. Birkhauser, Birkhauser, Basel. Basel. Although, to my taste, this book has an overenthusiastic preface, I enjoyed very much reading it, and I strongly strongly recommend it to anybody who wants to write something about non-Euclidian geometry in any any of Leonardo's notebooks or similar documents. For a "classic' on the subject, subject, see Klein 1928. TYLOR 1870, 1870, Early History of Mankind. London. In # 241, see figures for rotary devices to ignite wood. See also KLEMM, KulturWissenschaft, voi. II and HI. VAN der WAERDEN, Bartel L. L. 1901 1901 Science Awakening. Awaken ing. Oxford Oxfor d U.P.
71
WEYL, 1952 1952.. Symmetry. Sy mmetry. Princeton University Press, Princeton, New Jersey. Jersey. An excellent book which can be very useful (together with others mentioned in these refer ences) to anybody interested in the geometry of Leonardo da Vinci. WHITTAKER, E.T. 1944 1944.. A treatise on the analytical dynamics o f particles and rigid bodies. Dover, New York. (First edition in 1904). The chapter on kinematical preliminaries is highly recommended recommend ed for anybody interested in the science of kinematics, or its history. See, for instance, instance, the beautiful theorem o f Rodrigues Rodrigues and Hamilton, or the fundamental theorem that Chasles rediscovered in the years that kinematics was conceived as a new science, or the fascinating connection of rotations of a solid body in space and homographies in the plane (Cayley-Klein). WHEELER, J.A. 1962. 1962. Geometrod Geom etrodvnam vnamics. ics. Academic A cademic Press, New York
ANCIENT
ARISTOTLE -IV C. C. Problems of Mechanics. BOYER, C.B. C.B. 1956 History of Analytic Geomet Geo metry. ry. New York Boyer, p. 32-33. 32-33. Curves that were described kinematically: kinematically: spiral of Archimedes, quadratrix of Hippias, cissoid of Diodes (a cubic), conchoid of Nicomedes ( a quartic) (These two intro duced during earlier part of -2C). -2C). Much later Proclus (412-485) applied the kinematic kinematic approach to conics showing that any point of a segment with ends sliding on two perpendicular lines describes an ellipse. ellipse. Perseus ( uncertain period ) determined solid figures by rotating a circle around an axis in its plane and then intersecting with a plane (spiric curves, oval-shaped curves, and the the lemniscate of Bernoulli). Bernoulli). The cylindrical helix was known to Geminus, Geminus, Pappus and Proclus, and perhaps to Apollonius. Eudoxus apparently sought to represent the the motion of the planets with the hippopede. A little later the motion of the planets was repre sented by Apollonius and others as combinations of circular motions; this is tantamount to construction of epicyclic curves but the curves themselves escaped interest, even the cycloid may have escaped attention (although this may be the curve of double motion that according to Iamblichus was constructed by Karpis of Antioch to square the circle.) See also Medieval history of kinematics. COHEN, M. R. R. & J.E. DRABKIN 1948. 1948. A Source Book in Greek Science. McGraw-Hill McG raw-Hill Book Company, New York. CROMBIE, A. C. 1964. 1964. Von Aug ustinus ustinu s bis Galilei. Die E ma nzipati nzi pation on Naturwissenschaft. Kiepenheuer und Witsch, Kòln - Berlin.
der
DINGLER, DINGLER , H. H. 1952. 1952. Ueber die Geschichte Geschich te und das Wesens des Experimen Experi ments. ts. Miinchen. According to Dingier, there was in ancient Greece an incapacity to develop a theory of motion similar to the treatment of that of statics because of their notions of immutability and immobility of the Idea and the Form. They possessed a static concept of form.
72
MACHAMER, Peter, 1978, 1978, Aristotle Aristotl e on Natural Place and Natural Motion. Isis, voi. 69. 69. NEUGEBAUER, NEUGEBAUER, O. 1962 1962.. The Exact Sciences in Antiquity HarperTo Harp erTorchb rchbook ooks, s, New York. York. NEUGEBAUER, O. 1948 1948 The Astronomical Astron omical Origin Ori gin of the Conic Sections. Proc. of the Am. Philos. Soc. So c. v. XCII, p. 136-138. ( The shadows sha dows cast by sundials). ZEUTHEN, G. 1888 1888 Sur l'usage des coordonnées dans l'antiquité l'antiquité Kongelikes Danske Videskabemes Selskabs. Forhandlinger Forhandling er Oversigt p. 127-144 127-144.. Says that geometric algebra of antiquity closely resembles use of coordinates and that Fermat was an an almost immediate disciple discipl e of Apollonius Apollon ius and Papus. His views here conflict with those of Gunther in "Die Anfange..." (Boyer). (Boyer).
MEDIEVAL - RENAISSANCE
BRADWARDINE. BRADWARD INE. Ed. and transl. by CROSSBY, CROSSBY , H.L. 1955. 1955. Thomas Thom as of Bradwardine Bradw ardine.. His His Tractatus Tractatus de Proportionibus. Proportionibus. seu de proportionibus velocitatumin modbus mo dbus Its Significance Significance for the Development Development of Mathematical Physics. Physics. The University of Wisconsin Press, Madison. On p. 14 , Crossby explains the meaning of analogia or proportion. Translation Translatio n is given on on page facing Latin original. Crossby, on p. 48: " the clear clea r distinctio dist inction n which is drawn between the 'qualitative' and the 'quantitative' meanings o f the term veloc v elocitas. itas. together tog ether with a concentration upon the former, provided a most fruitful point of departure for the work of the following generation at Merton College". The complete title of this book is Thomae Bradwardini tractatus propordonum seu de propor tionibus velocitatum in modbus. This and other books studied by by the school of Wisconsin included in this bibliography are very useful in the study of the history of kinematics in late medieval times. CHASLES, Michel. 1837. 1837. Aper
Clagett completely ignores deformable bodies and fluid kinematics as part of the history of kinematics, as many others do. Regarding Leonardo da Vinci, Clagett expresses the opinion that he was was not familiar with the Merton Rule for uniform acceleration (p.105). CLAGETT, M. M. 1968. NICOLE ORESME and the Medieval Geometry Geom etry of Qualities and Motions. ('Tractatus de confìgurationibus qualitatum et motum ) The U. U. of Wisconsin Wisconsi n P., Madison. (QA 32 O 6813) DUHEM , Pierre 1906-13 Etudes sur Léonard de Vinci. Vinc i. Paris. 3 vols. See, according to Uccelli, voi. HI, p. 290 - on origins of kinematics. Uccelli refers to Ms 8680 BN, mentioned by Duhem, without saying much about it GRANT, E. 1960. 1960. Nicole Oresme and a nd his De proportionibus proportionum. Isis, voi. 51. GRANT, E. 1971. Physical Sciences S ciences in the Middle Ages. Ages. New York. York. GUNTHER, Siegmund. 1877 1877 Le origini ed i gradi gradi di sviluppo sviluppo del principio dell coordinate. coordinate. Bulletino di Bibliog. e di Storia delle Sc. Mat, e Fisiche. X, p. 363-406. HEYTESBURY, HEYTESBURY , W. 1494. 1494. Regule solvendi sophismata. sophism ata. Venice. See also The tribus predicamenti and HEYTESBURY, HEYTES BURY, W. W. By WILSON WILS ON , Curtis 1956. William Will iam Hevtesburv: Hevtesb urv: Medieval Medieva l Logic and and the Rise of Mathematical Physi Physics. cs. Madison. (Comments of Heytesbury Heytesbur y 1494). 1494). KARPINSKI, KARPINSK I, L.C. 1937 1937 Is there progress in mathematical mathem atical discovery and did the the Greeks have analytical Geometry ? Iris v. XXVII p. 46-52 (Rejects view that they had it, it, like Gunther and contrary to Coolidge and Heath, and Zeuthen). KRAZER, Adolf, Zur Geschichte der graphischen Darstellung von Funktionen. Festschrift published in Karlsruhe 1915 1915.. 31 pp. (Cogent summary of Oresme's work]. LINDBERG, David Davi d C. Ed. 1978. 1978. Science in the Middle Ages. Age s. The U. of Chicago Press, Press, Chicago. See John E. Murdoch and Edith D. Sylla, authors of The Science of Motion, Chapter 7. It is interesting to compare this study with those of Clagett for the same period. LEONARDO da VINCI. (See the the facsimiles of his manuscripts, of which there are are publica tions scattered along the last century. Some new versions are coming out presently.) LORIA, Gino, 1900 1900.. Le ricerche inedite di Evangelista Torricelli sopra la curva logaritmica. logaritmica. Bibliotheca Mathematica (3) 1,75-89 LORIA, Gino, 1902-1903) 1902-1903) Sketch of the origin and development of o f Geometry prior to 1850 1850 (translated (tran slated by Halsted). Monis M onistt v. XIII, p. 80-102, 218-234 , for anal. geom. see p. 94, 94, note. note.
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LORIA, Gino, 1923, Da Descartes et Fermat a Monge e Lagrange. Contributo alla storia della geometria analitica. Reale Acc. dei Lincei Atti. Mem, della classe dio se. fisiche, mat, e naturali (5) XIV, p. 777-845. LORIA, Gino, 1929-33 Storia delle matematiche. Torino, 3 vols. MAGAGNO, M. 1987. Geometry in Motion in the Manuscripts of Leonardo da Vinci. (Internal Report Biblioteca d’Arte & Istituto d'idraulica, Milano. In 1988, improved version as a Report of the Department o f Mathematics, University of Iowa). MACAGNO, M. 1990. Leonardo da Vinci and Transformation Geometry. Proc. Ill Congress History of Mathematics. La Crosse, Wisconsin, October 5-6, 1990. MACAGNO, M. 1991. Geometry in Motion in the Manuscripts of Leonardo da Vinci. Scheduled for publication in Raccolta Vinciana, Milano. MAIER, A. 1949. Die Vorlaufer Galileis in 14 Jahrhundert. Roma. MARCOLONGO, R. 1932 La meccanica di Leonardo da Vinci, Atti della Reale Accademia di Scienze Fis. e Matem. Napoli, voi. XX, ser. E, No. 9. MARINONI, A. 1982. La matematica di Leonardo da Vinci Arcadia Phillips Editions. MURDOCH, John E. 1967 . The rise and development of the application of mathematics in 14C philosophy and theology. Actes 4ème Cong. Intern, de Philos. Médiévale, U. de Montréal, Montréal, Canada . Pubi. Arts Libéraux et Philosophic au Moyen Age, 1969. Sect. I. The continuous and the infinite. Discusses the ideas of atomists (which were not like those of Democritus, Epicurus, Lucretius).The controversy was about mathematical atoms, not physical atoms. Sect. II. The quantification of motion. Shows that Brawardine "dynamics" is based on arith metic variation for velocities and geometric variation for the ratios of forces. See also discussion of velocity to be attributed to a body (rigid or deformable) with non-uni form velocity distribution ( II-B) Also studied the problem of a long rod falling into planet's diametral tunnel; cannot go through according to Brawardine's law. (EM: why did they not consider a needle and a magnet ? which is immediately experimentable ?). Also interesting the use of curvilinear angles. Velocitas and tarditas are used in medieval physics, always as scalars. ORESME, Nicole, 1966 De proportionibus proportionum & Ad pauca respicientes. Edited by E. Grant. The University of Wisconsin Press, Madison and London. See in this book the different passages and discussions on kinematics. The editor considers that Oresme has followed a traditional division between kinematics and dynamics (as described by Clagett 1959, p. 163) At the end contains an extensive Bibliography of sources and an Index of Latin mathematical terms and expressions. See also article in Isis by Grant 1960. ORESME, Nicole. 1968. Tractatus de configurationibus qualitatum et motuum CLAGETT Nicole Oresme and the Geometry of Qualities and Motions. The U. of Wisconsin P. , Madison 1968 75
There is much on kinematics in this book. Regarding Leonardo, Clagett expresses the opinion that he was not familiar with the Merton Rule for uniform acceleration (p.105). ORESME, Nicole. Tractatus de figuratione potentiarum et mensurarum difformitatum. Bibl. Nat. Latin collection, Ms 7371, transl. by Duhem. ORESME, Nicole. 1971. Tractatus de commensurabilitate vel incommensurabilitate motuum celi. Ed. by GRANT, Edward. Nicole Oresme and the Kinematics of Circular Motion. U. of Wisconsin Press, Madison 1971 Oresme studied the conditions for two or more bodies in circular uniform motion to conjunct in one or more specific points of one circle or in two or more concentric circles. To him the question of commensurability or inconmnensurability was crucial, but he left to Apollo to determine the answer. The god never came up with a pronouncement. This seems to reflect the uncertainty of the answer of a problem formulated in purely mathematical terms. The con tribution to kinematics as a physical science is really difficult to establish. PEDERSEN, et al. 1990. Le concept de vitesse en astronomie, en mathématiques et en phvisique dArchimède à Galilée.Collogue International, 8-10 Juin 1990, Nice, France. Had this been a colloquium concerning a concept in geometry, there would probably have been no need to put it in the context of other sciences. This is revealing of the general notion that kinematics was not pursued for its own sake until rather recent times. RANDALL, J.H. 1940. The development of scientific method in the school of Padua. J. of Ihg-History „Qf Ideas- V. I, pp. 177-206. Good analysis of intellectual currents of the time. RAVETS, J. 1961. The Representation of Physical Quantities in the Eighteenth Century Mathematical Physics. Isis, voi. 52. The approaches of Galileo, Euler and Atwood are discussed. According to the author. Galileo made possible the development of mathematical kinematics. RETI, L 1974a The Unknown Leonardo. MacGraw-Hill Pubi. Co., New York. RETI,L. 1974b Commentary to Codices Madrid, voi. DI, McGraw-Hill Pubi. Co., UK and Taurus Ediciones, Madrid, Espana. Reti describes what he considers the contributions of Leonardo to the systematic study of mechanisms. SHAPIRO, H. 1956 Motion. Time, and Place According to William Ockam. Franciscan Studies, voi. 16, 213-303, 319-372.
UCCELLI, Arturo 1940. Ed. Leonardo da Vinci, I Libri di Meccanica. Ulrico Hoepli, Milano.
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Contains interesting Critical Introduction and Study of the Sources. The presentation is some what similar to mine in the HHR Monographs. Many passages belong to kinematics. Uccelli does not want to refer to work by Leonardo as kinematics, which is really silly; perhaps because the term did not exist at the times of Leonardo (see p. XXVII and footnote, p. Cl: voler parlar di cinematica nel senso che noi ogi attribuiamo a questo ramo speciale della meccanica, non è forse il caso ). Then Clagett could not refer to Kinematics in Medieval times l! WALKER, Evelyn, 1932. A study of the Traité des indivisibles of Roberval. New York. WIELEITNER, Heinrich 1913. Per Tractatus de latitudinibus form arum' des Oresme" .Bibliotheca Mathematica (3 ), XIII, p. 115-145. Valuable commentary of the work of the most important precursor of analytic geometry (Boyer). THOMAS Alvarus, 1509 Liber de triplici motu. Paris.
MODERN ANGELOS, Jorge. 1988. Rational Kinematics. Springer-Verlag. New York. This is part of series edited by Truesdell, whose leadership the author follows in his treating kinematics as a geometric subject for which invariants are sought that exist under change of observers ( i.e., frames supplied with a clock). ARIS, R. 1962 Vectors. Tensors, and the Basic Equations of Fluid Mechanics. Prentice-Hall. Englewood, NJ. Excellent book. See Bibl. p. 97: 'the basic material on kinematics goes back to the 17th C. Full references can be found in the works of Truesdell..." [Truesdell 1954, I960]. ARONHOLD, S. 1872. Grundziige des kinematischen Geometrie. Verhandl. des Vereins zur Beford. des Gewerbefl. in PreuBen, 51. BALDIN, Alexandr. M. 1961. Kinematics of Nuclear Reactions. Oxford U.P. London. (Transl by R. F. Peierls). BATCHELOR, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge U.P., Cambridge. Chapter 2. Kinematics of the flow field, p. 71-130. See also Ch. 3, where elements o f kine matics associated with properties of the fluid can be found. Batchelor says, e.g. (p. 131): " As a preliminary piece of kinematics, we consider the changes in size and orientation of material volume, surface and line elements, due to the movement of the fluid." On p. 13, the "rates of change of material integrals are considered". An undefined property is considered; this looks to me as a study of a material geometry in motion; no dynamics is involved yet. I took this approach in my teaching. Look for isolated discussions which are essentially kinematical in the rest of this book.
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BECK, Theodor, 1900. Beitrage zur Geschichte des Maschinenbaues. Springer, Berlin. See also the papers in Z.VDI 1906 on Leonardo da Vinci. BENSON, Sidney, W. 1960. The Foundations of Chemical Kinetics. McGraw-Hill, New York. BERNOULLI, J. 1742. De centro spontaneo rotationis. Lausanne. A century before, Descartes said that when a curve C rolls over curve C, the normals to the paths to all points of C pass through the instantaneous points of contact. BIRD, R. B., W. E. STEWART, and E. N. LIGHTFOOT, 1960. Transport Phenomena. Wiley, New York. Like Levich 1962, this book must be studied carefully to extract the kinematics of the large variety of phenomena considered by the authors. Then it remains to trace the historical roots for each kinematics. Roughly speaking, this can be expected to have strong links with the kinematics of the continuum and of discrete systems of many particles. BLASCHKE, W. 1911. Euklidische Kinematik und nichteuklidische Geometrie. Zeits. Math. Phys. 60, 1911, pp. 61-91 and 203-204. BLASCHKE, W. and MULLER, H.R. 1956. Ebene Kinematik. Miinchen. BOTTEMA, 0.& B. ROTH 1979. Theoretical Kinematics. North-Holland. Amsterdam. Dover published a corrected edition in 1990. "Everything that moves has kinematical aspects". Some fields of application: animal locomo tion, art, biomechanics, geology, robots and manipulators, space mechanics, structural chem istry, surgery. Essentially, we are dealing with what mathematicians call transformation geometry. The authors state that when they say displacement, they imply no interest in how a motion actually proceeds: "we consider only the position before and after the motion". (Sounds like G. Martin !. But soon in the book f(time) is introduced, and so are velocities and accelerations ). BRICARD, Raoul 1926-1927. Legons de Cinèmatique. Vols. I-H. Paris. CAUCHY, A. L. 1829. Sur l'équilibre et le mouvement intérieur des corps considerés comme des masses continues. Exercises de mathématique. 4. (See also Oeuvres complètes. v. 9.). CHABRIER, 1820. Note relative à une mémoire sur les mouvements des animaux près à 1' Académie. Ms. École des P. et Ch., Paris. CISOTTI, Umberto, 1925. Lezioni di Meccanica Razionale. Libreria Editrice Politecnica, Milano. Second Edition. Three chapters are devoted to kinematics in this book. IL Cinematica del punto, dei moti rigidi. XIV. Deformarzione infinitesime dei sistemi continui.
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in. Cinematica
CORIOLIS, G. G. 1835. Mémoire sur les équations du mouvement rela tif des systèmes des corps. J. de l'Ecole Pol, 15. CRANDALL, Strephen H. and William D. MARK, 1963. Random Vibrations in Mechanical Systems. Academic Press, New York. See also "Random Vibration", a survey article in Applied Mechanics Reviews 12, 11. 1959 DECROSSE, Anne 1991. Toute l'eau du monde. Du May, Paris.(See p. 136). DUSTERHUIS, E. J. 1957. Archimedes. The Humanities Press, New York. EULER, Leonard 1748. Introductio in analvsi infinitorum. 2 vols. Lausanne 1748. (Also available in French and German.) FEDERHOFER, Karl 1932. Graphische Kinematik and Kinetostatik, p. 81-198, Ergebnisse der Mathematik und ihrer Grenzgebiete, voi. 1, Heft 2. FRANK, H. 1968. Ebene proiektive Kinematik. Diss. U. Karlsruhe. FRANK-KAMENETSKII, D. A. 1955. Diffusion and Heat Exchange in Chemical Kinetics. Princeton U. P.Princeton, New Jersey. See Chapter I for basic notions of chemical kinetics. FREDERICK, D. and Tien Sung CHANG 1969. Continuum Mechanics. Allyn and Bacon, Boston. See Ch. 3 , Analysis of deformation in a continuum. Well illustrated presentation of Lagrangian and Eulerian descriptions. Strain, rate of strain, Conservation of mass in well bal anced discussion. FRIEDLANDER, S.K. and L. TOPPER Ed. 1961. Turbulence. Classic papers on statistical theory. Interscience Publishers, New York. Sections on kinematics are to be found in some of the papers. Because diffusion is considered by some, there is application of the Lagrangian description of flow in this book. GARNER, René 1951. Géométries cinématiques cavlevniennes. Paris. GA RNER, René 1954-6. Cours de cinématique. Paris. Among other topics: extension of Euler-Savary formula to 3-D space. GIULIO, 1847. Elementi di cinematica applicata alle arti. Torino. HAGENDORN, R. 1963. Relativistic Kinematics. A Guide to the Kinematic Problems of High Energy Physics. W. Benjamin, New York - Amsterdam. HART, H. 1877. The kinematic paradox. Nature. 7,8, 16.
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HARTENBERG, R. & DENAVIT, J. 1964 Kinematic Synthesis of Mechanisms MacGrawHill Book Co. , New York. "The full story of the kinematics of mechanisms, doing justice to the many who practiced the art of mechanisms and contributed to the science of kinematics, is vet to be written." (p.22) A point is always reached at which one must move from the kinematics of a mechanism to its dynamics ( forces and torques involved), (p. 46-48). Cardan joint is called Hooke joint, although none of the two was actually the inventor. Hooke put it to use in 17 C [ p.50] The authors explain that we go from kinematic chain to mechanism by selecting a fixed link. There is no term for the input or driving link, which exists in German for a mechanism in which the driving link has been selected: Getriebe Kette means chain and Mechanismus is the equivalent of mechanism. There is no acceptable translation for Getriebe although drive and train have been used. Getriebe appears as mechanism when used as a noun; when used as adjective as in Getriebelehre (Lehre = theory or science of) is translated as kinematics or mech anisms. (p.54). See definition of jerk on p. 93 HARTMANN, W. 1890. Geometrie, Mechanik und Kinematik. VDI Z. , 34. HELMHOLTZ, Hermann von 1858. Ùber Integrale der hydrodynamischen Gleichungen, welche der Wirbelbewegung entsprechen. J. reine und angew. math.. 55, 25. (English transla tion in 1867, Philos. Mag. 33, 485. ). Helmholtz after introducing new important kinematical notions, proved very important theo rems on vortex flow. After Helmholtz, we can define for any fluid in motion a family of lines, the vortex lines, which accompany the fluid in its motion as a geometry that changes with time and elegantly defines one important aspect of the motion. JULIA, G. 1936. Cours de Cinématique. Gauthier-Villars, Paris. KOZZLOF, 1969. Max Jasper Johns. Abrams for Medirian Books, New York. See 34. Rotative demisphere by Marcel Duchamp, 1925.Copper disk with metal stand and electric motor. KRAUSE, Martin 1920. Analyse der ebenen Bewegung. Berlin und Leipzig. LEGENDRE, Robert 1965. Streamlines in a Continuous Flow. NASA Technical Translation F-405 (From La Recherche Aerospatiale. No. 105, 1965, p. 3-9). Characterization of the vicinity of singular points where the velocity is zero in 3-D flows. Useful for interpretation of complex flows. LEHMANN, H. 1967. Ziir Mobius-Kinematik. Diss. Univ. Freiburg, Br. LEVI-CIVITA, T. 1918. Corso di Meccanica Razionale. La Litotipo, Padova. LEVICH, Veniamin G. 1962. Physicochemical Hydrodynamics. Prentice-Hall. Englewood Cliffs, N.J. 80
The author studies a great number of processes in which motion is considered. The kinematics involved is not directiy apparent, but it seems possible to extract it after careful examination. This may very well be one of the difficult tasks ahead for the historian of kinematics in a num ber of fields. MACAGNO, Enzo 1971. Historico-critical Review of Dimensional Analysis, Journal of the Franklin Institute, voi. 276, no. 6 The historical difficulty in introducing kinematic and dynamic physical magnitudes derived from a few fundamental ones is discussed. MACAGNO, Enzo 1984/5. La rappresentazione del "flusso" prima e dopo Leonardo. Tracce. Immagini. Numeri. AST, Roma. In this article, the notion of "rheograms" is introduced to designate symbols used by painters along several millennia. Such rheograms have analogs in the elementary components of com plex flows developed only in the last few centuries by scientists. MACAGNO, Enzo. 1985. Hidrostàtica Vinciana en el Codice Hammer. Anales de la Universidad de Chile. Quinta serie, No. 8. MACAGNO, Enzo 1989a Unexplored Flow Studies in the Codex Arundel 263. IIHR Monograph No. 106. The University of Iowa, Iowa City, IA, USA. In Appendix I (43-49) a discussion of the conservation statements is included, which may be useful for the correct interpretation of what is kinematical and what is dynamical in such state ments. MACAGNO, Enzo 1989b Leonardian Fluid mechanics in the Manuscript M IIHR Monograph No. 109. The University of Iowa, Iowa City, IA, USA. MACAGNO, Enzo 1989c Leonardian Fluid mechanics in the Manuscript I IIHR Monograph No.111. The University of Iowa, Iowa City, IA, USA. (See References for papers and other monographs by E. Macagno.) MACAGNO, Enzo 1989d. Experimentation, Analogy and Paradox in Leonardo da Vinci. Raccolta Vinciana. fase. 23. Milano. MACAGNO, Enzo 1991. Lagrangian and Eulerian Descriptions in the Flow Studies of Leonardo da Vinci. Raccolta Vinciana, Milano. MACAGNO, Enzo & L. LAND WEBER, 1958. Irrotational Motion of the Liquid Surrounding a Vibrating Ellipsoid of Revolution. J. o f Ship Research, voi. 2. The paper is divided in three main sections: Geometry. Kinematics. Dy namics. This was, per haps, facilitated by the easiness in decoupling kinematics from dynamics in the problem con sidered. MAGGI, 1914-19. Geometria del movimento. Pisa, Spoerri. (See also Dinamica fisica 1921, Dinamica dei sistemi 1921). 81
MARCOLONGO, R. (1934) Il trattato di Leonardo da Vinci sulle transformazioni dei solidi. Atti della R. Accademia delle se. fisiche e matem. di Napoli. Voi. XX, serie 2a, n. 9. MAXWELL, J. C. 1867. On the dynamical theory of gases. Philos. Trans. Rovai Soc. London. A157. (See also Philos. Mag. 35, 129,1868, and Scient. Papers, voi. 2.) MEYER zur CAPELLEN, W. 1933. Einfache kinematische Probleme in schulmatemathischer Behandlung. Zeitschrift. fur den Mathem und Naturwissenschaftliche Unterricht. 64. MONTUCLA, J. F. 1758. Histoire des mathématiques. Chez. A. Jombert. Two volumes. There is a great deal of applied mathematics in this history. See also later edition in four vol umes. Although in need of some corrections nowadays, it is a valuable source of information. MULLER, Hans Robert 1962. Spharische Kinematik. Berlin. MULLER, Hans Robert 1970. Kinematische Geometrie. Jbr. D. Math. Ver. 72, 1970, pp. 143-164. MURDOCH, John E. 1967 . The rise and development of the application of mathematics in 14C philosophy and theology. Actes 4ème Cong. Intern, de Philos. Médiévale. U. de Montréal, Montréal, Canada . Pubi. Arts Libéraux et Philosophic au Moyen Age, 1969. Sect. I. The continuous and the infinite. Discusses the ideas of atomists (which were not like those of Democritus, Epicurus, Lucretius).The controversy was about mathematical atoms, not physical atoms. See also discussion of velocity to be attributed to a body ( rigid or deformable) with non-uni form velocity distribution ( II-B) O'MATHUNA, D. 1977. Highway traffic kinematics. Dept, of Transportation. Available through NTIS. ÓZGÒREN, Kemal. Optimization of Manipulator Motions. Preprints, Second CISM-ITFOMM Symposium On Theory and Practice of Robots and Manipulators. Warzaw, p. 27-36. PAGENDARM, H. G. et al. 1991. Computer graphics and flow visualization in computa tional fluid dynamics, von Karman Institute for Fluid Dynamics, Belgium.( A lecture series, September 1991.) PERIGAL, H. 1878. On a Kinematic Paradox - the Rotamer. Proc. London Math. Soc. 10. PIMENOV, Revol'st I. 1970. Kinematic Spaces, Seminars in Mathematics. V.A. Steklov Mathematical Institute. Leningrad. Consultants Bureau
Jackson Lear, under the title Man the Machine. We learn in this review that Hermann von Helmholtz who made outstanding contributions to the kinematics of deformable bodies, also contributed to the "science of work". RESAL, H. 1862. Traité de cinématique pure. Mallet-Bachellier, Paris. See the Preface for a brief history of the adoption of the point of view of Ampère in Paris and its consequences in both research and teaching. Resal attributes to Transon the notion o f the rate of change of acceleration, introduced in an article published in 1845. See Transon 1845 in Part II of this Monograph. REULEAUX, F. 1875. Theo retische Kine matik: Grundziige eines Theorie des Maschinenwessens. Friedrich Vieweg und Sohn, Brunswick. REULEAUX, F. 1876. The Kinematics of Machinery Macmillan. London. (There is a new edition by Dover, published in 1963 ) REULEAUX, F. 1900. Lehrbuch der Kinematik. Friedrich Vieweg und Sohn, Brunswick. REYNOLDS, Osborne., 1894. On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion. Philosph. Transactions A, 186. See also Papers, voi. II. ROBERVAL, G. 1730. Geometria motus. Mém. Acad. Sc.. Paris. ROSS, Shepley L. 1964. Differential Equations. Blaisdell, New York. See in Chapter 13 the definition of phase plane, and the many "flow" patterns around critical points introduced in the interpretation of the many cases of behavior of solutions of differential equations. Do not be mislead by the circumstances that some equations happen to be from mechanics; this is only incidental. ROUSE, H. and S.INCE, 1957. History of Hy drau lics. Iowa Institute of Hydraulic Research, The University of Iowa. Iowa City. SARD, Robert D. 1970. Relativistic Mechanics. W.A. Benjamin, New York. SCHOENFLIES, Arthur 1886. Geometrie der Bewegung in svnthetischer Darstellung. B.G. Teubner, Leipzig. SHAMES, Irving, H. 1960. Engineering Mechanics. Dynamics. Prentice-Hall, Englewood Cliffs, NJ. This author of a well written textbook, emphasizes the importance of learning kinematics before engaging in the study of dynamics, but he defines kinematics as the study o f the motion of particles and rigid bodies. This attitude, of overlooking all other kinds of motions, seems to be typical of mechanical engineers; of course, it is not found among aeronautical and hydraulic engineers.
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SHUTER, W .L .H. 1983 Kinematics, dynamics and structure of the Milky Way. Proc. Workshop on the Milkv Wav. Vancouver, Canada, May 17-19, 1982. Derdrecht-Holland, Boston. SINGER, Lothar 1966. Russian-English-French-German Dictionary. Set. Info. Consultants., London. SYLVESTER, J. J. 1875. History of the plagiograph. Nature. 12. THOMAS, Tracy Y. 1931. The Elementary Theory of Tensors, with Applications to Geometry and Mechanics. McGraw-Hill, New York. See Ch. IV Kinematics. TRUESDELL, C.A. 1954a The kinematics of vorticitv. Indiana U.P., Bloomington. The "peculiar and characteristic glory of three-dimensional kinematics" is the subject of this treatise. On p. 29: Continuum is a region in a Euclidean 3-D space, subject only to the proviso that the region be possessed of a positive volume. By a motion of a continuum we shall mean a one parameter family of mappings of the continuum onto other continua. The real parameter t we identify with the time - oo < t < + <», t = 0 is an arbitrary initial time. At t = O, let X be the coordinates of a typical point (or particle X ). Let the motion be the family of mappings: xj= xj ( X, t ), i = 1,2,3,
or
x = x (X, t )
TRUESDELL, C.A. 1954b Rational Fluid Mechanics. 1687-1763 Introduction to Euleri Opera Omnia (Lausanne MCMLIV), Orell Fiissli, Zurich. TRUESDELL, C.A. 1958. Recent Advances in Rational Mechanics. Science. 127, 3301. TRUESDELL, C.A. 1980. Studies in the History of Mathematics and Physical Sciences. TRUESDELL, C.A. and R. TOUPIN, 1960. The classical field theories, in Handbuch der Phvsik m/1. Ed. S Flugge. TRUESDELL, C.A. and BHARATHA, 1977. The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines Rigorously Con-structed upon the Foundation Laid by S. Carnot and F. Reech. Springer-Verlag, New York. Truesdell's opinion is that it is impossible to write the history of a science unless it possesses a structure that is clear, explicit and logical. (See Preface, p. vii). Because the theory of heat was in a state of confusion, he adds, we had bad historical essays. Of course, may I add?, bad essays can also result from not having a good knowledge of a sound theory. USHER, Abbott P. 1954. A History of Mechanical Inventions. Harvard University Press, Cambridge, MA. Revised edition. Published by Dover in 1988. This book appears as very valuable for the early history of empirical kinematics. The author points, for instance, that there is information one can obtain,concerning mechanisms and 84
devices long disappeared, by studying their products. Such is the case of the study of ancient textiles mentioned on pp. 50-51, which reveals much of the looms in which they were woven. It remains to determine, I believe, details of the kinematics of those looms. VELDKAMP, G.R. 1963. Curvature Theory in Plane Kinematics. Diss. Tech, Univ. of Delft. THOMSON, W. 1869. On Vortex Motion. Trans. Rovai Soc. Edinburgh. 25, 217. See also Collected Papers, 4, pp. 13-66. The very important kinematical concept of circulation was introduced by Lord Kelvin in this paper.
MACHINES - MECHANISMS
ADAMS, 1967. Moderne Kinematik im Textilmaschinenbau. Wirkerei- und SrickereiTechnik. 17, 1. ALLIEVI, L. 1895. Cinematica della biella piana. Francesco Giannini e Figli. Naples. BALL, Robert S. 1876. The Theory of Screws, a Study in the Dynamics of a Rigid Body. BALL, Robert S. 1900. A Treatise on The Theory of Screws. Cambridge (England). BEGGS, J.S. 1955. Mechanism. McGraw-Hill Book Co, New York. BEYER, Rudolf 1929. Reuleauxsche Kinematik. Maschinen Konstrukteur, 62, 19. BEYER, Rudolf 1963. Technische Raumkinematik. Springer Verlag OHG, Berlin. BEYER, Rudolf 1953. Kinematische Getriebesvnthese. Springer Verlag OHG, Berlin. English translation by H. Kuenzel, The kinematic Synthesis of Mechanisms. McGraw-Hill, New York, 1963. BRESSE, C. 1853. Mémoire sur un théorème nouveau concemant les mouvements plans. L de l'Ecole Polvtechnique. voi. 20. Bresse rediscovered the inflection circle (see de la Hire, L'Hospital). BRICARD, Raoul 1926-1927 Letpons de cinématique. Gauthier-Villars, vols. I- Cinématique théorique, II- Cinématique appliquée. CAYLEY, A. 1876. On Three-bar Motion. Proc. London Math. Soc. , voi. 7. CHEBYSHEV, P.L. 1853. Théorie des méchanismes connus sous les noms de parallélogrammes. See Oeuvres de P.L. Tchebvchef. 1899, voi. 1. Markoff et Sonin, St. Petersburg. Also in Modem Mathematical Classics; Analysis, ed. R. Bellman 1961, S730, Dover, New York.
85
CHEBYSHEV, P.L. 1878. Les plus simples systhèmes de tiges articulées. See Oeuvres de P.L. Tchebvchef. 1902, voi. 2. Markoff et Sonin, St. Petersburg. Reprint Chelsea Pubi. Co., New York , 1962. DUDLEY, D. W., 1969. The Evolution of the Gear Art. American Gear Manuf. Assoc. , Washington, DC. Data on gears during the period 3000 BC to 100 BC. FAUX, I.D. and PRATT, M J. 1978 Computational Geometry for Design and Manufacture. Ellis Horwood. HACHETTE, 1811. Traité complet de mécanique. Paris. He divided the parts of machines into six classes [Reuleaux, p. 10-11]. HARRISBERGER, L. et al. 1967. Three-dimensioned mechanisms, frontiers of kinematics. Mechanical Engin, Ngw$, 4, 4. HARTENBERG, R. S. 1969. Early Coupler-point Curves. J. of Mechanisms. 4. 2. HARTENBERG, R. S. and J. SCHMIDT, 1969. The Egyptian Drill and the Origin of the Crank. Technology and Culture. 10, 2. HINKLE, Rolland 1960. Kinematics of Machines. Prentice Hall. HIRE, de la, P. 1706 Traité des ro ul ettes.........Mém. math. phys. Ac. Roy. Sci. Paris Discussed inflection circle, apparently discovered by L'Hospital 1696. HRONES, J. A. and G.I. Nelson, 1951. Analysis of the Four Bar Linkage. MIT Press, Cambridge, MA. HUNT,K.H. 1978 Kinematic Geometry of Mechanisms. Oxford. KENNEDY, Alexander B.W. 1886. The Mechanics of Machinery. Macmillan, London. KENNEDY, Alexander B.W 1876. Kinematics of Machinery. Macmillan, London. KRAUSE, Martin 1920. Analyse der ebenen Bewegung. Berlin und Leipzig. LANZ, & BETANCOURT, 1808. Essai sur la compositions des machines. Paris These authors filled up the details of the master plan of Hachette (1806) based on Monge and Carnot's conception about the separation of mechanisms and machines. Monge had entitled the subject "Elements of Machines" (Remember Elementi mach inali of Leonardo), which he intended to be equivalent to the means to change the direction of motion. Monge considered the possible combinations of rectilinear alternative and continuous and circular continuous and alternative motions. In a 2nd ed. 1829 the classes of motions ( and I presume of mechanisms) were increased from 10 to 21 by adding other curvilinear motions.
86
LIGUINE. V. 1883. Liste des travaux sur les systèmes articulés. Bull, des Sc. Math, et Astron. 2e. s., 7. LEUPOLD, 1724-1739. Theatrum Machinarum. Leipzig. LEVITSKII, N.I. Analysis and Synthesis of Mechanisms. Amerind. LUOS PITAL, G.F.A. 1696. Analyse des infinitements petits . . . .Paris. Inflection circle (See de la Hire). MEYER zur CAPELLEN, 1969. Kinematik und Dynamik der Getriebe. Terminologisch Worterbuch: Deutsch, English, Franzòsisch, Russisch, Bulgarisch. Industrie-Anzeiger, 91. ROBERTS, Samuel. 1875 Three-bar motion in plane space. Proc. London Math. Soc. ,7. Showed that a coupler-bar point describes a curve of the sixth order. Roberts's theorem states that three different coupler-bar mechanisms describe identical coupler curves. There is another Roberts, Richard, who also made notable contributions, but he is not included in Hain's or in de Groot's bibliographies [ see Hartenberg 1964]. ROSENAUER, N. and WILLIS, A.N. 1967. Kinematics of Mechanisms. Dover. RYAN, Daniel L. 1981. Computer Aided Kinematics for Machine Design. Marcel Dekker ?. SMITH,Robert H. 1889. Graphics. Longmans, Green & Co., London. See also A New Graphic Analysis of the Kinematics of Mechanisms. Trans. Royal Soc, Edinburgh, voi. 32, 1882-83. SUH, C.H. 1978. Kinematics and Machine Design. Wiley. TRAENKNER, G. 1953. The significance of kinematics in the design of automatic machines. Maschinenbautechnik, 2, WILLIS, R. 1841. Principle of Mechanism. Parker, London. (2nd edition by Longmans, Green &Co., London 1870).
EXPERIMENTAL KINEMATICS
ALEXANDER, R.M. and G. GOLDSPINK Eds. 1977. Mechanics and Energetics of Animal Locomotion. Chapman and Hall, London. BEGGS, J. S. 1960. Mirror-image Kinematics. J. of the Optical Soc. of America, 50, 4. CUNDY, H. , MARTIN & ROLLET 1954, 1961. Mathematical Models. Oxford U.P. New York.
87
de JONGE, A. E. R. 1953. Graphical Methods. Kinematics. Trans, of the ASME..75. EDGERTON, Harold E. 1970. Electronic flash, strob. McGraw-Hill, New York. EDGERTON, Harold E. and J. R. KILLIAN 1939. Flash ! Seeing the unseen bv ultra high speed photography. Hale, Cushman & Flint. Boston. FELDMAN, Edmund B. Varieties of Visual Experience. Art as Image and Idea. Prentice-Hall, Englewood Cliffs, NJ. FISCHER, O. 1909. Zur Kinematik der Gelenke vom Typus des Humero-Radialgelenkes. Abhandl. der Koenigl. Sachsischen Gessel. der Wissenschaften, Leipzig. GOLDSTEIN, S (ED.) 1938. Modem Developments in Fluid Dynamics. Oxford, Clarendon Press. Two volumes. Together with the Handbuch der Experimentalphvsik [see Schiller 1932] these two volumes constitute a valuable summary of the status of flow science after the important developments at the beginning of this century. There is a good amount of experimental kinematics in these books. GOMBRICH, Ernst, 1966. The Story of Art. Phaidon Press, London. GROH, W. 1955. Kinematic investigation of the human knee joint. Archiv fur Orthopadische und Unfall-Chirurgie, 47. HARRISBERGER, L. 1963. Motion Programming. Machine Design. 35, 1. HARTENBERG, R. S. 1958. The Modellsprache in der Getriebetechnik. VDR Bench, voi. 29. HARTENBERG, R. S. and T. P. GOODMAN, 1960. Kinematics. A German-English Glosary. Mechanical Engineering. , 82. HARTENBERG, R. & DENAVIT, J. 1964. Kinematic Synthesis of Mechanisms MacGrawHill Book Co. , New York. See Figs. 12-11, 12-13, 12-16 which illustrate several three-dimensional models. HAUG, Edward J. 1989. Computer-aided kinematics and dynamics of mechanical systems. Allyn & Bacon, Boston, MA. HILLHOUSE, JOHN W. 1989. Deep structure and past kinematics of accretered terranes. Intern. U. of Geodesy and Geophysics, Washington D. C. HYZER, W. G. 1959. Measuring motion with high-speed movies. Machine Design. 31, 18. KENNEDY, A. 1876. The Berlin kinematic models. Engineering. 22. KREIGHBAUM, Ellen and K. M. BARTELS, 1990. Biomechanics: a qualitative approach for studying human movement. Macmillan, New York. 88
KUHLENKAMP, A. 1949. Computing Mechanisms. Feinwerktechnik. 53. LICHTENFELD, W. 1956. Theory of mechanisms in teaching and research. Abhandl. der Deutschen Akad. der Wissenschaften. 9. Berlin. LIGHTHILLJ. 1975. Mathematical Biofluid-dvnamics. Society for Industrial and Applied Mathematics, Philadelphia. LONGSTREET , JAMES R.1953. Systematic Correlations of Motions. Conference on Mechanisms. Reprint from Machine Design. Dec. 1953.
Trans. 1st.
Studies of motions in weaving machines. MACAGNO, E. 1969. Flow Visualization of Liquids. IIHR Report. The University of Iowa. MACAGNO, E. 1984/5. La rappresentazione del "flusso" prima e dopo Leonardo. Tracce. Immagini. Numeri. AST. Roma. Artists anteceded scientists by several millennia in representing elementary components of flow kinematics. The most abundant "rheograms" are those of waves and vortices. For a long time, water was really not depicted, or protrayed "realistically"; the region supposedly occupied by water was covered with one or several "rheograms". MARTIN, G. 1987 Transformation Geometry. Springer, New York. See section on the use of experiments involving motion in transformation geometry. MELCHIOR, P. 1955. Inertia and Lag as Kinematic Concepts. Konstruktion. 7. MEYER zur CAPELLEN, W. 1955. Feinwerktechnik. 59.
The area planimeter, a kinem atics study.
MEYER zur CAPELLEN, W. 1955. Die Totlagen des ebenen Ge lenkviereckcs in analytischer Darstellung. Forschungauf dem Gebiete des Ingenieurwesens. 22. Totlagen were already considered by Leonardo da Vinci in a truly remarkable page of the Codex Madrid I [Macagno M. 1991].
MUYBRIDGE, Eadweard, 1887. Animal Locomotion. The Muybridge work at the University of Pennsylvania . New York, Amo. NIKRASHEVL,PARVIZ E. 1988. Computer-aided analysis of mechanical systems. Prentice Hall, Englewood Cliffs, NJ Among other things, kinematics of data processing. PARKIN, Robert E. 1991. Applied robotic analysis.
89
PENROSE, J.C. 1874. On a method of drawing, by continued motion, a very close approxi mation of the parabola, proposed with a view to its application to figuring reflectors. Monthly Not, X Qthe RQy.aLAstrQnQmisal.SgiQ. 34. PRANDTL, L., K. OSWATITSCH and K. WIEGHARDT. 1969. Fiihrer durch die Sromungslehre. Vieweg 7 Sohn. Braunschweig. See chapter on Kinematics, and the numerous illustrations of experimental kinematics through out the book. Prandtl was a pioneer of flow visualization by modem techniques. In his col lected works and books there is much on theoretical and experimental fluid kinematics of the first half of this century. RAWLENCE, C. 1990. The missing reel. Collins, 306 pp. See comment by D. Coward in TLS, June 1-7 1990, p. 575. "The untold story of the lost inventor of moving pictures" . The retina retains the image for a split second ( 1/10 to 1/4 s ) before being stimulated by fresh light signals. In the 1820's, Faraday, Roget, and others noticed that when observed through an aperture of a rotating disk, motion could be accelerated positively or negatively, and even stopped. Some devices were invented to see figures in motion, but "living pictures' had to wait. Projected in Paris in 1881, Muybridge's sequences of horses in motion proved painters wrong because a galloping horse is never airborne. (See Gombrich's comments in his History of Art.) These pictures came from a series of cameras, rather than a single one. REID, R.R. and STROMBACK, D.R.E. 1949. Mechanical Computing Mechanisms. Product Engineering, 20. REULEAUX, F. 1876. The Kinematics of Machinery. Macmillan, London. (There is a new edition by Dover, published in 1963 ). Models used by Reuleaux are illustrated throughout his book. ROSHCHIN, G.I. 1955. Methods and instruments for the study of the walking of individu als in good health and with prosthesis. Akad. Nauk USSR Inst. Machinovedenia Trudv. 15, 59, SAYRE, M. F. 1934. Photograph method simplifies velocity and force calculations. Machine Design. 6. SCHILLER, Ludwig, ED. . 1932. Handbuch der Exp er im en tal ph vsik . Hydro- und Aerodynamik. 2 Teil, Widerstand und Auftrieb. Akademische Verlagsgesellschaft, Leipzig. Four volumes of this Handbook of Experimental Physics are devoted to fluid flow. [See also Goldstein 1938 ] 90
VOGEL, Steven 1981. Life in moving fluids. William Grant Press, Boston, Ma. The reader must find here and there the kinematics in this book. See, concerning experimental kinematics, Appendices I and II. VOLLER, Vaughan 1991. Editor of Video Journal of Engineering Research. This is a new development in which kinematics plays an important role together with other sci ences. According to the editor contributions to this new journal can take the form of computer animations, video recording of flow visualizations among other possibilities. WEBB, J. B. 1883. Reuleaux's kinematic models. Trans, o f the ASME. WILLIS, ROBERT 1841. Principles of Mechanism. 1st ed., Parker, London. (Second edi tion by Green and Co, London, 1870). WILLIS, ROBERT 1851. A System of Apparatus for the Use of Lecturers in Mechanical Philosophy. Especially in those Branches which are connected with Mechanism. John Weale, London. REVIEW ARTICLES AND BIBLIOGRAPHIES
The following list is revealing of the state of the art in history o f kinematics; almost all the following contributions are in the field of mechanical engineering. There seem to be no interest in the history, or the state, or the future of kinematics in other fields. ARTOBOLEVSKI, I.I. 1957. Work during twenty years of the Seminar on Theory of Machines and Mechanisms. Akad. Nauk USSR Inst. Machinovedenia Trudy. 17, 65. BEYER, R. 1931. Technische Kinematik. J. A. Barth Verlag, Leipzig. This is a good source for material prior to 1930 in the field of mechanisms. BOTTEMA, O. 1953. Recent work on kinematics. AMR. 6, pp. 169-170. BOTTEMA, O. & F. FREUDENSTEIN 1966. Kinematics and the Theory of Mechanisms. AMR, voi. 19, no. 4, p. 287 p.287. Whatever history we have in this article begins with A.M. Ampère's definition of kinematics: " the mathematical investigation o f the motions that take place in mechanisms and machines and the investigation of the means of creating these motions, namely, of the mecha nisms and machines themselves" (see de Jonge, ref. 70), and continues with developments in
the 19 C and early 20C. Extensive bibliography. Not much of a history of the subject.
91
p.289. Advice of Chevyshev to Sylvester: "Take to kinematics, it will repay you; it is more fecund than geometry; it adds a fourth dimension to space." DE GROOT, J. 1970. Bibliography on kinematics. I and II. Eindhoven University of Technology. Eindhoven, Holland. Contains about 7000 items. Initially intended for pure kinematics, it was expanded to include kinematics applied to mechanisms. What was understood by "pure kinematics" is not clear, but I could not find Euler, Lagrange, Truesdell (to mention just three who made impor tant contributions) included in this bibliography. With due respect to such great an effort, I think that the restricted area covered should have been indicated in the title. DE JONGE, A, E.R. 1942. What is wrong with "kinematics" and "mechanisms", Mechanical Engineering, 64. DE JONGE, A.E.R. 1943. A brief account of modem kinematics. Trans. ASME, voi. 65. DE JONGE, E.R. 1951. Are the Russians ahead in Mechanisms Analysis? Machine Design. 23, 9, p. 127, 200, 202, 204, 206, 208. FIAT Reports. HAIN, Kurt und W. MEYER zur CAPELLEN 1948. Kinematik in Naturforschung und Medizin Deutschland 1939-1946 fur Deutschland bestimmte Ausgabe der Fiat Review of German Science, Bd 7. Angewandte Mathematik , Teil V, S. 1-41. FREUDENSTEIN, F. und K. HAIN, 1958. Der Stand der Getriebe-Synthese im Schriftum des englischen Sprachegebietes. Konstruktion 10. p. 454 -458. FREUDENSTEIN, F. 1959. Trends in the Kinematics of Mechanisms. Appi. Mech. Revs. , voi 12, pp. 587-590. GOODMAN, T.P. 1958. Der Stand der Getriebe-Analyse im Schriftum des Englischen Sprachgebietes. Konstruktion 10. p. 451-454. HAIN, Kurt 1957. Mechanism Design in Germany. Fourth Conference on Mechanisms. Purdue University, p. 94-97. HAIN, Kurt 1959. Ungleichformige ubersetzende Getrieben. Schriftum Ubersicht der deutschen Arbeiten 1956-58. VDI Z. 101 (1959), S.255-259, 102 , (1960), S. 245-247. HAIN, Kurt 1960. Ungleichformige ubersetzende Getrieben. Schriftum Ubersicht der deutschen Arbeiten 1959 . VDI Z. 102 , (1960), S. 245-247. HAIN, Kurt 1967. Applied K inematics. McGraw-Hill, New York (English version of Angewandte Getriebelehre. Dusseldorf, 1961). Contains bibliography with about 2000 items. HARDING R. L. Kinematic Courier. Holden, Massachussetts.
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KANAYAMA, R. 1933. Bibliography on the Theory of Linkages , The Tohoku Math. Journal. 37 , 1933, pp. 294-319. LEVTTSKIJ, N.I. und L.S. GROOZENSKU. 1960. Die Synthese der Getriebe mit niederen Elementenpaaren. (Schriftumsubersicht). Maschinenbautechnik. 9, p. 432-440. MACAGNO, Enzo 1980. Fluid Mechanics of the Duodenum. Annual Reviews of Fluid Mech. voi. 12. Review paper of research in sixties and seventies. Aspects of the kinematics of the intestinal tube and of the chyle are covered in this review paper. The kinematics of the wall was treated stochastically while that of the chyle was mostly assumed to be deterministic. NERGE, G. 1959. Uber den Stand der Analyze und Synthese von Kurvengetriebe in den USA. Maschinenbautechnik. 8, p. 620-621. RAGULKIS, K.M. und W. RÒSSNER 1960.Die Behandlung der Kurvenscheibe in der sowjetischen Literatur. Maschinenbautechnik. 9, p. 2062-213. ROSENAUER, N. 1957. Eine kurze Ubersicht iiber die Russische Literatur in der Getriebetechnik. Konstruktion. 9, p. 359-361. RÒSSNER, W. 1957. Enw icklungsrichtun gen der deustschen Getriebetechnik. Maschinenbautechnik. 6. p. 457-462. ROTHBART, H. A. und HAIN, K. 1959. Die Kurvengetriebe und ihre Behandlung im Schriftum des englischen Sprachgebietes. Konstruktion. 11, p. 360-363. SHOENFLIESS A. & M. GRÙBLER 1902. Kinem atik. Enziklopàdie der Mathem Wissenschaften. , IV, 3 1902, pp. 190-277. Many references and much historical information. VAES, F. J. 1938. Kinematika aan de Technische Hogeschool te Delft. De Ingenieur. 53. VOLMER, J. 1957. Getriebetechnische Literatur. Maschinenbautechnik. 6, p. 571-575. WEIHE, Carl 1942. Franz Reuleaux und die Grundlagen seiner Kinematic. Abhandlungen un Berichte. Deutsche Museum, Miinchen.
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PART II
94
INCEPTION OF MODERN KINEMATICS by Enzo Macagno
INTRODUCTION
Kinematics, as an independent science, begins about two centuries ago in France with a gradual change in approach to both teaching and research of Mechanics.
Most of the docum ents in which one can trace this ince ption
emanate from the engineering schools of that time in the French capital, but there are also documents from other institutions like the Academy of Sciences, Collège de Fran ce, etc. It is true that in Par is itsel f, in En glan d, and in Italy there were previous imp ortant developm ents [Clagett 1959, M acagno M. 1987, 1991, and Appendix I in this monograph.] but they did not result in the establishment of a discipline with continuity and life of its own.
Kinematics, in its theoretical form, may be described as time dependent geometry, and like transformation geometry, the methods that are used are not only those of elementary geometry, but those of algebra, group theory , analytic geom etry, etc.
In addition, expe riments are not excluded for rather
complex mo tions, and this makes kinematics part of ph ysics .
From the point
of view of the phy sical dim ensio ns, geometry is con cerned with length , k inematics with length and time, and dyna mics with leng th, time and mas s. 95
One
could perhaps say that kinematics is halfway between geometry and dynamics both in th ei r develo pm ent and in th eir te ach in g. H ow ever, on e m ust be car ef ul not to take a sim plistic view.
Kinem atics is not ju st geom etry depe nde nt on a
para m ete r, or geom etry in m otion, as it may be su ggest ed by so me th eore tical studies of given displacements of given figures.
In this contribution, I will trace the development of the notion and the first steps in the development of an independent new science lying in between geom etry and ph ysics but clos er to the latter.
It is in fact akin to a geometry
in which the figures are parameter dependent, but with the added condition that the parameter is the time, and that we need to connect all the work with the requ isites of physics rather than those of mathematics.
Thus, although
superficially kinematics may appear very close to geometry, it is part of physi cs w it hout any doubt.
In ste ad, geo metry may be vie w ed as part of m ath-
ematics as long as mathematicians do not claim that it is very valuable in solving physical problems, because then it should be submitted to certain tests bas ed on experi m ents .
Usually, Ampère [1834] is credited with having launched (in 1834) modern kinematics, but in fact he should share this credit with several of his colleagues .
The trend tow ards the separa tion of kinem atics from geom etry and
dynamics, as an independent science is noticeable at least half a century befo re .
Ampè re him self , can be seen in his w rit in gs m ovin g in th at dir ec tion
years before the publication of his famous essay on the philosophy of science. I will summarize the results of examining the writings of Ampère and others.
96
[Bossut, Chasles, Bélanger, Coriolis, Carnot, D'Alembert, Deidier, du Buat, Fran coeu r, Girault, Hach ette, Hato n, Lagrange, La Hire,
Laplace,
Navier, Pare nt, Poin sot, P ois so n, P ro ny, Resa l, Sain t Ven an t].
97
Monge,
KINEMATICS BEFORE AMPÈRE
As stated in Part I of this mono graph , the study of motion is very old, and the firs t docum ents go back more than two millennia.
But wh atever kinematics
existed until late medieval times and until Leonardo da Vinci, it was always integrated into some other discipline.
Even the name, k in e m a tic s ,
app ears a little more then a cen tury and a ha lf ago.
only
In Pa rt I there is enough
information on previous developments by the schools of Oxford and Paris, and by Leonard o; th er e is th ere fo re no need to sa y mor e at this poin t.
In the seventeenth and eighteenth centuries, there were many contributions to mechanics which necessarily contained in some cases important aspects of theoretical kinematics, either because their authors had no alternative than ignoring the forces involved or saw great advantage in beginning by a study of motion in a purely geometric man ner, w ithout cons iderin g the caus es.
A very
good example of the latter case is found in some of E ule r's w ritings.
But
neither himself nor colleagues and followers thought of an independent discipline.
Euler wrote that
The investigation of the motion of a rigid body may be conveniently sepa rated into two parts, on e geom etrical, and the oth er mechan ical.
In the
f i r s t pa rt, the transfer en ce o f the body fro m a giv en po sit io n to any ot he r p o siti o n m u st be in ve st ig ate d w it h out re sp ec t to the ca us es o f m otion, and
98
m ust be represented by analytical form ula e, which w ill defin e the po sition o f each po int o f the bod y.
This investiga tion w ill therefore be referable
solely to geometry, or rather to stereotomy . It is clear that by the sepa ration o f this part o f the quest io n fro m
the
other, which belongs properly to mechanics, the determination of the motion fro m dynam ical princip les will be made much ea sier than i f the two parts were undertaken conjointly.
Although not so clearly expressed, D'Alembert shared similar ideas, as also did Kant according to Ha rtenbe rg [1964].
D 'Alem bert, in his D ynam iq ue ,
stated that all that we can see without doubts in the motion of a body is that it travers es a certain distance in a certain time.
He thou ght that one could d erive
from this simple fact all the principles of dynamics; in this he was less accurate than Ampère who put the pure study of motion on a less ambitious but firmer base.
Here is D' alembert statement:
Tout ce que nous voyo ns bien distinctem ent dans le mo uvm eent d 'un corps, c 'est q u 'il parcourt un certain espace, et q u 'il emploie un certain temps à le parcourir.
C est done de cette seule idée qu 'on doit tirer tous
les principes de la mécanique, quand on veut les démontrer d'une manière nette et précise ; ainsi on ne sera pas surp ris, qu 'en conséq uenc e de cette reflexio n, j'aie , po ur ainsi-dire, de tourné la vue des de ssus les causes motrices,
pour
n'envisager
uniquement
pro duis ent.
99
que
le
mouvement
qu'elles
Indeed, D'Alembert can hardly be called a modern kinematicist although there is in him an evident recognition of how powerful can be a deep under standing of motion.
D 'Ale m bert's principle for dynamics appears like a con
ception in which one uses kinematics to reduce dynamics to the classical statics' methodology.
In contrast, if we examine the famous M éc an iq ue A naly li que by La gran ge [1788], we find that the approach is the one that gives to the forces the pri m ordial role.
Sta tics, taking about one third of the volume , include s hydro-
and ae rosta tics.
The author states, in his A v v e rt is se m ent , at he beginning of
the book, that he deals in the second part with la Dynamique ou la Théorie du M o u vem en t , but there is no attempt made at separating the study of what we
call now kinem atics and dynam ics.
Of cou rse, L ag ran ge 's is not the only
bo ok to take th is appro ach whi ch se em s to be quite co mmon duting the eig h teenth centu ry, but the co ntra st with Euler is quite intere sting .
Incide ntally, it
is in this A vv erti ssem en t that Lagrange boasted that On ne trouvera point des Figures dans cet Ouvrage.
For a historical study, the M éc an ique A n aly ti q ue constitutes an example of the methodology discussed in Part I, when it becomes necessary to trace the kinem atics subme rged in a docum ent.
As soon as one begins the reading of
the first part of this book, one discovers that for Lagrange motion actually pla ys an im port ant ro le in Sta tics.
On pa ge 2 of his tr eati se, he st ate s tha t
there are three principles in statics: /. that of the equilibrium of the lever; 2. that o f the com position o f m otio ns: 3. that o f the virtual velocities.
100
I have
underlined two key words because they are extremely important to understand the position taken by Am père.
Statics may be about system s that appear to us
as static, but the theory of this discipline requires that we have a knowledge about motion.
Most of the teachers and writers of textbooks in the field of mechanics (either theoretical or practical) of the times of D'Alembert were people with experience in research. them.
Thus two genuine sources of innovation were within
They came to realize that for both teaching and researc h a separate
study of motion and its general indep end ent prop erties we re ess entia l.
Ampère
himself surely went through such a process, as reflected in his comments concerning kinematics in his
Ess ai .
His proposal was the resu lt of experience,
very well conceived and founded, and it was accepted by a number of contem pora ries and by poste ri ty .
Although at that time, mechanics was being taught and developed in several other places than Paris, the inception of modern kinematics appears to be esse ntially a Frenc h co ntribu tion, at least in the sense o f actually un dertak ing the writing of essa ys and books on the subject, and clearly definin g the pu r po se and go als of the new dis cip li ne.
Of cours e, much o f kin em atics alre ady
existed incorp orated in a large numbe r of mem oirs, pape rs, and boo ks.
In
Part I, I have already described, how ever succ inctly, the beginnings of mathematical kinematics in the astronomical works of Greece and Mesopotamia, and those of empirical kinematics much before since the dawn of civilization in
101
diff eren t lands. But all that kinem atics was not recognize d as a separate body of knowledge.
It is interesting that at the very moment when the formulation of physics in general and mechanics in particular was reaching a level that for macrophe nomena is still generally valid today, and when certain sections of those sci ences could have been very well absorbed into a more general scheme, as simple links of a well set methodological chain, mechanicists began to realize the need for kinematics to become an independent discipline emerging with undeniable force.
One aspect that is common to several authors of the period I am describing is that they grew uncomfortable with the introduction of forces ad-hoc in their studies of motion, and wished they could leave out what appeared to them as metaphy sical considera tions concerning my sterious causes of motion.
After
all, even today the teacher of mechanics does not have a way of rationally introd ucin g the grav itationa l force ! And pre ssu re in fluid m echanics is also introduced quickly and empirically (although one could make, at least for gases, a simplified appeal to the kinetic theory ).
At the close of the eighteenth and beginning of the nineteenth century, the École Polytechnique saw a band of teachers, who were very innovative, change the approach to the teaching of several subjects, among which theoreti cal and applied mechanics received, nor without controversy, an extraordinar ily pro gres sive reshaping and a revolutionary approach. Referring to the ele102
merits of machines, Monge proposed that the corresponding course dealt with the means by which the direction of any motion are changed.
He explained
this by saying that teaching should aim at showing how motion along a straight line, rotation around an axis, and back and force motion could be transform ed one into the other.
He prop osed that, since machines em body the
resu lt of com binations of certain m otions, a com plete enume ration of such motions be estab lished.
Hach ette prepared a chart in which he included i llus -
trations showing, for example, how circular continuous motion could be trans fo rm e d into re ctilin ea r alternating m oti on.
This cha rt was pre sen ted in 1806,
and it was follow ed by a book in 1811. By this time, Lanz and Bétan cou rt had already published Ess ai sur la com posit io n des mac hi ne s [1808].
Borgn is
(Italian engineer and professor at the University of Pavia) also proposed a classifica tion system that Co riolis simp lified.
These were w orks that are
important in the genesis of the notion of kinematics as an independent science, but are still eff orts to m ode rn iz e engin eering m ec ha nics ra th er than laun ch a new science. efforts,
Monge and also L. N.M .
especially
through
their
Ca rnot had a direc t influe nce in those
teaching
Polytechnique.
103
and
the
shaping
of
the
Ecole
AMPERE’S CONTRIBUTION
In the preface of his Essai sur la philoshopie des sciences, or Exposition analytique d'une classification naturelle de toutes les connaissances humaines, Ampère [1834] stated that already in 1829, when preparing his course to be delivered at the Collège de France, he was considering two important questions.
The first was a definition of general phy sics including how to dis tin-
guish it from other sciences.
The second was abou t the diffe ren t ( both exis-
tent and inexistent) branches of physics.
"En 1829, lorsque je préparais le cours de phy siq ue générale et expérimentale don t je su is chargé au Collège de France, il s 'o ffr ii d 'abo rd à moi deux questions à resoudre: lo .
Q u'est-c e que la ph ysiq ue générale, et par quel caractère précis est-elle
distinguée des autres sciences. 2o.
Quelles
son t
les
différen tes
branches
de la ph ys iqu e
circonscrite, q u'o n pe ut considerer, à volo nté,
générale
ainsi
comme autant des sciences
p a rti culi ères, ou co m me les div ers es pa rtie s de la sc ie nce plu s éten due dont il est ici question?
The book makes interes ting reading even today.
Ampère, ap parently knew
his Greek well enough to discuss his choice of new names for sciences still with out a name.
One of them was kinem atics.
He started with KlV£jJ,OC ,
meaning motion, and formed the adjective KlvejUOCTlKO^ , from which we
104
have now kinematics.
The reader will excuse me if I do not include the correct
accents in these two Greek words .
My point is to show a bit of Amp èr e' s
erudition, rather than my own.
A word is necessary to prepare the reader for the hierarchy adopted by Ampère, when he consi dered sciences of diff erent order.
I believe this will be
obvious if one of his tables is shown:
Sciences de lèr e ordre Sciences de 2ème ordre
Sciences de 3ème ordre
Cinématique Mécanique. éleméntaire Statique MECANIQUE Dynamique Mécanique transcendante Mécanique moléculaire
Ampère also mentions that long time before writing his Essai he had noticed that all books on mechanics omit, at their beginni ng, general con sid erations relative to motion. the third order.
Such consi derat ions sho uld constitute a science of
In parti al form somethi ng of this kind has been done by
authors like Carnot in his writings about motion cons ider ed geometrical ly, and in Lanz and Betancourt's Es sai sur la composition des ma ch ines .
105
Referring to kinematics, Ampère argued that kinematics should comprise all that there is to be said about the different kinds of motion, regardless of the forces that may prod uce them.
He added that it must cover all the co nsid era-
tions
in
about
emp loyed.
spaces
traversed
different
motions
and
about
the
times
It should deal with the calculation of the veloc ities attained
depe nding on the functions relating space and time in each case.
Then the dif-
feren t instru m ents , or mec hanism s, that can change one motion into another shou ld be studied.
The defin ition of machine should be changed to say that
they are instruments or devices by means of which one can change the direction and the velocity of a given motion, instead of continue to say that a machine serves to change the direction and intensity of a given force {For as collection of definition of machine, see Reuleaux 1876, Note 7].
Here are the
concepts of Ampère:
Cette science, doit renfermer tout ce qu'il y a dire des differentes sortes de mouvements, independamment des forces qui peuvent les produire. aux
Elle do it d ’abord s ’occuper de toutes les con sidérations relatives espaces
parcourus
dans
les
différens
mouvemens,
aux
temps
employés à les parcourir, à la determination des vitesses d'après les dive rses relations qui pe uv en t existe r entre ces espaces et ces temps.
Elle
doit ensu ite étudier les differen s instruments a la id e des quels om peut changer un mouvement en un autre; en sorte qu'en comprenant, comme c ’est Vusage , ces instrum ens sou s le nom de mac hines, il fau dra d éfinir une machine, non pas comm e on le fa it ordinairement, un instrume nt a
106
l'aide duquel on peu t changer la direction et l intensità d'u ne forc e donnee, mais bien un instrument a laide duquel on peut changer la direc tion et la vitesse d 'un mou vem ent donné.
He added that the consideration of forces will only serve as a distraction for whoev er tries to und erstand the mechanisms involved.
He gave the exam
ple o f the m ec ha ni sm in a watch whi ch dete rm in es the same ra tio o f velo citie s no ma tter how it is driv en , by the watch m otor or by hand .
He also said that a
treatise on the subject serait d'une extreme utilité dans linstruction', the reason be ing that the stu dent w ould over com e the dif fic ultie s in unders ta ndin g m ech anisms without the added hindrance of having to study also the
forces
involved.
Ampère insisted that kinematics must deal also with the ratios of velocities of different points of a machine, and more generally of any system of material poin ts .
This was re la te d to the dete rm in ation of the vir tu al velo citie s.
Ampère
be liev ed that the te achin g of the pri ncip le of vir tu al velo citie s, usually ha rd to grasp by the students, would become free of difficulties if they were already familiar with the kinema tical aspe cts.
Maybe he was refe rring to his ex pe ri
ences as a teacher, when he stated: ne leur présentera plus aucune difficulté.
In Ampère's plan, after kinematics, one would continue with the study of statics, motion.
where
he
says
that
one
considers
the
forces
independently
from
Statics must come only after kinem atics because motion is of imme-
107
diate perception and knowledge, tandis que nous ne voyons pas les forces que produisent le mouvem ens o b se rvés.
After studying motion without forces, and forces without motion, one should attack the problem in its totality; il reste à les considérer simultanément, à comparer les forces aux mouvements qu'elles produisent, et à déduire de cette comparaison les lois connues sous le nom de lois générales du m o u v e m e n t , ............ Thus, the purpose of dynamics is defined; it must estab lish general laws, and use them to predict motions given the forces, or serve to find the forces, given the motion
EMBRANCHEMENT S
SOU S -EMB RAN CHE ME>ITS
Sciences de lèr e ordre
Arithmologie Mathématiques propreme nt dite s Géometrie SCIENCES MATHEMATIQUES Mécanique Physico-mathématiques Uranologie
He
added
that
Uranology
should
Astronomy, Celestial mechanics.
108
comprise
Uranography,
Hel iostatics,
AFTER AMPERE
The idea of Ampère of creating an independent science dealing exclusively with the study of motion was well received by many; in fact it was already in the making by different people in 1834, when he published his Ess ai . Som e did not understand Ampère and his colleagues who never conceived kinematics as not supplemented by whatever other branches of science had to be applied, in the same way that the creators of geometry never thought that their science would alone suffice to deal with problems beyond geometry, like optical perspective, for example, or geo desics, or celestial mech anics. Astrono my with only geometry would never have gone beyond the Ptolemaic approach. Among the critics of the approach we have some French professors and also Franz Reuleaux in Germany .
Among those who m isund erstoo d his idea there are
also some in other cou ntries.
Reuleaux [1876] exp ressed criticism towards
"pure" kinematics, without realizing that by definition it was as "pure" as geometry. nihilism,
He thoug ht that true kinem aticists fell into Redtenbacher's and
cut
' Cinématique pure" from
"Cinématique
appliquée"
(Redtenbacher was Reuleaux's teacher at Karksruhe, where an engineering school had been establish ed takin g the École Poly tech niq ue as a model. ) . Reuleaux criticized Resal's Cinématique pure as an example of the sublimation of problems of kinematics into those of pure mechanics [Reuleaux 1876, p. 16]. In this passag e, o bvio usly, Reuleaux refers to his kind of kinem atics ( the restricted one of the mechanical engineer) and not the generalized one conceived by Ampère.
109
Concerning Ampère's proposal, Reuleaux [1875] considered the year 1830 as one that saw a great change, but I hope I will be able to show that such a change had been in the making for several decades before the Essai of Ampère. I believe that Reuleaux never fully understood the long term scope and also the limitations of A m père 's propo sal, and thought that Kinematics should only be concerned with mechanisms and machines and not be an absolutely isolated science.
In addition to kinem atics, dynam ics and other disc iplin es, are esse n-
tial for Reuleaux's purposes, which are much more restricted than those of Ampère who was not propounding what we call an engineering science but a new scientific disciplin e.
When Alex B.W. Kennedy [1876] was confron ted
with the choice of the title to be given to the translation of the famous book by Reuleaux, he opted for Kinematics of Machinery, and not for a literal translation of Theoretische Kinema tik, w hich would have been really misleading.
By
1876, it was already obvious to everybody that, in less than half a century, the idea of Ampère had prevailed without distortion.
After Ampère, kinematics was gradually accepted as a new discipline, and develop ed to ever higher levels. According to Resal, Po ncelet in his lectures at the Faculté des Sciences in Paris , from 1838 to 1840, put into practice A m pè re's plan. Among the imp ortant con tributio ns of an early period we must recognize those of Chasles and Poinsot both graduates of the École Poly techn ique who studied rigid geometric bodies in motion.
Of course , the
notion of instantaneous center had already been introduced by Johann Be rnou lli [1742]. But there were also impo rtant developm ents in the kinematics of elasticity and in that of fluid mechanics. 110
In the A vant-Pro pos of his Cinématique , Bélanger [1864] still finds necessary to make clear wha t is the role of kinem atics.
In a comm ent to
Laboulaye's Traité de Cinématique , he stated that one should consider kinematics as a science p o ssessin g its own th eo ries , in dependen tly fro m dyn am ic s and from the know ledge o f the physica l and experimental properties o f the moving body.
Bélang er, who taugh t at the École Poly tech niqu e from 1851 to
1860, compares algebra and geometry in an analogy with kinematics and dynamics, arguing that they can be taught separately but they do not need to be imparted at seque ntial times.
He mentions that this is prec isely w hat is done
successfully at the École Centrale.
The movement initiated in Paris soon spread over other countries, and soon papers and books ap pea red there. [W illis 1841 in England and Giu lio 1847 in Italy ], but this study is limited to the inception period.
Anyw ay, for some
time it was in France where most of the activity reside d.
[Ch asles , Po inso t,
Girault, Haton, Bélanger, Poncelet, Navier, etc. [See Reuleaux 1876]
In his preface to the English version of Reuleaux's book, Prof. Alexander Kennedy refers to Willis 1[841] as a treatment in which motion is considered merely for its own sake, witho ut referenc e to force or time. excluded, this should not be con sidere d as a book on kinematics.
If time is Kenn edy
says also that later w riters did not carry the analytic proce ss furth er. sume he refers to English books and articles only.
111
I pre -
CONCLUSION
Even after the preceding narrative, the idea of studying thoroughly kine matics without consideration of the causes of motion, before one studies dynamics, or even statics, may still appear to some readers as putting the cart before the ho rse.
It may se em that to stud y motion
p e r se
was unavoidable to
the astronomers of ancient times or to the engineers that were then developing different kinds of mechanisms and machines, but why such an approach should be continued after the equations of physics were finally brought to a reliable complete system, at least for macro-scale phenomena ?.
Howeve r, if
one considers problems beyond the elementary ones, the need for studying kinemati cs first and then dynamics becomes obvious in many cases.
It is true
that the historian of science, who is not at the same time a well trained physi cist or engineer, will have difficulty in discovering this point, but if one is familiar, for instance, with Eulerian and Stokesian fluid mechanics, the point emerges sponta neousl y with great clarity and force.
Eule r's equations for
hydromechanics ( which are equations for inviscid fluids) could be formulated with little general notion about motion and forces, but the forces due to vis cosity could not be easily expressed.
The complete constitutive equations for
viscous fluids are not easy to guess before the kinematics of rates of strain has been de veloped.
It is tru e that Newton ga ve a for mu la for an ext reme ly si mpl e
situation, but to do that also required a basic understanding of the kinematics of fluids .
The diffic ulty
becomes
monumental
for fluids
that are non-
Newtonian in th ei r rheological be ha vi or , and it took a gr eat effort to ac hi eve
112
general and intrinsically valid formulations for non-N ew tonian fluids.
In such
efforts, kinematics based on tensorial calculus was of paramount importance. There should be no doubt that Ampère was in the right track in propounding kinematics as an autonom ous science.
I am sure that those w ho know re lativ
ity better than I, would make a point similar to the above one to show how important was a deep knowledge of theoretical kinematics in the formulation of Eintein's theories.
The great historical role of the study of motion in all its aspects has been overlooked perhaps because the notion of kinematics as an independent science has been perceived and emphasized only in relatively recent times thanks to the simple and at the same time great conception of Ampère.
We can draw a pa ral
lel between the conc eptions o f Euclid and Ampère w ho, sepa rated by more than two millennia, tried to separate the mathematical and the physical aspects of two scienc es, a lthough one unde rtook to summ arize all the know ledge available while the other did not. value of studying geometry regard ing motion per se.
p e r
But what one perce ived a bout the great se ,
it was equally perceived by the other
In the same way that the histo ry of geom etry goes
back to tim es much befo re Eucl id, th at of kin em atics also go es ba ck to much earlie r times than thos e of Ampère.
We only need to think tha t the motion of
stars and planets was studied for millennia without regard to the forces involved, to realize how old kinematics really is.
In a future more detailed study, I plan to give more coverage to precursor movements of that of Ampère and his colleague s.
113
I think that one should
include with enough detail at least those of the kinematicists of Oxford and Paris and the kinematical studies of Leonardo da Vinci as foreshadows of modern kinem atics.
Most of the Medieval kinem aticists were truly unco n-
cerned abou t the causes of motion.
Leona rdo, at least in part of his stud ies,
clearly dealt with the study of the geometry of motion and not with dynamics, following in fact what is now Ampère's approach.
I feel, because of questions and comments at the occasion of lectures I have given on the subjec t of this mon ograph befo re historian s of science, that I must say a few words about the rationale of my study of the history of kinematics and abou t the role of kinematics itself.
Many con sider geom etry and
dynamics as subjects that are well and alive, but regarding kinematics there is a general notion that it is a closed chap ter in the developm ent o f ph ysics.
The
truth is that in several aspects it is still advancing and the object of intense researc h efforts in science, engin eering and art.
One strikin g exam ple is the
great flurry of activity in flow visualization only matched by computational kinem atics. The histor y of modern kinematics has not been written except for some partial aspects mainly related to mechanical engineering application s. will not be an easy task to do it in a comprehensive way.
114
It
ACKNOWLEDGEMENTS
This contribution (Part II) is mainly based on research carried out during two period s in Paris in the spring s of 1990 and 1991. The most helpfu l ins ti tution was the École Polytechnique in whose modern and modernly-run library I had the opportunity to collect a large amount of very useful information in a rather short time. I expre ss here my heartfe lt thanks to Mesdames Masson and Billoux and the helpful staff under them.
115
BIBLIOGRAPHY
This list contains more items than those mentioned in the text, because in this way some indication of the repercussions of the Paris movement can be surmised. AMPÈRE, André-Marie 1823. Mémoire sur quelques proprietés nouvelles des axes perma nents de rotation des corps et des plans directeurs de ces axes. Imp. royale. Brochure. Paris. This is certainly a contribution to the science of kinematics, a decade before the proposal in the Essai sur la philosophie des sciences.
AMPÈRE, André-Marie 1834 . Essai sur la philoshopie des sciences, or Exposition analvtique d'une classification naturelle de toutes les connaissances humaines. Bachelier, Paris. ARONHOLD, S. 1872. Grundziige der kinematische Geometrie. Verhandl. der Ver, z. Beford. des Gewerb. in PreuBen,, voi. 51. BÉLANGER, J.B. 1864. Traité de cinématique.pure. Dunod, Paris. In the Preface of this sixth edition of his treatise, Bélanger acknowledged the leadership of Ampère. There is a first part on general theoretical kinematics, while the second is devoted to kinematics applied to mechanisms and machines. There are also some notes of works preced ing his: Hachette and Lanz & Bétancourt in 1808, Poncelet ( 2nd. edition) 1845, Willis 1841, Laboulaye 1849. BOSSUT Charles 1810. Histoire générale des Mathématiques depuis leur origine iusqu'à l'année 1808. Tome Second. F. Louis, Paris. BOTTEMA, O. & F. FREUDENSTEIN 1966. Kinematics and the Theory of Mechanisms. AMR, voi. 19, no. 4, p. 287 Whatever history we have in this article, begins with A.M. Ampère's definition of kinematics and continues with developments, in the 19 C and early 20C, in mechanical engineering kine matics. Extensive bibliography. On p.289: Advice of Chevyshev to Sylvester: "Take to kine matics, it will repay you; it is more fecund than geometry; it adds a fourth dimension to space."
BOTTEMA, O AND B, ROTH, 1990. Theoretical Kinematics. Dover, New York. In their Preface, the authors state that kinematics has to do with everything that moves. In spite of that, and of listing many areas of application of kinematics, they seem blind to the existence of certain areas in science and technology in which deformation and flow occurs, from meteo rology to blood circulation, and from stellar dynamics to Brownian motion. 116
BRICARD, Raoul 1926-1927. Leyons de Cinèmatique. Vols. Ì-II. Paris. See historical notes. BURMESTER, L. 1874. Kinematische-geometrische Untersuchungen der Bewegung àhnlichverànderlicher ebener Systeme. Zeits. Math, und Pkvsik. 19, p. 145-169. (See also pp. 465-491). BURMESTER, L. 1888. Lehrbuch der Kinematik. BERNOULLI, J. 1742. De centro spontaneo rotationis. Lausanne. We must take into account that a century before, Descartes said that, when a curve C rolls over curve C, the normals to the paths to all points of C pass through the instantaneous points of contact. BOUR, E. 1865. Cours de mécanique et des machines professé a 1' École Polvtechnique. Gauthier-Villars.Paris. BRESSE, Jacques A. Ch. 1884. Cours de mécanique des machines professé a 1' École Polvtechnique. Gauthier-Villars, Paris. The first part of voi. I is on kinematics. CARNOT, Lazare 1783. Essai sur les machines. Dijon . Second edition in 1786. in Paris. According to C. C. Gillispie, this is an application of the science of motion to the study of the principles governing the operation of machines, and it was written as what we call now an engineering science. Carnot introduced the idea of "geometric motions", a kind of proto-vector analysis. CARNOT, Lazare 1803. Principes fondamentaux de Téguilibre et du mouvement. Deterville, Paris. First edition in 1783, under the title Essai sur les Machines en général. In the Preface of the second edition (1803), Carnot discusses his approach by saying that there are two possibilities: either to consider mechanics as the theory of the forces which produce the motion, or as the theory of the motions themselves. He adds that the first approach is followed almost by every body. We have, in this preface, a recognition of a new approach CAUCHY, A.L. 1827 Exercises de Mathématiques. II. Paris, (See also Oeuvres, (2), VII, p. 94.) CAYLEY, A. 1875-6. On three-bar motion. Proc. London Math. Soc. VII, pp. 136-166. CHASLES, Michel 1831. Note sur les propriétés générales du système de deux corpes semblables entre eux, placés d'une manière quelconque dans l'espace; et sur le déplacement fini, ou infinitement petit, d'un corps solide libre. Bull, des Sc. Math, de Férussac. XIV, pp. 321-336. Also Comp. Ren. Ac. Sc. 1843, XVI, p. 1420. 117
According to Bricard and Whittaker, Chasles rediscovered a theorem already established by Mozzi 1763. (See also Cauchy 1827), CHASLES, Michel. 1875. Aperyu historique sur l'origine et le developement des méthodes en géométrie. Paris. 2nd. ed. Biased in favor of Descartes, as sole inventor of analytic geometry (Boyer). But regarding kinematics there are very useful passages. He refers frequently to kinematics, mechanical construction of curves, etc. CLAGETT, Marshall 1959. The Science of Mechanics in the Middle Ages. The U. of Wisconsin Press, Madison. [See Part n, Medieval Kinematics ]. Clagett deals with the origins of kinematics, but he completely ignores the kinematics of deformable bodies as part of its history, as many others do. Regarding Leonardo da Vinci, Clagett expresses the opinion that he was not familiar with the Merton Rule for uniform accel eration (p.105), but he overlooks most of the kinematical work of Leonardo. D'ALEMBERT, 1743. Traité de dvnamique. David l'ainé, Paris. Second edition in 1796. DARBOUX, G. 1881. Sur le déplacement d'une figure invariable. Comp. Rend, de l'Ac. des Sc.XCII, p. 118-121. DEIDIER, 1' Abbé 1741. La mécanique générale contenant la statique. la airométrie. rhvdrostatique et l'hvdraulique. Jombert, Paris. DE JONGE, A.E.R. 1943. A brief account of modem kinematics. Trans. ASME, voi. 65, pp. 663-683. DELAUNAY, N 1899. Die Cebyschevschen Arbeiten in der Theorie der gelenkmechanismen. Z. Mathem. Phvsik. 44 p. 101-111. Du BUAT, L.J. 1824. Mémoires sur la mécanique. Firmin Didot, pére et fìls, Paris. DURRANDE, 1864. Essai sur le déplacement d'une figure de forme variable. GauthierVillars. Brochure. Paris. FERGUSON, E.S. 1962. Kinematics of Mechanisms from the Time of Watt. U.S. Natl. Museum Bull 228, Paper 27, pp. 185-230. (Superitend. of Doc., Washington D.C.) FINCK, P. J. E. 1864-65. Mécanique rationnelle. Gauthier-Villars. Tome I: Cinématique pure. FOURIER, J. B. Legons d'analvse et de mécanique professés à 1' École Polvtechnique. Ms. de Tauteur. FRANCOEUR, L.B. 1807. Traité élémentaire de mécanique adopté dans l'instruction publique. Bernard, Paris, 4ème edition. 118
FREUDENSTEIN, F. 1959. Trends in the Kinematics of Mechanisms. Appi. Mech. Revs. , voi 12, pp. 587-590. GILLISPIE, C.C. 1971. Lazare Carnot. Savant. Princeton University Press, Princeton. Contains facsimiles of unpublished writings on mechanics and calculus, and interesting com ments and discussions. HACHETTE, 1808. Programme d'un cours élémentaire des machines. Ecole Polytechnique, Paris. HACHETTE, 1811. Traité complet de mécanique. Paris. HACHETTE, 1836. Correspondance sur lEcole Rovale Polvtechnique à l'usage des élèves de cette école. Paris. HABICH, E. 1879. Études cinématiques. Gauthier-Villars. Paris HALPHEN, M. 1882. Sur la théorie du déplacement. Nouv. Annales de Math. 3 ,1, pp. 296299. Contains Halphen's theorem on the composition of two general displacements (See also proof by Burnside, Mess, of Math. XIX. p.104, 1889). HARTENBERG, R. & DENAVIT, J. 1964. Kinematic Synthesis of Mechanisms MacGrawHill Book C o., New York. Chapter 1: An Outline of Kinematics to 1900 This chapter is excellent but the authors vision is very narrow; they see the history of kinematics as that of the growth of machines and mecha nisms and the related mathematical developments. On p. 22: "The full story of the kinematics of mechanisms, doing justice to the many who practice the art of mechanisms and contributed to the science of kinematics, is yet to be written". See also Ch. 3 on kinematic models, which includes some historical data. See also Reuleaux. Great historical figures listed in Ch. 1 KOENIGS, Gabriel 1897. Leyons de cinématique. Avec des notes par G. et E. Cosserat. Paris. LAGRANGE (de), 1811-15. Méchanique analvtique. Nouv. édition. Veuve Courrier, Paris. In his Avvertissement for this edition, Lagrange stated: On a déja plusieurs Traités de Mechanique, mais le plan de celui-ci est entièrement neuf. He could not foresee that not many years afterward there would be still more innovative approaches under the leadership of Ampère. From another statement in this Avvertissement is evident that his innovation was to convert mechanics into algebra rather than geometry: On ne trouvera point des Figures dans cet Ouvrage. Les méthodes que j'y expose demandent ni constructions, ni raisonnements géometriques ou méchaniques, mais settlement des operations algébriques ........ ( underlining
added). He ends by claiming that he has made mechanics a branch of (mathematical) Analysis. LANZ, & BETANCOURT, 1808. Essai sur la compositions des machines. Paris 119
They implemented the master plan of Hachette (1806) based on Monge and Carnot's concep tion about the separation of mechanisms and machines. Monge had entitled the subject "Elements of Machines" (Remember Elementi macininoli of Leonardo), which he intended to be equivalent to the means to change the direction of motion. Monge considered the possible combinations of rectilinear alternative and continuous and circular continuous and alternative. In a 2nd ed. of 1829 the classes of motions ( and I presume of mechanisms) were increased from 10 to 21 by adding other curvilinear motions. MAGGI, 1914-19. Geometria del movimento. Pisa, Spoerri. (See also Dinamica fisica 1921, Dinamica dei sistemi 1921). MOIGNO, M. l'Abbé, 1868. Le ons de Mécanique Analvtique. Redigées principalement d'après les méthodes d'Augustin Cauchv. et étendus aux travaux plus récents. GauthierVillars, Paris. 9
Nearly a century after Lagrange, Moigno sticks to the Lagrangian approach. In another contri bution, more will be included about this book; suffice it to quote one paragraph from the long Preface of this book: On trouvera étrange que le nom de M. Poncelet, le législateur en France de la Mécanique Appliquée, ne soit pas prononcé dans cet ougrage; cela vient de ce que je dus faire ici de Vanalyse, de Véquilibre et du mouvement virtuel, tandis que Poncelet est le chef d'école de la synthèse, du mouvement et du travail. .
MOZZI, Giulio 1763. Discorso matematico sopra il rotamento momentaneo dei corpi. Naples. LA HIRE, 1645. Traité de mécanique. Imprimeries royales, Paris. McCORD. G.W. 1883. Kinematics of Machines. New York. MANNHEIM, A. 1894. Principes et Dévelopements de Géométrie Cinématique. Paris. NAVTER, Louis M. H. 1831-34. Le ons de Mécanique donnés à 1' École Polvtechnique. lère et 2éme Années. 9
Navier did also research into the fundamental theory of viscous fluids, and there he proved to be a powerful kinematicist when formulating equations which were also established by Cauchy, St. Venant and Stokes; they are now called the Navier-Stokes equations. In spite of this; his approach as a teacher was rather conservative, compared with that of St. Venant. OLIVIER, TH. 1842. Théorie géométrique des engrenages. Paris. OSTROGRADSKY, 1838. Mémoire sur les déplacements instantannés des svstèmes assujetis a des conditions variables. Brochure, St. Petersburg. PETERSEN, J. 1884. Kinematik. Kopenhagen. POINCARE, Henry 1899. Cinématique et mechanismes. Le ons à la Sorbone,. G.Carré, Paris. 9
POISSON, S. D. 1811. Traité de mécanique. Mme. Veuve Courcier, Paris. 120
From Preface: .... je ne me suis assujéti exclusivement, ni à la méthode synthètique, ni à la marche analytique. . . . Poisson took the same approach in the 1833 edition of this book, i.e. he does not appear to have had early ideas about kinematics as an independent science. PRONY, R. De la composition et de la décomposition des mouvements circulaires. plus une note sur l'axe spontanée de rotation. Ms. Ecole des P. et Ch. , Paris. PRONY, R. 1795 (VII). Mécanique Philosophique ou Analyse Raisonné des diverses parties de la science de l'équilibre et du mouvement. Imprimerie de la République, Paris. PRONY, R. 1797 (IX). Pian raisonné de la partie de l'enseignement de fècole Polvtechnique qui à pour objet l'équilibre et du mouvement des corps. Imprimerie de Courcier, Paris. PRONY, R Cours de mécanique à l 'École Polvtechnique. Ms École des Ponts et Chaussées, Paris. PRONY, R. 1810-1815. Leyons de Mécanique Analvtique donnés à V École Impériale Polvtechique. Imprimerie de l 'Ecole Impériale des Ponts et Chaussées, Paris. RESAL, H. 1862. Traité de cinématique pure. Mallet-Bachellier, Paris. See the Preface for a brief history of the adoption of the point of view of Ampère in Paris and its consequences in both research and teaching. REULEAUX, F. 1876. The Kinematics of Machinery. Macmillan, London. Translation of Theoretische Kinematik: Grundziige eines Theorie des Maschinenwessens, 1875. (There is a new edition by Dover, published in 1963 ) Of historical interest are the Introduction pp. 1-25, and Ch. VI, Sketch of the History of Machine Development, pp. 201-246. ROBERTS, Samuel 1875. On three-bar-motion in plane space. Proc. London Math. Soc. Vii, 14-23. St. VENANT, B. de 1850. Principes de mécanique fondés sur la cinématique. Bachellier, Goeurg, Litographie. Paris. See annotations for Navier 1831. The title of Chapter 2, Cinématique, ou étude géométrique du mouvement , is revealing of St. Venant's approach. SCHOENFLIES, Arthur 1886. Geometrie der Bewegung in Svnthetischer Darstellung. Leipzig. (French version: La Géométrie du Mouvement. Paris 1893.) SICARD, H. 1902. Traité de cinématique théorique. Paris. STUDY, E. 1903 Die Geometrie der Dvnamen. Leipzig. TRANSON, 1845. Journal des Mathématiques pures et appliquées. (See Resai 1862). WILLIS, 1841 Principles of Mechanism. London/ 121
Willis' approach differs from that of Monge. The scheme of Lanz, notwithstanding its apparent simplicity, must be considered a merely popular arrangement. Reuleaux [1876] is somewhat critical of Willis, especially of his idea of considering only rigid elements and exclude water wheels, windmills and other fluid machines. Fascicule Ouinquenal de la Societé Amicale des Anciens Elèves de TEcole Plovtechnique. From the 1973 issue, I have extracted the following list of teachers of Mechanics at the E. P. from 1794 to 1903. I believe that it speaks for itself, and that it may be very useful for the study of developments affecting kinematics during the 19th century. I have not been able yet to compile a list of those who taught courses on mechanisms and machines. Lagrange (1794-98). Fourier (1795-97). Gamier (1798-1801). Lacroix (1799-1808). Poisson (1802-14). Ampère (1809-27). Mathieu (1828-32). Navier ( 1833-36). Duhamel (1837-39). Sturm (1840-50). Bélanger (1851-60). Delaunay (1851-71). Bour (1861-65). Phillips (1866-79), Resal (1872-95). Bresse (1880-83). Sarrau (1884-1903).
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APPENDIX
I
LEONARDO DA VINCI's SCIENTIFIC AND TECHNICAL KINEMATICS
by Matilde Macagno*
Leonardo's studies in geometry as a part of physics have been only superficially explored although there are a good number of investigations published regarding classical aspec tsfsee,
e.g.
M arin on il[982].
The most interesting
notes and drawings on geometry left by Leonardo are those in which he considers motion as a means of deriving geometrical results, and geometry as means of studying motion itself.
There is a full spec trum o f notes going from
pure ly geom etric al questions [S ee , e .g ., L eo n ard o 's Theore m in Martin 19 82 ], to con figura tions in art and science [Macagno, M. 19 87 ],and to the analy sis of engine ering kinem atical problem s [Macagno, M. 1988].
Leo nardian kinem at-
ics comprises from the study of motion of a single material point to that of the turbulent flow of fluids, passing through questions of rigid body motion and perform ance o f connec te d rigid and fl exib le ele m ent s.
*Department of Mathematics, University of Iowa. Prof. Em. Matilde Macagno presented this paper at the Meeting of the Midwest Junto of the History of Science, Ames, IA, 1989. 123
Leonardo's notes on geometry in motion and scientific and technical kine matics are spread in the thousands of folios he left in the hands of his last discip le,
Franc esco Melzi,
when he died in France
in
1519.
Of those
manuscripts only about half are now available. ( the originals are in Italy, Fran ce, Spa in, England and USA ).
In mo st of the extant man uscripts one
finds an imp ortant part devoted to question s of motion.
He dealt with such
questions from different points of view and with varying methodology, depending on Leonardo himself approaching the topic in question as an artist, or an eng ineer, or a scien tist.
Fo r this contribu tion, I have chosen a number
of rep resen tative accomplishm ents in his studies of motion.
Prese ntly I will
offer only a summary; a paper has been accepted already for publication with a more comprehensive treatment, including the Mathematical Laboratory studies I have performed to gain insight into the work of Leonardo [Macagno M. 1988].
1. Leonardo applied the finite motion of figures and its parts to gain knowl edge about the equivalence of areas and volumes of figures of different shape. This became important for science and engineering when he related this result to wh at we call now the equ ation of contin uity .
This is in fact a misn om er and
a more adequate name would be the equation for conservation of volume.
2. Although Leonardo did not recognize the existence of the instantaneous center of rotation in plane motions of figu res, that was going to be discovered by J. B ern oulli, he was aw are of the re la tion betwee n the velo ci ties of two poin ts in ri gid -body m otion, as sh own by his study o f the cla ss ic al co nnec ting
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rod mechanism.
This finding was, pe rhaps, more the resu lt of tactile than
visual experimentation, as demonstrated by a laboratory study, in which a mechanical model is operated by the two hands, one driving a nearly uniform circular motion and the other, as passive as possible, following the motion of the other end of the rod.
3. An interesting study of Leon ardo comp rises the an alysis of the motion of string and pulleys in a variety of tackles (systems of pulleys,strings and weig hts).
His mo tivation was the statics of these system s, but he did it by
studying their kinematics, believing that the ratios of forces was related to that of displacements.
4. Another important resu lt is that of the basic elements o f what we call now the theory of deform able bodies.
Leonardo was really assuming that the tran s-
formations were linear ( linearization is an assumption or an approach often pre se nt in L eon ard o's note s) .
His sk et ch es of def orm ation of plane fi gure s
are strongly reminiscent of the theory developed centuries after by Helmholtz. (See the notable sketches in Folio 72R of Codex Madrid II).
5. I favor the hyp othesis that Leona rdo owed much of his deep u nders tandin g of transform ation geometry, to his appre nticeship in V erro ch io's art shop, and to the wor k with his hands in his own studio .
But there is also the evidence
of a profound interest in geometry as a science (which he seems to have had difficulties in learning as handed down by tradition) that he appears to have mastered when approached in his own way, d riven by his own m otivation s.
125
I
think that there is a link between the almost obsessive study of the subdivision of figures of differe nt kinds, as attested, for instance, by a num ber of sheets in the Codex Atlanticus, and his advanced understanding of the geometry in motion of deform able bod ies.
I find that is fortunate that one of the sheets
assembled by Pompeo Leoni in Codex Atlanticus ( CA 602R) should contain an extraordinary depiction of deformation in the inner space of a figure, thus offering a proof of how far Leonardo went in his exploration of the geometria che si prova col moto .
In closing this contribution, I want to share some of the insight I have gained on Leo nard o and his stud ies of motion.
I see him in three phase s that
may have happ ened all nearly at the same time.
I believ e that he began with
some study of elementary geometry, apparently not very successful, but from there he developed what seems to be an obsessive interest in the figures of equivalent area and volume, and then moved from static ways of doing this work to displace m ent and motion of figu res. which
he
called
appropriately
Thus he developed a geometry
kinematical geometry
.
To
this
way
of
approaching geometry surely contributed also his ability, as an artist, to take a port io n of any m ater ial and sh ape it in any desi re d fo rm . The fr oze n no tion of fixed form prevalent in his inherited knowledge of geometry was thus replaced by the study of chan gin g fo rm , either by steps or in a continuous pro ce ss . The second phase is Leonardo's strong interest in the subdivision and aggregation of figures, which one cannot separate from his artistic motivations, but that is intrinsically a geometric procedure and it was for him an important sou rce of ideas and also a tool . We mu st think that, if one de form s any of
126
these complex figures, one visualizes the internal changes together with those of the perimeter, and this must have played a great role in his studies of the geometry of deformation.
Finally, the engineer in Leonardo was concerned
with curves described by moving particles and bodies, by parts of different mechanisms, and by the amazingly complex families of curves that water constructed before his curious eyes.
But this was also a source o f geometry in
motion for him, because, in turn, only geometry could provide a frame to arrive at some und erstan ding of flow phenom ena.
Thus Leo nardo m ust have
learnt from art, nature and technology, a geometry much more interesting than that of Pacio li and, closing the circle, used it to study complex p henom ena of all kinds.
Leon ardian geometry is not only geometry done with m otion is also
the geometry of motion.
127
