Kim H. Veltman

To Kenneth D. and Mary Keele 

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

       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.

       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.

       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 influence9, 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.

       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 Physics16 or to given propositions in Euclid. At least twenty propositions ranging from books 1 to 5 of the Elements are cited explicitly17. In his study of ancient weapons in the Manuscript B he cites a wide range of ancient authors: Pliny18, Virgil19, Lucretius20, Aulus Gellius21, Livy, Plautus, Flavius22, Lucan, Caesar23, Quintilian, Varro24, Plutarch25, Hermes Trismegistus, Pompeius Festus26, and Ammianus Marcellinusi27. Elsewhere he cites Plato28 and Vitruvius29. Mediaeval sources include Swineshead30, Thabit ibn Qurra31, Peckham32, 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 Pavia37 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.

       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 students41. 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 190v45 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.

       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 elsewhere46 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 following55," 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 bottom59, i.e. 66v. On G75r he adds two notes: "Here follows what is on the opposite page60," 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 mirrors63, 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 treatises66.

       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

       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

       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 number88. 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 F90v90, F45v91 (c. 1508) and E12r92 (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 hollow95

       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 M99, 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.

       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 universe100: 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).

Fig. 2: 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 machine114," 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 flowers118, 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 notes119 and is limited almost entirely to arithmetical proportion, practical problems deriving from the abaco school120 and some computations. Words in terms of language121 and literature122 (cf. figure 3) interest him more and constitute roughly 4% of his notes.

                                            Mental World

                                                Language

            Elements                  Transformations              Combinations

                        Ingredients      Notes     Letters      Stories

Allegory Bestiary Emblems Aphorisms Anecdotes Fables Jests Philosophy

Fig. 3: Branches of Leonardo's work in language and literature.

       Much more important are his studies of geometry. When Leonardo praises mathematics he usually means geometry123 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. 4). Pacioli leads him to study Euclid125, whose Elements deal mainly with geometry in two dimensions.

                                Mental World

                                Geometry

Principles     Elements (2D)     Basic Forms (3D)     Transformations

(Proportion)     (Euclid)              (Polyhedra)              (Lunules,Chords)

Fig.4. 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 game126 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 shade128 by means of which he can deal with human forms129, drapery130, trees and plants131, and geometry provides him with the principles of transforming their shapes (fig. 5).

               Represented world

Optics Geometry Perspective

Light and Shade Diminution of form Colour Linear Motion

Human form Drapery Trees and plants

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

Man made world

Constructed world

Architectural Mechanical Instrumental

Military Civil Weapons Machines Tools Instruments

Fig. 6: Branches of Leonardo's studies of the constructed world.

       Yet the focus of Leonardo's attention is in the constructed world which includes architecture132, mechanics and instruments (fig. 6). 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 architecture133 involves some significant innovations, his weapons134 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 machines136 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 Reti137 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 machine138 (see fig. 7).

Earth

Hoisting..Dragging Rolling Digging Weaving Pounding..Weighing..Measuring..-Space..-Time

Crane.....Winch.....Cart.....Shovel...Textile...............File......Balance...............Compass.Clock

Water Air Fire

Sailing Raising Flying Burning Mirror

Boat Screw of Archimedes Mechanical.Bird

Fig 7: Leonardo's machines and instruments seen in the context of basic actions and the four elements.

       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 clock142, balance143, 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):

       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 friction161 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 Alhazen163 (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 sight164," 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 instruments165 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 which167, 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 infinity183, 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 proposition200," a "fifth proposition201," the "43rd proposition of the first book202," 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 Archimedes211, Theodosius212, an unidentified Zenofonte213, 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.

                                PRINCIPLES

                                    FORM

                        (Transformational Geometry)

                                NATURE

                            (Four Powers)

                        INTERPRETATION

                                (Optics)

                        REPRESENTATION

                                (Perspective)

UNDERSTAND               CONSTRUCT          REPRESENT
UNIVERSE                       MAN MADE             UNIVERSE
CREATED BY GOD        WORLD                    RECREATED BY MAN

Fig. 8: 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. 8). 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.

       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 follows215," 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 Maccagno219 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 2038229 which was originally part of Manuscript A. When an experiment has been carried out Leonardo explicitly writes: "experimented". there are examples in Madrid Codex II230, Codex Arundel231 and particularly Forster II232, 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 it235," 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.

Fig. 1 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 order246 with respect to chapters in a book. By 1490 on CA86va he is referring to a rule of pulleys247 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 Euclid251 or Pythagoras.252 But elsewhere he uses the term specifically in connection with experiment as on CA153vd (14593-1495): "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 forces260," 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 CA268va263 (1493-1495) and CA155vb264 (1495-1497). He pursues this theme on Mad I 60r mentioning what to do "if you wish to make a general rule265," on Mad I 77r where he notes that he has "experimented and it is a general rule"266, and on Mad I 170v267 and 171v268 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 forward269," 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 object274

       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 studies275. 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 equal276

    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

       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 this278, 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 thickness281

       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 out285 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 widens286

       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 problems287 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 sagittas288

       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 parts289, governed in turn by basic powers of nature290. 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 time29

       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 descriptions294, 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 straight297

       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 plane298

       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 quantity299

       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 dense300

       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 weight303

       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 weight304

       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 weight305

       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 cord306

       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.

       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 tomorrow316," 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 request323," 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 presses337 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

       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 note341 about architecture342, water343, painting344 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 end347" and five notes instructing one to invert the book348 (i.e. read it in a mirror) followed by another: "N.B. This writing is inverted and is to be read in a mirror349," which phrase is repeated at the beginning of Forster III350 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 Sketchbook353. 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 Munich355, and has brought attention to various sixteenth century manuscript copies of the Treatise of Painting356. 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 Leonardo358; 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 Leonardo359.

       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 Lencker362; to Kassell, where Bürgi developed a particular version which became so linked with his name through publications by Hulsius (pl. 30)363 and Bramer364 that this instrument is still frequently assumed to have been his invention365. Probably via Mordente, word about this compass also reached Antwerp where Coignet366 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).

       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 consequence370. A third involved the way in which astronomy was studied. Leonardo's conviction that each plan