Kim H. Veltman

LEONARDO'S METHOD

To Kenneth D. and Mary Keele

Table of Contents

Preface and Acknowledgments

Text

  1. Introduction
  2. Sources and Contacts
  3. Treatises
  4. Plans for Books
  5. Themes
  6. Method
  7. Plans for Publication
  8. Influence
  9. Historiography
  10. Conclusions

Appendix: Computers as an Historical Tool
Notes
Illustrations

Preface

In 1973-1974 the late Dr. Kenneth D. Keele, M.D., F.R.C.P. and the author reconstructed some of Leonardo's descriptions of perspective in order to determine whether these had an experimental basis. It was found that they did. The possibility that they had simply been thought experiments was excluded because some of his claims were so unlikely that they had to be tested in order to make sense. The experiments led to a long term cooperation: first two years together as Senior Research Fellows at the Wellcome Institute for the History of Medicine (London); then with intermittent visits during the seven years while the author was at the Herzog August Bibliothek (Wolfenbüttel). As work progressed Leonardo's method became increasingly evident. The challenge of communicating this method inspired Dr. Keele to write Leonardo da Vinci's Elements of a Science of Man and led the author to write his Leonardo Studies I-II.

There have also been several attempts to make the results of these studies more accessible. In 1981 there was a published lecture on Visualisation and Perspective at the world conference on Leonardo e l'età della ragione (Milan). Here there was great enthusiasm for the sequences of diagrams, which showed a methodical approach, but the criticism was made that the order had been imposed after the fact. Rearranging Leonardo's notes did not prove that he was not chaotic: it heightened the suspicion that he was. Further study led to new evidence that this method was not merely wishful thinking, but very much a part of Leonardo's approach. This led, in February, 1984, to three lectures on Leonardo's method at Brigham Young University, organized by the kind efforts of Professor Dan Blickman. The next impulse came unexpectedly in December 1989 through an invitation to organize, with Dr. Michael Sukale, a section at the European Forum (Alpbach) on Leonardo's Laws of Vision and of Nature. By way of preparation the notebooks were reread and this brought to light Leonardo's lists with their systematic play of variables. An essay was written in haste, which was not suited to the format of the proceedings in Alpbach. So it was distributed to a few friends for criticism and allowed to mature. What follows is the result.

Acknowledgments

The final version of this essay was written in ten days, but the work on which it is based covers nearly 20 years. This would not have been possible without an extra ordinary amount of support. At the outset a doctoral fellowship from the Canada Council permitted me to work in London (1971-1975). This was followed by a senior Research Fellowship from the Wellcome Trust (1975-1977). The seven years of research at the Herzog August Bibliothek in Wolfenbüttel (1977-1984) saw a series of fellowships from the Volkswagen, A. Von Humboldt, Thyssen and Gerda Henkel Foundations. Next came a year at the Getty Centre for Art History and the Humanities (1986-1987), since which time there has again been support from the home base through a Canada Research Fellowship from the Social Sciences and Humanities Research Council of Canada (1987-1992). I am very grateful to these bodies both for their individual support and for the cumulative effects which their help has brought. I am grateful also to the Institute for the History and Philosophy of Science and Technology in Victoria College at the University of Toronto, which has been my base for the past seven years.

Those who are not inclined to write by nature need encouragement and sometimes persuasion to arrange their material for a public audience. Hence I am thankful to the organizers of the 1981 conference in Milan, to Professor Dan Blickman who organized the three lectures at Brigham Young; Dr. Michael Sukale who was so kind in making arrangements for the six lectures at Alpbach and to Ing. De Toni for making possible this publication. I am deeply indebted to my friend Dr R. N. D. Martin who painstakingly read through the draft and helped transform a series of lectures into a statement.

1. Introduction

Leonardo da Vinci (1452-1519) has evoked two fundamentally different responses: one sees him as central to early modern science, another dismisses him as an eccentric with no influence. Both views were found while he was still alive. For instance, Pacioli (1509) praised him as being among the most perspicacious of architects and engineers, an assiduous inventor of new things, famous for sculpture and painting, for his construction of the horse, the Last Supper and for his writings: that he was working on "an inestimable work on local motion, percussion, weights and all the forces, that is, accidental weights, having already with great diligence finished a worthy book on painting and human movements."[1] Aspects of this view were kept alive by Venturi (1797)[2], Solmi (1905)[3], Uccelli (1940)[4], Reti (1974)[5] and Keele.[6] On the other hand, Castiglione (1528)[7] criticized him indirectly for frittering away his time on useless mathematical speculation. Serlio (1545) made a different claim: that Leonardo was too much of a perfectionist and that this kept him from publishing.[8]

Twentieth century scholars such as Marie Boas Hall argued that because he never published he had no influence[9], while one recent scholar has dismissed him as "an ingenious empiricist working in an intellectual vacuum."[10] Was Leonardo merely a recluse working on his own out of touch with the great traditions; an over ambitious amateur making notes without any underlying method and structure? This paper shows that Leonardo was widely read and in contact with some of the major scholars of his day. A survey is made of his extant treatises to confirm that these are much more structured than is at first apparent. His plans for books are examined to discern how he intended to arrange his material. It is shown that for all his universality Leonardo focussed on a surprisingly small number of basic themes; that although Leonardo's study of the natural world includes physics, biology and botany he treats them all in terms of mechanics. A detailed reading of his notebooks reveals that he was guided (and inevitably sometimes misguided) by a clear method. The notebooks also contain proof that Leonardo did not write them solely for private study; that he specifically intended them for other readers and had plans for publication. Finally, it will require a survey of historiography to understand how all all this could have been forgotten, with the result that scholars have claimed that he was a peripheral figure, when in fact Leonardo's method is of central importance to the western tradition.

2. Sources and Contacts

The evidence of Leonardo's notes and correspondence is of a man with wide contacts and reading. Sometimes he cites oral reports. In the Madrid Codex he reports that Julius had seen a case of a wheel being worn out while in Germany.[11] He also considers letters. In one of his prophetic statements he mentions "Writing letters from one country to another. Men will speak from very remote countries to one another, and reply."[12] He writes business letters to his employers such as the Duke of Milan and the Pope. Letters are also one of the ways he searches for evidence, as when he reminds himself to write to Bartholemew the Turk on the question of tides and specifically about the Caspian sea.[13]

Leonardo is also a reader of books and often he cites the evidence that he finds in these. Sometimes he refers generally to Aristotle's De caelo[14] or Euclid's Elements.[15] Sometimes he refers to a specific book and chapter of Aristotle's Physics[16] or to given propositions in Euclid. At least twenty propositions ranging from books 1 to 5 of the Elements are cited explicitly[17]. In his study of ancient weapons in the Manuscript B he cites a wide range of ancient authors: Pliny[18], Virgil[19], Lucretius[20], Aulus Gellius[21], Livy, Plautus, Flavius[22], Lucan, Caesar[23], Quintilian, Varro[24], Plutarch[25], Hermes Trismegistus, Pompeius Festus[26], and Ammianus Marcellinusi[27]. Elsewhere he cites Plato[28] and Vitruvius[29]. Mediaeval sources include Swineshead[30], Thabit ibn Qurra[31], Peckham[32], a Treatise on the abacus[33], as well as Biagio Pelacani da Parma.[34] Contemporary sources include Leon Battista Alberti.[35] We know that he had a personal collection of 119 books.[36] We know also that he did not limit himself to the books he owned himself. There are numerous references to books in other collections. For instance he takes note of a copy of Witelo which is said to be in the library at Pavia[37] and a copy of Archimedes in from the Bishop of Padua[38]. This is not exactly an intellectual vacuum, especially in 1494, four decades after the advent of printing.

In addition to books and correspondence there were his direct contacts with contemporaries. He studied and wrote in the margin on seven pages in one of Francesco di Giorgio Martini's manuscripts (Florence, Cod. Ashburnham 361). In 1490, he travelled with this engineer to Pavia to visit Fazio Cardan, the learned editor of Peckham's Optics and father of Jerome Cardan. In 1494 he bought a copy of Luca Pacioli's Summa and subsequently worked together with this mathematician for three years at the court of Milan (1496-1499). At the same court, Leonardo was "as a brother"[39] with Jacopo Andrea da Ferrara, a leading architect and Vitruvian commentator of the time. As an employee of Cesare Borgia he was a colleague of Machiavelli. In the period 1508-1510 he appears to have worked with Marc Antonio della Torre a professor of anatomy at Milan. In Rome (1513-1515) he worked for the Pope on a sixteenth century version of the star wars project, using burning mirrors as described by Archimedes as a means of defence. The fame of his activities caught the attention of the King of France who persuaded Leonardo to move to France for an early "retirement". The curriculum vitae of a recluse would look somewhat different.

3. Treatises

Both the size and contents of the notebooks vary considerably. There are tiny pocket-size booklets such as Forster III (9 x 6.7 cm) or Manuscript H (10.3 x 7.2 cm)[40] and large folio sheets such as the Codex Atlanticus and Windsor. In terms of contents the notebooks fall into three different kinds: travel notes, study notes and draft treatises.

When he travels Leonardo likes to make notes and he recommends this practice to his students[41]. Manuscript L is a good example. Here Leonardo does some surveying, sketches the lay of the town at Cesenatico, notes architectural features at Urbino etc.[42] A second category of study notes is based on both experiments and books, and involves gathering material for his basic themes. Some of this material was unbound. It is likely that Leonardo kept two large piles of unbound notes in his study: one devoted primarily to the man made world (machines, inventions and architecture), now the Codex Atlanticus; the other dealing mainly with nature (anatomy, botany and geology) now in Windsor. This division was not strict. After Leonardo's death, Pompeo Leoni attempted to sharpen the distinction between these two piles by cutting out various portraits and caricatures from the Codex Atlanticus and adding these fragments to the Windsor collection, as Carlo Pedretti has so elegantly shown.[43] There is evidence that Leonardo planned to have what is now the Windsor Corpus[44] bound, but this did not happen during his lifetime. Around 1508 the materials for what is now the Codex Arundel were also in an unbound state, although a note on Arundel 190v [45] mentions that he plans to have this bound also.

The number of topics dealt with on these loose sheets varies. A few contain many themes. A number of sheets contain two or three topics. The majority of sheets are dominated by or deal exclusively with one topic. There is, however, a spectrum of ways in which a theme is treated. In early stages of formulation, diagrams and texts are scattered indiscriminately (e.g.pl.1ab). In a further stage, there is some order. Then there are sheets with some evidence of numbering. Next a pattern of texts with illustrations or vice-versa emerges. Finally we find sheets showing a single example sometimes with an accompanying text. Some of these are presentation drawings intended to be shown separately (cf. pl. 2ab), while others become parts of treatises. In the notebooks which are bound, most folios deal with a single theme and this usually continues over a number of pages. Indeed the majority of the manuscripts are dominated by only a few themes. As Leonardo's ideas evolve he copies them or has them copied into another manuscript, crossing out the original passage involved (pl. 3-4). A note at the beginning of the Codex Arundel suggests that it was drawn up in this way. In the case of the Treatise of Painting, Melzi added a sign alongside passages that he copied from manuscripts such as BN 2038, A, E and G.

Plate 1ab
Image 4a. Image 4a. Image 4b. Image 4b.
Plate 2ab
Image 5a. Image 5a. Image 5b. Image 5b.
Plate 3
Image 6a. Image 6a. Image 6b. Image 6b.
Plate 4
Image 7a. Image 7a. Image 7b. Image 7b.

Leonardo's earliest bound notebooks, the Codex Trivulzianus (1487-1490, 21 x 14 cm), Manuscript B (c. 1488-1490, 31 x 22 cm) and Manuscript A (c. 1492, 21 x 14 cm) are large in size,and generous in their use of paper. But paper must have become increasingly scarce. On rare occasions, such as CA206ra (549r, c. 1497), Leonardo proceeds in palimpsest fashion, writing over a passage with a new topic. Elsewhere, especially in the Windsor Corpus and the Codex Atlanticus, we find instances where he seems determined to use every inch of space. This is partly because he keeps returning to a sheet in order to add further notes on a given topic, which explains why the dates of these two manuscripts range from the beginning to the end of his career.

Some of the bound manuscripts can be described as study notes insomuch that the order of the folios remains implicit. Manuscript A, which contains his treatise on perspective is an example. The treatise begins on A36v. There is a number 4 at the bottom of the folio, which is repeated at the top of A37r. Similarly at the bottom of A37r there is a 5 which is repeated at the top of A37r. These are the only formal clues of sequence. But as I have shown elsewhere[46] the argument proceeds methodically from 36v through to 42v, thus comprising a short, thirteen-page treatise. A second such treatise is found in Forster II. It goes backwards beginning at 158v and ending at 65v. There is an independent numbering to help us. Hence, folio 158v is 1, 157v is 2...68v is 91, 67v is 92, 66v is 93 and 65v is 94. This accounts for the cryptic note in Latin: "Most powerful mechanics, beginning at the end."[47] Another instance is found in Manuscript M, where a discussion on motion and percussion continues in sequence from 94r to 93v and so on to 90r, thus making a brief treatise of nine pages.

There are also notebooks in which order remains mainly implicit except for isolated passages which give hints of a larger plan. An example is Manuscript F which contains a draft of a chapter for his treatise on cosmology. A note on F94v outlines the overall purpose of the book:

My book sets out to show how the ocean along with the other seas, with the help of the sun, makes our planet reflect light the way the moon does and from a greater distance appears like a star, and this I prove.[48]

By way of introduction Leonardo feels he must establish that the eye is not being deluded when it looks at the sky. So he writes a chapter on optics. This begins on F95v. At the bottom of the paragraph he writes: "It is not possible to define this here for lack of paper, but go to the beginning of the book [i.e., chapter] at folio 40 where this is defined."[49] At folio 40 the treatise continues and proceeds backwards to 39v,39r and so on through to 28r. I have made a complete analysis of this treatise elsewhere.[50] It is important for our purposes here to note that Leonardo also uses this method with respect to other themes discussed in the same manuscript. On F13v, for instance, he writes "Turn the page."[51] On F26v he writes "Here follows the proof of that which is said on the page opposite."[52] On F52r he notes "Go to page 59."[53] All this is not simply because he is being obtuse. These were times of war. Paper was in great shortage. There were many interruptions. These were his working notes. But even so, he too wanted order.

Similar notes are found in two other manuscripts. In the case of Manuscript E, the page sequence again is frequently the opposite of his argument, i.e. he starts at the back and works forward. Hence, when on E75r he writes "Here is finished what is lacking three pages before this,"[54] We need to go to 77r to find the relevant passage. Manuscript G contains at least seven notes of this kind. On G44v Leonardo writes "And this is drawn in the margin at the bottom four folios following[55]," i.e. G48r. On G46r he writes: "Here follows what is lacking on the page opposite"[56], i.e. 45v. On G46v he notes: "Read page 45[r]."[57] On G51v he writes "go to page 44."[58] On G67r he explains that the text continues on the page opposite at the bottom[59], i.e. 66v. On G75r he adds two notes: "Here follows what is on the opposite page[60]," i.e., 74v at the top, and also "the round beam is drawn on the page opposite."[61] Finally on G80r there is a similar note.[62] The sequence of his argument is much more erratic in G than elsewhere. He is working in Rome at the time (1513-1515), and developing his pyramidal law with respect to mirrors[63], precisely the kind of information that the German industrial spy, Johannes, the mirror maker, was trying to steal. I suspect that Leonardo was in this case consciously giving a superficial impression of chaos.

In a third kind of manuscript Leonardo uses part or even a whole manuscript to gather material related to a specific topic. This kind of treatise confirms that Leonardo is capable of more coherent and systematic presentation. An early example is Manuscript C (c.1490-1491), which deals with light and shade. The most interesting example is Forster I (c. 1505). A note in mirror script on Forster I 3r informs us that this is a "book entitled on transformation of one body into another without diminution or augmentation in material."[64] This note has been rewritten in ordinary script by a later reader. A note on 3v, which has again been rewritten in ordinary script informs us that this manuscript was "begun by me, Leonardo da Vinci on the 12th of July 1505."[65] The actual treatise begins on 39v with a proposition numbered "1st." On 39r a 2nd and 3rd proposition follow. These continue in order until proposition II on 35r. From 34v through 28v (pl. 7-8) there is a second series of 13 numbered propositions. From 28r through 20r there is a third series of 20 numbered propositions. His numbered list of 28 geometrical transformations cited earlier gives us another glimpse of the order he had in mind. The latter part of Forster I, namely folios 40v through 55r, deals with a distinctly different topic, hydraulic machines. Here the diagrams are much rougher and the general impression is more chaotic. If we look closely, however, we find that beginning on folio 45r the diagrams are numbered 1, 2 and respectively. On 45v we find the numbers 4, 5 and 6. This continues in orderly fashion until numbers 41 and 42 on 53r.

This principle of numbering the illustrations by way of establishing the sequence of his ideas recurs in Madrid Codex I, this time in the context of weights and balances (pl. 5-6). On Mad I 190r Leonardo adds to his illustrations the numbers 3, 4, 5 and 6. This sequence of numbers continues in the opposite direction of pages such that we find illustration 100 on folio 172r. This sequence then begins a fresh on this same folio 172r with figures 1 and 2 and continues until figure 90 on folio 158r. In Manuscript K, beginning on 79r, this time in the context of geometrical diagrams we find two numbered propositions which continue in sequence until 14 on folio 73r. A late example of this approach is Manuscript D (c.1508), on problems of vision.

As will become evident, by 1492 Leonardo had developed an explicit method for presenting his ideas that was reminiscent of the form Euclid established for classical geometry: a proposition (i.e. a claim), followed by demonstrations (i.e. examples based on experiment or at least experience), frequently accompanied by illustrations to show different possibilities. In the notebooks, the propositions increasingly serve as headings for demonstrations in paragraph form accompanied by diagrams, often in the margin. This procedure is seen clearly in Codex Leicester (now Hammer) and Manuscripts E, F and G. Sometimes the margins give summary versions of the proposition, as in Manuscript D. Hence, in addition to his travel and study which are frequently without a planned order, Leonardo has a clear method of presentation when he begins to organize these with a view to creating formal treatises[66].

Besides this physical evidence there are clear references in Leonardo's notebooks to specific books and propositions. Some of these are in the Codex Atlanticus. On CA384ra (1493-1495), for instance, Leonardo mentions: "I stated in the 7th conclusion how percussion...."[67] On CA155vb (1495-1497) he asks us to "look at the 7th [proposition] of the fifth [book] of the axle and the wheel."[68] On CA2ra (1515) we find a precise reference to his book on machines discussed earlier:

Since without experience one cannot give true science of the power by means of which the drawn wire resists that which draws it, I have drawn here, on the side, these four motor wheels of the perpetual screws, marking the degrees of power alongside each one. These powers are true as is proved in the 13th [proposition] of the 22nd [chapter] of the elements of machines written by me.[69]

Another late note on CA287re (c. 1514-1515) informs us that "Mr. Battista dall'Aquila, private secretary to the pope, has my book in his hands."[70] In the Codex Arundel, on folio 12r, Leonardo refers us to "the 5th of the 7th"[71] in connection with weights. On Arundel 25r, he mentions "as proved in the 4th of my [book on] perspective."[72] On Arundel 25v he refers to "book 9 of water".[73] Sometimes the references are laconic as on K30r to a "sixth book"[74] or in Manuscript F where there are at least four references to "book 9 on water" on folios 5r, 24v, 72v and 88r respectively, two references to a "book 10 on water" on folios 4v and 24v, plus headings on 35r for "Book 42. On rain"[75] and on 37r for "Book 43. On the motion of air included below water"[76], as well as a simple note on 66v: "Beginning of book [of water]."[77] On Manuscript I 72v there is also a: "Beginning of book on water."[78] On E59v (1513-1514) there is a: "Beginning of this book on weights"[79] and on E27v there is a note: "proved by the ninth of percussion."[80] On W19061r (K/P 157r) we read: "proved by the 5th on force."[81] Leonardo also refers on W19061r (K/P 154r) to the "order of the book [of anatomy]....Hence with these 15 entire figures the cosmography of the microcosm will be shown to you with the same order as Ptolemy used before me in his Cosmography".[82] Thus Ptolemy's method of arranging the macrocosm serves as a direct model for how Leonardo arranges the microcosm. On W19009r (K/P 143r) Leonardo refers once more to his book on machines:

Make sure that the book on the Elements of Machines with its practice comes before the demonstration of motion and force of man and other animals, and by means of these you can prove all your propositions.[83]

4. Plans for Books

In addition to the above evidence of books actually written or at least in progress, there is considerable evidence of plans for books. On rare occasions such as Mad I 173v (c. 1499), we find him making a note on method: "You will put the whole text together and then you will divide it and add the commentary."[84] Then there were plans for specific subjects: painting, perspective, cosmology, transformational geometry and machines. The book on machines included his work on the four powers of nature (force, motion, percussion and weight). From a note on CA117re (1490) we know that he had begun planning this work at an early stage in his career: "First you will deal with weight, then with motion which gives birth to force, then you will deal with force and finally with percussion."[85] A few years later on CA149rb (1493-1495) he elaborates on his plan:

Beginning of the nature of weights.
The plan of your book will proceed in this form: first the simple beam, then supported from below, then partly suspended, then entirely, then these beams will support other weights.[86]

In the decades that follow this evolves into a major treatise which deals with the theory and practice of machines (cf. pl. 25-26) and their relation to the four powers of nature, all of which serves to introduce his treatise on human and animal movement. A separate book was planned for the flight of birds. The Codex on the flight of birds now in Turin is but a fragment of the projected work as we learn from a passage on K3r (after 1504):

Divide the treatise on birds into 4 books. The first will be on flying by flapping their wings. The second will be on flight without flapping thanks to favourable winds. The third will consider principles of flight common to birds, bats, fish, animals and insects. The final book will deal with instrumental flight.[87]

Plate 25
Image 63a. Image 63a. Image 63b. Image 63b.
Image 64a. Image 64a. Image 64b. Image 64b.
Plate 26
Image 68a. Image 68a. Image 68b. Image 68b.
Image 69a. Image 69a. Image 69b. Image 69b.

Instructive in this context is Madrid Codex I where we find a number of references to specific books and propositions. On Mad. I 105v-106r, for instance we find a series of (almost) consecutive propositions. Sometimes he provides the name of the book in addition to book and proposition number as when he refers to "Bk.5.3 of motion and percussion" (moto e colpo) on Mad. I 69v; "Bk.7.5 of motion and force" (moto e forza) on 94r; "Bk.7.5 and 9.7 of his theory" (teorica) on Mad. I 140v. Sometimes he simply refers to a proposition number without reference to book number as in "5th of theory" on Mad. I 147v, "5th" on Mad. I 71v, "6th" on Mad. I 87r and "7th" on Mad. I 140v. In 33 cases he gives book and proposition number[88]. If Leonardo were truly as chaotic as he is generally assumed to be there would be little incentive to refer so often to specific books and propositions. On F41v (c. 1508) Leonardo outlines a slightly different plan:

To speak of such material you need in the first book to define the nature of resistance of air. In the second the anatomy of the bird and its feather. In the third the operation of such feathers through different motions of their own. In the fourth the value of wings and tail without flapping the wings with the aid of different headwinds in steering with different movements.[89]

Water was another theme about which Leonardo planned to write a major work. On F90v [90], F45v [91] (c. 1508) and E12r [92] (1513-1514) he makes notes concerning the order of topics in this work. On F87v he describes his general plan:

First write everything about water in each of its motions and then describe all the surfaces over which it flows and their materials always adding the propositions of the aforesaid waters and let it be in good order otherwise the work will be confused.[93]

Much more elaborate plans are found in the Codex Atlanticus. These are striking because they again reveal the sytematic play of variables that we have identified as an essential element of his method. On CA79ra (c. 1505-1506), for instance, Leonardo makes a list headed:

Book on the percussion of water with various objects
Encounters of water with permanent objects of different shapes that overcome the water
Encounters of water with immobile objects covered by water
Encounters of water with mobile objects covered by water
Encounters of water with permanent objects that overcome the water
Encounters of water with pliable objects that are overcome by water
Encounters of water with objects which fall with a circular motion such as wheels of aquatic instruments.[94]

On CA74v (1505-1506) Leonardo makes further lists, among them one on different kinds of eddies:

Eddies which are superficial
Eddies which rise from the bottom to the surface
Eddies which go from the surface to the bottom
Eddies which move with the course of the stream
Eddies which change direction, as those in ebbs and tides of rivers
Eddies which are lateral and continuous
Eddies which are lateral and discontinuous
Eddies which are wide above and narrow below
Eddies which are narrow above and wide below
Eddies which are straight from bottom to top
Eddies which are oblique from bottom to top
Eddies which are very large
Eddies which are small
Eddies which have gurgles
Eddies which are pipe-like
Eddies which are screw-like
Eddies which are hollow and filled with air
Eddies which are not hollow[95]

Leonardo made further such lists both in the Codex Atlanticus[96], Codex Arundel[97] and the Codex Leicester (now Hammer).[98] Indeed, as Carlo Pedretti has claimed Leonardo made a series of references to a now lost treatise, Codex M[99], which dealt with problems of water. What emerges, therefore, is a much more coherent picture than is usually ascribed to Leonardo. We shall see that this applies equally to the basic themes on which he focusses his attention, and the method which he uses in dealing with these themes.

5. Themes

Far from being just a wild enthusiast making notes about everything possible, Leonardo is surprisingly specialized in his studies. Moreover, the basic themes which he chooses are guided by systematic principles. It is striking that only about 10% of Leonardo's extant notes are about the natural world. Nearly 90% of his notes are concerned with man-made worlds which can be divided into mental, represented and constructed worlds. Of these the mental world receives about 15% of his attention, the represented world approximately 20%, while the constructed world receives approximately 65% of his attention, if we judge on the basis of extant notes.

Leonardo's study of nature focusses on three aspects: physical, biological and botanical. With respect to the physical world, he is guided by two interests: cosmology and physics. He wishes to write a major treatise on the nature of the universe[100]: to show that the moon reflects the sun as does the earth; that the moon has oceans like the earth (pl. 32) and that from a greater distance both the earth and moon look like stars. This is why he spends at least 20 pages on the problem of the sun's image in water. Leonardo insists, moreover, that the earth is in the centre of its elements rather than in the centre of the universe. Thus he can argue that the moon is in the centre of its elements and that the same applies to the other planets. In so doing he challenges objections that water and other elements on the moon should fall back to earth. This rejection of the geocentric model of the universe before Copernicus is of interest in its own right, but for the moment we must limit ourselves to the structure of his thought. In order to be certain about the nature of the earth requires some attention to geography, geology, the nature of tides and meteorology. To be certain of the nature of the heavens requires some attention to astronomy. Here Leonardo focusses interest on the moon: its appearance, substance, its phases. To certify that he is not being deluded in what he sees, requires study of optical principles. His studies of the eye in Manuscripts D and F were intended as chapters in the treatise on cosmology. His studies of optical instruments, notably mirrors, eyeglasses and a prototype of the telescope, were also part of this enterprise.[101]

Leonardo's second motive for studying the physical world lies in his physics. Here he is guided by his concept of four powers of nature (force, weight, motion and percussion), which he treats mechanically and by means of which he intends to gain a new understanding of the four elements: earth, water, air and fire. With respect to the natural world he focusses on two powers, motion and percussion, in conjunction with one element, water. This is no coincidence. Given his principle of limiting himself to study of visible phenomena, water provides him with the best medium for studying both motion and percussion. Water is of practical interest with respect to canals, irrigation, etc. It is also of theoretical interest. Leonardo sees water as a slow motion version of air. As early as 1490 he claims of CA361va: "Wind has similarity with the movement of water."[102] This has consequences for his study of both the natural and the man made world. On M83r (before 1500) he notes "swimming shows the way of flying"[103] a thought he restates in CA66rb (c. 1505). "Swimming in water teaches men how birds do it in air."[104] "He elaborates on this line of reasoning on E54r (1513-1514):

In order to give a true science of the motion of birds in the air it is necessary first to give the science of winds which you prove through the motions in water and this science of a sensible nature will serve as a ladder in gaining cognition of birds in air and wind.[105]

These same assumptions have important consequences for his biological studies, where he studies man, horses, some other animals (e.g. dog, donkey, bear), birds, insects and fish, but largely from a viewpoint of their underlying mechanical principles of motion. Instead of making catalogues of birds, he studies how they fly. Instead of making catalogues of fishes, he examines how they swim. So too with horses: he focusses attention on how they run. Moreover, all these studies have an ulterior motive, aside from providing him with subjects for painting: to improve man's ability to move through the elements with mechanical equivalents of running, swimming and flying (see fig. 2).

Earth Water Air
Man * * *
Horse *
Fish *
Bird *
Motion Running Swimming Flying
Mechanical Equivalent Cart Boat Mechanical Bird

Fig. 1: Links between elements, biological studies, motions, and machines.

Once these connections are recognized, Leonardo's emphasis on human motion in his anatomical studies takes on new meaning. So too does his decision to preface these studies with his work on machines and the four powers, as does his concern with particular machines such as carts, boats and mechanical birds. Leonardo is neo-Platonist insomuch that he is fascinated by the traditional microcosm-macrocosm analogy. At the same time he is guided by what might paradoxically be termed a mechanical anthropomorphism which helps us to understand other features of his work. On A55v (1492) he compares the body of a man with the body of the earth.[106] On A56r (1492) he compares the veins of man with the underground rivers of the earth.[107] Even much later, on CA5260ra (c. 1508-1510), he compares the lungs of man with those of the earth.[108] Cited out of context, as they often are, such passages make Leonardo look thoroughly committed to outdated classical and mediaeval organic cosmological metaphors. But this ignores the mechanical context, which he assumes for both man (microcosm) and world (macrocosm).

In the Windsor Corpus he refers repeatedly to the human body as an instrument as on W19029r (K/P 71r) where he mentions the "wondrous instrument invented by the consummate master"[109], or W19037r (K/P 81v), where he announces that he will "demonstrate this instrumental figure of a man in 24 figures."[110] Leonardo's view of birds is analogous, as is clear from a passage on CA161ra (c. 1505): "The bird is an instrument operating by mathematical laws, which instrument it is within the power of man to make with all its motions but not with as much power."[111] Similarly, Leonardo looks upon the earth as a machine. On W19147-8r (K/P 22r) he mentions it as a "terrestial machine."[112] On CA269ra (1490) he states that as a result of "various opinions concerning the size of the spherical terrestial machine, I have become concerned to create or rather to construct an instrument which will adopt this form."[113] Here he is speaking of a surveying instrument. On CA252rb (1490-1492) he refers to this "terrestial and worldly machine[114]," and on A59v (1492) to "the universal machine of the earth."[115] In short, while maintaining some traditional organic metaphors concerning the microcosm-macrocosm analogy, Leonardo treats them mechanically rather than organically.

In addition to anatomy and principles of motion, two other aspects of man are of concern to Leonardo: the senses and reproduction. With respect to the senses he makes some mention of all five. But by far the greatest attention is on sight and this for two reasons. First, sight is the sense that gives access to visible things and the visible is his standard for truth. The study of optics is thus crucial to ensure against illusion and to certify that his experimental observations are as objective as possible.[116] Secondly sight is directly connected with perspective, which serves to demonstrate measurable relationships between what is seen, what is represented and the natural world. Perspective also serves as a bridge between abstract mathematics and the concrete world. In addition it plays a significant role in his painting.[117] Hence optics and perspective remain leitmotifs throughout Leonardo's writings.

Leonardo also has some interest in the problem of sexual reproduction and devotes a few pages to the relevant male and female organs, and to the problem of a foetus in the womb. This constitutes such a small fraction of his work that it would not deserve mention here were it not for a curious analogy which Leonardo sees between the umbilical cord of a newborn child and the flowers and blossoms of certain plants. Indeed his attention to the botanical world, which includes trees, plants and flowers[118], focusses in very large part on the question of plant reproduction: blossoms, flowers, fruits, seeds. As in the biological world instead of making catalogues of genera and species, Leonardo's attention is focussed on a specific problem. And once again it is guided by his study of man and woman.

Hence, although Leonardo's study of the natural world includes physics, biology and botany, he treats them all in terms of mechanics. Indeed, he focusses on man and the universe to create a mechanical version of the microcosm-macrocosm analogy. This is one of his central concerns even if it involves only a small fraction of the notebooks.

Leonardo's interest in the mental world is primarily in terms of principles of communication which he would see as threefold: numbers (arithmetic), words (language) and diagrams (geometry). His work in arithmetic amounts to about 1% of his notes[119] and is limited almost entirely to arithmetical proportion, practical problems deriving from the abaco school[120] and some computations. Words in terms of language[121] and literature[122] (cf. figure 2) interest him more and constitute roughly 4% of his notes.

Fig. 2: Branches of Leonardo's work in language and literature.

Much more important are his studies of geometry. When Leonardo praises mathematics he usually means geometry[123] and sometimes geometry in combination with mechanics. Through his study of perspective (c. 1488-1492) Leonardo becomes interested in both geometry and geometrical proportion. When Pacioli's compendium on the subject is published in 1494 Leonardo buys his own copy. From 1496 through 1499, Leonardo draws illustrations for Pacioli's Divine proportion later published in Venice (1509). Like Pacioli, he sees proportion as a key to nature. But Leonardo is more concerned with earthly proportion. As he states on K49r (after 1504): "Proportion is found not only in numbers and measures, but equally in sounds, weights, times and sites and every power that exists."[124] Even so, proportion is but one of the branches of geometry that interests him (cf. fig. 3). Pacioli leads him to study Euclid[125], whose Elements deal mainly with geometry in two dimensions.

Fig.3. Branches of Leonardo's geometrical studies.

Leonardo's study of perspective prompts him to explore three-dimensional treatment of geometrical forms. In his treatise on the geometrical game[126] he limits himself mainly to the five Platonic solids. Elsewhere he explores most of the 13 Archimedeian solids. For Leonardo transformational geometry involves an infinite variety of shapes and he sees them almost literally as building blocks of reality. Moreover, because these changes are reversible and repeatable they serve to demonstrate his concept of science.[127]

One of the reasons why Leonardo's work is not in a vacuum is because it is related to his professional concerns as a painter. Hence, in addition to the mental world with its branches of arithmetic, language and geometry there is the represented world to which he devotes his Treatise of painting. Leonardo sees painting and science as intimately connected because painting creates bridges between geometry and nature and helps to record visible evidence, which is his key to truth. Perspective plays a central role in this process, while optics and geometry are also significant. Optics provides him with the laws of light and shade[128] by means of which he can deal with human forms[129], drapery[130], trees and plants[131], and geometry provides him with the principles of transforming their shapes (fig. 4).

Fig. 4: Branches of Leonardo's studies of the represented world.

Fig. 5: Branches of Leonardo's studies of the constructed world.

Yet the focus of Leonardo's attention is in the constructed world which includes architecture[132], mechanics and instruments (fig. 5). In his architectural studies we find him playing systematically with basic geometrical forms in the ground plans of his designs for churches (pl. 21-22) in Manuscript B as early as 1490 before he develops the idea of applying a play of variables to nature's powers. Military concerns play some role in his exploration of the constructed world but are more peripheral than one might expect. If his military architecture[133] involves some significant innovations, his weapons[134] are surprisingly traditional. As noted in our discussion of sources, Leonoardo makes a detailed study of ancient military authors and studies contemporaries such as Francesco di Giorgio Martini.[135] This results in his weapons being almost entirely dramatic representations of existing warfare rather than radically new devices.

His originality lies in his treatment of machines[136] and instruments. If we leaf through the Codex Atlanticus or the Madrid Codices with no understanding of his method, our first impression is an endless variety of mechanical devices. This is not the case (pl. 25-28). As Reti[137] has shown, Leonardo considers 21 mechanical elements. He is not disordered. Indeed if we examine actual machines we find six basic types that he studies in some detail: pulley, crane (including crane shovel), winch, cart, textile machine and file machine. With respect to water he has three further machines: boat, archimedeian screw and fountain. With respect to air he has his flying machine[138] (see fig. 6).

Earth Water Air Fire
Hoisting Dragging Rolling Digging Weaving Pounding Weighing Measuring Space Time Sailing Raising Flying Burning
Crane Winch Cart Shovel Textile File Balance Compass Clock Boat Screw of Archimedes Mechanical Bird Mirror

Fig 6: Leonardo's machines and instruments seen in the context of basic actions and the four elements.

Plate 27
Image 74b. Image 74b. Image 73a. Image 73a.
Plate 28
Image 73b. Image 73b. Image 74a. Image 74a.

Leonardo is trying to catalogue basic mechanical actions with respect to the four elements. Each of these actions involves combinations of the four powers (motion, force, weight and percussion). Hence his study of the constructed world is guided by a simple, underlying purpose: to establish the mechanical principles of the four powers with respect to the four elements of nature. He is also concerned with the principles governing: a) the four powers (pyramidal law); b) geometrical forms (transformational geometry); c) interpretation (optics); and, d) representation (perspective. He is inspired by three goals: to understand the natural world created by God; to construct new man made dimensions of the natural world and to represent new man-made worlds.[139]

If machines inspire him to look for universal principles they do not suffice to demonstrate them. For this he needs instruments, and it is surely no coincidence that he has at least three times as many notes on instruments as on machines. Some are surveying instruments, which we find him discussing in the context of settling disputes and certainty as on CA269va (727r, c. 1490):

Having seen various opinions of the size of this orb, the terrestial machine, I judged that since among so many disputants there were as many opinions, certain truth must be quite distant from them since, if the truth had come to their minds, all would be of one opinion. And given this great diversity of opinion, I decided to create or compose an instrument in this form....[140]

But again most of his energies are focussed on four instruments: mirror (concave and convex as well as plane),[141] clock[142], balance[143], and compass.[144] The regularity of these precision instruments permits him to begin testing his intuitions about the universality of nature's mechanisms in a systematic way, because he can now, under controlled repeatable conditions, check the effects of changing one or more variables. Instruments thus become models for testing whether nature's powers are as regular as he thinks they are. Each of the instruments has its own special use in this process. Mirrors serve to explore laws of light and also, in the case of concave mirrors, heat, both of which Leonardo considers as instances of percussion. Clocks (cf. pl. 4, 25) serve to demonstrate percussion, through the striking action of the lock mechanism, as well as weight, motion and force. Balances are particularly suited to studying properties of weights. Pulleys which are a variant form of balance, allow the study of weight, force and motion. Study of these instruments leads him to think in universal terms. On CA321re (882r, 1493-1495), for instance, he explores "how all wheels are of the nature of a balance."[145] On CA396rd (1102r, 1495), he reminds himself to "make mention of the general rule about the contact of axles and all weights."[146] At the same time instruments offer a way of testing his ideas about all four powers. By the mid 1490's he is planning to write on this as he mentions on CA155vb (421v, c.1495-1497):

First speak of motion, then weight, because it is born of motion, then of force which is born of weight and motion, then of percussion which is born of weight, motion and often of force.[147]

A passage on CA267ra (721r, 1495) confirms that he is thinking in terms of a general rule for at least two of the powers of nature and considering Pythogorean music as an integrating principle: "General rule of percussion. General rule of force. In these two rules, that is, of percussion and force, one can adopt the proportion which Pythagoras uses in his music."[148] Another passage on CA20va (66r, 1493-1495) shows that he is seeking: "To make a general rule of the difference that there is between simple weight and weight with percussion caused by different motions and forces."[149] He pursues this approach on CA 120vc (330av, 1497-1498): "Just as you find a rule to diminish weight with respect to a motive force, you will also find a rule to increase time with respect to motion."[150] This leads to the systematic list in the Madrid Codex cited below.[151]

Meanwhile, as of 1492, he has been developing his laws of perspective. They begin as quantitative demonstrations of systematic changes in the visual pyramid when it is intersected at various points by an interposed plane. These principles apply equally to representation and thus become the basis of his new perspectival laws of painting. This gives him a way of testing changes in visible images. The regularity with which the visual pyramid grows and diminishes becomes, for Leonardo, a model of nature's regularity. He develops a pyramidal law. By about 1500, on CA151ra (407r) he is combining this pyramidal law with his concept of the four powers of nature:

All the natural powers have to be or should be said to be pyramidal, that is, that these have degrees of continuous proportion towards their diminution as towards their growth. Look at weight, which in every degree of descent, as long as it is not impeded, acquires degrees in continuous geometrical proportion. And force does the same in levers.[152]

By about 1503, Leonardo is referring, on CA335vd (915br) to "a treatise of mine on local motion, force and weight," in which he emphasizes the use of instruments and speaks of their particular use and value in producing claims which are confirmed by experience.[153] In a paragraph on CA271re (732br, c. 1508) headed, "On local motion" Leonardo supports his claim by reference to "the fifth of the ninth which states...[154]," from which we can infer that his treatise is by now organized into books and propositions. This is almost certainly the treatise to which Pacioli in his publication of 1509 refers as nearing completion.

Leonardo continues working on these problems and makes plans to incorporate them into a treatise which also deals with more complex interplays of the four powers as we learn from CA81vb (220v, 1508-1510) where in a note "On the elements of machines" he outlines his new plan: [To study] "weight proportioned to the power which it moves one has to consider the resistance where such a weight is moved and of this a treatise will be done,"[155] (cf. pl.27-28). Meanwhile he has been collecting material on each of the individual powers. On CA298rb (818r, 1495) he refers to a fifth [proposition] of the seventh [book] with respect to weights.[156] On CA283vab (771v, 1517-1518) he refers to an "eighth book on weight."[157] This applies also to other powers. On CA384ra (1062r, 1493-1495) he refers to a "seventh proposition"[158] concerning percussion. By the time he is established in Rome he has enough material to organize it into book form as we learn from CA241ra (657v, 1513):

Divide percussion into books of which, in the first one there is demonstrated the percussion of two bodies of which one moves, the percussor to an immobile object; [in] the second book percussor and percussed move reciprocally, one against the other. A third is of liquid materials; a fourth of pliable objects; a fifth...[159]

On CA241vb (657v), i.e. elsewhere on the same folio, following a discussion of weight, Leonardo adds an important note: "The book on impetus precedes this and before impetus goes motion."[160] Impetus is another term for force in Leonardo's scheme. Hence we can infer that by 1513 Leonardo's work has resulted in books on each of the four powers which he intends to arrange in the order: motion, impetus [i.e. force], weight and percussion. About this time we also find him on CA374ra (6043v, 1515) planning to write a book on friction[161] in machines which presumably is intended as a further section in his Elements of machines.

Leonardo has in mind an even bigger picture as becomes evident from two lines of text on CA58ra (161r, 1503-1505): "Of two cubes, of which one is double, the other, as is proved in the fourth part of the Elements of machines composed by me."[162] In other words the Elements of machines which deals with principles of the four powers also deals with principles of geometry and geometrical problems.

At this juncture a digression is necessary. By way of context we need to examine developments in optics and perspective. Ever since Antiquity there had been discussions concerning how one could be certain that the eye was not being deceived. Ptolemy (c. 150) had explored criteria for this. These were examined in much greater detail by Alhazen[163] (fl. 1000-1030). In the Latin West, Witelo (fl. 1260-1280) worked in this tradition and considered astrolabes and quadrants as "instruments for the certification of sight[164]," the assumption being that the eye is readilly deceived and instruments are needed to insure against this. This philosophy was consciously in the minds of those responsible for the great proliferation of scientific instruments[165] in the latter fifteenth and throughout the sixteenth centuries. In this context the perspectival window was, in one sense, merely another instrument for the certification of sight. At the same time it introduced a new factor: an ability to record the image involved in a systematic way. To do this, however, required the use of ruler and compass. In short, perspective not only transformed the way pictures looked by giving them coordinated vanishing points: it did something basic to the process of representation by linking it in a fundamental way with instruments.[166] Moreover, instruments such as the compass had traditionally been linked with proportion and problems of geometry. Hence perspective brought into play a nexus of five unlikely elements: certification of sight, representation, instruments, proportion and geometry.

It was not until the period 1490-1510 that this nexus became apparent largely through the efforts of two individuals. Luca Pacioli played an important role in publishing his great Compendium of geometry, proportion, proportionality (1494), as well as emphasizing religious and metaphysical dimensions in his book on Divine proportion (written 1496-1499, published 1509). Meanwhile Leonardo played a more fundamental role. By 1492 he had demonstrated that perspective involved a systematic play of images which[167], he realized, were geometrical. Hence, visual transformations and geometrical transformations represented on the picture plane were recognized as being the same problem with a common solution: using instruments such as the compass.

By 1500, Leonardo is studying Euclid's Elements in some detail.[168] We know from a much later note on CA174v (476v, c. 1517-1518) that he wants to use Euclid to transform geometrical shapes.[169] Yet his goals are quite different from Euclid. On CA183rb (502r, c. 1500) Leonardo states: "I want to make of a circle an infinite variety of curvilinear figures of equal capacity."[170] At the outset he proceeds as if arithmetical and geometrical approaches are interchangeable in pursuing this goal. On CA141ra (386r, c. 1500-1505), for instance, he notes that "In equally diminishing one and the other extreme of each proportion arithmetically, the geometrical proportion will always increase accordingly."[171] Later he is conscious that there are exceptions, as on CA174va (475v, c. 1517-1518), where he observes: "But this calculation wants to be geometrical because, if you wished to do it by means of arithmetic it would be impossible."[172]

The study of square roots makes him more aware of the value of geometrical proportion. He comes to it relatively late. In 1504 we find him writing on CA120rd (331v): "Learn the multiplication of square roots from Master Luca Pacioli."[173] Four years later he is giving instructions on how to reach a solution and arrive at a rule for both square and cube roots on CA159ra (428v, c. 1508): "With the circle br you will make a rule of the square roots up until 20 and then, with another [circle] you will make another rule of cube roots to twenty and you will see the differences that there is from one rule to the other."[174] Later he simply gives the rule on CA102va (281r, 1516-1517): "If you wish the square root of any number, this is the rule..."[175] (There are hints that Leonardo intended to combine these studies of square and cube roots with the operations of his proportional compass, but no concrete evidence of this is found in manuscripts until some forty years after his death. Printed versions appeared in 1584, 1604, 1605,1606 etc. This nexus of mathematics and instruments goes hand in hand with the rise of trigonometry and is reflected in a title of a book by Bramer: Trigonometria planorum mechanica, 1617).

While Leonardo uses instruments such as the compass from the outset it is noteworthy that he only gradually accepts their validity in arriving at rules which he considers true. On CA231ra (629r, C. 1505), for instance, he makes a geometrical construction with a note: "Make a large one and you will see with greater certainty whether this is true."[176] On the same folio he remarks: "The mechanical proof is true even it if is with difficulty that one finds this truth."[177] Here he is dealing with the problems of two mean proportionals and duplication of the cube. His reference to geometry in his book of machines, it will be recalled, involves this same problem. So Leonardo's Elements of machines clearly has two programmes: one to give principles governing instruments and machines in terms of the four powers, a second to provide principles by means of which instruments can be used to represent geometrical truths and transform them systematically. All this is of particular interest to him because it links up with his standard of the visible and because, in being reversible and repeatable, it exemplifies his concept of true science.[178]

And so a nexus evolves which links instruments, geometry, proof and science. Sometimes, as on CA218va (587v, c. 1503-1505) he simply notes: "here the mechanical proof is given"[179], a phrase which reminds us that Leonardo's reference to mechanics being "the paradise of the mathematical sciences"[180] is something much more than an engineer's enthusiasm for machines and gadgets. It reflects a conviction that mechanical instruments provide new keys to mathematical demonstration and proof. By 1508, on CA220vb (593v), Leonardo is referring to a "geometrical rule."[181] Sometimes he carefully records the date of a new insight as on CA239v (627r) when, in connection with falcates (curved sections of circles used in his transformational geometry) he refers to a "first [proposition] which is found in this rule...on the third of March 1517."[182]

The rules to which he keeps referring become increasingly universal in their scope. On CA103va (285r, c. 1515-1516), for instance, he mentions how the rule in question goes on to infinity[183], a phrase which he also uses in connection with the four powers.[184] A year or two later he is confident enough to speak of general rules as on CA107va (297v, c. 1517-1518): "And concerning this diminution or augmentation one will give a general rule which, as we shall see with precision, has a clear note of the truth."[185] Systematic augmentation and diminution are again terms he uses in connection with the four powers. What is most striking about this passage, however, is the way in which the precision of instruments is implicitly associated with his concepts of general rule and visible truth, through constructed geometrical diagrams. It is hardly surprising, therefore, that Leonardo gradually uses geometrical diagrams as synonymous with the term demonstration.[186]

Proportion plays an ever greater role in this nexus of interests. As noted earlier he bought Pacioli's Compendium of geometry, proportion and proportionality when it was first published in 1494, and like Pacioli, he was convinced that proportion extended to all realms of nature.[187] A note on CA177rb (483ar, c. 1508-1510) mentions "and this is proved in the eighth of proportion"[188], which suggests that he is writing a book on this topic also. Later, on CA166vb (454r, c. 1515) he refers to "a rule to know the value and proportion of many curvilinear parts."[189] Indeed, by 1513-1515 he has invented his own compasses of proportion as is confirmed by two notes on CA 157vb (425v)[190] and CA385ra (1064r).[191] In a third note on CA248ra (672r, c. 1513) he refers specifically to "a compass of proportionality" showing it both in profile and face on, noting that its central axis is moveable and that "this works in irrational proportionality."[192] By about 1515, on CA83va (225v) Leonardo notes that "With this [proposition] of the Elements one can give any proportion of a circle, rational as well as irrational."[193]

Here the reference to Elements is once again to his magnum opus, which began as Elements of machines. As Leonardo's work on geometrical transformation progresses he refers to individual books and sections by a variety of names. On CA128ra (353r, c. 1508), for instance, he refers to a "Book of equations"[194] (in the sense of equivalent shapes). On a number of occasions he refers to treatises "On transmutation" (i.e. transformation)[195] and "On the geometrical game."[196] A note on CA45va (124v, 1515-1516) records the beginning of this latter project: "Having finished giving various means of squaring the circle, i.e. giving quadrates of equal size to those of the circle, and having given rules to proceed to infinity, at present I am beginning the book On the Geometrical Game and I shall once again give the means of infinite progression."[197] Later, on CA167r (455r, c. 1517-1518), he refers to a "Treatise on continuous quantity."[198] As early as 1508-1510, however, he refers to a work on "Curvilinear geometrical elements."[199] By 1513 he is referring to specific propositions in a work entitled Elements: a "first proposition[200]," a "fifth proposition[201]," the "43rd proposition of the first book[202]," the "last proposition of the second book."[203] All this points unequivocally to a much more systematic approach than has thus far been suspected. And this is confirmed by a passage on CA170ra (463v, c. 1516):

Many of these curvilinear circles of mine can be squared in themselves with the transmutation of their proper parts within their whole and there are many which cannot be squared with their own parts, but with parts taken from other surfaces produce quadrates equal to themselves. And with this I am composing my last work of 113 books in which 33 different ways are given of making rectilinear quadrates equal to circles, i.e. equal in quantity.[204]

An extraordinary picture thus emerges. Leonardo, who is frequently pictured as a chaotic amateur or even dismissed as a craftsman working in an intellectual vacuum, was engaged in a project the scope, coherence and system of which had never before been seen. The regularity of machines convinced him that there were a limited number of principles which could be identified. He found 21. Reuleaux, more than three and a half centuries later, was able to find one more.[205] While Leonardo was searching for these principles he became convinced that there were four basic powers underlying these: weight, motion, force and percussion. At first he limited his study mainly to static conditions, focussing on weight, using instruments to create model situations by means of which to test claims made within the abaco tradition. This led to his studies in Forster II.

Meanwhile, as his study of anatomy progressed, he developed a method of presentation based on Ptolemy's Geography. Just as Ptolemy started with the world followed by the provinces, so too did Leonardo begin with the whole human body followed by its parts.[206] Wishing to account for the principles of human movement, Leonardo focussed attention on two other powers: motion and force. By 1503, he was writing his treatise "On local motion, force and weight"[207] and this evolved into his Elements of machines, which he planned to serve as an introduction to his anatomical studies.[208]

As the scope of his vision widened so too did his search for original sources. In the period 1505-1508 we find translations from Jordanus Nemorarius' Elementa, De ratione ponderis and Liber de ponderibus, on CA154ra-va (416rv).[209] The latter of these texts is also cited elsewhere as on CA124ra (342r, c. 1508).[210] In this period he also studied Archimedes[211], Theodosius[212], an unidentified Zenofonte[213], and continued, of course, with his studies of Euclid. By this time he became aware also that if instruments were fundamental in providing model cases for testing propositions concerning nature, instruments were equally crucial in actually representing and demonstrating geometrically the principles involved. So what had begun as his Elements of machines led to a new branch involving Elements of geometry which was quite distinct from Euclid. Where Euclid used theoretical propositions in which diagrams were of incidental significance, Leonardo emphasized practical propositions in which diagrams played a fundamental role, functioning as demonstrations and sometimes even replacing verbal claims. Hence, whereas Euclid focussed on idealized verbal propositions, Leonardo emphasized constructed visible demonstrations. Euclid's aim was to catalogue the rules of static geometrical shapes. Leonardo's goal was to discover the systematic laws of how geometrical shapes could be transformed. He was not just interested in finding some handy solution. He wanted to find all possible solutions and, as we have seen, found 33 alternatives.

Fig. 7: Basic motivations underlying Leonardo's studies.

Leonardo's work on the Elements of machines and what might be termed his Elements of mechanical geometry thus became two parts of a single vision: to explain the created universe in terms of a constructed universe, that was simultaneously mechanical, geometrical, visible and therefore experimentally testable and capable of being both recorded and represented (fig. 7). That this second part of this project alone involved 133 books, in the sense of chapters, gives a sense of the enormity of this plan. The modern mind may see something manic in Leonardo's project and be tempted to dismiss it as over ambitious. To do so, however, would be to overlook the extraordinary optimism that made possible the Renaissance.

6.Method

In Leonardo's case this optimism sprang from his awareness that he had his own method for approaching science. Both experience and experiment (as in French the same word is used for both terms in modern Italian, although Leonardo sometimes distinguishes between them) are very much a part thereof. On CA125ra (c. 1490-1492) for instance, he notes "I find by experience that...[214]" On A47r (1492) he advises "Experiment as follows[215]," a phrase that returns on CA117va (1495) and D4v (1508). Sometimes he describes what is to be experimented as on CA338va (c. 1490): "Experiment on motion, the cause of the blow"[216] or CA151va (c. 1500) where he sets out: "To experiment the proportion of the intervals of descent."[217] Sometimes he describes precisely the means to be used, as in D3v: "To do an experiment how the visual power receives the [multiplication of] species of objects from its instrument, the eye, let there be made a sphere of glass five eights of a braccio in diameter."[218]

The interpretation of such passages has been a source of misunderstanding and controversy. In modern science there is a distinction between thought experiments carried out in one's mind and actual experiments using instruments and physical apparatus. Some scientists believed that Leonardo's ideas about science were purely theoretical, and thus assumed that he must have conducted only thought experiments. In 1972 this view was so strong that when the late Dr. Kenneth Keele and the author set out to examine whether Leonardo's claims about perspective had an experimental basis, the project met with considerable scepticism. This scepticism remained even when the evidence of the reconstructions established clearly that Leonardo must have carried out actual experiments. Since then the important work of Maccagno[219] has shown that a number of Leonardo's claims in the realm of hydraulics are also confirmed by actual experiment.

By 1490, the principles of classical geometry are a basic part of his experimental approach as we learn from CA109va: "Make simple propositions and then demonstrate them with figures and letters."[220] This he restates in more detail on A31r (1492): "I remind you that you should make your propositions and that you illustrate the things written above with examples. If you did so with propositions it would be too simple."[221] Hence we find in Manuscript A phrases such as "proposition proved by experience"[222], "proposition confirmed by experience"[223], "proved by experience"[224] or paraphrases such as "proof"[225], "the cause of the proposition"[226], "this case is seen manifestly"[227] or "this is demonstrated clearly."[228] Analogous phrases are found in BN 2038[229] which was originally part of Manuscript A. When an experiment has been carried out Leonardo explicitly writes: "experimented". there are examples in Madrid Codex II[230], Codex Arundel[231] and particularly Forster II[232], which contains no less than sixteen such cases.

We also find the equivalent of thought experiments, where the conditions are considered beforehand because Leonardo is conscious that there can be debate over that which constitutes a good experiment as on CA126va (1487-1490):

And if you say that this is not a good experiment since water in itself is a unified and continuous quantity and millet is discrete and discontinuous, at this point I reply to you that I wish to take the license that is common to mathematicians, that is, just as they divide time into degrees and from a continuous quantity make it discontinuous, I shall do the same in comparing millet or gravel to water.[233]

Experiment becomes, for Leonardo, linked specifically with things which are visible and can be represented. On CA86ra (1490-1492), for instance, he notes that "experience, interpreter of artifice filled nature, demonstrates that this figure is necessarily constrained not to operate in ways other than is here represented."[234] On CA274vb (c. 1495), he adds: "Make this figure return in experience before you judge it[235]," an idea which he expresses slightly more forcefully on F91v (c.1508): "all these figures have to come out of experience."[236] In the Codex Arundel he notes laconically: "I tested it myself, drawing it."[237] Underlying this connections between experiment and figures is a more fundamental conviction on Leonardo's part that figures and illustrations constitute visible evidence which is the basis of science. Indeed we find him gradually developing an opposition between visible and invisible as summarized in figure 1.

Visible Invisible
Concrete mathematical (mechanical) Abstract mathmatical
Practical Purely Theoretical
Coporeal Incorporeal
Physical Mental
Material Spiritual
Dynamic Static

Fig. 8 Contrasts between visible and invisible characteristics.

Leonardo's studies of perspective brought this distinction into focus. On a perspective window visible objects can be traced; invisible objects cannot. A measured relation between object and image is only possible if the object is visible. Perspective thus called for a distinction between visible objects which could be recorded, represented and measured on a picture plane and invisible objects which could not, and the quest became to bring things into the realm of the visible. Here models played an important role. Leonardo dealt with mathematical forms in terms of physical models. At the same time he sought to deal with both organic forms and abstract concepts in terms of these same kinds of physical models.[238] The challenge became to distinguish visual and non-visual reality.

In the case of motion, for example, which Aristotle had defined in general terms, Leonardo uses his criterion of the visible to cut through various meanings on CA203va (1495-1495): "But let us say that the kinds of motion are of two natures, of which the one is material, the other is spiritual because it is not understood by the sense of sight, or let us say that the one is visible and the other is invisible."[239] Leonardo uses the same criterion with respect to weights on CA93vb (c. 1513): "I have found that these ancients were led astray in this judgement of weights and this deception arose because in part of their science they used corporeal poles and in part [they used] mathematical poles, that is, mental or incorporeal ones."[240] Similarly, he uses this criterion to distinguish between abstract mathematics and concrete mechanics on CA200r (c. 1515): "Between the mechanical and mathematical point there is infinite difference because this mechanical point is visible and consequently has continuous quantity."[241] Consistently Leonardo is concerned with focussing on visible knowledge. In this context his oft quoted phrase on Manuscript E8v (1513-1514): "Mechanics is the paradise of the mathematical sciences"[242], takes on deeper meaning. As a result of this approach he devotes passages in the Windsor Corpus to show that visual knowledge through figures is superior to verbal description.[243] On CA221vd (c. 1490) he notes: "These rules are to be used by checking the figures."[244] Diagrams and figures become a basic aspect of his method as is clear from a comment on CA274ra (c. 1495): "I make many figures in order that you know all the cases which are subjected to a single rule."[245]

Leonardo's use of the term rule in the context of this nexus of figures and experiment is no coincidence. One of his earliest uses of this term on CA149rb (c. 1487-1490) is in the sense of order[246] with respect to chapters in a book. By 1490 on CA86va he is referring to a rule of pulleys[247] and, on CA119va, he is also articulate about his use of experience and where his rules stand in relation to this:

Many believe that I should reasonably start again, alleging that my proofs are against the authority of some men who are greatly esteemed with their inexpert judgments, not considering that my things are born of simple and mere experience which is a true mistress.

These rules are a ground to make you know the true from the false which thing permits that men promise themselves things which are possible and with more moderation and that you do not hide ignorance which would lead to not having effect and in your desperation, give yourself melancholy.[248]

This idea he restates on CA337rb (c. 1493-1495): "Effect of my rules....They hold a bridle to engineers and investigators not to let them promise to themselves or to others things which are impossible and make themselves either mad or cheaters."[249] It is significant that this same quest to avoid false promises also enters into his discussion of experience on CA154rb (1508-1510): "Experience never fails. Only your judgments fail, promising of some effect that which is not caused in our experiments."[250]

On occasion Leonardo uses "rule" in referring to the work of Euclid[251] or Pythagoras.[252] But elsewhere he uses the term specifically in connection with experiment as on CA153vd (1493-1495): "Test and make a rule of the difference that there is between a blow that is given with water onto water and water which falls on a hard surface"[253] or on CA337rb (1493-1495): "Again make a rule of the different trajectories of the ball."[254] This approach is restated on Mad I 51r (c. 1499): "Make experiments and then the rule."[255] In the same manuscript he speaks of applying the same rule that one uses for dragging for the study of pushing.[256] Sometimes as on CA271vb (1508) he refers laconically to: "rule."[257] By the late period the term, rule, has acquired another connotation, reversibility, as on CA130va (1517-1518):

If a rule divides a whole in parts and another rule recomposes these parts into such a whole, then both rules are valid. If by a certain science one transforms the surface of one figure into another figure, and this same science restores such surface into its first figure then such a science is valid. The science, which restores a figure to the first shape from which it was changed, is perfect.[258]

Notable here is Leonardo's geometric model for science. By this time, rule, science and reversibility, in the sense also of repeatability, have become well established in his method. Meanwhile, Leonardo has also been developing a concept of a general rule which he defines succinctly on Mad I 129r: "When a rule is confirmed by two different reasons and experiments, then this rule is said to be general."[259] One of his earliest references to this concept comes in a note on CA20va (1493-1495): "To make a general rule of the difference there is between a simple weight and a weight with percussion of different motions and forces[260]," and on CA82rb (1493-1494) which again deals with weights.[261] A note on CA253va (1493-1495) links this concept with a systematic quantitative approach:

General rule: to know about a beam tied to the extremity of a cord, which is drawn from a single place, and is lifted at its base, and to know how to say, in all the degrees of its raising, how much weight there is in its motor.[262]

Further notes occur on CA268va [263] (1493-1495) and CA155vb [264] (1495-1497). He pursues this theme on Mad I 60r mentioning what to do "if you wish to make a general rule[265]," on Mad I 77r where he notes that he has "experimented and it is a general rule"[266], and on Mad I 170v [267] and 171v [268] where he simply notes that a "general rule" is involved. Read together these passages leave little doubt that, while Leonardo is concerned with practical experience and experiment, his quest is also to find a theoretical set of rules. As he states on Mad I 164v: "this demands practice, but remember to put the theory forward[269]," an idea which he expresses afresh on CA147va (c. 1500): "No effect in nature is without a cause. Understand the cause and you do not need experience."[270] Indeed Leonardo explicitly develops a concept of laws of nature in a passage on Mad I 152v:

See what a wondrous thing it is to consider what (this) nature adopts in all its objects and with what laws it has terminated the effects of all the causes, the least part of which it is impossible to change."[271]

How was it that Leonardo became so convinced that nature had rules and even laws? I have shown elsewhere that in the case of linear perspective he arrived at an understanding of its basic laws by a systematic play of three basic variables: eye, picture plane and object.[272] Kenneth Keele has demonstrated the importance of perspective for Leonardo's anatomical studies and has called perspective Leonardo's gateway to science.[273] Indeed his study of perspective convinced him that if he could apply his concept of systematic variation to both mathematics and nature he would arrive at the laws of science. In this quest Leonardo resorted to a particular kind of list making which is important because it confirms that he is systematically playing with variables in a manner basic to early modern science. One of the earliest of these lists, on CA116rb (1495-1498), concerns light sources and objects (pl. 13, cf. pl. 23-24):

Several lights with one object
One light with several objects
Several lights with several objects
Several lights above one object[274]

Plate 23
Image 49a. Image 49a. Image 49b. Image 49b.
Plate 24
Image 50a. Image 50a. Image 50b. Image 50b.

In isolation this list would have limited interest. But it becomes highly significant when we discover that Leonardo's notebooks contain many diagrams without text that exemplify precisely this approach. This method of playing with variables guides Leonardo in conceiving his Seven Books on Light and Shade and indeed all his optical studies[275]. However sceptics may rightly object that the existence of diagrams which can be arranged by others in a systematic fashion does not prove that Leonardo was systematic, or that he even intended such order. We need his word for it and fortunately it exists in the form of lists in various domains of his work. These confirm that Leonardo consciously plays with variables. On CA147va (c. 1500), for instance, he applies this principle to counterweights under the heading:

The regular natures of counterweights which press against the reservoir are 9, i.e.
Wider than the reservoir and heavier
Wider than the reservoir and lighter
Wider than the reservoir and equal
Narrower than the reservoir and heavier
Narrower than the reservoir and lighter
Narrower than the reservoir and equal
Equal to the reservoir and heavier
Equal to the reservoir and lighter
Equal to the reservoir and equal[276]

Here the essential elements of his method can be seen clearly. Leonardo takes one variable, in this case size, keeps it constant, while considering three kinds of weight (heavier, lighter, equal), then chooses another size and again holds it constant as he changes the weight variable. It is typical of Leonardo that he applies this systematic play of variables equally to declensions of verbs (pl. 9)[277]. Perhaps inspired by the work of grammarians he uses the same method to illustrate combinations of vowel sounds as on W19115r (K/P 114v, 1506-1508). Here he begins with the vowel "a", adds this to each consonant ofthe alphabet then does the same with "e" and the other vowels (pl. 10):

a e i o u
ba be bi bo bu
ca ce ci co cu
da de di do du
e
fa fe fi fo fu
ga ge gi go gu
la le li lo lu
ma me mi mo mu
na ne ni no nu
pa pe pi po pu
qa qe qi qo qu
ra re ri ro ru
sa se si so su
ta te ti to tu
Plate 10
Image 56a. Image 56a. Image 56b. Image 56b.

Such a list could readily be seen as an amusing game. But it is not in isolation and the way in which such list making is systematic becomes more apparent when we examine how Leonardo applies this method to geometry. There was a well established Renaissance interest in transformations of geometrical shapes known as the geometrical game (de ludo geometrico). Alberti had written a book on this[278], which Leonardo studied, as we know from a note on Arundel 66r.[279] On CA99vb Leonardo defined the geometrical game as giving "a process of infinite variety of quadratures of surfaces of curved sides."[280] But it soon became much more than a game. Leonardo saw it as a key to all systematic transformations of forms. On Arundel 154r (c. 1505), for instance, he explores basic transformations involving a pyramid.

a pyramid [is] extended to a given length
a pyramid [is] shortened to a given lowness
from a pyramid one makes a cube
from a cube one makes a pyramid
from a cube one makes a pyramid of a given height
from a pyramid of a given height one makes a cube

from a pyramid one makes a table of a given thickness
from a pyramid one makes a table of a given width
from a pyramid one makes a table of a given width and thickness[281]

Leonardo also makes lists of different kinds of transformations possible in geometrical objects. On CA245vb (1505-1506), for instance, he mentions: to shorten, lengthen, make fat, make thin, widen, restrict.[282] These he crosses out and then makes a list of twelve kinds of simple transmutation.[283] Eleven kinds of composite transmutation follow.[284] Again he crosses these out[285] and on Forster I 12r-11v (c. 1505) he uses these ideas as the basis for an extraordinary list of twenty eight kinds of transformation, the first twelve of which correspond to the simple kind, whose one aspect does not change and the remaining sixteen of which are composite, i.e., where all the aspects change (pl. 11):

1 shorten as much as one widens without changing the size
2 shorten as much as one thickens without changing the width
3 lengthen as much as one squeezes without changing the size
4 lengthen as much as one makes thin without changing the width
5 fatten as much as one squeezes without changing the length
6 fatten as much as one shortens without changing the width
7 thin as much as one widens without changing the length
8 thin as much as one lengthens without changing the width
9 widen as much as one thins without changing the length
10 widen as much as one shortens without changing the size
11 squeeze as much as one thickens without changing the length
12 squeeze as much as one lengths without changing the size
13 shorten and fatten as much as one widens
14 shorten and thin as much as one widens
15 shorten and widen as much as one size
16 shorten and squeeze as much as one fattens
17 lengthen and fatten as much as one squeezes
18 lengthen and thin as much as one widens
19 lengthen and widen as much as one thins
20 lengthen and restrict as much as one fattens
21 fatten and widen as much as one shortens
22 fatten and restrict as much as one lengthens
23 thin and widen as much as one lengthens
24 thin and restrict as much as one lengthens
25 fatten and lengthen as much as one restricts
26 fatten and shorten as much as one widens
27 thin and lengthen as much as one squeezes
28 thin and shorten as much as one widens[286]

This list comes at the end of a treatise with three books of numbered propositions cited earlier. We know, moreover, that Leonardo continued to work on these problems[287] In the last years of his life, on CA136ra (1517-1518) for instance, he makes another systematic chart relating to geometrical transformation (pl. 12):

Equal sagittas and chords have equal arcs
Equal sagittas and arcs have equal chords
Equal chords and sagittas have equal arcs
Equal chords and arcs have equal sagittas
Equal arcs and sagittas have equal chords
Equal arcs and chords have equal sagittas[288]

The regularity of these geometrical transformations led Leonardo to use them as a model for his concept of science (cf. above). Hence, both his transformational geometry and science became based on principles that were universal, reversible and repeatable.

The universality of this enterprise became apparent as he applied it to his study of nature. As we have shown Leonardo developed a mechanical model of nature. His study of machines convinced him that nature involved a surprisingly small number (21) of physical parts[289], governed in turn by basic powers of nature[290]. By 1492, Leonardo had become convinced that there were four such underlying powers of nature: force, motion, gravity and percussion. He described a series of preliminary experiments involving these powers in Manuscript A.[291] It is not until Mad I 152v (1499-1500), however, that we find evidence of systematic study which he prefaces with a brief, explicit statement that he is here making a thought experiment in trying to ascertain the laws of nature:

I have 4 degrees of force and 4 of weight and, similarly, 4 degrees of motion and 4 of time. And I wish to make use of these degrees and as necessary, I shall add or subtract in my imagination to find out what is required by the laws of nature.[292]

Leonardo then takes three of his powers of nature, plus the factor of time, and presents them as a systematic play of variables (pl. 18):

2 of weight and 4 of force and 4 of motion require 2 of time
2 of weight and 2 of force and 4 of motion require 4 of time
2 of weight and 2 of force and 2 of motion require 2 of time

2 of force and 4 of weight and 4 of motion require 8 of time
2 of force and 2 of weight and 4 of motion require 4 of time
2 of force and 2 of weight and 2 of motion require 2 of time

2 of motion and 4 of force and 4 of weight require 2 of time
2 of motion and 2 of force and 4 of weight require 4 of time
2 of motion and 2 of force and 2 of weight require 2 of time

2 of time and 4 of force and 4 of weight require 2 of motion
2 of time and 2 of force and 4 of weight require 1 of motion
2 of time and 2 of force and 2 of weight require 2 of motion

1 of force and 4 of weight and 4 of motion require 16 of time
1 of time and 4 of motion and 4 of weight require 16 of force
1 of motion and 4 of weight and 4 of force require 1 of time
1 of weight and 4 of motion and 4 of force require 1 of time[293]

For our purposes the question whether these calculations are correct is of less interest than the conviction that a systematic approach will inevitably reveal the laws of nature. Leonardo never uses modern algebra in this process. It is significant, however, that he sometimes treats these basic variables as abstract symbols. On CA212vbb (1502-1504), for example, he considers power (p), a variant name for force; space (s); motion (m); and time (t) and in addition to his verbal descriptions[294], produces a chart which summarizes these variables, underlining a different one each time (pl. 17):

p s m t
p s m t
p s m t
p s m t
p s m t

He makes another list for power, weight (g, i.e., gravita), motion and time.[295] He develops similar lists on CA355va (1502-1504) adding quantitative values to the symbols: e.g. s2 and t2.[296] We must take care not to read Galilean physics into this. Yet Leonardo's approach helps us to reconstruct the context which made Galileo's enterprise possible.

Leonardo pursues this theme by applying the same systematic play of variables to individual powers of nature. In the case of motion, for instance, he makes lists pertaining to different kinds thereof on CA165va (c. 1500-1503) (pl. 15):

On simple and composite
Straight , curved and straight
Curved, straight and curved
Curved and straight, straight
Straight and curved, curved
On composite
Curved and straight, straight and curved
Curved and curved, straight and straight
Straight and straight, curved and curved
Curved and curved, curved and curved
Straight and straight, straight and straight[297]

Similarly Leonardo makes a list of different kinds of mobile objects and surfaces on CA193rb (c. 1500), once again applying his method of systematic play with variables (pl. 16):

Hard mobile with a hard plane
Soft mobile with a soft plane
Hard mobile with a soft plane
Soft mobile with a hard plane
Rough mobile with a polished plane
Polished mobile with a rough plane
Rough mobile with a rough plane
Polished mobile with a rough plane[298]

Leonardo uses the same method with respect to percussion, another of his four powers of nature when, on CA74vb (1506-1508) he makes a list of possible kinds of percussion in water.

Encounters of water equal in power and in quantity
Encounters of water equal in power and not in quantity
Encounters of water equal in quantity and not in power
Encounters of water not equal in power and not in quantity[299]

This systematic approach to percussion is even more evident in his plan on CA65va (c. 1508) to study (pl. 14):

Percussion of rare in rare
Percussion of rare in dense
Percussion of dense in rare
Percussion of dense in dense[300]

The Windsor Corpus provides evidence that Leonardo is collecting these ideas in a systematic fashion, with the explicit purpose of writing a book. On W19141v (K/P 99v, 1506-1508), for instance, he notes: "In this 4th book I have to treat of six things as instruments, that is, the axle, round beam, lever, cord, weight and motor."[301] On the same folio he outlines the elements necessary to study: "the nature of the working parts required for the functioning of the capstan."[302] Directly beneath this is another of his charts with six variables (pl. 19a):

Given the axle, round beam, lever, cord and weight one seeks the motor
Given the round beam, lever, cord, weight and motor one seeks the axle
Given the lever, cord, weight, motor and axle one seeks the round beam
Given the cord, weight, motor, axle and round beam one seeks the lever
Given the weight, motor, axle, round beam and level one seeks the cord
Given the motor, axle, round beam, lever and cord one seeks the weight[303]

On the same folio Leonarto considers another combination, this time of five variables (pl. 19b):

Given the lever and counterlever, fulcrum and weight one seeks the motor
Given the counterlever, fulcrum, weight and motor one seeks the lever
Given the fulcrum, weight, motor and lever one seeks the counterlever
Given the weight, motor, lever and counterlever one seeks the fulcrum
Given the motor, lever, counterlever and fulcrum one seeks the weight[304]

He pursues these problems on W19143r (K/P 101r, 1506-1508) where he outlines the elements involved in a screw (pl. 20a):

Given the screw, screwthread, number, lever and weight one seeks the motor
Given the screwthread, number, lever, weight and motor one seeks the screw
Given the number, lever, weight, motor and screw one seeks the screwthread
Given the lever, weight, motor, screw and screwthread one seeks the number
Given the weight, motor, screw, screwthread and number one seeks the lever
Given the motor, screw, screwthread, number and lever one seeks the weight[305]

And on the same folio he makes a corresponding list pertaining to the parts of pulleys (pl. 20b):

Given the diameter, number, axis, weight and cord one seeks the motor
Given the number, axis, weight, cord and motor one seeks the diameter
Given the axis, weight, cord, motor and diameter one seeks the number
Given the weight, cord, motor, diameter and number one seeks the axis
Given the cord, motor, diameter, number and axis one seeks the weight
Given the motor, diameter, number, axis and weight one seeks the cord[306]

This is followed by a note which leaves little doubt that Leonardo is proceeding with a systematic plan in mind:

The parts of the pulleys given above are the diameter of the wheels of these pulleys, and the number of the wheels and the thickness of the axle which is within every wheel and the quantity of weight which is sustained by the pulleys and the thickness of the cord which pulls the weight, and the motor of this weight, which said things are six. Now five of them are given and the sixth is sought. This is indeed subtle investigation and will never be made without its theory, that is, the definition of the four powers, as weight, force, motion and percussion.[307]

This passage reveals why Leonardo is at such pains to study systematically the characteristics of weight, force, motion and percussion. These four powers of nature have become the basis of his theory of nature. Theory, moreover is here used in a special sense. Leonardo is claiming that one needs theory to provide a structure for, and to organize, the practical experience and experiments at one's disposal. Theory and practice are now interdependent. By contrast, in Antiquity and throughout most of the Middle Ages there had been an tendency to oppose theory and practice. This grew out of an assumption, supported by neo-Platonism, that theory was noble and practice was base. Hence, Plato's Timaeus was, for instance, replete with abstract thoughts and claims in isolation, with minimal reference to practical experience and no records of practical experiments. Lucretius' theory of the universe was presented in poetic form, and even the treatise of a practicing architect Vitruvius gave instructions in abstract terms without mention of practical variants. Vitruvius was concerned with how an Ionic column should look and did not discuss whether this was confirmed by examples of Ionic columns in Rome or Athens. For Vitruvius and his classical colleagues it was a question of theory versus practice. Leonardo's work convinces him of the need for a fundamentally different approach in which practical experience, experiment and testing using the controlled conditions of machines (pl. 25-28) will provide a basis for his theory.

Leonardo's paragraph is headed with a brief note: "The exercise and nature of the parts of pulleys and their relationships - 4th book."[308] Mention of the 4th book (in the sense of a chapter), confirms that this is intended to be part of the work cited above. A further note on W19060r (K/P 153r, c. 1509-1510) describes the contents of the book of which this was to have been a part.

On machines
Since nature cannot give motion to animals without mechanical instruments as I demonstrate in this book on the motive works of nature made in animals. I have, for this reason, composed the rules in the 4 powers of nature without which nothing can give local motion to these animals.[309]

Elsewhere on W19070v (K/P 113r, c. 1508-1510) Leonardo tells us that "the book of the science of machines precedes the book of the movements."[310] Is this book on the science of machines the same book as that to which Pacioli referred as being near completion in 1509 in the passage cited earlier? Of this we cannot be certain. There can be no doubt, however, that Leonardo was working methodically, that his lists of variables provided him with a means of studying controlled situations systematically. When applied to his transformational geometry this led to the treatise in Forster I which became a basis for later writings. When applied to his concept of the four powers of nature (weight, force, motion and percussion), this same method of listing variables which were to be experimentally tested, inspired further books. The next step, as was suggested above, was to combine these two sets of findings into a new synthetic vision. Hence the systematic play of variables which grew out of perspectival studies not only furnished Leonardo with a method. It persuaded him that he had something to say; was the reason for his notebooks and why he hoped to present his ideas in published form.

Seen in the context of centuries, Leonardo's work could be seen (indirectly) as a first draft for Descartes' Discourse on Method. It could also be seen as more. Leonardo's programme called for a systematic experimental catalogue of mechanical powers which for him constituted nature's principles. It took half a century before there were enough instruments around for this programme to become universal and another fifty years before the instruments were sufficiently accurate for this universality to attain the level of precision which made possible the syntheses of Kepler, Galileo and Descartes. The goal of explaining nature's principles could then be joined with a long standing goal of a systematic encyclopaedia of nature's contents, that is usually remembered as Baconian science.

7. Plans for Publication

It would be misleading to assume that the notebooks are solely treatises waiting for a publisher. The notebooks also contain very different kinds of material some military, some personal, some effectively lab notes and in these cases Leoanrdo is obviously less interested in communicating his ideas.

His military notes are almost always secretive, although when he writes to Ludovico Sforza, the Duke of Milan he offers to teach him "my secrets."[311] In everyday work he is guarded. In the Codex attanticus, for instance, he makes a note to himself to "make this secret."[312] He has good reason to be cautious. While he was Rome (1512-1515) working on burning mirrors, considered to be of great military use, there was a German competitor who practiced an early form of industrial espionage, trying to steal his ideas, not balking at writing hate letters to the pope.[313]

Some of Leonardo's notes are personal. Sometimes it is to remind himself that he has done something as when he notes "On the first day of August 1499, I wrote here on weight and motion."[314] Sometimes it is to remind himself to do something: "Tomorrow make the figures descending through the air, of various forms of carton, falling from our little bridge and then draw the figures and the motions which descents each one makes in various parts of their descent."[315] Sometimes this idea is put more succinctly as: "Experiment of tomorrow[316]," and elsewhere a larger time frame is involved: "Here one will make a record of all those things which have to do with the bronze horse which is presently in preparation."[317] In the personal notes we also find confirmation that he is concerned with spreading his ideas. For instance, in the midst of his studies of birds there is a revealing little note on CA214rd (c. 1507-1508). "Tomorrow look at all these cases, then copy them and cancel the originals and leave them in Florence, in order that if you should lose those which you are carrying with you, the invention will not be lost."[318] With comments such as this, it comes as no surprise that Leonardo's notes give evidence that he was writing to be read. On numerous occasions Leonardo refers specifically to readers. The most famous example is in the Madrid Codex I:

Read me, o reader, if you delight in me, because they are very rare the times that I am reborn into the world. Because the patience of such a profession is found in few who wish to recompose anew similar things once more. And come, o men, to see the miracles which by such studies are discovered in nature.[319]

There are other instances in the Codex Atlanticus. One is headed "On motion and weight: But make sure, o reader, that in this case you know to take into account the air."[320] Another refers to ancient philosophy: "Now observe, o reader, that which we can believe of our ancients who wished to define what kind of a thing is soul and life, unprovable things, while those things which at any hour can be known clearly and tested have been ignored and falsely believed for so many centuries."[321] In a third case Leonardo writes: "I request you, o reader, that when I speak of beam, that you understand that I wish to say a piece of equal length and weight, that is a body which has a length of equal weight and thickness."[322]

On other occasions Leonardo gives instructions to specific readers. When he writes to Diodarius of Soria, the lieutenant of the sacred Sultan of Babylon, "Do not be dismayed, O Diodarius, by the tardiness of my reply to your desirous request[323]," an imaginary reader may be involved. But elsewhere the persons addressed sound hardly fictive. In the Madrid Codex, for instance, Leonardo writes: "I remind you, o constructor of instruments."[324] In BN 2038 Leonardo refers to what the painter must consider, in the third person.[325] But on one occasion at least he shifts to the second person: "Hence, since you, o painter, know."[326] Similarly he writes "When you, o draughtsman, wish to make a good and useful study."[327] Elsewhere this becomes a plural: "When you, o draughtsmen, wish"[328] and in like fashion: "If you historians or poets or other mathematicians had not seen things badly with the eye..."[329] In the Windsor Corpus there is further direct discourse: "o observer of this machine of ours, do not be saddened that through the death of another you give knowledge but rather, rejoice that our author has fixed the intellect on such an excellent instrument."[330] In Manuscript E (1513-1514) he refers again to painters: "remind yourself, o painter, that the shades of shadow are as varied..."[331] and on the next folio: "O anatomical painter".[332] In a late passage in Manuscript G (1515-1516) Leonardo notes: "O observer of things do not praise yourself for knowing things which nature ordinarily conducts on its own, but take delight in knowing the cause of those things which are drawn in your mind."[333]

There is a larger context which makes these references to specific readers more important, namely, the hundreds of passages written in the second person. As we have noted, a few of these are Leonardo's reminders to himself. But many unequivocally assume a reader, as for instance a passage in the Codex Atlanticus where Leonardo writes: "I stated in the 7th conclusion how percussion....Now you for yourself experiment how the stick..."[334] Sometimes it is in the form of a question: "I ask you."[335] As we have seen above, this is part of his method. Many times instructions are intended to help readers repeat his experiments. Other passages confirm that he specifically planned to publish his work. In the Windsor Corpus, for instance, he makes a plea:

But through this very concise way of drawing it [i.e. the human body] in its various aspects one will give a complete and true knowledge and in order that this benefit reaches men, I teach the ways to print it methodically and I pray ye, o successors, that avarice not constrain you from printing it.[336]

Leonardo designed his own printing presses[337] and in the Madrid Codex there is a fascinating passage where he describes his method:

Of casting this work in print.
Coat the iron plate with white lead and eggs and then write on it lefthanded, scratching the ground. This done you shall cover everything with a coat of varnish, that is, a varnish containing giallolino or minium. Once dry, leave the plate to soak, and the ground of the letters, written on the white lead and eggs, will be removed together with the minium. As the minium is frangible, it will break away leaving the letters adhering to the copper plate. After this, hollow out the ground in your own way and the letters will stay in relief on a low ground. You may also blend minium with hard resin and apply it warm, as mentioned before, and it will be frangible.In order to see the letters more clearly, stain the plate with fumes of sulphur which will incorporate itself with the copper.[338]

This method would have given right way round printing and raises a fascinating possibility. Leonardo's notebooks contain a number of particularly clear drawings combined with a very careful handwriting. Were these drafts for the method described above? If so the very mirror script that is usually cited to prove that Leonardo was secretive and obtuse, may be evidence to the contrary.

Leonardo continued trying to get his work published and these attempts continued after his death as we learn from Vasari:

N.N., a painter of Milan, also possesses some writings of Leonardo, written in the same way, which treat of painting and of the methods of design and colour. Not long ago he came to Florence to see me, wishing to have the work printed. He afterwards went to Rome to put it in hand, but I do not know with what result.[339]

If this was the Treatise of painting, then we know in retrospect that it was not until a century later, namely, 1651, that the text was published.[340]

8. Influence

Vasari's claim about Leonardo's notebooks being read by others has an unexpected confirmation. There is physical evidence of actual readers in the notebooks themselves. The notebooks are written in mirror script. But in the Codex Trivulzianus, for instance, we find at least a half dozen instances where someone has written in ordinary script that this is a note[341] about architecture[342], water[343], painting[344] or a battle.[345] In Forster I, we find another note written in ordinary script: "This is a book entitled on transformation, that is from one body into another without diminution or augmentation of material."[346] Forster II contains a similar note in Latin: "Most powerful mechanics beginning at the end[347]" and five notes instructing one to invert the book[348] (i.e. read it in a mirror) followed by another: "N.B. This writing is inverted and is to be read in a mirror[349]," which phrase is repeated at the beginning of Forster III[350] and then repeated in abbreviated form another half dozen times.[351] Manuscript B has at least 49 notes in Spanish written right side round identifying the subject matter.[352]

But can we prove that Leonardo influenced others? There is some direct evidence. We know that Dürer had access to at least two folios of Leonardo's anatomical studies which he copied in reverse form into his Dresden Sketchbook[353]. Professor Putscher has drawn attention to parallels between Leonardo's anatomy and a series of drawings published by Titian, who may have provided a link with Vesalius.[354] Professor Pedretti has noted copies of mechanical drawings in Florence and Munich[355], and has brought attention to various sixteenth century manuscript copies of the Treatise of Painting[356]. Leonardo's instruments were studied by the clockmaker, Lorenzo della Golpaia,[357] who copied a number of them in his manuscripts.

Some of the evidence is indirect and more in the manner of smoking guns than the kind which would necessarily convince a jury. We know that from 1515 until the time of his death in 1519 Leonardo was in France where he served the king as mathematician, engineer, and in other capacities. It is therefore of some interest to note that there are close parallels between Leonardo's transformational geometry and the work of Claude de Boissière who was a mathematician to the king of France in the generation after Leonardo[358]; or similarly that a surveying instrument which Leonardo describes in the Codex Arundel, should have a close parallel in an instrument developed by Abel Foullon who was also an engineer to the king of France after Leonardo[359].

Leonardo had a particular interest in compasses and several types are known to have been copied directly by Lorenzo della Golpaia. Another type compass explored by Leonardo was developed into the Mordente compass.[360] As already noted above, in the Codex Atlanticus there is also a new kind of adjustable compass which Leonardo designates as a proportional compass (pl. 29a)[361]. Two generations later this compass of proportion found its way to Nürnberg, where it became part of a manuscript on perspective attributed to Lencker[362]; to Kassell, where Bürgi developed a particular version which became so linked with his name through publications by Hulsius (pl. 30)[363] and Bramer[364] that this instrument is still frequently assumed to have been his invention[365]. Probably via Mordente, word about this compass also reached Antwerp where Coignet[366] developed them in manuscripts that spread to Brussels, Paris, Madrid, Modena, Florence, Rome and Naples. The Coignet manuscripts are of further interest for two reasons. They acknowledge that some individuals at the time associated the instrument with Michelangelo although the original inventor of this instrument was already forgotten. Secondly these manuscripts contain an alternative form of this instrument which corresponds to that which Galileo claimed to have invented (1606)[367]. It is perhaps instructive to note that Galileo made analogous claims about having invented the telescope which is another instrument concerning which there is evidence in Leonardo's notes (pl 31-32)[368]. Galileo's fame is also firmly linked with his inclined plane experiments yet another theme that Leonardo explored a century earlier (pl. 33-34).

Plate 34
Image 71a. Image 71a. Image 71b. Image 71b.

Some scholars might claim that such isolated examples of technology transfer have nothing to do with science in a deeper sense. In this context it is important to recall a fundamental shift in method that Leonardo helped to bring about: whereby geometry and science were linked through representation and construction, made possible through instruments and whereby one needed instruments to demonstrate geometrical principles. One consequence of these profound changes was that transformational geometry using instruments became part of the perspectivists' task. Danti's (1583) description of perspective in terms of transformational geometry reads like a direct paraphrase of Leonardo's own goals:

Since beyond the description of rectilinear figures it is very useful for the perspectivist to know how to tranform one figure into the other, I wish...to show the normal way, not only to transform a circle and any other rectilinear figure that is wished into another but also move to expand and diminish it in any proportion that is desired, in order that in this book the perspectivist will have all that is required for such a noble practice.[369]

The rise of universal measuring devices including various kinds of proportional compasses was a second consequence[370]. A third involved the way in which astronomy was studied. Leonardo's conviction that each planet is at the centre of its own elements, led him to the study the elements of earth, air, fire and water in relationship to centres of gravity. Astronomy and cosmology became for him problems of both geometrical and physical models. So he developed instruments to observe the heavens and he built orreries to explore the relationships of various planets. In the next generation this idea of model making was taken further by Peter Apian, author of the Astronomicum Caesareum (Ingolstadt, 1540), which used perspective to create three dimensional views of sections of the heavens and employed elaborate movable volvelles to picture relationships of planets and stars. He was, of course, visualizing the Ptolemaic world view and the details of his approach were occasionally "wrong." But the model making impulse remained alive and Jost Bürgi, a member of the Brahe circle at Kassell who studied the book, made his own physical models of the universe linked up with clock mechanisms[371]; while Kepler took Plato's abstract theories from the Timaeus and tried to construct physically a model of the universe involving the five regular solids, which he described in detail in his Mysterium Cosmographicum (1596). Ultimately it was the discrepancy between this instrumental model of the universe and the evidence of heavens which he observed by means of instruments, that led Kepler away from the Ptolemaic world view. In other words, the so-called paradigm shift of the Copernican revolution was not simply an abstract decision for which we have simply to imagine theoretical philosophical, psychological or sociological explanations. It was inspired by a new confrontation of evidence from instruments of observation, with that of instruments of model making: precisely that nexus that Leonardo brought into focus.

In this context Leonardo's work on compasses and telescopes can no longer be dismissed as amusing toys, or neat gadgets. His emphasis on instruments such as balances, automatons and clocks (cf. pl. 4, 25, 26) provoked much more than mechanical metaphors for the natural world. They provided the very framework that made it possible to think of the world in a scientific way. They established a visible standard which permitted one to insist on observation and experiment. And by means of instruments Leonardo set European culture on a quest for laws concerning four powers of nature: weights (pl. 27-28), which Tartaglia, Benedetti and Guidobaldo del Monte would take much further; motion (pl. 33-34), on which Galileo would build his reputation; percussion, which became the basis of Huygen's philosophy and force which in some senses had to wait for Newton. In short, Leonardo did have a method in his work and the questions he asked gave basic directions to the research programmes of the centuries that followed.

9. Limitations

Why then have these contributions never been recognized? Why is Leonardo regularly dismissed as a chaotic amateur? It is mainly because hardly anyone reads the notebooks. Artists feel that they can limit themselves to his paintings. Scientists believe that they need not read them because they assume that unpublished works had no effect. The few individuals who have nonetheless read the notebooks have usually taken the surface chaos at face value. Three notable exceptions have been Venturi, Solmi and Keele mentioned at the outset.[372] Their work inspired the present study.

It is easy, however, to overemphasize this structure and method, and necessary to note very important limitations on various fronts. Some are practical. Had paper not been so scarce, had Leonardo had more space on which to write his work, he would not have needed to be so cramped in his writings, sometimes writing in the margins, sometimes having to skip a few, or even a great number of pages, to find the next empty space to pursue his idea. Some limitations are more subtle. He may have a mechanical view of the universe, but the microcosm-macrocosm analogy lingers. His mechanical anthropomorphism leads him to consider mechanical birds, but not airplanes. Some are problems of classification. A modern reader who finds a page (e.g. A1v) with discussions of images hitting the eye, sounds hitting the ear and hammers hitting the ground may assume this is pure chaos. In Leonardo's mind it is not: for him all three are instances of percussion. Some limitations are technical: Leonardo sets out to make lenses in order to make the moon appear large. But he could not hope to see what Galileo did a century later with better equipment. Some are procedural. While praising the mathematical sciences, Leonardo views mathematics mainly in terms of geometry. He does not treat geometry and arithmetic together in the systematic way that begins in the 1580's. He has effectively no algebra. He has no conception of trigonometry, let alone calculus.

Unlike his mediaeval prodecessors who were frequently concerned with learning and keeping secrets to themselves, Leonardo wants to communicate his ideas. He teaches, noting "If you do not teach you will only be excellent."[373] He writes with a view to being read, and also wants to be published in order to be more widely read. If the limitations of printing at the time made infeasible the publication of his many illustrations, his new visual demonstrations, this was not his fault. He lived in a time when war and politics repeatedly forced him to move. So the chaos of his notes is partly the chaos of his times. And for all these reasons he is Leonardo and not Galileo, Newton, Helmholtz or Einstein. To acknowledge these limitations does not threaten his unique place in the history of early modern science. He was one of those giants on the shoulders of whom later scientists stood, whose greatness lay in focussing attention on four fundamental principles, which introduced to the myriad impressions of nature herself a new sense of structure and method.

10.Historiography

That Leonardo's contributions have never been properly acknowledged is not simply due to limitations in his presentation. It is also a question of fashions in historiography that we need to examine briefly if we wish to re-assess Leonardo's position. For there is more at stake than deciding on the study habits of a well known individual. There is a more fundamental debate on the origins of early modern science: whether there was a gradual evolution from the mediaeval period to the present or whether there was a scientific revolution, a sudden paradigm shift. And as we shall show these debates have become heated because they are no longer about who invented or discovered what or where. They have become debates about method: about the how and why of invention and discovery. And ultimately they are about something even more basic: whether history counts. To understand these developments we need to go back to the nineteenth century.

The history of science as pictured by nineteenth century scientists such as Whewell[374] was straightforward. There was progress and there were great men. The middle ages had basically been a millenium of darkness. Then a genius such as Leonardo brought about a scientific revolution which was carried further by the next genius, Galileo, and so on. This was so obvious that there was no need to document the details of how it had occurred. This view remained largely unquestioned until the autumn of 1903 when Duhem found irrefutable evidence that Leonardo had studied mediaeval sources such as Jordanus of Nemore.[375] Duhem soon stated his case more forcefully: "There is no essential idea in the mechanical works of Leonardo da Vinci which does not derive from the mediaeval geometers."[376] Soon, even sceptics accepted that there must be some continuity between mediaeval and renaissance science, a gradual evolution rather than some sudden event[377]. This evidence was particularly taken up in the United States. Sarton included mediaeval science in his great Introduction[378], while individuals such as Thorndike[379], Benjamin[380] and Clagett[381] founded mediaeval science as a proper field of study. Their studies established that the scope of mediaeval science was much larger than had been assumed, that knowledge of mediaeval sources in fifteenth century Italy was much more widespread than had been imagined. So Leonardo was not alone. But while Duhem concluded that Leonardo had acquired all his ideas from this mediaeval tradition, Clagett decided that Leonardo's knowledge of it was incomplete at best. Clagett's students were less generous and became convinced that Leonardo's knowledge of mediaeval sources was scanty at best. Hence even if there was a continuity Leonardo was not really part of the story.

Meanwhile, in Europe, the quest to understand the continuity of early modern science led Olschki to write his great two volume history of early technical-scientific literature.[382] This had various effects. It firmly established the concept of artist-engineers, which Gilles[383] subsequently took up. Leonardo now became one in a long line of engineers from Guido da Vigevano through Taccola, Francesco di Giorgio, pointing to the work of Agricola, Besson, Ramelli and ultimately individuals such as Leupold. The overall thrust of Olschki's work was to focus attention on technology rather than science and while confirming beyond doubt that there existed a continuity at the level of technology, it implicitly raised doubts whether something different might be the case in terms of science.

Olschki's work was used in very different ways. The marxist Zilsel[384], concerned with the sociological roots of science, cited it to support his thesis that there were three distinct strata of intellectual activity from 1300 to 1600: university scholars, humanists and artisans and that the first two strata were uninteresting. Zilsel assumed that science was synonymous with causality and that "craftsmen were the pioneers of causal thinking in the period."[385] They used quantitative methods but lacked methodical intellectual thinking:

Thus the two components of the scientific method were separated by a social barrier: logical training was reserved for upper-class scholars; experimentation, causal interest and quantitative method were left to more or less plebeian artists. Science was born when, with the progress of technology, the experimental method gradually overcame the social prejudice against manual labour and was adopted by rationally trained scholars. This was accomplished about 1600 (Gilbert, Galileo, Bacon).[386]

In Zilsel's scheme Leonardo became one of a list of craftsmen which included: Ghiberti, Piero della Francesca, Alberti, Biringuccio, Dürer, William Bourne, Robert Norman, William Borough and Palissy.[387] Having assumed the existence of a social barrier until 1600, Zilsel or his followers had no incentive to look for evidence of both strands in Leonardo a century earlier.

Nor were these ideas considered only by marxists. Drake and Drabkin[388], for instance, shared none of Zilsel's ideological assumptions, yet also argued for the importance of this practical tradition. They accepted a continuity of mediaeval mechanical ideas in the universities, but held that this was not the source of Galileo's discoveries: that Galileo owed most to a tradition of men outside the universities which went back to the time when these began publishing. This had begun with Tartaglia. It now appeared that there was a scientific revolution which began in the 1540's. The year 1543, when Tartaglia produced his vernacular edition of Euclid, was also the year that Vesalius and Copernicus published their great works. Leonardo, having been dismissed from the continuity thesis on the assumption that he was a craftsman, in spite of his 119 books, was now excluded from the craft tradition because he did not publish.

Meanwhile the work of both Duhem and Olschki had inspired interest from a very different front. The neo-Kantian philosopher Cassirer was interested in Leonardo for his own reasons. He admitted some continuity, citing Duhem concerning Leonardo's study of Cusa[389], but not Jordanus of Nemore, partly because he wished to emphasize Leonardo's break with the past, his challenge against traditionand authority and to show that, just as Cusa had developed a concept of lay piety, Leonardo established a concept of lay knowledge. Cassirer identified its key elements as a reliance on experience, proportion, measurement and ultimately mathematics. These scientific insights, he claimed, derived from his art:

The scientific theory of experience, in the version to be given it by Galileo and Kepler, will base itself on the basic concept and on the basic requirement of exactness as formulated and established by the theory of art. And both the theory of art and the theory of exact scientific knowledge run through exactly the same phases of thought.[390]

Cassirer cited Panofsky[391] who believed that he had demonstrated how Renaissance artists such as Dürer had discovered principles of descriptive geometry long before the mathematicians. In the hands of more popular writers these specific claims were translated into general notions that artists are precursors of science. De Santillana[392], for instance, used Brunelleschi as a key example. While art historians could become enthusiastic about Brunelleschi's technical skills, there was no serious evidence to link him with fundamental developments in science. This of course made nonsense of Cassirer's original claim and, more important for our purposes, meant that scholars looked to Brunelleschi[393] for things that could not be found there and once again ignored Leonardo.

One important thrust of Cassirer's claims was that Renaissance science involved more than Galileo's discovery of laws of motion, that there were basic shifts in philosophical framework involved. Cassirer himself explored these in general terms in Substance and Function.[394] This possibility excited Burtt (1924)[395] who made his own claims: that the Platonic and Pythogorean traditions involved metaphysical speculations asserting a cosmological status of mathematics which provided both a foundation and justification for science. In so doing, he drew attention to the importance of the new astronomy, and early modern science was associated with Copernicus Kepler, Galileo, Descartes, Gilbert, Boyle and Newton. He was challenged by Strong (1936) who claimed that:

The meaning of concepts employed by mathematicians and scientists in their work was found to be established in the limited operations and subject matter constituting the science. The conclusion finally driven home was the conviction that the achievements of Galileo and his predecessors were in spite of rather than because of prior and contemporary metaphysical theories of mathematics.[396]

In terms of sources, Strong focussed on mathematics from Tartaglia through Cataneo, Clavius and Veglia to Galileo. But there was more to this debate than the interpretation of specific texts. Strong was insisting that the history of science had special problems of its own which could, indeed should be, studied in isolation. This set the stage for what would later be termed an internalist approach. On the other hand, what Burtt was arguing, and what Cassirer had assumed, was that history of science was but one manifestation of a larger cultural framework. This would later become the externalist approach.[397]

The quest to understand the history of Renaissance science in a larger context led in other directions also. At Oxford, Crombie[398] set out to demonstrate that some of the key terms of philosophy necessary for experimental method which were used by Galileo and other seventeenth century thinkers already existed in the thirteenth century. As he presented the evidence it appeared as if Grosseteste had effectively articulated all the key terms of early modern science. That the actual context within which these terms were used might have changed entirely in the meantime was not discussed. Scholars such as Boas-Hall[399] were more careful, and while eager to acknowledge some continuity from the mediaeval period, insisted that there was something fundamentally different about the period 1450-1630. Although focussing on astronomy, Boas-Hall emphasized the significance of other fields such as cartography, botany, biology, medicine and mathematics.

Meanwhile, Burtt's claims about the importance of Platonism were taken up by Koyré‚ who became convinced that there were special connections between metaphysics and measurement.[400] He linked these concepts firmly with the Copernican revolution.[401] Gradually the Copernican revolution in astronomy could be seen as synonymous with the scientific revolution.[402] If we stop for a moment to consider these various theories, we find at least ten different claims about when the Renaissance began. Such a list is instructive because it suggests how concerns with specific kinds of problems inevitably led most scholars to overlook the possible role of Leonardo. Indeed, the only exception was Cassirer and his assumptions did not require him to make a detailed study.

DATES FIGURES FIELDS SCHOLARS
1200-1250 Grosseteste Philosophy Crombie
1300-1350 Ockham, Buridan Philosophy, Theology Duhem
1400-1425 Brunelleschi Art Santillana
1450-1630 Cusa, Leonardo Philosophy Cassirer
1450-1630 Cusa, Copernicus, Astronomy Boas-Hall
Brahe, Porta, Digges, Cartography
Kepler, Galileo Biology, Botany
1540-1700 Copernicus, Kepler, Metaphysics Burtt
Galileo, Descartes,
Gilbert, Boyle, Newton
1540-1630 Tartaglia, Cataneo, Mathematics Strong
Clavius, Galileo
Tartaglia, Guidobaldo Mechanics Drake
Benedetti, Galileo
1540- Copernicus Astronomy Koyré, Kuhn
1600-1630 Galileo, Gilbert Craftsmen Zilsel
Bacon Mechanics
Surveying

Fig. 9: Ten claims about when the Renaissance in science began.

There is something else which this cursory review of historiographical trends brings to light: a gradual shift from asking questions about when and where the scientific revolution occurred to problems of how and why. Indeed, what increasingly seemed essential was no longer the discovery of a systematic approach to the universe, but rather how one systematic explanation was replaced by another, or, as Kuhn has termed it, how one paradigm replaced another.[403] Hence the assumption has spread that only changes in thought structure count: that it is really a question of a shift in mentality. The versions of these assumptions are several. In mild cases it is simply a matter of mentioning Foucault or Derrida. Others have subtle formulations about seeking to discover how the scientific mind works; that one must use new conceptual tools, above all that of the thematic content of science. So history becomes a series of case studies and quick probes. There is no longer a cumulative picture to be understood, and in this context the Renaissance and Leonardo are too early to be relevant.

There are also more radical versions which argue more strongly. Since these are universal problems, what individuals wrote on given pages of specific documents is too trivial. So one is saved the trouble of learning old languages. Indeed one can, for the most part do without texts, and can certainly spare oneself the bother of looking at manuscripts, troublesome archives and other outdated, irrelevant modes of communication. In extreme cases there is a conviction that history, if approached properly, should stop worrying about the past (which is over anyway) and concentrate instead on philosophy, which explains the logic of basic ideas; psychology which enables us to see structures of the mind and sociology which provides a social context for changes in those structures. In this extreme view history is about universal ideas, not individual opinions; about objective manifestations of truth, not subjective samples of biased creatures who do not reflect the norm. This extreme view is one that is overheard in conversation with graduate students or found expressed in their essays. If such views seem so outlandish that they are felt to have no place here, then it is sobering to recall the case of a not unknown scholar, John Hermann Randall, Jr. who, while openly admitting that he had never read the notebooks, felt that he could safely "lay down" three propositions: 1) Leonardo was not a scientist, 2) the notebooks did not contain "a single theoretical scientific idea that is essentially new or that was unknown in the organized scientific schools of his day," 3) even if Leonardo had had original ideas in scientific theory, "they remained unknown until the Paris Codici were published 1881-1891 and the Codice Atlantico in 1894."[404]

11. Conclusions

From this there emerges a picture quite different than that of an ingenious empiricist working in an intellectual vacuum. The evidence of Leonardo's notebooks confirmed that he was widely read and had many contacts. His extant treatises revealed much more structure than has generally been assumed. Moreover, they evidenced a number of clear plans for books. Examination of his entire extant corpus brought to light another unexpected feature: for all their universality the notebooks are focussed on a surprisingly small number of basic themes: crucial among these are his studies of transformational geometry and a mechanical approach to nature, which uses as a point of departure his concept of four powers (weight, force, motion and percussion), and serves ultimately to integrate both his study of the microcosm (anatomy) and the macrocosm (astronomy) within a single grand plan. It was shown that these studies were guided by a distinct method of listing variables systematically and playing with them experimentally. It was claimed that the results of this enterprise inspired him to write treatises and led him to make serious plans for publication. While these plans were unsuccessful, there is nevertheless evidence, mainly indirect, that Leonardo, who was at the centre of action in the major cities of the high Renaissance (notably Florence, Milan and Rome), was not without influence in the century that followed. To explain why Leonardo was subsequently forgotten, the limitations of his approach were considered, as were the limitations of historiography in our own century.

All this was guided by two further purposes: a) to establish that Leonardo and the sixteenth century context leading to Galileo require more careful study, and b) to suggest that new combinations of earlier approaches are required. It is possible, for instance, to accept that there was some continuity between the thirteenth and the seventeenth centuries without assuming that nothing happened between the time of Grosseteste and Galileo. If Leonardo's knowledge of mediaeval and classical sources was not exhaustive, it was almost certainly for a reason. He decided that they could not provide him with an adequate picture. So he studied these sources and challenged them also. This process continued into the seventeenth century. In 1625, for example, Accolti[405], whom one might have expected to cite Kepler, still acknowledged the thirteenth century writer, Witelo, as the most important authority on optics. At the same time Accolti treated Witelo critically and challenged him. The continuity question offers a means of documenting how specific experiences and experiments were gradually seen as legitimate tools to challenge first individual passages and finally the very authority of traditional sources. This is of interest, but in our view there are more pressing issues.

While accepting some continuity we have suggested that there was a breakthrough in science during the period 1490-1510. Interestingly enough this period coincides with the high renaissance in art and thus confirms in an unexpected way Cassirer's claims concerning links between science and art. But whereas he focussed on philosophical context, we have shown that the practical context of machines and instruments played a central role. That the history of technology offers a key to understanding developments in the history of science is perhaps the most important issue raised by Leonardo's work. It means that marxists such as Zilsel were right about the importance of craftsmen. It also means that Zilsel's claims need revision on two counts. First, Leonardo's life confirms that there was no invisible social barrier preventing different classes from meeting prior to 1600. In theory, as an illegitimate son Leonardo had little social standing, yet he was as a brother with Jacopo Andrea da Ferrara[406], a leading Vitruvian commentator, he was friends with dukes, popes and the King of France. Secondly, as we have shown, Leonardo combined both of Zilsel's ingredients of science, causal thinking plus systematic organization, in the period 1490-1510, over a century before Gilbert, Galileo and Bacon. Indeed, as we have noted, Leonardo was engaged in writing this in the form of a treatise in his Elements of machines and was close enough to completion that Pacioli could mention it in print in 1509. Hence, whereas Zilsel and everyone since has assumed that the sixteenth century could provide only isolated examples of causal thinking, we now have firm evidence of both causal thinking and systematic method in the first decade of the century. In the midst of the high renaissance it is highly unlikely that insights of such magnitude would simply be forgotten. So we need to look afresh, and much more closely[407] at the sixteenth century if we are to see in a proper framework the contributions of Tartaglia, Benedetti, Guidobaldo del Monte, Galileo and later thinkers; to establish a chronology of when, which discovery or invention was made where, in order that individual scientists and craftsmen are given due credit.

It is important to recognize that Leonardo's example also raises deeper questions of method. For it suggests that questions of why and how the scientific revolution took place are not simply abstract problems of philosophy, psychology or sociology, but intimately connected with the historical evidence itself. If Leonardo had only an astrolabe, a quadrant and a few isolated gadgets, he could not have dared to make his claims about machines in universal terms. The great number of machines and instruments with which he dealt was a vital ingredient in making the universality of his claims possible and credible. Hence, aside from individual characteristics of given machines and instruments, upon which historians of technology have traditionally focussed their attention, there is a cumulative dimension to their development which makes science possible and which requires study. Needed is a history of how, what were originally seen as a great variety of individual procedures, techniques and methods, were gradually recognized as part of a single, cumulative programme of mathematical sciences, how trigonometry gradually became an integrating tool. This goes beyond Strong's arguments about procedures and requires more than internalism. Thus far we have encyclopaedias of techniques in terms of first occurrences and latest developments. We need a record of what came in between if we are to make some map of this cumulative interplay between technology and science that is unique to the West. This is not to deny the significance of specific references in the manuscripts and printed texts of the period to Archimedianism, Platonism, Pythogoreanism, Vitruvianism, hermeticism, mysticism and other metaphysical influences. Nor is it to question the value of philosophy, psychology and sociology as tools for the history of science. What the discovery of structure and method in Leonardo's notebooks suggests rather, is that we have not looked closely enough even at the basic facts and that a deeper understanding of the history of science will require that we begin focussing on its history.

In the 1490's an important development in the use and study of machines and instruments provided one of the key shifts that made possible what is now remembered as the scientific revolution. It was accompanied and partly inspired by a fresh examination of historical sources which printing had recently made newly accessible. In the 1990's an analogous mechanical revolution may be occuring in the spread of electronic devices. The latest developments in computer memory make feasible for the first time a history of scientific instruments and techniques which will include not only the first and the last but also stages in between.

Paradoxically, without the help of computers and related modern instruments (CD-ROM, videos, etc.) a systematic history of these developments in instrumentation connected with Leonardo is not feasible; the enormity of the evidence connected with the mechanical-instrumental revolution of the 1490's is only becoming possible with the new mechanical means of the 1990's. Perhaps our new scientific revolution will again be accompanied by a parallel revolution in access to historical knowledge, and possibly this more systematic approach to the past will lead to a new appreciation of Leonardo and his method.

APPENDIX

Kim H. Veltman in collaboration with Eric Dobbs

Computers as a Historical Tool

  1. Introduction
  2. Animation
  3. Alternative Constructions
  4. Geometry, Arithmetic and Algebra
  5. Play
  6. Plans
  7. Conclusions

1. Introduction

The quantitative advantages of computers are obvious. They enable access to hitherto unthinkable amounts of knowledge. In terms of historical material this is of great significance because it will make the search for sources (ad fontes) available to individuals who do not have access to the very few remarkable libraries (e.g London, Paris, Vatican, Washington) where this level of research has traditionally taken place. In the foregoing text we have suggested that there is more to this quantitative dimension than sheer numbers of documents or size of databanks. Without such enormous amounts of hitherto scattered materials many questions can scarcely be addressed with any depth: how surveying instruments spread across Europe; what methods they employed; to what extent these methods were supplemented by textual descriptions; to what extent the local variations decreased as individuals became aware that methods developed in Urbino, Nürnberg, or Antwerp could be applied elsewhere; to what extent one can map changing relations between practice and theory? Related to this are questions of how materials can be presented in new ways: to what extent can one use combinations of lists, maps, and images to bring into focus relations among different aspects of knowledge which were hitherto invisible? All these are long term goals.

Meanwhile there are also short term possibilities: computers offer many new qualitative methods for the interpretation of historical material. This paper focusses on some specific examples in the context of Renaissance art and science which are being explored under the auspices of the Perspective Institute at the McLuhan Centre (University of Toronto) and outlines some future plans. An IBM compatible AT using AutoCAD 10 in conjunction with D BASE III Plus is being used to make visible in a new way the genesis of perspectival methods from Alberti to Leonardo and demonstrate links with transformational geometry (de ludo geometrico), conic sections, principles of square and cube roots and other aspects of Renaissance mathematics, science and art. The chief characteristics of this approach can be described in terms of animation, alternative constructions; geometry and number as well as play, each of which will be considered in turn.

2. Animation

One of the greatest problems in the understanding of mathematical diagrams is that they represent the conclusion of a series of steps which, especially when the case is complex, provide very little clue about the steps taken in arriving at this conclusion. Traditionally this has been assumed to be the purpose of the accompanying text. One reads through a laborious proof, retraces the steps taken by the mathematician in question and thus arrives at his conclusions. This problem has been compounded by the fact that the heritage of Euclidean geometry has favoured abstract methods of presentation. Hence the diagrams frequently show three-dimensional spatial situations in two-dimensional terms. This heritage continued with the Renaissance artists and mathematicians who developed linear perspective. As a result those who discovered a new method of three dimensional representation (ironically) codified their findings in abstract two dimensional diagrams. Using AutoCAD 10[1], Eric Dobbs has shown how one can make these geometrical diagrams in the early treatises on perspective much more comprehensible by retracing step by step the various stages of a perspectival construction. This has the great advantage of making visible the process as well as a the end product. Thus far reconstructions have been made of key diagrams by Alberti, Antonio Averlino (Filarete), Piero della Francesca (pl. 35-36), Francesco di Giorgio Martini, Leonardo da Vinci (pl. 39-40) and Egnatio Danti.

These animated reconstructions have other advantages. They permit a modern viewer to see three-dimensional consequences of diagrams and thus resolve ambiguities which the Renaissance artists and mathematicians did not see because they drew their diagrams in only two dimensions. For instance, traditional debates concerning the legitimate construction (costruzione legittima) and the distance point construction are clarified if their principles are demonstrated in terms of three-dimensional situations.

3. Alternative Constructions

Alberti, who wrote the first treatise on perspective (1435) presents a special case. He appears to have avoided diagrams altogether and relied entirely on verbal descriptions. (In any case the earliest known manuscripts of his De pictura are without illustrations). Alberti's text has given rise to at least three different interpretations. By reconstructing these three different versions of the same text, it is possible to see and assess their relative merits.

Alternative constructions are sometimes useful even when the text and accompanying diagram are unambiguous. Piero della Francesca's De prospectiva pingendi offers a case in point. When Piero described the construction of a foreshortened pentagon he relied on the presence of a static diagram and thus drew a series of lines the significance of which remained unclear until the diagram was nearly complete. Using a computer one can retrace these steps precisely (pl. 37). In addition, with the aid of coloured lines and the use of a different sequence of steps one can make visible Piero's method in a new way (pl. 38).

4. Geometry, Arithmetic and Algebra

During the Renaissance there were frequently two alternative demonstrations for basic mathematical principles. For instance, in the case of square roots there was both a geometrical solution which illustrated the principle using a diagram and an arithmetical solution which dealt with the problem in terms of numbers. To a certain extent the two methods were interdependent. The development of geometrical methods made visible and measureable a set of relations which could then be summarized numerically. By the late sixteenth century this numerical arithmetic solution was gradually replaced by an algebraic solution, while the geometric solution was increasingly forgotten. Paradoxically, this advance hid the visual phase of the experience, which had made possible the algebraic abstraction. As a result historians of mathematics have tended to write the history of their subject as a gradual liberation from visual demonstrations and an evolution towards abstraction. Computer animations allow one to move back and forth easily between geometrical, numerical and algebraic versions of the same principles and thus understand connections among these in a new way.

5. Play

As we look more closely at Renaissance mathematics, art and architecture we discern that underlying a seemingly overwhelming variety of forms, there are a surprisingly small number of basic geometrical shapes. Leonardo da Vinci's architectural studies offer a fascinating case in point. Most of his ground plans for churches involve circles, semi-circles, squares, rectangles and octagons. Indeed many of these begin with an octagon around which these other geometrical elements are added in some combination. Systematic drawing tools such as AutoCAD can be used to catalogue these combinations and identify which subset of these were actually used by Leonardo (e.g. pl. 41-42). This same method can be applied to Leonardo's studies of the geometrical game (de ludo geometrico), which again involves a surprisingly small number of basic forms combined in various ways. Using a computer one can reconstruct these shapes, see their equivalences both visually in terms of geometry and mathematically in terms of computed areas. Moreover one can see which subset of potential combinations he actually studied. A computer thus permits more than a simple recreation of the activities of historical individuals: it enables us to picture their horizons and see the context thereof in a new light.

6. Plans

Implicit in these reconstructions are new possibilities of entering into mental spaces of the past. In terms of mathematics, computers can show us correspondences between geometry and number and help us to understand historical trends in the changing interplay of geometry, arithmetic and algebra: i.e. the interplay of visualization and abstraction. In terms of science, if we study the texts and instruments available at a given period we can gain a reasonable picture of the parameters of methods open to individuals living at that time. In the case of surveying instruments we can explore links between more accurate instruments and growing levels of precision in topographical views and maps. In the case of telescopes it is possible to gain some idea of how much is visible with the use of various lenses. A co-ordination of recorded sightings with historical facts about the history of lenses will thus enable us to create an approximate map of how the limits of the visible universe changed with time. A similar approach can be used in all the domains of science affected by the concept of quantification: we can explore the interplay between the parameters of instruments to measure and the horizons of measurement. And as we have noted earlier a census of instruments available for measurement will be an important means of exploring this phenomenon, for it is not just the act of measuring that is important. Over 100 publications about the proportional compass in six European languages within 50 years is very different from one person using a proportional compass in isolation. This new kind of census taking will offer further insights into the cumulative dimensions of knowledge that set Europe apart from the rest of the world. It will also bring to light dimensions of the interdependence of science and technology that are characteristic of the west.

In terms of art it will be possible to catalogue themes and subjects treated as well as to trace the history of individual spatial and other elements. The continuity of images will thus come into focus in a new way. If the evidence of painting practice is compared with that of the theoretical literature on art (for which von Schlosser provided an essential guide), then we shall be able to explore the changing interplay of practice and theory. In the context of fields such as ornament and architecture where variations are to a certain extent predictable, it will become possible to map the extent to which individuals in the past have employed these forms in their designs of the man made world. In short we shall be able not only to reconstruct essential steps in the creativity of historical figures but also to gain some sense of ways in which the horizons of creativity have changed with time. With the aid of GIS technology[2] it is foreseen that these developments can be linked to both historical and contemporary maps.

7. Conclusions

Every major breakthrough in civilization has involved a reorganization of knowledge. Greek civilzation introduced not just the Academy but also the idea of storing knowledge in written form. Arabic civilization at Gundishapur in the eighth century began with a more systematic approach to collecting and translating the great texts known at the time. High mediaeval culture which saw the remarkable Summa of Saint Thomas Aquinas would not have been thinkable without access to a much greater corpus of knowledge by means of a great manuscript collection in Paris. In the Renaissance, the printing press enabled Leonardo to have 119 books and have access to many more through his wide circle of learned contacts. As we have tried to show, this access plus his systematic experiments with machines were the essential ingredients that made his new insights possible.

In our day computers are obviously marvellous new tools for storing historical facts in a more compact form. There is every reason to believe that they will also prove to be much more than this. Using animation techniques will allow us to recostruct earlier methods and to demonstrate alternative versions, such that we not only understand the value of these earlier explanations more clearly, but also learn why it was necessary to go beyond them. Computers are leading to both a reorganization and new interpretation of our cultural heritage. If there is some predictability of historical trends we are at the interstices of a new breakthrough in civilization. It might be appropriate if one of the insights of this new stage were a fresh understanding of the vision that inspired Leonardo five centuries ago.

  1. The Perspective Institute at the McLuhan Centre is grateful to have been chosen as a test site for Autodesk products, of which AutoCAD is a registered trademark.
  2. The Perspective Institute at the McLuhan Centre is equally grateful to have been chosen as a national test site for applications to education of Generation 5 GIS technology.

Notes

INTRODUCTION

1. Fra Luca Pacioli, Divina proportione, Venice: Paganinus de Paganinus, 1509 (Reprint Vienna: Verlag von Carl Graeser, 1889), p. 33:

in compagnia deli perspicacissimi architecti e ingegnieri e di cose nove assidui inventori Leonardo da Venci nostro compatriota Fiorentino qual de scultura getto e pictura con ciascuno il cognome verifica. Commo ladmiranda e stupenda equestre statua. La cui altezza dala cervice a piana terra sonno braccia 12 cioe 37 4/5 tanti dela qui presente ab. e tutta la sua ennea massa alire circa 200000 ascende che di ciascuna loncia communa fia el duodecimo ala felicissima invicta vostra paterna memoria dicata da linvidia di quelle defidia e Prasitele in monte cavallo altutto aliena. Colligiadro de lardente desiderio de nostra salute simulacro nel degno e devoto luogo de corporale e spirituale refectione del sacro templo dele gratie de sua mano penolegiato. Al quale oggi de Apelle Mirone Policreto e glialtri conviene che cedino chiaro el rendano. E non de queste satio alopera inextimabile del moto locale dele percussioni e pesi e dele forze tutte cioe pesi accidentali (havendo gia con tutta diligentia al degno libro de pictura e movimenti humani posto fine) quella con ogni studio al debito fine attende de condurre.

2. J.-B. Venturi, Essai sur les ouvrages, phisico-mathématiques de Leonard de Vinci, Paris: Chez Duprat, 1797.
3. E. Solmi, Studi sulla filosofia naturale di Leonardo da Vinci, Mantua: stab. Tip. G. Mondovi, 1905.
4. Leonardo da Vinci, I libri di meccanica nella ricostruzione di Arturo Uccelli, Milan: Ulrico Hoepli, 1940.
5. Ladislao Reti, ed., The Unknown Leonardo, London: Hutchinson, 1974, particularly pp. 264-287.
6. Kenneth D. Keele, Leonardo da Vinci's Elements of the science of man, New York: Academic Press, 1983.
7. Baldesar Castiglione, The Book of the courtier, trans. George Bull, Harmondsworth: Penguin Books, 1967, p. 149:

"Another, one of the world's finest painters despises the art for which he has so rare a talent and has set himself to study philosophy; and in this he has strange notions and fanciful revelations that, if he tried to paint them, for all his skill he couldn't."

8. Sebastiano Serlio, Il Primo [-secondo] libro d'architettura, Venice: Giovanni Battista et Marchio Sessa, 1544-1568, Bk. II.3. The following is a translation from the first English edition (London: Printed for Robert Peake, 1611), fol. 8v:

Therefore the most notable paynter Leonardus Vinci, was never pleased nor satisfied with anything that he made, bringing but little worke to perfection, saying, the cause thereof was that his hand could not effect the understanding of his mind. And for my part, if I should do as he did, I should not, neither would I suffer any of my works to come forth: for (to say the truth) whatsoever I make or wryte, it pleaseth me not: but (as I sayd in the beginning of my worke) that I had rather exercise in worke that small talent, which it hath pleased God to bestow upon me, then suffer it to lye and rot under the earth without any fruit.

9. Marie Boas Hall, The Scientific renaissance, 1450-1630, New York: Harper and Row, 1962, p. 30.
10. David C. Lindberg, Theories of vision from Al-Kindi to Kepler, Chicago: University of Chicago Press, 1976, p. 168: "As Leonardo's various confusions so clearly reveal, the problem of sight was not to be solved through a fresh start by an ingenious empiricist working in an intellectual vacuum."

2. SOURCES AND CONTACTS

11. Mad. I 12v: "Dice Giulio aver visto nella Magni[a] una di queste rote essere consumate dal polo m."
12. CA 370va (1033v, c. 1497-1500): "Dello scriver lettere da un paese a un altro Parleransi li omini di rimotissimi paesi d'uno all'altra, e risponderansi."
13. CA 260ra (697r, c. 1508-1510): "Scrivi a Bartolomeo Turco del frusso e refrusso del mar di Ponto, e che intenda se tal frusso e refrusso e nel mar Ircano, over nel mare Caspio."
14. CA 97va (266v, c. 1515-1516): "Vedi Aristotile de cielo e mondo." Cf. 289vc (785bv, c. 1487-1490): "Aristotile nel terzo dell'Etica." Estimates concerning the extent of Leonardo's reading have varied enormously. One of the most thorough studies of possible sources remains Solmi as in note 3 above.
15. CA 183va (503r, c. 1517-1518): "e provato nelli elementi d'Euclide."
16. CA 83vb (226v, c. 1508): "Omnis motu mensuratur tempore...in 2 de 8o fisicae."
17. CA 221va (596r, c. 1517-1518): "e questa regola e nata dalla 14a e ultima del 2o delli elementi d'Euclide."

A list of specific references to Euclid's Elements found in the Codex Atlanticus follows:

c. 1500, CA 169rb (462ar), 285vc (776dr) 5 Postulates, 11 Definitions
c. 1514, 90va (244v), I.1
c. 1515-16, 44va (122v)
c. 1517-18, 174v (476v)
c. 1500-05, 184va (506br) I.5
c. 1500, 177vd (483bv) I.7
c. 1500, 169vb (462bv) I.12
c. 1500, 169vc (462av), (483br) I.13
c. 1500, 169rc (462br)
" "
" "
" "
I.17
I.18
I.19
I.20
c. 1500, (776dv)
"
I.21
I.22
c. 1503-05, 203rb (544r)
" "
I.34
I.43
c. 1500, 184va (506ar)
" "
" "
221va
I.46
I.47
I.48
II.14, II.15
c. 1508-10, 269vb, 726r
" "
V.6
V.8

Cf. CA 259vb (696v, 1515), where he refers to "una d'Euclide" but cites the second of the common notions, or CA 96va (264v, c. 1500) where he states simply "E qui disse Euclide."

18. B 8r: "catapulta come dice nonio e plinio," Cf. G 48v: "Dicie Plinio nel secondo suolibro a 103 capitoli."
19. BN 2037 7v (formerly part of Ms. B): "virgilio dicie."
20. Ibid 8v: "Lucretio nel terzo delle cose naturale."
21. B 8v: "rhomphea...secondo aulo gelio."
22. Ibid 9r: "dicie livio nel settimo delaguera cartaginese secondo trovo inuna comedia di plauto...dice flavio."
23. Ibid 30v: "dicie lucano ciesare che..." Cf. 41v: "ciesere ne fa mentione nel secondo delli sua comentarii."
24. Ibid 41r: "sicome vuole quintiliano...plinio nel VI libro de naturale istorie...ecome vole varone...secondo lucano nel nono."
25. Ibid 41v: "delaqual fa mentione Plutarcho nella vita di graccho."
26. Ibid 43r: "ecquesto erma varone festo ponpeo ne testimonia diciendo."
27. Ibid 45v: "sechondo amiano marciellino."
28. E.g. F 27r: "A Platone si risponde," which begins a commentary on his treatment of regular solids in the Timaeus.
29. E.g. L 53v: "Dice vetruvio" or G 54v: "Vetruvio ne pone uno nella sua opera darchitectura."
30. M 8r: "Suisset."
31. M 11r: "Tebit."
32. On CA 203ra (543r, c. 1489-1490) Leonardo translates the opening passage of Peckham's Perspectiva communis. For an English translation see the author's Leonardo Studies, Vol. I, Linear Perspective and the Visual Dimensions of Science and Art, Munich: Deutscher Kunstverlag, 1986, p. 56. CA 277ra (750r, c. 1513): "E questo solu e bastante alla detta pr[o]va la quale a data nella prespettiva comune eccetera."
33. E.g. Mad. I 171v: "Le lettere dell'abbaco da'pesi in su non parlano del peso che ttira le corde anzi di quello che ssi sscarica sopra le gifelle." Cf. BN 2038 8r: "Onde conquesto che ffacto cholla sperienza. subito poi legiere la lettera dellabaco che tocha dal filo essai la verita delpeso."
34. E.g. BN 2038 2v: "dicie il pelachane." Cf. Mad. I 133v: "Pruova contro al Pellacane."
35. E.g. Arundel 263 31v: "Dice batista alberti nuna sua opera mandata al signore malatesta da rimjni come quando la bilanca" or 66r: "Dice battista albertj nuna sua opera titolata exludis rerum mathematicarum..."
36. See Ladislao Reti, in Burlington Magazine, London, vol. CX, 1968, pp. 81-89 with supplements in vol. CX, 1968, pp. 406-410, vol. CXI, 1969, p. 91. Cf. Leonardo's list on CA 210ra, concerning which see Leonardo da Vinci, Scritti letterari, ed. A. Marinoni, Milan: Rizzoli, 1974, pp. 279-257.
37. CA 225r. There are other references to Witelo on B 58r and CA 247r.
38. L 2r. There are a number of other notes of this kind scattered throughout the manuscripts. On BM132v he refers to [Amerigo] Vespucci giving him a book on geometry. On Forster III 2v he mentions that "Maestro Stefano Caponi, a physician lives at the piscina and has Euclid's De ponderibus or on Forster III86r he notes that "the heirs of Maestro Ghiringello [a professor at Padua] have the works of Pelacano".
39. Luca Pacioli, as in note 1, p.33.

3.TREATISES

40. These dimensions are taken from Richter, as in note 13, vol. 1, pp. 108-109.
41. Cf. CA 199v.
42. On this topic see Nando de Toni, "I rilievi cartografici per Cesena ed Urbino nel manoscritto `L' dell'Istituto di Francia," Lettura Vinciana, No. 5, 15 April 1974, Florence: Giunti 1974.
43. Carlo Pedretti, Fragments at Windsor Castle, London: Phaidon, 1957.
44. W 19070v (K/P 113r): "fa legare li tua libri dj noa."
45. Arundel 190v: "legare il mil libro."
46. Leonardo Studies I, as in note 32, pp. 57-60.
47. Forster II 64r: "Mechanica potissimum in fine incipiendum."
48. F 94v: "Libro mio sastende a mostrare come locean colli altri mari fa mediante ilsole splendere il nostro mondo a modo di luna e a piu remoti pare stella ecquesto provo."
49. F 95v: "Nonsi puo difinire qui per carestia di carta ma va inverso il principio dellibro ha carta 40 cheli edifinita."
50. See the author's Leonardo Studies II-III, Continuity and discovery in optics and astronomy, (Munich: Deutscher Kunstverlag) (awaiting funds for publication).
51. F 13v: "Volta carta ennota il retrosa accidentale essua govamento."
52. F 26v: "Qui si segue la prova di quel che detto nella opposita facca."
53. F 51r: "va acarte 59."
54. E 75r: "Qui sifiniscice quel che mancha nella terza charta innanti a quessta."
55. G 44v: "Ecquesto edisegnato in margine della quarta carta dapiedi."
56. G 46r: "Qui seghue quelche mancha disocto arri scontro."
57. G 46v: "Leggi in carta 45."
58. G 51v: "Va alle 44 charte di quessto."
59. G 67r: "Per quel che errischontro dappie e choncluso."
60. G 75r: "Qui seguita quelche nella contrapposta carta."
61. Ibid.: "El subbio e fighurato nella contrapposta faccia."
62. G 80r: "Qui si finisscie quel che mancha qui dirieto acquesto lato della charta cioe dappie in margine."
63. For an analysis of these passages see Leonardo da Vinci Studies I, as in note 32, pp. 257-268.
64. Forster I 3r: "Libro titolato de strasformatione, cioŠ d'un corpo `n un altro sanza diminuizione o acresscimento di materia."
65. Forster I 3v: "Principiato da me Leonardo da Vinci addi 12 luglio 1505."
66. As I plan to present this some day as an independent study I do not document the foregoing paragraphs in detail.
67. CA 384ra : "Io dissi nella 7a conclusione come la percussione."
68. CA 155vb: "Guarda nel 7o del quinto de polo e rota."
69. CA 2ra (10r, c. 1515):

Perche sanza la sperienza non si puo dare scienzia vera della potenzia colla qual resiste il ferro trafilato al suo trafilatore, io ho fatto qui da parte queste quattro rote motrici delle vite sanza fine delle quali ciascuna ha a riscontro segnato il numero de'gradi che ha la sua potenzia. Le quali potenzie son vere, comŠ provato nella 13a della ventiduesimo delli elementi macchinali da me composti.

70. CA 287ra (780r, c. 1514-1515): "Messer Battista dall `Aquila, camerier segreto del papa, ha il mio libro nelle mani."
71. Arundel 12r: "Qui per la 5a del 7o, il peso..."
72. Arundel 25r: "Come e provato nel 4 della mja prosspectiva."
73. Arundel 25v: "Libro 9e dellacqua de poci permanente e dellacqua fugitiva."
74. K 30 [29]r: "Sesto libro."
75. F 35r: "Libro 42. Delle Pioggie."
76. F 37r: "Libro 43. Del moto dellaria inclusa sotto lacqua."
77. F 66v: "Principio dellibro."
78. I 72 [24v]: "Principio dellibro dellacque."
79. E 59v: "Principio di questo libro de pesi."
80. E 27v: "Pruovasi per la nona deperchusione cheddicie."
81. W 19064r (K/P 157r): "e per la 5a de forza e provato quel che dj sopra sicontiene."
82. W 19061r (K/P 154r): "ordine del libro... Adunque quj con 15 figure intere ti sara mosstro la cosmografia del mjnor mondo colmedesimo ordjne che inanzi ame fufatto dattolomeo nella sua cosmografia.."
83. W 19009r (K/P 143r): "Fa chellibro delli elementi machinalicolla sua praticha vada inantj ala djmostratione del moto e forza dellomo e altrj anjmali e medjante quelli tu potrai provare ognj tua propositione."

4. PLANS FOR BOOKS

84. Mad. I 173v: "Farai tutto il testo insieme e poi piu oltre lo dividi col suo commento."
85. CA 117rc (324r,c. 1495): "Tratterai prima del peso, poi del moto che partorisce la forza e po'd'essa forza e in ultimo del colpo."
86. CA 149rb (403r, c. 1493-1495):

"Principio della natura de'pesi. La regola del tuo libro proceder… in questa forma :prima l'aste semplice, poi sostenute di sotto, poi sospese in parte, poi tutte, poi esse aste fieno sostenitori d'altri pesi."

87. K 3r:

Dividi il trattato delli uccelli in 4 libri il primo sia dellor volare per battimento dalie il secondo del volo sanza batterali e perfavol divento il terzo del volare incomune come ducelli, pipistrelli, pessci, animali, insetti ultimi ultimo dellmoto strumentale.

88. A list of specific books and propositions cited in Madrid Codex I follows:
1.4 73v
1.5 87r, 114v
1.7 127v
2.5 65r, 122r, 141r
2.9 144r
2.20 19v, 97r
2.28 153v
2.penultimate 26r,148v
3.5 102v, 114v
3.penultimate 121r
4.3 144r
4.7 69v
4.penultimate 153v
5.3 74r
5.penultimate 172v
6.5 26r, 47v, 70r, 118r
7.5 73v, 127v, 139v
7.60 43r
9.5 443r, 73v, 139v
9.7 73v, 143r
89. F 41v:

Aparlare dital materia ti bisognia nel primo libro difinire lanatura della resistentia dellaria nel 2o lanotomia dello uccello e delle sua penne nel terzo la operation dital penne per diverse moti dasse nel quarto la valitudine de lalie e coda sanza battimento dalie confavore divento aversi a guidare perdiversi moti.

90. F 90v: "Ordine dellibro... Poni nel principio cochepofare un fiume."
91. F 45v: "Ordine del libro."
92. E 12r: "Ordine del primo libro delle aque."
93. F 87v:

Scrivi inprima tutta lacqua inciasscuno suo moto dipoi desscrivi tutti lisua fondi elle lor materie senpre al legande le propositioni delle predette acque efia buono ordine altrementi d'opera sarebbe confusa."

94. CA 79ra (III 214br,c. 1505-1506):

Libro delle percussione dell`acqua in diversi obbietti

Scontri dell'acqua inobbietti permanenti di diverse figure che superanol'acqua
Scontri dell'acqua inobbietti immobili coperti dell'acqua
Scontri dell'acqua inobbietti mobili coperti dell'acqua
Scontri dell'acqua inobbietti permanenti che suprano l'acqua
Scontri dell'acqua inobbietti piegabili superati dell'acqua
Scontri dell'acqua nelliobbietti piegabili che superano l'acqua
Scontri dell'acqua nelliobbietti che cadono con moto circulare, come sono le rote dellistrumenti acquatici

Cf. CA 74vb (III 201v) for a related list.

95. CA 74va (III 201v,c. 1505-1506):

Delle cose percosse dalle acque
Delle cose che percotano l'acque
Retrosi superfiziali
Retrosi che si leva<n> dal fondo alla superfizie
Retrosi dalla superfizie al fondo
Retrosi che si movan col corso del fiume
Retrosi scambievoli nellilor raggiramenti, come son quelli de'refrussi e frussi de'fi<u>me
Retrosi laterali continui
Retrosi laterali discontinui
Retrosi larghi di sopra e stretti in fondo
Retrosi stretti di sopra e larghi in fondo
Retrosi diritti dal fondo al disopra
Retrosi obbliqui dal fondo al bisopra
Retrosi grandissimi
Retrosi brieve
Retrosi de'bollori
Retrosi a canne
Retrosi a vite
Retrosi vacui e ripien d'aria
Retrosi non vacui

At least some of this material found its way into a manuscript attributed to Leonardo in the Biblioteca Barberini in Rome, entitled Trattato della natura, peso e moto delle acque, e osservazioni sul corso de' fiumi which was published as Del moto e misura dell'acqua, ed. Francesco Cardinali, Bologna: a spese di F. Cardinali, 1826 (Raccolta d'autori italiani che trattano del moto dell'acque, tom. 4).

96. See, for instance, CA 74ra,va,vb (201rv, 1505-1506).
97. E.g. Arundel 35rv, 45r, 122r.
98. E.g. Leicester 5r, 9r, 15v. These and a number of the above passages have been translated by Richter, as in note 13, vol. 2, pp. 141-167.
99. See: Carlo Pedretti, Commentary: The Literary works of Leonardo da Vinci, London: Phaidon, 1977, vol. 2, pp. 140-145.

5. THEMES

100. F 94v as in note 48.
101. I have reconstructed his treatise on cosmology showing where his work on optics, instruments and astronomy fit into this in Leonardo Studies II-III, as in note 50.
102. CA 361va (1007v, c. 1490): "Il vento a similitudine col movimento dell'acqua."
103. M 83v: "el notare mostra il modo del volare."
104. CA 66rb (186r, c. 1505): "Il notare sopra dell'acqua insegna alli omini come fanno li uccelli sopra dell'aria." Cf. CA 214rd (571ar, c. 1507-1508): "Scrivi del notare sotto l'acqua e arai il volare dell'uccello per l'aria."
105. E 54r:

Per dare vera scientia del moto delli uccielli infrallaria e neciessario dare prime lasscientia deventi laqual proverren mediante di moti dellacqua insemedesima he equesta tale isscientia sensibile fara di se scala aper venire alla chognitione de volatili infrallaria elvento.

106. A 55v: "Adunque se'l corpo della terra non avesse similitudine coll'omo." See Richter, as in note 13, no. 917 for the complete passage.
107. A 56r: "Dico che siccome il naturale calore tiene il sangue nelle vene alla sommità dell'omo... similmente le vene che vanno ramificando per il corpo della terra..." See Richter, as in note 13, no. 969 for the complete passage. Cf. his no. 965 where he cites Arundel 263v.
108. CA 260ra (697r, c. 1508-1510):

è l'attrazione e respirazione dell'aria nelpolmon dell'omo. Ora s'è l'attrazione dell'acqua, che farebbe la terra in 12 ore col frusso e refrusso, ci poterebbe mostrare la grandezza del polmon della terra inquesto modo.

109. W 19029r (K/P 71r): "instrumento mirabile inventionato dalsomo, maesstro."
110. W 19037r (K/P 81v): "Questa figura strumentale dellomo djmonsterremo in <24> figure."
111. CA 161ra (434r, c. 1505): "L'uccello e strumento oprante per legge matematica, il quale strumento è in potesta dell'omo poterlo fare con tutti li sua moti, ma non con tanta potentia."
112. W 19147-8r (K/P 22r): "della terresste macchina al suo cientro."
113. CA 269va (727r, c. 1490): "le vari opinioni... della grandezza de l'orbiculare macchina terreste... o preso ardire creare, over comporre un istrumento, il quale adoperai in questa forma."
114. CA 252rb (681r, c. 1490-1492): "Questa terrestre e mundiale macchina."
115. A 59v: "universal macchina della terra." For examples of Leonardo's attempts to treat the four elements in quantitative ways so that they too can fit into his mechanical model see, for instance, CA 79vb (214ar, 1505-1506) and CA 72ra (197r,1508-1510).
116. See Leonardo Studies II, as in note 50.
117. See Leonardo Studies I, as in note 32.
118. Cf. William A. Emboden, Leonardo da Vinci on plants and gardens, Portland: Dioscorides Press, 1987.
119. The manuscripts in which arithmetic is discussed are mainly: Forster II, Manuscript L, CA, and Mad. II.
120. See, for instance, CA 69ra (189r, c. 1498).
121. The main manuscripts with respect to language are: Trivulzianus, H, I and Windsor.
122. The main manuscripts for literature are H, I, CA and Windsor.
123. Geometry is found in Forster I, II, Manuscripts A, B, E, G, I, K, M, Arundel, CA and Mad. II.
124. K 49 [48 et 15]r: "La proportione no solamente nelle numeri emisure fia ritrovata ma etiam nelli suoni, pesi, tempi essiti ecqualunche potentia sicia."
125. The extent to which Euclid and indeed mathematics generally was important to Leonardo is a matter of considerable debate. For another side see A. Marinoni, La matematica di Leonardo da Vinci, Milan: Arcadia Edizioni (Philips S.P.A.), 1982.
126. We have already mentioned his treatise on this subject in Forster I. A note on CA 45va (II 124v, c. 1515-1516) confirms that he was going to write anew on this topic over a decade after the Forster I:

Avendo io finito li conto vari modi di quadrare li circoli, cioè dare quadrati di capacità equali alla capacità del circolo e date le regole da procedere in infinito, al presente comincio il libro De ludo geometrico, e d• ancora modo di processo infinito.

127. CA 130va(360r, c.1517-1518) as in note 178.
128. Cf. Treatise on Painting (Codex Urbinas 1270), trans. A. Philip McMahon, Princeton: Princeton University Press, 1956, vol. II. Facsimile. Quinta Parte. D'ombra e lume.
129. Ibid., Terza parte. Comincia di vari acidenti et movimenti dell'huomo e prima delle mutationi delle misure dell'huomo, pel movimento delle membre a diversi aspetti. This section is as much inspired by geometry and anatomy as light and shade, however.
130. Ibid., Quarta parte, De panni.
131. Ibid., Sesta parte, D'alberi et verdure.
132. The chief manuscripts for military architecture are CA, B, L, Mad. I, II. Civil and ecclesiastical architecture are chiefly in CA.
133. The standard work on this topic is Pietro Marani, Architettura fortificata negli studi di Leonardo da Vinci, Florence: Olschki, 1984.
134. Most of his weapons are in Manuscript B and CA.
135. Leonardo wrote on the margins of 7 pages of Francesco di Giorgio's Cod. Ash. 361 (Florence, Laurenziana).
136. Most of Leonardo's machines are in Forst. I (hydraulic), CA and Mad. I.
137. See note 5.
138. This is discussed mainly in Manuscript B with drafts in CA and the Turin Codex.
139. Representation in this sense is much more than an idle pastime. It is man's key to understanding God's action as a creator: a re-creation with glimpses into God's mysteries. Hence painting and science combine to provide Leonardo with a new version of natural theology: a fusion of science and art which shines as an underlying unity through the chaotic surface of his notebooks.
140. CA 269va (727r, c. 1490):

avendo io visto infra i matematici speculatori le varie openioni della grandezza de l'orbiculare macchina terreste, ho giudicato ch'essendo infra tanti desputatori tante varie sentenzie, che la certezza della verita sia assai lontana da loro, imper• che, se il vero fussi pervenuto ai lori ingegni tutti sariano d'una medesima sentenzia. E sopra questa tanta diversità d'openioni ho preso ardire creare ovver comporre uno istrumento in questa forma.

141. The chief notebooks on mirrors are CA and Arundel.
142. Clocks are considered in Mad. and CA.
143. The chief notebooks on balances are Forster II, CA, E, L, and Arundel.
144. Compasses are considered mainly in B and CA.
145. CA 321ra (X 882r, c. 1493-1495): "Come tutte le rote sono di natura di bilancia."
146. CA 396rd (XII 1102r, c. 1495): "Fa menzione e regola generale sopra il contatto de' poli e di tutti i pesi."
147. CA 155vb (421v, c. 1495-1497): "Parla prima del moto, poi del peso, perch‚ nasce dal moto; poi della forza che nasce dal peso e moto, po' della percussione che nasce da peso, moto e spesso dalla forza."
148. CA 267ra (721r, c. 1495): "Regola generale del colpo. Regola generale della forza. In queste 2 regole, cioè di colpe e di forza, si può adoperare la proporzione che Pittagora usò nella sua musica." Leonardo also compares perspective and music in terms of proportions on A 103r (BN 2038 23r, TPL 31).
149. CA 20va (66r, c. 1493-1495): "Per fare regola generale della differenzia ch'è da peso semplice a peso col colpo di diversi moti e forze."
150. CA 120vc (330av, c. 1497-1498): "Si come tu trovi qui regola a diminuire il peso al motore, ancora troverai la regola di crescere il tempo al moto."
151. Mad. I 152r. See note 293 below.
152. CA 151ra (407r, c. 1500):

Tutte le potenzie naturali hanno, ovver sono da essere dette piramidali, con ciò sia che esse abbino gradi in continua proporzione inverso il suo diminuire, come inverso il loro

accrescimento. Vedi il peso, che'n ogni grado del suo discenso, essendo libero, acquista gradi in continua proporzione geometrica. El simile fa la forza nelle lieve.

153. CA 335vd (915br, c. 1503):

Molti strumenti si potrebbono allegare i quali in simili cavamenti si sono paragonati colle carette... come si mostra in un mio trattato di moto locale e di forza e peso... e a questa ho ordinato il figurato strumento, del quale quanto sia la sua utilit… e valitudine, le ragioni di quelle assegnate non daranno la sentenzia, le quali sempre fien conferme dalla sperienzia.

154. CA 271re (732cr, c. 1508): "De moto locale... E per la quinta del nono che dice."
155. CA 81vb (220v,c. 1508-1510):

Elementi macchinali:
Del peso proportionato alla potenzia che'l move s'ha a considerare della resistenzia dove tal peso e mosso, e di questo si farà un trattato.

156. CA 298rb (818r, c. 1495): "Se tu tirarai per la linia bf, il tirare si rende difficile per la quinta del settimo ma per la nona del decimo, tirando per la linia Mb..."
157. CA 283vb (771v, c. 1517-1518): "la qual obbliquità è difinità nel'ottavo libro del peso, dove
si divide la gravità accidentale dalla gravità naturale."
158. CA 384ra (1062r, c. 1493-1495): "Io dissi nella settima conclusione come la percussione."
159. CA 241ra (657ar, c. 1513):

Dividesi la percussione in libri de quali nel primo si dimostra la percussione de due corpi, de' quali l'uno move il percussore al percosso immobile; l'altro muove li percussori e li percossi scambievolmente l'un contro all'altro; terzo e delle materie liquide; quarto delli corpi piegabili; quinto...

160. CA 241vb (657v, c. 1513): "El libro dell'impeto va inanti a questo e inanti all'impeto va il moto." Cf. Mad. I 103r: "Quess[t]a 8a e allegata nellibro dell'impeto. Adunque quesste figure vanno in esso libro."
161. CA 374ra (1043r, c. 1515): "Come e la concavità che in se riceve il polo de 'circumvolubili, si debbe provvedere contro alla sua dilatazione. E di questo si tratterà nel libro de confregazione." There is a further reference to this work on Mad. I 121r under the heading:

Della vite e confregazione: Ho mostrato nella penultima del 3o come tutte le varie quantita delle confregationi che po fare un medesimo peso sopra una medesima natura di resistentia over sostentaculo, essere tutti d'equal fatica al suo motore."

162. CA 58ra (151r, c. 1503-1505): "De 'due cubi i quali son doppi uno all'altro come si prova nel quarto delli Elementi Machinali da me composto."
163. This optical tradition has been considered in the author's Leonardo Studies II, as in note 50.
164. Witelo, Vitelonis Thuringopoloni opticae libri decem, Basel: Episcopios 1572, Liber Quintus, 57:

Possibile est speculum unum planum in camera propia taliter sisti, ut in ipso videantur ea, quae geruntur in domo alia vel in vicis vel in plateis... quod totum potest fieri per astrolabium sive quadrantem vel aliud instrumentum certificationis visuum.

165. See, for instance, Jacques Bassentin, Amplification de l'usage de l'astrolabe, ed. J. Focard, Paris, 1551, p. 91:

Et pource qu'il n'est pas du tout possible que le sens et la raison puissent bien connoitre le vraye quantit‚ de l'anglet aigu et variable, par ains8i il seroit tres difficile de naturellement comprendre la certaine quantit‚ d'une chose, par la science de la perspective seulement. A ceste cause les anciens geometriciens et mesureurs ont invent‚s certains instrumens artificiels: et par le moyen d'iceux ont donn‚ facilement a connoistre les quantitez des choses avec la certitude d'icelles. Mais pource qu'il y ha plusieurs et divers instrumens servans et faits pour cest art comme sont un cadran, un triangle geometrique, taculus Iacob, umbraculum visorium, verge geometrique, horloge manuel, quilindre et autres....

166. For a discussion of these connections see the author's Sources of perspective, Munich: Saur, 199?
167. The relevant passages have all been discussed in the author's Leonardo Studies I, as in note 32.
168. Cf. note 17.
169. CA 174v (476v, c. 1517-1518): "Io voglio della mezza porzione di circolo abcd fare una transmutatione in abd coll'aiuto d'una d'Euclide ne'sua Elementi."
170. CA 183rb (502r, c. 1500): "Io voglio fare d'un circolo infinite varietà di figure curvilinee d'equale capacit…."
171. CA 141ra (386r,c. 1500-1505): "Diminuente equalmente aritmetice l'uno e l'altro extremo di ciascuna proporzione, sempre cresce la proporzione geometrica che prima era."
172. CA 174va (475v, c. 1517-1518): "Ma questa calculazione vole essere geometrica, perchè se la volessi fare per arismetrico, sarebbe impossibile."
173. CA 120rd (331r, c. 1504): "Impara la multiplicazione delle radice de maestro Luca."
174. CA 159ara-va (428v, c. 1508): "Col cerchio br farai una regola di radice quadrate insino in venti, e poi con un altro cerchio farai un' altra regola di radice cube insino in venti, e vedrai la differenzia che è dall'una regola all'altra."
175. CA 102va (281r,c. 1517-1518): "Se voli qualunche radice di numero si voglia, questa e la regola."
176. CA 231ra (629ar, c. 1505): "Fanne una grande e vedrai con piu certezza se questa regola e vera."
177. Ibid.: "La pruova meccanica è vera, benchè con fatica si trovi essa verità..."
178. CA 130va (360r, c. 1517-1518):

Se una regola divide un tutto in parte e un'altra d'esse parte rincompone tal tutto, allora l'una e l'altra regola e valida. Se per cierta scienzia si transforma una superfizie d'una in altra figura, e che la medesima scienzia restituisca tal superfizie nella sua prima figura, allora tal scienzia e valida... Quella scienzia che restituisce la figura nella prima forma da lei variata, ha perfezione.

179. CA 218va (587v, c. 1503-1505): "Qui accade la prova meccanica."
180. E 8v: "La mechanicha e il paradiso delle scientie matematiche."
181. CA 220vb (593v, c. 1508): "Geometrica regola."
182. CA 239vba (627r, c. 1516): "La prima che è trovata in questa regola. Addi 3 di marzo 1516." Cf. Mad. II 112r for another example.
183. CA 103va (285r, c. 1515-1516): "E questa regola va in infinito."
184. E.g. CA 81vb (220v, c. 1508-1510): "e cosi ti bisognierebbe procedere inverso lo infinito. Adunque sta bene la regola di prima."
185. CA 107va (297v, c. 1517-1518): "E sopra questa tal diminuzione o accrescimento si darà regola generale, che con precisione verreno a chiara notizia del vero."
186. E.g. CA 221vab (596r, c. 1517-1518): "In questa dimostrazione," refering to a figure.
187. See note 124. For a discussion of this passage see Ernst Cassirer, The individual and the cosmos in renaissance philosophy, tr. Mario Domandi, New York: Harper and Row, 1963, p. 50 (originally published 1927).
188. CA 177rb (483ar, c. 1508-1510): "E questo si prova nell'ottavo de proportione."
189. CA 166vb (454r, c. 1515): "Regola da sapere la valuta e proporzione di molte parte curvilinie di vari cerchi."
190. CA 157vb (425v,c. 1515): "Sesto proporzionale."
191. CA 385ra (1064r, c. 1513-1514): "Sesto della proporzione."
192. CA 248ra (672r, c. 1513): "Sesto de proporzionalità in profilo. Sesto di proporzionalità in faccia; e il suo pole e mobile... Questo vale nelle proporzionalità inrazionale."
193. CA 83va (225v, c. 1515): "Con questa delli Elementi si può dare qualunche proporzione di circolo, così inrazionali come razionale."
194. CA 128ra (353r, c. 1508): "Libro d'equazione." Cf. Mad. II 112r: "Scientia di equiparantia."
195. E.g. CA 157rb(425r, c. 1515): "De trasmutazione di superfizie rettilatere in superfizie curvilinie e de converso." Cf. CA 160rb (432r,1515-1516): "De trasmutazione" and CA 167rb-a (455r, c. 1515): "De trasmutazione d'equali superfizi rettilinie in varie figure curvilinie, e così de converso" or CA 242rb (660r, c. 1517-1518): "De trasmutazione delle superfizie."
196. E.g. CA 184vc (505v, c. 1516): "De ludo geometrico" or CA 174v (476v, c. 1517-1518): "De ludo geometrico." For a full discussion of this topic see the dissertation by McCabe, as in note 158.
197. CA 45va (124v, c. 1515-1516):

Avendo io finito di cont<r>o vari modi di quadrare li circoli, cioè dare quadrati di capacità equali alla capacità del circolo e date le regole da procedere in infinito, al presente comincio il libro De ludo geometrico e d• ancora modo di processo infinito.

198. CA 167r (455r, c. 1515): "Trattato de quantità continua." Cf. Mad. I 0v: "Libro titolato de quantita e potentia."
199. CA 139rab (381r): "Elementi geometrici curvilini."
200. E.g. CA 303rb (828r, 1513-1514): "per una delli Elementi segnasta in margine." Also CA 378vab (1053r, c. 1513): "una delli Elementi geometrici" which, as Marinoni notes in his new edition of CA corresponds to Euclid I.37.
201. E.g. CA 374va (1043v, c. 1513): "Per la quinta delli Elementi geometrici."
202. CA 249rba (673r, 24June 1518): "43 del primo delli Elementi geometrici."
203. CA 242rb(660r, c. 1517-1518): "E la regola di questo si fa coll'aiuto della ultima del secondo delli Elementi geometrici. Per la ultima del secondo delli Elementi geometrici."
204. CA 170ra (463v, c. 1516):

Di queste mia superfizie curvilinie molte ne son quadrabile in se medesime colla trasmutazione delle sue propie parte nel suo tutto, e molto ne son che colle sue propie parte sono in quadrabile, ma si da quadrati equali loro, tolti d'altre superfizie. E con queste si compone l'ultima mia opera di cento 13 libri da me composti nella quale è 33 modi variati di dare quadrati rettilini equali a circoli, cioè equali in quantità.

205. Cf. Reti as in note 5. See Franz Reuleaux, Theoretische Kinematik. Grundzüge einer Theorie des Maschinenwesens, Braunschweig: Vieweg, 1875.
206. See note 82.
207. See note 152.
208. See note 83.
209. Cf. Pierre Duhem, Les origines de la statique, Paris 1905-1906. E.A. Moody and M. Clagett, The medieval science of weights, Madison: University of Wisconsin Press, 1960.
210. CA 124ra (342r, c. 1508): "Elli e provato nel primo De ponderibus."
211. Cf. CA 118va (325v, c. 1508-1510): "per la regola d'Archimede." On CA 153rbc (413r, n.d.) he has copied out a passage from a manuscript of De insidentibus in humido, concerning which see: Marshall Clagett, "Leonardo da Vinci and the medieval Archimedes," Physics, 1969, p. 141. Cf. also Mad. II 105rv: "piu vicino al vero che Archimede."
212. E.g. CA 176rd(481v, 1505-1508) where he copies definitions from both Euclid and Theodosius concerning the sphere.
213. E.g. CA 196ra (528r): e similmente e quadrabile la lunola b per Zenofonte or CA 201vb (540v, c. 1509): "aro fatto il bisogno di Zenofonte."

6. METHOD

214. CA 125ra (345r, c. 1490-1492): "Io trovo per isperienzia che..."
215. A 47r: "La ssperientia farai inquesto mo[do]."
216. CA 338va (924v, c. 1490): "Sperimento del moto causa del colpo."
217. CA 151va (407v, c. 1500): "Per isperimentare la proporzione delli intervalli del discenso."
218. D 3v: "Per fare sperientia come la virtu visiva ricie[v]a le spetie delli obbietti dallochio suo strumento esara fatto una palla di vitro di cinque ottavi dibraccio perdiamitro."
219. See for instance the following articles by Enzo Maccano:

"Analogies in Leonardo's studies of flow phenomenon", Studi Vinciani in onore di Nando di Toni, Brescia: Ateneo di scienze, lettere ed arti, 1986, pp.19-49.

"Multichannel tabulation in the notes on flow in the French manuscripts on Leonardo da Vinci", Raccolta Vinciana, Milan, vol. 22, 1987, pp.213-237.

"La nocion de presion en la mecanica de fluidos vinciana", Raccolta Vinciana, Milan, vol. 22. 1987, pp.239-263.

In addition Enzo Maccagno has made a series of careful studies analysing the major codices of Leonardo, demonstrating their experimental basis. In chronological order these are:

"Mechanics of fluids in the Madrid Codices", Scientia, Milan, 1982, 99.333-374.

"Hidrostatica vinciana en el Codice Hammer", Anales de la universidade de Chile, Santiago de Chile, 5ta ser. n.8, agosto 1985, pp71-76.

Leonardian fluid mechanics in the Codex Atlanticus, Iowa: University of Iowa, 1986. (Iowa Institute for Hydraulic Research, Monograph no. 100).

Leonardian fluid mechanics: what remains to be investigated in the Codex Hammer, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 101).

Leonardian fluid mechanics: unexplored flow studies in the Codices Forster, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 102).

Leonardian fluid mechanics in the Manuscript C, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 104).

Leonardian fluid mechanics in the Codex Atlanticus, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 105).

Leonardian fluid mechanics in the Manuscript H, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 106).

Leonardian fluid mechanics: unexplored flow studies in the Codex Arundel, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 107).

Leonardian fluid mechanics in the Manuscript L, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 108).

Cf. Matilde Macagno, Geometry in motion in the manuscripts of Leonardo da Vinci, Iowa: University of Iowa, Division of Mathematical Sciences, Department of Mathematics, 1987.

220. CA 109va (303v, c. 1490): "Fa le proposizione semplice e poi la dimostrazione configure e lettere."
221. A 31r: "Io ti richordo chettu facci le tue propositioni echettu alleghi lessopra scritte cose peresenpli e non per propositioni chesarebe tropo senplice edirai chosi."
222. A 11r: "questa propositione si pruova perrisperientia."
223. A 57r: "Questa propositione si pruova chiaramente... per ragione chonferma dalla issperienza."
224. A 45v: "Quessto si pruova perissperientia." Cf. Mad. I 87v: "Pruova e conclusione ultima di questo tale moto, provato per lo sperimento della bilancia di sopra."
225. A 13v: "Propositione... Pruova." Cf. A 15r.
226. A 45v: "La ragione della propositione."
227. A 46v: "Questa ragione si vede manifestamente."
228. A 47r: "Questa si dimostra chiaramente."
229. BN 2038 12v: "la propositione di sopra e molto bene dimostrata e chonferma dalla sperienza"; 15r: "la ragione promessa di sopra chiara mente apare per isperienza"; 15v: "Questa...propositione chiara mente apare essiconferma dala esperienza."
230. Mad. I 77r, 78r.
231. Arundel 263 32v, 67v.
232. Forster II 73r, 95r, 97v, 98r, 99r, 99v, 100r, 104r, 120v, 135r, 146r, 147v, 148r, 153v, 154r.
233. CA 126va (348r,c. 1490-1492):

E se tu dicessi questa non essere bona sperienza, perche l'acqua in se è quantità unita e continua, e'l miglio è disunito e discontinuo a questa parte io ti rispondo che io vo'pigliare quella licenza che'è comune ai matematici, cioè, siccome loro si dividano il tempo a gradi, e di quantità continua la fanno discontinua, ancora io faro il simile, dando col miglio o renella comparazione all'acqua.

It is interesting to note that Leonardo sometimes records conditions which are not suitable for experiment as on CA 151va (407v, c. 1500):

"E se volessi sperimentare con quantità continua, nessuna cosa liquefatta con foco non è bona perché la prima parte si raffreda e sicongela, quando l'ultima e ancora liquida. Se volli fare questa prova, la cerbottana non e bona, perché..."

On rare occasions he states clearly that he is skeptical as regards a solution as on CA 75va (205r, 1506-1508): "Ma di questa non veggo nello umano ingegno modo di darne scienzia."

234. CA 86ra (234r, c. 1490-1492): "La sperienza interpetre della artefiziosa natura ne dimostra questa figura essere per necessità constretta a non altre menti oprare che qui figurata sia." This idea of interpreters of nature recurs on CA 117rb (323r, c. 1490): "E da essere giudicati e non altrimenti stimati li omini, inventori e'nterpreti tra la natura e gli omini."
235. CA 274vb (739v, c. 1495): "Fa che questa figura ritorni nella sperienza, inanzi tu giudichi altro di lei."
236. F 91v: "Tutti queste figure anno ausscire dalla ssperientia."
237. Arundel 19r: "O fatto pruova io medesimo disegnandole."
238. Leonardo Studies I, as in note 32, pp. 202-239.
239. CA 203va (543v, c. 1495-1497):

"Ma direno solamente i moti essere di 2 nature, delle quali l'uno e materiale e l'altro spirituale, perche non e compreso dal senso del vedere, overo direno d'uno essere visibile e l'altro invisibile."

240. CA 93vb (257r, c. 1513):

"Dove la scienza de'pesi è ingannata dalla pratica."O trovato essi antichi essersi ingannati in esso giudizio de'pesi, e questo inganno è natoperche in gran parte della loro scienza anno usati poli corporei, e in gran parte polimatematici, cioè mentali, o vero incorporei, le quali inganni pongo qui di socto.

241. CA 200r (537r, c. 1515):

"Dal meccanico punto al matematico e varietà infinità perchè esso meccanico è visibile e per conseguenza quantita continua e ogni quantità continua e divisibile in infinito."

242. E 8v, as in note 179 above.
243. These passages have been translated and discussed in the author's Leonardo Studies, vol. 1, as in note 32, p. 223.
244. CA 221vd (597br, c. 1490): "Queste regole sono da usare solamente per ripruova delle figure."
245. CA 274rd (738r, c. 1495): "Io fo molte figure perchŠ tu conosca tutti i casi, che son sotto posti a una sola regola."
246. CA 149rb (403r, c. 1493-1495): "La regola del tuo libro proceder… in questa forma: prima l'aste semplice, poi sostenute di sotto, poi sospese in parte, poi tutte, poi esse aste fieno sostenitori d'altri pesi."
247. CA 86va (234v, c. 1490-1492): "Regola de'sostentaculi traversi."
248. CA 119va (327v, c. 1490):

Molti mi crederanno ragionevolmente potere (mi) riprendere, allegando le mie prove esser contro all'alturità d'alquanti omini di gran reverenza apresso de'loro inesperti judizi, non considerando le mie cose esser nate sotto la semplice e mera sperienza, la quale e maestra vera.

Queste regole son cagione di farti conoscere il vero del falso la qual casa fa che li omini si promettano le cose possibili e con piu moderanza, e che tu non ti veli di ignoranza che farebbe che, non avendo effetto, tu t'abbi con disperazione a darti malincolia.

249. CA 337rb (922r, c. 1493-1495): "Effetto delle mie regole... elle tengon la briglia all'ingegneri e investigatori a non lasciare promettere a se medesimo, o ad altri, cose impossibili, e farsi mattu o giuntatore." Cf. 368rc: "Queste regole fanno le operatori solleciti perch‚ scoprano e loro errori."
250. CA 154rb (417r, c. 1508-1510): "La sperienza non falla mai, ma sol fallano i vostri giudizi prommettendosi di quella effetto tale che in e nostri experimenti causati non sono."
251. CA 221vd (597br, c. 1490): "e questa regola e nata dalla 14a e ultima del 2o delli elementi d'Euclide."
252. G 42v : "colla reghola della penultima di pittagora."
253. CA 153vd (415r, 1493-1495): "Pruova e fa regola della differenzia ché da colpo dato coll'acqua sopra l'acqua, a l'acqua che cade sopra cosa dura."
254. CA 337rb (922r, c. 1493-1495): "Ancora farai regola de'diversi viaggi che fa la ballotta."
255. Mad. I 51r: "Fanne sperientia e poi regola."
256. Mad. I 148v: "Quella regola che ttu usi a trovare la natura del tirare ancora userai nella natura dello spingere."
257. CA 271vb (732ar, c. 1508): "Regola."
258. CA 130va (360r, c. 1517-1518), as in note 177 above. Cf. another version of this passage on CA 108rb (IV 300r, 1517-1518):

Se una regola ti trasmuta la figura d'una superfizie 'n un'altra figura e che la medesima regola restituissi la prima figura a tal superfizie, certo tal regola è perfetta, come si vede a presso alli aritmetrici ne'numeri partiti per un altro numero e poi rimultiplicati per il numero che lo partì, rifaccia il primo numero, eccetera.Come si vede al 4 partire il 12 in 3 parte e rimultiplicar da poi el 4 per 3; rifa il 12.

259. Mad. I 129r: "Quando una regola fia conferma da 2 varie ragione e ssperientie quella regola fia allora detta generale."
260. CA 20va (66r, c. 1493-1495): "Per fare regola generale della differenzia ch'è da peso semplice a peso col colpo di diversi moti e forze."
261. CA 82rb (222r, c. 1493-1494):

"Ricordo come tu debbi fare sperienza del reggere, a vero quanto peso po sostenere uno filo di ferro; alla quale sperienza terrai in questo modo... e fa di ciascuna cosa regola generale."

262. CA 253va (682v, c. 1493-1495):

"Regola generale: a sapere una trave legate nello stremo da una corda, che sia da uno solo loco tirata e levata in più, e saper dire, in tutti li gradi del suo levarsi, di quanto peso sia al suo motore."

263. CA 268va (723v, c. 1493-1495):

"Regola generale come ogni trave sospesa per li sua stremi da perpendiculari corde, darà di sè equale peso a ciascuna corda."

264. CA 155vb (421v, c. 1495-1497): "Fammi regola generale di dare occhetto che move ogni rota."
265. Mad. I 60r: "Se volle fare regola gienerale infra 2 mobili."
266. Mad. I 77r: "Sperimentate ed e regola gienerale."
267. Mad. I 170v: "Regola gienerale de'pesi sopra li stremi braci delle bilancie."
268. Mad. I 171v: "Questa e gienerale regola."
269. Mad. I 164v: "Questa si dimanda pratica, ma ricordati di mettere la teorica dinanzi."
270. CA 147va: (398v, c. 1500): "Nessuno effetto ‚ in natura sanza ragione; intendi la ragione, e non ti bisogna sperienza."
271. Mad. I 152v:

"Vedi che mirabil cosa e a considerare [come] questa natura adopera in tutte sue cose e con che legie ell'ha terminato li effetti di tutte le cause i quali e impossibile mutare inalcuna minima parte."

272. Leonardo Studies I, as in note 32, pp. 56-86.
273. Kenneth Keele, as in note 6, pp. 43-92.
274. CA 116rb (320r,c. 1495-1498):

Piu lumi con un corpo
Un lume con piu corpi
Piu lumi e piu corpi
Piu lumi sopra un corpo

275. Leonardo Studies II, as in note 50.
276. CA 147va (398v, c. 1500):

Le nature regulari de'contrapesi, che premiano i bottini sono 9 cioe

Piu larghichel bottino e piu grevi
Piu larghi e piu lieve
Piu larghi ed equali
Piu stretti e piu grevi
Piu stretti e piu levi
Piu stretti ed equali
Equali e piu grevi
Equali e piu lievi
Equali ed equali

Leonardo returns to this problem on F 96 [48]r (1508) under the heading:

Contrapesi regulari dando di loro gravita comparazione allacqua:
piu larghee di materia piu grave
piu larghae piu lieve
piu larghaed equale
piu strettae piu grave
piu strettae piu dieve
piu strettaed equale
equale e piu grave
equale e piu lieve
equale ed equale

277. See, for instance, H 3v-4r (1494) where he declines the verb "amo" systematically. The roots of this approach can be traced back to his long lists of verbs in the Codex Trivulzianus, which appear to be connected with his efforts to learn courtly language. A thorough study of these lists has been made by Augusto Marinoni, Gli appunti grammaticali e lessicali di Leonardo da Vinci, Milan: Castello Sforzesco, 1944, p. 46 etc.

By c. 1497 we find Leonardo making lists of action verbs such as that on CA 213vb (VII 569r):

Percotere Divellere Dimenarsi Fregare
Cadere Diradicare Levarsi
Saltare Precipitare Abassarsi
Gittare Ruinare Piegarsi
Battere Isbalzire Dirizarsi
Urtare Disbattere Torciersi
Spingiere Sdracciolare Divincolarsi
Tirare Scorrere Treittare
Stracinare Fugire Crollare
Rotolare o burlare Sfuggire Ristringiersi
Rivolgiere Deripare Allargarsi
(Bulare) Dimostrare Sgambettare
Movere Spanarsi Calcitrare
Correre Scotersi Sollevarsi
Sbattere

Shortly afterwards on L 87v (1497-1502) he makes lists of all different kinds of trees. Here we catch a first glimpse of a more systematic approach in considering the alternatives:

Trees low, tall, sparse, dense with leaves, dark, bright, yellow, red, branching upwards, branching towards the eye, branching below, with white trunks, transparent in air, not transparent in air, those which are narrow, spread out. ("Alberi bassi alti rari spessi cioe difoglie scuri ciari ciali rossi ramifichati in su chidirita all ochio che ingiu gambi bianchi a chistra par laria alcinno chietrito di poste, chieraro.")

I am suggesting that his study of language and particularly the systematic grammatical treatment of conjugations was one of the early incentives in provoking Leonardo to think in terms of listing variables and their possible combinations. Yet the idea of a fully systematic approach emerges in his perspectival studies as has been shown in Leonardo Studies I and is first evident in tabular form as in a list of weights in CA 152vb (410v, c. 1490-1495), where balances are shown corresponding to each of the fractions 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2, 0/0. In short, it was his conviction of some underlying mathematical regularity that changed his habit of exhaustive list making into the beginnings of a systematic programme for understanding nature, now frequently thought of as Baconian science.

278. Leon Battista Alberti, Ludi matematici, ed. S. Timpanaro, Florence: 1926. Cf. Luigi Vagnetti, "Considerazioni sui Ludi Matematici," Studi e documenti di architettura, Florence: Teorema Edizioni, n. 1, 1972, pp. 175-259.
279. Arundel 66r: "Dice batissta albertj nuna sua opera titolata ex ludis rerum mathematicarum che quando la bilancia."
280. CA 99vb (273br, c. 1515): "De ludo geometrico, nel quale si dà il processo l'infinite varietà [di] quadrature di superfizie dilati curvi."
281. Arundel 154r:

la piramide astesa a una data lunghezza
la piramide acortata a una data bassezza
della piramidesi facci il cubo
del cubo si facci la piramide sua
del cubo si facci la piramide a una data altezza
duna data altezza di piramide se ne facci il cubo

duna piramidesi facci una tavola duna data grossezza
duna piramidesi facci la tavola di data larghezza
duna piramidesi facci la tavola di data larghezza e lunghezza

282. CA 246vb (667vb, c. 1505-1506): "Accortare, Allungare, Ingrossare, Assottigliare, Allargare, Restringere."
283. CA 246vb (VIII 667v, 1505-1506): Trasmutatione semplice
racortisi a una data distantia sanza mutatione di larghezza quanto singrossa
racortisi
a una data lunghezza sanza mutatione di grossezza quanto salarghera
allunghisi a data lunghezza sanza diminutione di larghezza quanto sasottigliera
allunghisi
a data lunghezza sanza diminutione di lunghezza quanto sallargera
ingrossisi a data grossezza sanza mutatione di larghezza quanto s'accortera
ingrossisi
a data grossezza sanza diminutione di lunghezza quanto si ristrignera
assottiglisi a data grossezza sanza crescer di larghezza quanto si distendera
assottiglisi
a data grossezza sanza astensione di lunghezza quanto s'allargera
allarghisi a data larghezza sanza mutatione di grossezza quanto racortera
allarghisi
a data larghezza sanza mutatione di lunghezza quanto ingrossira
risstringhisi a data larghezza sanza mutatione di grossezza quanto s'alunghera
risstringhisi a data larghezza sanza mutatione di lunghezza quanto s'ingrossera
284. CA 246vb (VIII 667v, c. 1505-1506): De trasmutazion composite

Raccortisie ristringasia dati termini :quanto s'ingrossera
Raccortisie allarghisia dati termini :quanto ingrossera o assotigliera

Ristringasie assottiglisia dati termino :quanto s'allungher…
Ristringasie ingrossisia dati termini :che far… la lunghezza

Allunghisie allarghisia dati termini :quanto s'assottiglierà
Allunghisie restringasia dati termini :quanto s'assottiglierà oingrossera

Allarghisie assottiglisia dati termini :che far… da sua lunghezza
Allarghisie ingrossisia dati termini :quanto si raccorterà

<Ingrossisi>e raccortisi a dati termini :quanto <s'>allarghera o ristrignera
<Ingrossisie allar>ghisia dati termini :quanto s'accorterà
<Ingrossisie allar>ghisia dati termini :che mutazione ara la lunghezza

285. When Leonardo crosses something out, it means he has developed it somewhere else, frequently in a different manuscript.
286. Forster I 12r-11v:

1. accortae non mutare grosseza quanto s'allarga
2. accortae non mutare largheza quanto s'ingrossa

3. allungae non mutare grosseza quanto si restrigne
4. allungae non mutare largheza quanto s'asottiglia

5. ingrossa e non mutare lungheza quanto si sstrigne
6. ingrossa e non mutare largheza quanto si racorta

7. assottiglia e non mutare lungheza quanto s'allarga
8. assottiglia e non mutare largehza quanto s'allunga

9. allargae non mutare lungheza quanto s'asottiglia
10. allargae non mutare grosseza quanto si racorta

11. resstrigni e non mutare lungheza quanto s'ingrossa
12. restrigni e non mutare grosseza per quanto s'alunga

13. acortisi e ingrossisi quanto s'allarga
14. acortisi e assottiglisi quanto allarga
15. acortisi e allarghisi qual fia la grossezza
16. acortisi e restringasi quanto s'ingrossa

17. allunghisi e ingrossisi quanto si restrigne
18. allunghisi e assottiglisi quanto s'allarga
19. allunghisi e allarghisi quanto s'assottiglia
20. allunghisi e resstringasi quanto s'ingrossa
21. ingrossisi e allarghisi quanto s'acorta
22. ingrossisi e restringasi quanto s'alunga
23. assottiglisi e alarghisi quanto s'alunga
24. assottiglisi e restringasi quanto s'allunga
25. ingrossisi e alunghi quanto restrigne
26. ingrossisi e acortisi quanto s'allarga
27. assottiglisi e alunghisi quanto restrigne
28. assottiglisi e acortisi quanto s'alarga

287. See, for instance, James Edward McCabe, Leonardo da Vinci's De ludo geometrico, (Ph.D., University of California at Los Angeles), 1972.
288. CA 136ra:

Saette e corde equali anno equali archi
Saette e archi equali anno equali corde
Corde e saette equali anno equali archi
Corde e archi equali anno equali saette
Archi e saette equali anno equali corde
Archi e corde equali anno equali saette

There is a related list on CA 242vb (660v, 1517-1518) where he also abbreviates the terms saette corda and archo as sa, cor, ar in the following form:

sa cor
sa ar
-----------
cor sa
cor ar
-----------
ar sa
ar cor

289. See Ladislao Reti, as in note 5, p.
290. See Keele, pp. 93-130 and the author's Leonardo Studies II as in note 50.
291. Ibid.
292. Mad. I152r:

"Io mi trovo 4 gradi di forza e 4 dipeso e similmente 4 gradi di moto e 4 ditempo. E voglio con questi gradi adoperare e secondo le necessita agiungere o llevare colla imaginatione e trovare quello che natura insua legge ne vole."

293. Ibid:

2 di pesoe 4 di forza e 4 di moto vole 2 di tempo
2 di pesoe 2 di forza e 4 di moto vol 4 di tempo
2 di pesoe 2 di forza e 2 di moto vol 2 di tempo

2 di forzae 4 di peso e 4 di moto vol 8 di tempo
2 di forzae 2 di peso e 4 di moto vol 4 di tempo
2 di forzae 2 di peso e 2 di moto vol 2 di tempo

2 di motoe 4 di forza e 4 di peso vol 2 di tempo
2 di motoe 2 di forza e 4 di peso vol 4 di tempo
2 di motoe 2 di forza e 2 di peso vol 2 di tempo

2 di tempoe 4 di forza e 4 di peso vol 2 di moto
2 di tempoe 2 di forza e 4 di peso vol 1 di moto
2 di tempoe 2 di forza e 2 di peso vol 2 di moto

1 di forzae 4 di peso e 4 di moto vol 16 di tempo
1 di tempoe 4 di moto e 4 di peso vol 16 di forza
1 di motoe 4 di peso e 4 di forza vol 1 di tempo
1 di pesoe 4 di moto e 4 di forza vol 1 di tempo

294. CA 212vb (565v, c. 1502-1504):

Se una potenzia move un grave una quantità di spazio in tanto tempo, la medesima potenzia moverà la metà di quel grave 2 tanti quello spazio nel medesimo tempo tutto.Ovvero essa intera potenzia tutto quel grave nel mezzo tempo la metà ditale spazio; ovvero nella metà dello spazio, della intera potenzia e tempo dui tanti quel grave; overo l'intera potenzia nella metà del temp<o>, tutto quel peso nella metà di quello spazio.

295. CA 212vb (565v, c. 1502-1504):

pg t s
pg t s
pg t s
pg t s2
tg p

296. CA 355va (982v, c. 1503-1505):

p g t s2 p g t s

p g t s p g t2 s

p g t s p g t s

p g t s p g t

297. CA 165va (V 449r, c. 1500-1503):

De semplici e composi
Retto, curvo e retto
Curvo, retto e curvo
Curvo e retto, retto
Retto e curvo, curvo
De composti
Curvo e retto, retto e curvo
Curvo e curvo, retto e retto
Retto e retto, curvo e curvo
Curvo e curvo, curvo e curvo
Retto e retto, retto e retto

For a related list see: CA 98rb (III 269r, c. 1500):

E moti composti sono:
retto curvo e retto
curvo retto e curvo
curvo e retto retto
retto e curvo curv<o>

Cf. Madrid I 131v (1499-1500): "Divisione del moto naturale"

Senplicenaturale
Senpliceacidentale
Senpliceneutrale

Senplicecirculare d'equidiacente moto
Senplicecirculare d'equidiacente polo
Senplicecirculare neutrale

Flessuoso rettilineo
Flessuoso curvilinio

Angulare rettilinio
Angulare curvilinio

Moticompossti
Naturale circulare
Accidentale circulare
Neutrale circulare

Naturale flessuosorettilinio e curvilinio
Naturale flessuosocurvilinio e rettilinio
Naturale fressuosorettilinio e curvilinio

Circulare flessuoso curvilinio d'equidiacente moto
Circulare flessuoso rettilinio d'equidiacente polo
Circulare flessuoso neutrale

Circulare naturale e fressuoso rettilinio
Circulare acidentalee fflesuoso

Angolorettilinio flessuoso
Angolocurvilinio e fflessuoso

298. CA 193rb (VI 525r, c. 1500):

Mobile duro con piano duro
Mobile tenero con piano tenero
Mobile duro con piano tenero
Mobile tenero con piano duro

Mobile aspro con piano pulito
Mobile polito con piano aspro
Mobile aspro con piano aspro
Mobile polito con piano aspro

He considers these problems in a different form on CA 198va (VI 532v, 1506-1508):

Sono le confregazioni de'corpi de 4 sorte, delle quali
La prima è quando due corpi sono puliti e piani, come qui è proposto

La seconda è quando il corpo stracinato e'l piano dove si move, è aspro
La terza è quando il corpo stracinato è aspro e'l piano ove simove, è polito

Il quarto modo è quando il corpo stracinato e'l piano dove si stracina è aspro

299. CA 74vb (III 201v, c. 1505-1506):

Scontri d'acquaequali in potenzia e in quantita
Scontri d'acquaequali in potenzia e none in quantita
Scontri d'acquaequali in quantita e none in potenzia
Scontri d'acquainequali in potenziae in quantita

300. CA 65va (III 183v, c. 1508):

Della percussione del raro nel raro
Della percussione del raro nel denso
Della percussione del denso nel raro
Della percussione del denso nel denso

301. W 19141v (K/P 99v): "Io o per istrumenti dj questo 4ø libro a mannegiare 6 cose coe polo, subbio, lieva, corda, peso e motore."
302. Ibid.: "Travagliamento e natura de membri apartenenti allusito dellargano."
303. Ibid.:

dato il polosubbio lieva corda e peso si ricercha il motore
dato il subbio lieva corda peso e motore il polo
dato il lievacorda peso motore e polo subbio
dato il cordapeso motore polo e subbio lieva
dato il pesomotore polo subbio e lieva corda
dato il motore polo subbio lieva e corda peso

304. Ibid.:

dato la lieva e contralieva polo el peso si ricercha il motore
dato la contralieva polo peso e motore la lieva
dato il polo peso motore e llieva ___________ la contralieva
dato il peso motore lieva e contralieva ___________ il polo
dato il motore lieva contralieva el polo ___________ il peso

Leonardo had, of course, begun examining these problems much earlier. See, for instance, his comments on A 62r (1492).

305. W 19143r (K/P 101r):

Discretion de membri della vite ellor travagliamenti
Dato la vite dente numero lieva peso si cerche il motore
Dato il dente numero lieva peso elmotore la vite
Dato il numero lieva peso motore e vite dente
Dato la lieva peso motore vite e dente numero
Dato el peso motore vite dente numero lieva
Dato el motore vite dente numero lieva peso

On CA 381rb (1480-1482) we find another earlier example of his systematic approach with respect to screws:

Per fare pruova della forza delle viti
Un pane alla femmina e farai pruova con 2 e con 3 e con 4 e 5 e 6 e col medesimo peso e lieva, e attendi alla variazione, E farai pruova se la vite di sotto abbrevia il tempo o no col pari pane alle femmine dell'una che della altra, e cosi si vuole fare pruova d'ogni ragion vite, cioè in 2 doppi, in 3 in 4, in 5, in 6.

306. Ibid.: Esercitatione e natura membri delle taglie ellor circunstantie - libro 4o

Dato il diametro numero polo peso e corda si cercha il motore
Dato il numero polo peso corda e motore_________ il diamitro
Dato il polo peso corda motore e diamjtro_________ il numero
Dato il peso corda motore diametro e numero________ il polo
Dato la corda motore diametro numero e polo_________ il peso
Dato il motore diamjtro numero polo e peso _________ la corda

As Kenneth Keele has noted in his edition of the Corpus of Anatomical Drawings in the Collection of Her Majesty the Queen (1977-1980) there are related passages on CA 381r, Mad. I 4v, Forst. II and B 70v.

307. Ibid.:

Lequalj cose dette sono 6 ora esenda 5 ericerchasi la natura della sessta la quale invero essottile invesstichatione e nonsi fara maj sanza la sua teoricha coe della difinjtione della 4 potentie come peso forza moto e percussione.

A skeptical reader might insist that all of the above passages are exceptions to a lack of rule in Leonardo. To establish that this pattern is more basic to Leonardo it may be useful simply to cite in chronological order other obvious examples in the Codex Atlanticus. On CA 69rb (192r, 1502), for instance, we find a mathematical list in terms of different kinds of square roots:

rotti per rotti
sani e rotti per rotti
sani e rotti per sane rotti
sani e rotti per rotti
rotti per sani e rotti
sani e rotti per sani
sani per sani e rotti
sani per rotti
rotti per sani

On CA165va (449r, 1500-1505) Leonardo makes a list involving moving objects:

Retto il moto del motore e circulare il mobile
Circulare il motore e retto il mobile
Circulare il motore e l'mobile
Retto il motore e'l mobile: quandoil caval tira da barca.

On CA 241vb (657v, c. 1513) there is a list involving another of Leonardo's four powers:

Del percosso immobile
Del percosso veloce quanto il percussore
Del percosso men veloce che'l percussore
Della percussion del circunvolubile colla parte incidente
Della percussion del circunvolubile colla parte refressa

On CA 182rc (499r, 1517-1518) Leonardo applies this method of playing with variables to transformational geometry:

Le parte d'un tutto diviso in parte rifanno il loro tutto essendo infra loro ricongiunte
Le parte d'un tutto, rincongiunte a tutte le lor parte, sempre rifanno il lor tutto
Le parte d'un tutto diviso in parte, essendo ricongiunte, rifanno il lor tutto

In addition to the above there are also lists such as those on CA 29rb (82r, 1499-1500), CA 202vb (542r, c. 1500), CA 105ra (290r, c. 1500), CA 79va (214av, 1505-1506) which are not entirely systematic. There are also systematic listings of multiplication tables as on CA 110va (307v, 1517-1518) and of geometrical proportion, which I have discussed in Leonardo Studies I, as in note 32, pp. 240-277.

308. See note 305.
309. W 19060r (K/P 153r):

Delle machine
Perche natura non puo dare moto allianjmalj sanza strumenti machinali chome prima sidjmosstra in quessto libro nellopere motive daessa natura fatta nelli animali e per questo io ho chonposto le reghole nelle 4 potentie dj natura sanza lequali niente per essa po dare moto lochale a essi animali.

310. W 19070v (K/P 113r): "il libro della scientia delle machine va inanzi al libro degovamenti. fa legare litua libri dj novo."

7. PLANS FOR PUBLICATION

311. CA 391ra, (1082r, c. 1482): "farmi intender da vostra excellenzia, aprendo a quella di secreti mei."
312. CA 102ra (279v, c. 1503-1505): "Fa la segreta."
313. Cf. the drafts of a letter to Giuliano de'Medici on CA 247vb (671r, c. 1514-1515) and 283ra (768r, c. 1515) and 182vc (500r, c. 1515) and 92rb (252r, c. 1514-1515). These are translated in The Literary Works of Leonardo da Vinci, ed. Jean Paul Richter, London: Phaidon, 1970, vol. 2, pp. 336-339.
314. CA 104rb (289r, c. 1493-1495): "A di primo d'Agosto 1499 scrissi qui de moto e peso."
315. CA 375rc (1047r, c. 1513-1514): "Fa domane figure discendenti infrall'aria, di varie forme di cartone, cadenti dal nostro pontile; e poi disegna le figure e li moti, che fanno li discensi di ciascuno in varie parte del discenso."
316. M 38r: "sperientia di domane." Cf. 220ra (VII 591r, 1506-1508): "Vedi domattina se l'uccello che gira venendo conto al vento."
317. Mad. II 157v: "Qui si fara ricordo di tucte quelle cose le quali fieno al proposito del cavallo de bronzo del quale al presente sono inn opera."
318. CA 214rd (571ar, c. 1507-1508): "Vedi doman tutti questi casi e li copia e po'cancella li originali e lasciali a Firenze, acci• che se si perdesse quelli che tu parti con teco che non si perda la invenzione."
319. Mad. I 6r:

Legimi lettore setti diletti di me, perch‚ son rarissime volte rinata al mondo. Perchè la patientia di tale professione si trova in pochi che vogliono di novo riconpore simile cose di novo. E venite e omini e vedere i miracoli che per questi tali studi si scopre nella natura.

Cf. Mad. I 0r: "Petitione ch'io dimando a'mia lettori."

320. CA 108ra (299r, c. 1508): "De moto e peso... Ma intendi, lectore, che in questo casu s'a (sa) a fare conto coll'aria."
321. CA 119va (327v, c. 1490):

Or guarda, lettore, quello che noi potremo credere ai nostri antichi i quali anno voluto difinire che cosa s[ia a]nima e vita, cose inprovabili q[uando] quelle che con isperienzia ognora si possono chiaramente conoscere e provare sono per tanti seculi ignorate e falsa mente credute.

322. CA 268va (723v, c. 1493-1495): "Io richiedo a te, lettore, quando io nomino trave, che tu intenda io volere dire perso d'equale lunghezza e d'equale peso, cioŠ corpo, che … lunghezza d'equale peso e grossezza."
323. CA 145va-vb (393v, c. 1500): "Al Diodario di Soria, loco tenente del sacro Soltano di Babilonia....Non ti dolere, o Diodario, del mio tardare a dar risposta alla tua desiderosa richiesta."
324. Mad. I 152v: "Io ricordo a te conponitore di strumenti."
325. BN 2038 10r: "Il pittore debbe prima....sempre il pittore debbe chonsiderare."
326. BN 2038 2r: "Adunque chonosciendo tu pittore."
327. BN 2038 27v: "quando tu disegniatore vorai fare bono e utile studio."
328. BN 2038 21v: "Quando vorete ovoi disegnatiori."
329. BN 2038 19v: "Se voi storiografi opoeti oaltri matematici nonnavessi visto le cose male."
330. W 19075r (K/P 179r): "ho sspechulatore dj questa nosstra machina non ti contristara perche collaltrui morte tu ne dja notitia ma rallegrati che il nostro altore abbia fermo lo intelletto a ttale ecellentja djstrumento."
331. E 18v: "Richordati opictore chettanto son varie le osscurita dellonbre."
332. E 19v: "O pictore natomista."
333. G 47r: "Ho sspechulatore del le chose nonti laldaro di conossciere le cose che ordinariamen te perse medesima la natura... chontucie. Ma rallegrati dicho nossciere il fine di quellelle chose che son disegniate dalla mente tua."
334. CA 384ra (1062r, c. 1493-1495): "Io dissi nella la conclusione come la percussione...Ora tu da te sperimenta il bastone."
335. Mad. I 149r: "Io ti dimando." On occasion we find a more complex relationship where Leonardo uses a dialogue form with the reader as on CA 211rb-vc (VII 562r, 1495-1497):

Io voglio sol vedere il peso col quale tu, appicandolo in mezzo alla equidiacente aste, le dai certa pa<r>te di curvit…. Di poi tocca l'asta dove ti piace i io ti dir• che peso bisogna appiccare in essa parte a volere dare la medesima curvatura a essa aste.

In addition he has a dialogue in indirect speech in the form "the adversary says...to which it is replied," which derives from mediaeval and ultimately classical sources. There is also the phrase "Tell me" (Dimmj) which occurs on numerous occasions as listed by Carlo Pedretti in his Commentary on Richter, London: Phaidon, 1977, vol. 2, pp. 310-311.

336. W 19007v (K/P 139v):

Ma per questo brevissimo modo di figurarli per diversi aspetti se ne dara piena e vera notitia e acco che tal benefitio chiodo allominj io insegno il modo di ristamparlo con ordjne e priego voj osucesori che llavaritia non vj cosstringha affare le stampe.

337. Cf. CA 357rf (991v, 1480-1482), CA 358rb (995r, 1480-1482), and CA 372rb (1038r, c. 1497) which shows a moveable printing press.
338. Mad. II 119r:

Del gittare in istampa questa opera.
Metti la piastra di ferro di biaca a uovo e poi scrivi a mancina sgraffiando tal campo. Fatto quessto e ttu metti di vernice ogni cosa, cioŠ vernice e giallolino o mmin[i]o. E sseco che è, metti i'molle, e'l campo delle lettere fondato sulla biaca a uovo fia quello che ssi lever…insieme col minio, il quale, per esser frangibile, si romper… e llascier… le letteri apicate al rame. E poi cava il canpo con modo tuo e tti rimar… le letere di rilievo e'l canpo basso. E poì ancora mistare il minio con pece greca e così calda darla, come di sopra dissi, e sarà più frangibile. E perchè meglio si veghino de lettere, tigni la piastra col fumo del zolfo che ss'incorpora col rame.

See Ladislao Reti, "Leonardo da Vinci and the graphic arts: the early invention of relief-etching," Burlington Magazine, London, vol. CXIII, n. 817, April 1971, pp. 188-195.

Another clear reference to Leonardo's acquaintance with printing methods is found in CA 263va (710br, 1510-1515): "E se tu li volessi gittare di piombo ovver della materia che si gittano le lettere da stampa, li pezzi sarebbono il medesima di sopra." Cf. CA 83vb (226v, c. 1508): "Stampala a corpo premanente, a parte a parte facendo tale lista d'un pezzo."

339. Vasari, as in note 72, vol. 2, p. 157.
340. Leonardo da Vinci, Traité de la peinture, trans. Roland Fréart de Chambray, Paris: Impr. de I. Langlois, 1651.

8. INFLUENCE

341. Triv. 28r: "notta." Cf. Triv. 36r: "del sono del martello con la incudine."
342. Triv. 8r: "architettura notta."
343. Triv. 37r: "notto de laacqua."
344. Triv. 38v: "notta di pittura" and Triv. 39r: "pittura notta."
345. Triv. 53v: "di bataglia nota."
346. Forster I 3r: "hoc est libro intitolato de trasformatione coe d'un corpo n'un altro senza diminutione o accrescimento di materia."
347. Forster II 64r as in note 47.
348. Forster II 2r, 3v, 7v, 24v, 30v.
349. Forster II 160v: "N.B. Haec scriptum inversa et in speculo legenda est."
350. Forster III 1r.
351. Forster III 2v, 13v, 32r, 36v-37r, 40v, 48r-47v.
352. See: B 5v, 6r, 7v, 8r, 10r, 10v, 11r, 11v, 15v, 20r, 21r, 23v, 26v, 29v, 34r, 36v, 38r, 47v, 49v, 50v, 51v, 52v, 53r, 53v, 54r, 54v, 55v, 56r, 57r, 57v, 58v, 62r, 62v, 64r, 64v, 65r, 65v, 66r, 68v, 71r, 72v, 75v, 76r, 81r, 81v, 88v, 89r, 90v.

This adds a new context to the remarks of Giorgio Vasari, The Lives of the painters sculptors and architects, trans. A.B. Hinds, London: Dent, (1927) 1963, vol. 2, p. 157: "Many designs for these notions [of engineering] are scattered about and I have seen numbers of them" and p. 163

"he wrote notes in curious characters, using his left hand, so that it cannot be read without practice and only at a mirror...Whoever succeeds in reading these notes of Leonardo will be amazed to find how well that divine spirit has reasoned."

353. On the problem of Dürer copying from Leonardo's W12613 in the Dresden Sketchbook, fol. 130v and 133v, see A Weixgärtner in Mitteilungen der Gesellschaft für vervielfältigende Kunst, vol. II, 1906, pp. 25-26 and Lord Kenneth Clark's edition of The Drawings at Windsor Castle, London: Phaidon, 1968, vol.1, pp. 126-127.
354. See: Marilene Putscher, "Ein Totentanz von Titian: die 17 grossen Holzschnitte zur Fabrica Vesals (1538-152)", Metanocite, 1984, 23-40. This dicusses a work entitled: Notomie di Titiano. The National Union Catalogue, vol. 65, p.378. lists an edition of c.1650 edited by Domenico Maria Bonavera noting that:

Bonavera includes the 3 skeletal plates and the 14 muscle plates (without description) re-engraved on copper from Calcar's work for the Fabrica of Vesalius and presents them as the work of Titian.

355. Carlo Pedretti, Commentary [on Jean Paul Richter, The Notebooks...], London: Phaidon, 1980, pp. **
356. Ibid., pp.**
357. Cf. Carlo Pedretti, Documenti e memorie riguardanti Leonardo da Vinci a Bologna e in Emilia, Bologna: Editoriale Fiammenghi, 1953.
358. Claude de Boissière Delphinois, L'art d'arithmétique contentant toute dimension tant pour l'art militaire que par la géométrie et autres calculations, revue et augmentée par Lucas Tremblay, Paris: Guillaume Cavellat, 1561, particularly pp. 53r, 72v.
359. Abel Foullon, Usaige et description de l'holometre pour scavoir mesurer toutes choses qui sont soubs l'estendue de l'oeil, Paris: P. Béguin, 1555.
360. Fabrizio e Gaspare Mordente, Il compasso, Antwerp: Plantin, 1584. These developments will be explored at greater length in the author's Mastery of Quantity.
361. CA 157vb (425v), 358ra (1064r), 248ra (672r). Cf. notes 190-191 above. For further evidence of the proportional compass in a manuscript dated 1509 cf. Anonymous, Breve corso di matematica, Modena, Biblioteca estense, Ms. It. 211=a.W.6.22.h.
362. Hans Lencker [attributed to], Perspectiva, Chicago, Newberry Library, Ms.B 128, particularly fol. 55v.

Cf. the Landgraf of Hesse's instrument in: Wilhelm IV, Landgraf Hessen, (attributed to), Circini proportionalis descriptio, Biblioteca apostolica Vaticana, Reg. lat. 1149, fol. 2r:

1. Datam rectam lineam iuxta datam proportionem dividere
2. Datam lineam circularem in propositas partes secare
3. Datam superficiem in similem superficiem multiplicare aut minuere
4. Datam corpus in simile corpus multiplicare aut minuere
5. Rationem cuiuslibet diametri ad suam circumferentiam invenire
6. Superficiem circularem aut quadratam in aliam transferre
7. Datum globum et quinque corpora regularia in sese invicem transferre.

363. Levinus Hulsius, Dritter Tractat der mechanischen Instrumenten Levini Hulsii, Beschreibung und Unterricht dess Jobst Burgi Proportional Circkels... Frankfurt: Levini Hulsii, 1604; Philip Horcher, Libri tres in quibus primo constructio circini proportionum edocetur, Mainz: Apud Balthasarum Lippium, 1605.
364. Benjamin Bramer, Bericht und Gebrauch eines Proportional Linials neben kurtzem Underricht eines Parallel Instruments, Marburg: P. Egenolff, 1617.
365. Cf. Ivo Schneider, Der Proporzionalzirckel. Ein universelles Analogrecheninstrument der Vergangenheit, München: Oldenbourg Verlag, 1970 (Deutsches-Museum. Abhandlungen und Berichte. 38 Jg., 1970, Heft 2).
366. Concerning Coignet see Paul Lawrence Rose, "Origins of the Proportional Compass from Mordente to Galileo", Physis, Florence, vol. 10, no. 1, 1968, pp. 53-69.
367. Galileo Galilei, Le operazione del compasso geometrico e militare, Padua: Casa dell'Autore, Per Pietro Marinelli 1606. On Galileo see: Stillman Drake, "Galileo and the First Mechanical Computing Device," Scientific American, New York, vol. 234, no. 4, April 1976, pp. 104-133.
368. This evidence has been considered in detail in the author's Leonardo Studies II, as in note 50. For a standard view on the history of the telescope see: Albert van Helden, The invention of the telescope, Philadelphia: American Philosophical Society, 1977. (Transactions of the American Philosophical Society , vol. 67, n.4, 1977).
369. See, for instance, Giacomo Barozzi il Vignola, Le due regole della prospettiva pratica, ed. E. Danti, Rome: Zanetti, 1583, Prob. XI, Prop. XL, p.49:

Perche oltre alle descrittione delle figure rettilinee, apporta gran commodita al prospettivo saperle trasmutare d'una nell'altra, ho voluto in queste tre seguenti propositioni mostrare il modo secondo la via commune non solamente di trasmutare il circolo e qual si voglia figura rettilinea in un altra, ma ancho di accrescerle e diminuirle in qual si voglia certa proportione, accio in questo libro di prospettiva habbia tutto quello, che a cosi nobil pratica si fa mestiere.

It is instructive to note that Leonardo's verb for transformation (trasmutare) is also used by Danti.

370. For examples see Edmond R. Kiely, Surveying instruments, their history and classroom use, New York: Bureau of Publications, Teachers College, Columbia University, 1947 (National Council of Teachers of Mathematics, Yearbook 19). For the underlying philosophy see the author's "Mesure, quantification et science," in: L'Epoque de la Renaissance, 1400-1600, Budapest: Akademiai Kiado, vol. 4, 199? (in press).
371. For an introduction to the context at Kassell see: L. von Mackensen, Die erste Sternwarte Europas mit ihren Instrumenten und Uhren, 400 Jahre Jost Bürgi in Kassel, Munchen: Callwey Verlag, 1982, particularly pp. 89-114. For two standard works see Klaus Maurice, Die Deutsche Räderuhr, München: Beck, 1976; Klaus Maurice und Otto Mayr, Die Welt als Uhr: Deutsche Uhren und Automaten 1550-1650, München: Deutscher Kunstverlag, 1980. For more specialized studies see: Hans von Bertele-Grenadenberg, "Eine mechanische Mondanomaliendarstellung auf Basis der copernicanischen Sekundären Epizyklen", Der Globusfreund, Nr.21-23, 1973, S.162-168; John H. Leupold, Die grosse astronomische Tischuhr des Johann Reinhold, Augsburg 1581 bis 1592, Luzern, 1974.

9. LIMITATIONS

372. See notes 2, 3 and 6.
373. CA 333v (909r, c. 1485-1487): "Non insegnare e sarai solo eccellente."

10. HISTORIOGRAPHY

374. William Whewell, History of the inductive sciences from the earliest to the present time, London: J.W. Parker, 1837.
375. On this and the complex religious, political and social context of the time see R.N.D. Martin, "The genesis of a mediaeval historian: Pierre Duhem and the origin of statics," Annals of science, London, vol. 33, 1976, pp. 119-129.
376. Pierre Duhem, Les origines de la statique, Paris: A. Hermann, 1905-1906, vol. 1, p. 192:

"Il n'est, dans l'oeuvre mécanique de Léonard de Vinci, aucune idée essentielle qui ne soit issue des géomètres du moyen age et, particulièrement du trait‚ de ce grand mécanicien que nous avons nommé‚ le Précurseur de Léonard."

Duhem subsequently published Études sur Leonard de Vinci. Ceux qu'ils a lus et ceux qui l'ont lu. Paris: A. Hermann, 1906-1913, 3 vol. His most comprehensive treatment was in the Système du monde, Paris: Hermann, 1913-1959, 10 vol.

377. Important in this context was Anneliese Maier, Studien zur Naturphilosophie der Spätscholastik, Rome: Edizioni di storia e litteratura, 1951-1955, 3 vol.
378. George Sarton, Introduction to the history of science, Baltimore: Pub. for the Carnegie Institute by Williams and Wilkins, 1927-1931 [i.e. 1950] 3 vol. in 5. Particularly II, pt. 1-2, III, pt. 1-2.
379. Lynn Thorndike, A history of magic and experimental science, New York: Macmillan, 1923-1958. 8 vol. particularly vol. 2-3.
380. Francis S. Benjamin Jr.'s most important contribution was establishing the Benjamin Data Bank of medieval scientific manuscripts now directed by Professor Wesley Stevens (Winnipeg).
381. Marshall Clagett, The Science of mechanics in the middle ages, Madison: University of Wisconsin Press, 1959.
382. Leonard Olschki, Geschichte der neusprachlichen-wissenschaftlichen Literatur, Heidelberg: Winter, 1919-1922, 2 vol.
383. Betrand Gilles, Les ingénieurs de la renaissance, Paris: Hermann, 1964.
384. Edgar Zilsel, "The sociological roots of science," American journal of sociology, Chicago, vol 47-48, no. 4, 1942, pp. 544-562.
385. Ibid., p. 544.
386. Ibid.
387. Ibid., p. 561.
388. Stillman Drake and I.E. Drabkin, Mechanics in sixteenth century Italy, Madison: University of Wisconsin Press, 1969.
389. Cassirer, as in note 3187, pp. 48-51.
390. Ibid., p. 163.
391. Erwin Panofsky, Idea, Ein Beitrag zur Begriffsgeschichte der Älteren Kunsttheorie, Studien der Bibliothek Warburg, V, ed. Fritz Saxl, Leipzig: B. G. Teubner, 1924.
392. Giorgio De Santillana, "The role of art in the scientific Renaissance," Critical problems in the history of science, ed. M. Clagett, Madison: University of Wisconsin Press, 1969, pp. 33-65.
393. There has been an incredible explosion of literature on Brunelleschi, particularly in terms of the science-art connection. For a recent bibliography see: Corrado Bozzoni, Giovanni Carbonara, Filippo Brunelleschi, Saggio di una bibliografia, Rome: Università degli studi, Istituto di fondamenti dell'architettura, 1977.
394. Ernst Cassirer, Substance and Function, trans. William Curtis Swabey, Marie Collins Swabey, Chicago: Open Court Publishing Company, 1923 (New York: Dover Publications, 1953).
395. Edwin Arthur Burtt, The Metaphysical foundations of modern science, New York: Doubleday and Co., 1924. Cf. particularly p. 28 re:Cassirer.
396. Edward W. Strong, Procedures and metaphysics, Berkeley: University of California Press, 1936, pp. 10-11.
397. For an introduction to some leading exponents of the two factions see: George Basalla, ed., The Rise of modern science, external or internal factors, Lexington (Mass.): D.C. Heath and Co., 1968.
398. Alistair Crombie, Robert Grosseteste and the origins of experimental science, 1100-1700, Oxford: Clarendon Press, 1953.
399. Marie Boas, The Scientific renaissance, 1450-1630, New York: Harper and Row, 1962 (Rise of modern science, vol. II).
400. Alexandre Koyré, Metaphysics and measurement. Essays in the scientific revolution, trans. R.E.W. Maddison, London: Chapman and Hall, 1968.
401. Alexandre Koyré, Etudes galiléennes, Paris: A. Hermann, 1939, 3 vol.; Ibid., The astronomical revolution, trans. R.E.W. Maddison, Paris: A. Hermann, 1973.
402. Thomas S. Kuhn, The Copernican revolution: planetary astronomy in the development of western thought, Cambridge, Mass: Harvard University Press, 1957.
403. Thomas S. Kuhn, The structure of scientific revolutions, Chicago: University of Chicago Press, 1962 (2nd ed. 1970).
404. John Hermann Randall, Jr., "The place of Leonardo da Vinci in the emegence of early modern science" in: Roots of scientific thought, ed. P. Wiener, A. Noland, New York: Basic Books, 1957, pp. 207-218.

11. CONCLUSIONS

405. Pietro Accolti, Lo inganno de gl'occhi, Florence: Pietro Cecconcelli 1625, p. 116 where Witelo is called "unico, & principal capo della scuola de perspettivi."
406. Luca Pacioli, as in note 1, p. 33.
407. It is striking that notwithstanding the monumental work of Ernst Zinner, Deutsche und niederländische astronomische Instrumente des 11-18 Jahrhunderts, München: Beck, 1972, which dealt mainly with astronomy, but touched on manuscripts on surveying, gauging and related fields, we still have no proper census even of the source materials. The same is true with respect to instruments although a great contribution has been made through the team led by Helmut Minow, Historische Vermessungsinstrumente. Ein Verzeichnis der Sammlungen in Europa, Wiesbaden: Chmielorz Gmbh, 1982. For an excellent survey of gauging literature another of the important themes see: Menso Folkerts, "Die Entwicklung und Bedeutung der Visierkunst als Beispiel der praktischen Mathematik der fruhen Neuzeit," Humanismus und Technik, Berlin, Bd. 18, Heft 1, 1974, pp. 1-41.