Kim H. Veltman

LEONARDO'S METHOD

Originally a lecture at the Ateneo di Brescia, "Struttura e metodo nei manoscritti di Leonardo" sponsored by the Centro di studi Leonardiani, Brescia, April, 1991. Published as a book: Brescia: Centro Ricerche Leonardiane, 1993.Revised March/April 2004.

Table of Contents

Preface Acknowledgments

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

List of IllustrationsNotesIllustrations

Preface

This book is dedicated to the late Kenneth D. and Mary Keele.

In 1973-1974, 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.

1

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.

Postscript 2004This new version of the book largely follows the text of the original. There are three changes. A series of coloured plates have been added. When appropriate, references to specific folios are relegated to footnotes. The accompanying essays have been dropped.

AcknowledgmentsThe final version of the 1993 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 sometimes need 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. My colleague, Johan van de Walle kindly made suggestions for improvement of the revised text.

2

1. Introduction

Leonardo da Vinci (1452-1519, plate 1) 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 (1983).6

There have, of course, also been very different views of Leonardo from the outset. Castiglione (1528)7 criticized him indirectly for frittering away his time on useless mathematical speculation. Serlio (1545) claimed that Leonardo was too much of a perfectionist and that this kept him from publishing.8

1Notes

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

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

2 J.-B. Venturi, Essai sur les ouvrages, phisico-mathématiques de Leonard de Vinci, Paris: Chez Duprat, 1797.3 E. Solmi, Studi sulla filosofia naturale di Leonardo da Vinci, Mantua: stab. Tip. G. Mondovi, 1905.4 Leonardo da Vinci, I libri di meccanica nella ricostruzione di Arturo Uccelli, Milan: Ulrico Hoepli, 1940.5 Ladislao Reti, ed., The Unknown Leonardo, London: Hutchinson, 1974, particularly pp. 264-287.6 Kenneth D. Keele, Leonardo da Vinci's Elements of the science of man, New York: Academic Press, 1983.

3

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 out of touch with the great traditions; an over ambitious amateur making notes without any underlying method and structure? This essay shows that Leonardo was widely read and in contact with some of the major scholars of his day. A survey of his extant treatises confirms 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.

We shall show that Leonardo began as an engineer by building on the work of earlier writers and contemporaries to create his own versions. Leonardo’s work on perspective led him to a systematic approach for representation. Applied to anatomy this led to a new programme for visualisation in terms of viewpoints and layers. His quest to visualise invisible currents in air led him to study currents in water.

Leonardo’s interest first in human growth and later in all transformations of shapes led him to study Euclid’s Elements and in 1505 to develop a Geometric “game” (de ludo geometrico) which equated geometry with instrumentation and was a profound attempt to account for all physical changes of form. These efforts explain his interest in compasses, reduction compasses and proportional compasses that led in the course of the sixteenth centtury to the evolution of the sector. Galileo’s publication (1606) made famous this first universal measuring device, which was their equivalent of a computer.

Leonardo’s fascination and near obsession with transformation led him to focus on a systematic play of variables as an essential part of his method. He applied this approach to his studies of light and shade particularly with respect to light sources and pinhole apertures (camera obscuras). His genius was to apply this approach to what he considered the four powers of Nature (force, motion, percussion and weight). From this, as we shall show, emerged a grandiose vision of a series of tomes that would link the microcosm and macrocosm, using the study of human vision and water as a starting point to explain properties of the moon and the planets.

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.

9 Marie Boas Hall, The Scientific Renaissance, 1450-1630, New York: Harper and Row, 1962, p. 30.10 David C. Lindberg, Theories of Vision from Al-Kindi to Kepler, Chicago: University of Chicago Press, 1976, p. 168: "As Leonardo's various confusions so clearly reveal, the problem of sight was not to be solved through a fresh start by an ingenious empiricist working in an intellectual vacuum."

4

A survey of historiography will help to understand why all this was forgotten and why scholars have claimed that he was a peripheral figure. Leonardo da Vinci had a clear scientific method and this makes him of central importance to the western tradition.

2. Sources and Contacts

The evidence of Leonardo's notes and correspondence is of a man with wide contacts and reading. Born in Vinci near Florence (plate 2), he studied with Verrocchio in Florence before moving to Milan around 1481 where the Castello Soforzesco became his “office” or rather his lab for most of the next two decades (plate 3). Thereafter there were times in Florence, back in Milan, Rome and finally Amboise.

Sometimes Leonardo 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 caelo14or 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 explicitly.17 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 abacus33, 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 from the Bishop of Padua38. 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.39 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"40 with Jacopo Andrea da Ferrara, a leading architect and Vitruvian commentator of the time. As an employee of Cesare

5

Borgia, he was a colleague of Machiavelli. In the period 1508-1510, he appears to have worked with Marc Antonio della Torre, a professor of anatomy at Milan. In Rome (1513-1515) he worked for the Pope on a sixteenth century version of the star wars project, using burning mirrors as described by Archimedes as a means of defence. The fame of his activities caught the attention of the King of France who persuaded Leonardo to move to France for an early "retirement". The curriculum vitae of a recluse would look somewhat different.

3. Treatises

Both the size and contents of the notebooks vary considerably. There are tiny pocket-size booklets such as Forster III (9 x 6.7 cm) or Manuscript H (10.3 x 7.2 cm, plate 4a)41 and large folio sheets such as the Codex Atlanticus and Windsor (plate 4b). 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 students42. Manuscript L is a good example. Here Leonardo does some surveying, sketches the lay of the town at Cesenatico (plate 5), and notes architectural features at Urbino.43 More detailed sketches from his travels sre found in the Windsor Corpus: as when he diligently depicts the ferry at Vaprio d’Adda (plate 6) near the home of his assistant Francesco Melzi.

A second category of study notes is based on both books and experiments, and involves gathering material for his basic themes. Sometimes Leonardo simply copies or rather adapts examples from his contemporaries such as the hoist that Brunelleschi used for the Cathedral in Florence (plate 7); the device for cleaning canals by Francesco di Giorgio Martini (plate 8); or a device for hoisting cannon by military authors such as Frontinus and Valturius (plate 9). It is noteworthy that Leonardo typically creates improvements rather than simply copying his contemporaries. Moreover, where his older contemporary Francesco di Giorgio Martini draws complete machines, Leonardo draws only key mechanisms that bring to light their underlying mechanical principles (plate 10, cf. plate 81). This approach evolves into a conscious method in his mature work. He uses exploded views to show relations of parts, then combines them and then shows them in context in the case of a milling machine (plate 11) or spinning wheels (plates 12-13).

Some of the results of this material were 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 the Windsor Corpus.

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.44 There is evidence that Leonardo planned to have what is now the Windsor Corpus45 bound, but this did not happen during his lifetime. Around

6

1508 the materials for what is now the Codex Arundel were still in an unbound state, although a note on Arundel 190v 46 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 a single 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. There is no clear theme and the first impression is complete chaos (e.g. plate 14). These are the kinds of folios, which are usually associated with Leonardo. However, they represent only the first stage of his work.

In a second stage, there are few recognizable themes and there is some order (plate 15). In a third phase, there is a single theme even if the order still seems haphazard. Not infrequently this phase has paragraphs with lines scratched through them to indicate that he has copied these notes elsewhere (plate 16). In a fourth phase, a pattern of texts with illustrations or vice-versa emerges (plates 17-20). Then there are sheets with evidence of numbering. In some cases these are still in rough state (plate 22). Elsewhere, these pages are neatly drawn (plate 23).47 Finally, we find sheets showing a single example sometimes with an accompanying text (e.g. plate 20).

Once we recognize this underlying order of the notes we discover new connections between his sketches and the evolution of his ideas. We shall give three examples. First, in the case of a siege machine, this development may be fairly straightforward from a rough sketch on one folio, which is developed on a second and becomes the dominant theme on a third folio (plate 24a-c). In the case of a clock mechanism this process is a little more complex. He begins with a sheet clearly devoted to the theme (plate 25). This is then developed elsewhere in the Codex Atlanticus (plate 26). As Leonardo's ideas evolve he copies them or has them copied into another manuscript, crossing out the original passage involved (plate 27a-b) and resulting in a clear structure (plate 28). 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.

A third example of how his ideas evolve is the case of a vice. Again he begins with messy folios where the vices appear among other subjects (plate 29a-b). Elsehwere this becomes the sole theme (plate 30) and then evolves as into beautiful presentation drawings (plate 31a-b).

A number of these presentation drawings are intended to be shown separately and cover a range of topics including botany (plate 32); weapons (plate 33); town plans such as the famous map of Imola (plate 34) and maps of regions (plate 35). Other presentation drawings become parts of treatises such as the Codex Madrid I-II.

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. Leonardo's earliest bound notebooks, the Codex Trivulzianus

7

(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, Leonardo proceeds in palimpsest fashion, writing over a passage with a new topic.48 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 elsewhere49 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."50 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 and 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.”51

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."52 At folio F 40 the treatise continues and proceeds backwards to F 39v, F 39r and so on through to F 28r. I have made a complete analysis of this treatise elsewhere.53 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."54 On F26v he writes "Here follows the proof of that which is said on the page opposite."55 On F52r he notes "Go to page 59."56 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

8

pages before this,"57 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 following58," i.e. G48r. On G46r he writes: "Here follows what is lacking on the page opposite"59, i.e. 45v. On G46v he notes: "Read page 45[r]."60 On G51v he writes "go to page 44."61 On G67r he explains that the text continues on the page opposite at the bottom62, i.e. 66v. On G75r he adds two notes: "Here follows what is on the opposite page63," i.e., 74v at the top, and also "the round beam is drawn on the page opposite."64 Finally on G80r there is a similar note.65 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 mirrors66, precisely the kind of information that the industrial spy, Johnny the German (Giovanni Tedesco), the mirror maker, was trying to steal. In this case Leonardo was probably 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."67 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."68 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 (figure 13) 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 to establish the sequence of his ideas recurs in Madrid Codex I, this time in the context of weights and balances. On folio 190 r

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.

9

As is becoming 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 (Hammer, now Gates) and Manuscripts E, F and G. Sometimes the margins give summary versions of the proposition, as in Manuscript D. Hence, in addition to his travel and study, which are frequently without a planned order, Leonardo has a clear method of presentation when he begins to organize these with a view to creating formal treatises.69

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. In an early note (1493-1495), Leonardo mentions: "I stated in the 7th conclusion how percussion...."70

A little later (1495-1497) he asks us to "look at the 7th [proposition] of the fifth [book] of the axle and the wheel."71 Elsewhere in the same manuscript (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 13 th [proposition] of the 22nd [chapter] of the elements of machines written by me.72

Another late note (c. 1514-1515) states that "Mr. Battista dall'Aquila, private secretary to the pope, has my book in his hands."73 In the Codex Arundel, Leonardo refers us to "the 5th of the 7th"74 in connection with weights. He also mentions "as proved in the 4 th of my [book on] perspective,"75 and he refers to "book 9 of water".76] Sometimes the references are laconic as in Manuscript K 30r to a "sixth book"77 or in Manuscript F where there are at least four references to "book 9 on water"78; two references to a "book 10 on water"79

plus headings for "Book 42. On rain"80 and for "Book 43. On the motion of air included below water"81, as well as a simple note: "Beginning of book [of water]."82

1 Complete body

2 Bones and veins

3 Bones and nerves

4 Bones83

10

1 Ramification of the bones

2 Bones sawn across

3 Bones complete

4 Nerves which rise from the nape of the neck

5 Bones and veins and where they ramify

6 Muscles

7 Skin and their proper proportions

8 Woman's body84

1 Veins and arteries

2 Membranes

3 Tendons, muscles

4 Bones85

5 Bones and nerves

6 Woman's body86

1 Bones

2 Ramification of the bones

3 Bones sawn across

4 Bones complete

11

5 Bones and nerves

6 Nerves which rise from the nape of the neck

7 Bones and veins and where they ramify

8 Veins and arteries

9 Membranes

10 Ligaments

11 Tendons

12 Muscles

13 Skin and their proper proportions

14 Complete Man’s Body

15 Complete Woman's body87

Figures 1-3. Leonardo’s plans for four, eight, and six sets of drawings of the human body.Figure 4. Reconstruction of his 15 views of the body (microcosm) following the example of Ptolemy’s maps of the earth (macrocosm).

In Manuscript I there is also a: "Beginning of book on water."88 In Manuscript E59v

(1513-1514) there is a: "Beginning of this book on weights"89 and a note: "proved by the ninth of percussion."90 In the Windsor Corpus we read: "proved by the 5th on force."91

Elsewhere in the same manuscripts 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.”92 In the Windsor Corpus Leonardo also refers 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."93

Such comparisons between man (microcosm) and the earth (macrocosm) have often been cited as proof that Leonardo remained rooted in Ancient and Mediaeval analogies. However, there is serious evidence to the contrary. Ptolemy's method of arranging the macrocosm served as a direct model for how Leonardo arranges the microcosm. Leonardo had a clear plan for demonstrating the body as a whole. There would be sets of

12

drawings. Each set would have three drawings: a frontal view to show height and position; a profile to show the depth of the whole and the parts, and a demonstration of the back parts.94 An early plan called for four of these sets (figure 1). Leonardo revised this and drew up a new plan, which would have eight of these sets (figure 2). He also outlined a third plan, which involved six sets (figure 3).

On encountering these three different lists, a first impression is that Leonardo kept changing his mind and that he had no coherent plan. Indeed this has been generally assumed. If we look more closely, we note that the lists do not contradict each other. Each focusses on different aspects: the first simply on the body and skeleton; the second adds muscles and skin; the third includes other levels. If these be combined an integrated approach of 15 sets emerges (figure 4).95 The incremental and cumulative dimension of his approach that leads finally to his plan for 15 complete drawings of the human figure thus emerges.96 Since each of these is to be seen from the front, behind and the side, this means that 45 basic drawings are required.97

While the complete set of this programme is not extant it is not difficult to discern traces of systematic approach in his treatment of the human head. Leonardo draws a frontal view of the skull (plate 36a), which he then shows in cross-section (plate 36). He draws perspectival cross sections to indicate where the optic chiasma passes through the skull (plate 37a-b). Next he draws the optic chiasma linked with the ventricles as separate entities (plate 38a-b). He then shows them in the context of the head (plate 39 a-b) and finally makes a see through drawing and an exploded view in order to reveal how these elements relate to one another (plate 40 a-b).

In the case of the hand, he does side views (plate 41 a-b) and frontal views of the skeletal bones of the hand (plates 42 a-b) and then makes a series of six numbered drawings to show the various layers of the skin (plates 43-45). Similar systematic treatment is seen in his studies of arms (plates 46-47). As Kenneth Keele demonstrated in his masterful Elements of a Science of Man and in his authoritative three-volume study of the Windsor Corpus such anatomical studies were much more than careful dissections of the human body. They served as a portal to his scientific approach. Leonardo’s study of the human body also led to studies of animal anatomy, particularly horses (plate 48) which were also important for his painting practice (plate 49).

4. Plans for Books

In addition to the above evidence of books actually written or at least in progress, there is considerable evidence of plans for books. On rare occasions (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."98 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 an early note (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

13

force, then you will deal with force and finally with percussion."99 A few years later (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.100

In the decades that follow this evolves into a major treatise which deals with the theory and practice of machines 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 has detailed pictures of weights and balances (plates 50-51) which at first sight seem to have nothing to do with flight until we realize that he is intent on seeing all flight as a problem of balances (plate 52); in visualizing flight patterns as different kinds of geometrical spirals (plate 53) and visualizing even air currents as patterns of balance (plate 54). All this leads to aviomorphic flight machines (plates 55-59). Ultimately the Turin Codex is but a fragment of the projected work as we learn from a passage in Manuscript K (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.101

Instructive in this context is Madrid Codex I where we find a number of references to specific books and propositions. On Madrid I 105v-106r, 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"102; "Bk.7.5 of motion and force"103; "Bk.7.5 and 9.7 of his theory"104.

Encounters of water with permanent objects of different shapes that overcome water“ “ with immobile objects covered by water“ “ with mobile objects covered by water“ “ with permanent objects that overcome the water“ “ with pliable objects that are overcome by water105

Figure 5. List of chapters for a book on the percussion of water (cf. figure 16).

Sometimes he simply refers to a proposition number without reference to book number as in "5th of theory"106, "5th"107, "6th"108 and "7th"109. In 33 cases he gives book and proposition number.110 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. In Manuscript F (c. 1508) he outlines a slightly different plan:

14

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.111

Water was another theme about which Leonardo planned to write a major work. In Manuscripts F and E (1508, 1513-1514) he makes notes concerning the order of topics in this work,112 inlcuding 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.113

Water became a fundamental theme for two reasons. First, it plays an essential role in his quest for visualization. Currents and forces in the air were invisible. Water served as a medium where such effects could be studied in slow motion. Second, he becomes convinced that the moon has oceans. Hence after 1500 water acquires a key role in his work on astronomy and cosmology. This explains why the themes of water and astronomy are so intricately connected in the Codex Leicester (Hammer, now Gates).

Much more elaborate plans for his work on water 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. Around 1505-1506, for instance, Leonardo makes a list headed: “Book on the percussion of water with various objects” (figure 5). In the same period he makes further lists, among them one on 18 different kinds of eddies.114 Leonardo made further such lists.115 Indeed, as Carlo Pedretti has claimed Leonardo made a series of references to a now lost treatise, Codex M116, which dealt with problems of water. What emerges, therefore, is a much more coherent picture than is usually ascribed to Leonardo.

5. Themes

This more coherent picture 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.

15

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 plans to write a major treatise on the nature of the universe117: to show that the moon reflects the sun as does the earth (plates 109-110); that the moon has oceans like the earth (plate 111) and that from a greater distance both the earth and moon look like stars. He considers the possibility that the moon might simply function as one convex mirror, but dismisses this possibility because the sun’s image would have been too small (plate 112). He explores the possibility that the moon has oceans, wherein every wave functions as a convex mirror. For this reason he spends at least 20 pages on the problem of the sun's image in water (e.g. plates 113-114). 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 Aristotelian and Mediaeval 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. Moreover, the nature of the heavens requires attention to astronomy. Here Leonardo focusses interest on the moon: its appearance, substance, its phases (plate 115).

To certify what he sees, requires study of optical principles. Hence his studies of the eye in Manuscripts D and F were intended as introductory chapters for the treatise on cosmology (plates 107-108). His studies of optical instruments, notably mirrors, eyeglasses and a prototype of the telescope (plate 120), were also part of this enterprise.118Leonardo'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.

Earth Water AirMan * * *Horse *Fish *Bird *Motion Running Swimming FlyingMechanical Equivalent Cart Boat Mechanical Bird

Figure 6. Links between elements, biological studies, motions, and machines.

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

16

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 that: "Wind has similarity with the movement of water."119 This has consequences for his study of both the natural and the man made world. Before 1500, he notes "swimming shows the way of flying"120 a thought he restates (c. 1505). "Swimming in water teaches men how birds do it in air."121

He elaborates on this line of reasoning in the Manuscript E (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.122

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 figure 6).

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 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. In Manuscript A (1492) he compares the body of a man with the body of the earth.123 He also he compares the veins of man with the underground rivers of the earth.124

Much later, (c. 1508-1510), he compares the lungs of man with those of the earth.125 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. He mentions the "wondrous instrument invented by the consummate master."126 At one point he announces that he will "demonstrate this instrumental figure of a man in 24 figures."127 This corresponds to his second programme (figure 2) of eight tiems three views: front, side and rear.

In the Codex Atlanticus, he has an analogous view of birds: "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."128 Similarly, he looks upon the earth as a "terrestial machine."129 In the Codex Atlanticus (1490) he states that as a result of "various opinions concerning the size of the spherical terrestial machine, I have become

17

concerned to create or rather to construct an instrument which will adopt this form." 130

Here he is speaking of a surveying instrument.

Elsewhere in the same treatise (1490-1492) he refers to this "terrestial and worldly machine,”131 and in Manuscript A (1492) he speaks of "the universal machine of the earth."132 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.133 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.134 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 flowers135, 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.

Figure 7. Branches of Leonardo's work in language and literature.

18

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 sees as threefold: numbers (arithmetic), words (language) and diagrams (geometry). His work in arithmetic amounts to about 1% of his notes136 and is limited almost entirely to arithmetical proportion, practical problems deriving from the abaco school137 and some computations. Words in terms of language138 and literature139

(cf. figure 7) interest him more and constitute roughly 4% of his notes.

Much more important are his studies of geometry. When Leonardo praises mathematics he usually means geometry140 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 in the Manuscript K (after 1504): "Proportion is found not only in numbers and measures, but equally in sounds, weights, times and sites and every power that exists."141 Even so, proportion is but one of the branches of geometry that interests him (figure 8). Pacioli leads him to study Euclid142, whose Elements deal mainly with geometry in two dimensions. Leonardo's study of perspective prompts him to explore three-dimensional treatment of geometrical forms. In his treatise on the geometrical game143 he limits himself mainly to the five Platonic solids. Elsewhere he explores most of the 13 Archimedeian solids.

19

Figures 8-10. Branches of Leonardo's geometrical studies; studies of the represented world and of the constructed world.

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.144

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 shade145 by means of which he can deal with human forms146, drapery147,

144 CA 130va (360r, c.1517-1518) as in note 196.

20

trees and plants148, and geometry provides him with the principles of transforming their shapes (figure 9).

Yet the focus of Leonardo's attention is in the constructed world, which includes architecture149, mechanics and instruments (figure 101). In his architectural studies we find him playing systematically with basic geometrical forms in the ground plans of his designs for churches (plates 50-52) 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 architecture150 involves some significant innovations, his weapons151 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.152 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 machines153 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. As Reti154]

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 four further machines: boat, fountain, Archimedeian screw and dredger. With respect to air he has his flying machine.155 With respect to fire he has burning mirrors. (figure 11).

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) veridity and 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 (figure 12).156

EarthHoisting Dragging Rolling Digging Weaving Pounding WeighingCrane Winch Cart Shovel Spinning File Balance

Machine WaterSailing Spouting Raising DredgingBoat Fountain Archimedes’ Dredger

ScrewAir FlyingMechanical Bird

21

Fire Burning Mirror

Figure 11. Leonardo's machines and instruments seen in the context of basic actions and the four elements.Figure 12. Basic motivations underlying Leonardo's studies.

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 in the Codex Atlanticus (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....157

22

But again most of his energies are focussed on four instruments: mirror (concave and convex as well as plane),158] clock159, balance160, and compass.161 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 believes.

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. plates 11-14) 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. In the Codex Atlanticus (1493-1495), for instance, he explores "how all wheels are of the nature of a balance."162 He reminds himself (1495) to "make mention of the general rule about the contact of axles and all weights."163 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:

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.164

Another passage (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."165 He is seeking (1493-1495): "To make a general rule of the difference that there is between simple weight and weight with percussion caused by different motions and forces."166 He pursues this approach (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."167 This leads to the systematic list in the Madrid Codex cited below (figure 22).168

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, 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,

23

acquires degrees in continuous geometrical proportion. And force does the same in levers.169

By about 1503, Leonardo is referring in the Codex Atlanticus, 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.170 In another paragraph (c.1508) headed, "On local motion" Leonardo supports his claim by reference to "the fifth of the ninth which states...”,171 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 a note (1508-1510) "On the elements of machines" where 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,"172 (cf. plates 27-28). Meanwhile, he has been collecting material on each of the individual powers. He refers (1495) to a fifth [proposition] of the seventh [book] with respect to weights.173

Later (1517-1518) he refers to an "eighth book on weight."174 This applies also to other powers. In an early note (1493-1495) he refers to a "seventh proposition"175 concerning percussion. By the time he is established in Rome he has enough material to organize it into book form as we learn from a note of 1513:

Divide percussion into books of which, in the first one there is demonstrated the percussion of two bodies of which one moves, the percussor to an immobile object; [in] the second book percussor and percussed move reciprocally, one against the other. A third is of liquid materials; a fourth of pliable objects; a fifth...176]

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."177

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 (1515) planning to write a book on friction178 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 a note (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."179 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

24

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 Alhazen180 (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 sight181," 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 instruments182 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.183 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 which184, 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.185 We know from a much later note (c. 1517-1518) that he wants to use Euclid to transform geometrical shapes.186 Yet his goals are quite different from Euclid. Already c. 1500, Leoanrdo states: "I want to make of a circle an infinite variety of curvilinear figures of equal capacity."187

At the outset he proceeds as if arithmetical and geometrical approaches are interchangeable in pursuing this goal. In a note 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."188 Later (c. 1517-1518) he is conscious that there are exceptions, when he observes: "But this calculation wants to be geometrical because, if you wished to do it by means of arithmetic it would be impossible."189 It would take more than a century before reliable mathematical notation bridging discrete (arithmetic) and continuous quantity (geometry) had emerged.

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: "Learn the multiplication of square roots from Master Luca Pacioli."190 Four years later (c. 1508) he is giving instructions on how to reach a solution and arrive at a rule for both square and cube roots: "With the circle br you will make a rule of the square roots up until 20 and then, with

25

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."191 Later (1561-1517) he simply gives the rule: “If you wish the square root of any number, this is the rule..."192

There are hints that Leonardo intended to combine these studies of square and cube roots with the operations of his proportional compass (cf. plates 116-118), 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 (cf. plate 119). 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: Mechanical Trigonometry of Planes (1617).193

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. In the Codex Atlanticus (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."194

On the same folio he remarks: "The mechanical proof is true even it if is with difficulty that one finds this truth."195 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.196

And so a nexus evolves which links instruments, geometry, proof and science. Sometimes, (c. 1503-1505) he simply notes in the Codex Atlanticus: "here the mechanical proof is given"197, a phrase which reminds us that Leonardo's reference to mechanics being "the paradise of the mathematical sciences"198 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, Leonardo is referring to a "geometrical rule."199 Sometimes he carefully records the date of a new insight 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."200

The rules to which he keeps referring become increasingly universal in their scope. In the Codex Atlanticus (c. 1515-1516), he mentions how the rule in question goes on to infinity201, a phrase which he also uses in connection with the four powers.202 A year or two later he is confident enough to speak of general rules (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."203 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

26

constructed geometrical diagrams. It is hardly surprising, therefore, that Leonardo gradually uses geometrical diagrams as synonymous with the term demonstration.204

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.205 A note in the Codex Atlanticus (c. 1508-1510) mentions "and this is proved in the eighth of proportion"206, which suggests that he is writing a book on this topic also. Later, (c. 1515) he refers to "a rule to know the value and proportion of many curvilinear parts."207 Indeed, by 1513-1515 he has invented his own compasses of proportion as is confirmed by two notes in the same treatise.208 In a third note, (c. 1513) he refers specifically to "a compass of proportionality" showing it both in profile and face on (plate 116), noting that its central axis is moveable and that "this works in irrational proportionality."209 By about 1515, Leonardo notes that "With this [proposition] of the Elements one can give any proportion of a circle, rational as well as irrational."210

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. In the Codex Atlanticus (c. 1508), he refers to a "Book of equations"211 (in the sense of equivalent shapes). On a number of occasions he refers to treatises "On transmutation" (i.e. transformation)212 and "On the geometrical game."213 Another note in the Codex Atlanticus (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."214 Later, in the same treatise (c. 1517-1518), he refers to a "Treatise on continuous quantity."215 As early as 1508-1510, however, he refers to a work on "Curvilinear geometrical elements."216 By 1513 he is referring to specific propositions in a work entitled Elements: a "first proposition,”217 a "fifth proposition218," the "43rd proposition of the first book219," the "last proposition of the second book."220 All this points unequivocally to a much more systematic approach than has thus far been suspected. And this is confirmed by another passage in the Codex Atlanticus (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.221

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

27

centuries later, was able to find one more.222 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.223

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"224 and this evolved into his Elements of machines, which he planned to serve as an introduction to his anatomical studies.225

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.226 The latter of these texts is also cited elsewhere (c. 1508).227 In this period Leonardo also studied Archimedes228, Theodosius229, an unidentified Zenofonte230, 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.

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 (figure 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.

6. Method

28

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. In an early note he mentions (c. 1490-1492): "I find by experience that...”231 He advises (1492): "Experiment as follows,”232 a phrase that returns elsewhere.233 Sometimes he describes what is to be experimented (c. 1490): "Experiment on motion, the cause of the blow"234 or when (c. 1500) he sets out: "To experiment the proportion of the intervals of descent."235 Sometimes he describes precisely the means to be used, as in Manuscript D: "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."236

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 Maccagno237 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 a note in the Codex Atlanticus: "Make simple propositions and then demonstrate them with figures and letters."238 This he restates in more detail in Manuscript A (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."239 In the same manuscript we find phrases such as "proposition proved by experience", "proposition confirmed by experience", "proved by experience" or paraphrases such as "proof", "the cause of the proposition", "this case is seen manifestly" or "this is demonstrated clearly."240 Analogous phrases are found in BN 2038241 which was originally part of Manuscript A.

Visible InvisibleConcrete Mathematical (Mechanical) Abstract MathematicalPractical Purely TheoreticalCoporeal IncorporealPhysical MentalMaterial SpiritualDynamic Static

Figure 13. Contrasts between visible and invisible characteristics.

29

When an experiment has been carried out Leonardo explicitly writes: "experimented". There are examples in Madrid Codex I, Codex Arundel and particularly Forster II, which contains no less than sixteen such cases.242 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 in an early note (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.243

Experiment becomes, for Leonardo, linked specifically with things which are visible and can be represented. For instance, he notes (1490-1492), that "experience, interpreter of artifice filled nature, demonstrates that this figure is necessarily constrained not to operate in ways other than is here represented."244 A few years later (c. 1495), he adds: "Make this figure return in experience before you judge it245," an idea which he expresses slightly more forcefully (c.1508): "all these figures have to come out of experience."246 In the Codex Arundel he notes laconically: "I tested it myself, drawing it."247 Underlying these 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 13. 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.248 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 in the Codex Atlanticus (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." 249 Leonardo uses the same criterion with respect to weights (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."250 Similarly, he uses this criterion to distinguish between abstract mathematics and concrete mechanics (c. 1515): "Between the mechanical and

30

mathematical point there is infinite difference because this mechanical point is visible and consequently has continuous quantity."251

Consistently Leonardo is concerned with focussing on visible knowledge. In this context his oft quoted phrase on Manuscript E 8v (1513-1514): "Mechanics is the paradise of the mathematical sciences"252, 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.253 In the Codex Atlanticus (c. 1490) he notes: "These rules are to be used by checking the figures."254 Diagrams and figures become a basic aspect of his method as is clear from a comment (c. 1495): "I make many figures in order that you know all the cases which are subjected to a single rule."255

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 (c. 1487-1490) is in the sense of order256 with respect to chapters in a book. By 1490, he is referring to a rule of pulleys 257

and 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.258

This idea he restates (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."259 It is significant that this same quest to avoid false promises also enters into a later discussion of experience (1508-1510): "Experience never fails. Only your judgments fail, promising of some effect that which is not caused in our experiments."260

On occasion Leonardo uses "rule" in referring to the work of Euclid261 or Pythagoras.262

But elsewhere he uses the term specifically in connection with experiment as in the Codex Atlanticus (1493-1495): "Test and make a rule of the difference that there is between a blow that is given with water onto water and water which falls on a hard surface"263 and: "Again make a rule of the different trajectories of the ball."264 This approach is restated in Madrid I (c. 1499): "Make experiments and then the rule."265 In the same manuscript he speaks of applying the same rule that one uses for dragging for the study of pushing.266 Sometimes (1508) he refers laconically to: "rule."267 By the late period (1517-1518) the term, rule, has acquired another connotation, reversibility:

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

31

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.268

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 Madrid I 129r: "When a rule is confirmed by two different reasons and experiments, then this rule is said to be general."269 One of his earliest references to this concept is in the Codex Atlanticus (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 forces270," and (1493-1494) which again deals with weights.271

Another note (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.272

Further notes occur elsewhere in the same manuscript (1493-1497).273 He pursues this theme in Madrid Codex I mentioning what to do "if you wish to make a general rule"; noting that he has "experimented and it is a general rule"; or simply that a "general rule" is involved.274 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 in the Madrid Codex: "this demands practice, but remember to put the theory forward275," an idea which he expresses afresh in the Codex Atlanticus (c. 1500): "No effect in nature is without a cause. Understand the cause and you do not need experience."276 Indeed Leonardo explicitly develops a concept of laws of nature in the Madrid Codex I:

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."277

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.278 Kenneth Keele has demonstrated the importance of perspective for Leonardo's anatomical studies and has called perspective and the four powers Leonardo's gateway to science:279 Leonardo’s believed 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 he 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 in the Codex Atlanticus (1495-1498), concerns light sources and objects (figure 14, cf. plate 60).

32

In isolation this list would have limited interest. But it becomes important 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 his Seven Books on Light and Shade and indeed all his optical studies.280 It is also very significant that this list is at least three to five years after he has done his experiments with light and shade in the Manuscript C. (cf. plates 61-68). In other words he makes empirical experiments and subsequently becomes articulate about the need for a systematic approach.

Sceptics might rightly object that the existence of diagrams which can be arranged by others in a systematic fashion does not prove that Leonardo even intended such order, let alone that he was systematic. We need his word for it and fortunately it exists in the form of numerous lists in various domains of his work. These confirm that Leonardo consciously plays with variables. In the Codex Atlanticus (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 (our figure 15).

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 (plate 76) including the well known Latin verb: to love: amo, amas, amat (plate 77).281 Perhaps inspired by the work of grammarians he uses the same method to illustrate combinations of vowel sounds in the Windsor Corpus (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 (figure 17 and plate 78).

In isolation, such a list could readily be seen as an amusing game. But there many other examples. We have already noted his plans for a treatise on water (figure 5). In the Codex Atlanticus (c.1505-1506) he gives a systematic list of encounters of water in terms of quantity and power (figure 16). A larger systematic programme comes to light when we examine more closely his geometrical studies. 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 this282, which Leonardo studied, as we know from a note in the Codex Arundel.283

Several Lights with One object One Light with Several objects Several Lights with Several objects Several Lights above One object284

Wider than the reservoir and heavier

33

Wider than the reservoir and lighter

Wider than the reservoir and equal

Narrower than the reservoir and heavier

Narrower than the reservoir and lighter

Narrower than the reservoir and equal

Equal to the reservoir and heavier

Equal to the reservoir and lighter

Equal to the reservoir and equal.285

Encounters of water equal in power and in quantity

Encounters of water equal in power and not in quantity

Encounters of water equal in quantity and not in power

Encounters of water not equal in power and not in quantity.286

a e i o uba be bi bo Buca ce ci co cuda de di do duefa fe fi fo fuga ge gi go gula le li lo luma me mi mo muna ne ni no nupa pe pi po puQa qe qi qo qura re ri ro rusa se si so suTa te ti to tu

34

Figures 14-17. Examples of a systematic approach using light sources; counterweights pressing against a reservoir; kinds of water; treatment of sounds.

A pyramid [is] extended to a given length

A pyramid [is] shortened to a given lowness

From a pyramid one makes a cube

From a cube one makes a pyramid

From a cube one makes a pyramid of a given height

From a pyramid of a given height one makes a cube

From a pyramid one makes a table of a given thickness

From a pyramid one makes a table of a given width

From a pyramid one makes a table of a given width and thickness.287

Figure 18. Transformations of a pyramid in the Codex Forster.

Elsewhere, Leonardo defines the geometrical game as giving "a process of infinite variety of quadratures of surfaces of curved sides."288 This process soon became much more than a game: Leonardo saw it as a key to all systematic transformations of forms. In the Codex Arundel (c. 1505) he explores basic transformations involving a pyramid (figure 18). Leonardo also makes lists of different kinds of transformations possible in geometrical objects. In the Codex Atlanticus he explores equivalents of chords, arcs and sagittas (figure 19, plate 79). In the same treatise (1505-1506), he mentions: to shorten, lengthen, make fat, make thin, widen, restrict with respect to length, width and length (plate 80).289

These he crosses out and then makes a list of twelve kinds of simple transmutation.290

Eleven kinds of composite transmutation follow.291 Again he crosses these out292 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 (figure 20).

As we have suggested earlier, the regularity of these geometrical transformations led Leonardo to use them as a model for his concept of science. 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. Leonardo developed a mechanical model of nature. His

35

study of machines convinced him that nature involved a surprisingly small number (21) of physical parts293, governed in turn by basic powers of nature294.

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.295 In the years that follow he makes a series of systematic lists concerning individual powers such as kinds of percussion (figure 20) and combinations of different kinds of motion (figure 21).

He also makes a number of visual demonstations involving thin and thick weights and gear devices (plates 82-86); systematic examples of wheels (plates 87-88) and examples of motion where elements are substituted (plate 89). Such variations are also summarized systematically both as lists with respect to motion (plate 90) or as composite visual diagrams with respect to percussion (plate 91a-b).

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 sagittas296

1 Shorten as much as one widens without changing the size2 Shorten as much as one thickens without changing the width3 Lengthen as much as one squeezes without changing the size4 lengthen as much as one makes thin without changing the width5 fatten as much as one squeezes without changing the length6 fatten as much as one shortens without changing the width7 thin as much as one widens without changing the length8 thin as much as one lengthens without changing the width9 widen as much as one thins without changing the length10 widen as much as one shortens without changing the size11 squeeze as much as one thickens without changing the length12 squeeze as much as one lengths without changing the size13 shorten and fatten as much as one widens14 shorten and thin as much as one widens15 shorten and widen as much as one size

36

16 shorten and squeeze as much as one fattens17 lengthen and fatten as much as one squeezes18 lengthen and thin as much as one widens19 lengthen and widen as much as one thins20 lengthen and restrict as much as one fattens21 fatten and widen as much as one shortens22 fatten and restrict as much as one lengthens23 thin and widen as much as one lengthens24 thin and restrict as much as one lengthens25 fatten and lengthen as much as one restricts26 fatten and shorten as much as one widens27 thin and lengthen as much as one squeezes28 thin and shorten as much as one widens297

Figures 18-19. Systematic play with variables in the Codex Atlanticus and the Codex Forster.

Percussion of rare in rare

Percussion of rare in dense

Percussion of dense in rare

Percussion of dense in dense.298

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

37

Straight and straight Straight and straight.299

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 time300

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

38

Figures 20-23. Systematic treatment of percussion; motion; combination of force, weight, motion and time; and power (p), a variant name for force; space (s); motion (m); and time (t).

Given lever counterlever fulcrum and weight one seeks the motorGiven ounterlever fulcrum weight and motor one seeks the leverGiven the fulcrum weight motor and lever one seeks the counterleverGiven weight motor lever and counterlever one seeks the fulcrumGiven motor lever counterlever and fulcrum one seeks the weight301

Given screw screwthread number lever and weight one seeks the motorGiven screwthread number lever weight and motor one seeks the screwGiven number lever weight motor and screw one seeks screwthread Given lever weight motor screw and screwthread one seeks the leverGiven weight motor screw screwthread and number one seeks the cordGiven motor screw screwthread number and lever one seeks the weight302

Given axle round beam lever cord and weight one seeks the motorGiven round beam lever cord weight motor one seeks the axleGiven lever cord weight motor axle one seeks the round beamGiven cord weight motor axle and round beam one seeks the leverGiven weight motor axle round beam and level one seeks the cordGiven motor axle round beam lever and cord one seeks the weight303

Given diameter number axis weight and cord one seeks the motorGiven number axis weight cord and motor one seeks the diameterGiven axis weight cord motor and diameter one seeks the numberGiven weight cord motor diameter and number one seeks the axisGiven cord motor diameter number and axis one seeks the weightGiven motor diameter number axis and weight one seeks the cord304

Figures 24-27. Further examples of systematic play of variables.

There was long standing mediaeval tradition studying weights and balances which included thinkers such as Jordanus of Nemore whom Leonardo studied. Not surpisingly therefore, Leonardo’s studies of weights are particularly detailed and reveal clearly his approach. In one sketch we see a series of interlinked balanaces with an accompanying series of number which become proportionately larger: 6, 36, 216, 1296 and 7776 (plate 92).

39

In the Madrid Codex he draws a composite drawing of a bow in a number of positions as a result of increasing weights (plate 93). This idea is pursued in another drawing where weights are moved laterally (plate 94). This principle is then applied to bows of differing sizes (plate 95). In the Codex Atlanticus he makes lists of various combinations (plate 96). He then makes a composite diagram indicating how these weights vary as one shifts them (plate 97).

Such examples lead to more systematic folios (plate 98) where he draws a series of cases with different weights which are then summarized in a composite diagram at the bottom of the page. A more detailed look at the series of sketches in the left hand column (plate 99, cf. plate 98), shows a bar with two small weights in the top diagram. The weights are increased in size and the bar moves higher. In a third diagram the weights are heavier and the bar is higher. In a fourth diagram the bar has moved as high as it can go. These four cases are then combined in a composite diagram and then redrawn more abstractly on another folio (plate 100).

Another folio of the Codex Atlanticus again shows two related columns with systematic variation of weights (plate 101) which are again recapitulated in the bottom diagrams. Elsewhere Leonardo develops the diagram on the bottom left (plate 102). As might be expected he also develops diagram on the bottom right (plate 103). Further folios (plates 104-105) confirm that this is part of a larger systematic treatement. A next logical step in this process would be systematic combination of these powers. This we find on Codex Madrid I (1499-1500). Here he prefaces his study with an explicit statement that he is making a thought experiment in order 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.”305

Leonardo then takes three of his powers of nature, plus the factor of time, and presents them as a systematic play of variables (figure 22). 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. However, it is significant that he sometimes treats these basic variables as abstract symbols. In the Codex Atlanticus (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 descriptions306, produces a chart which summarizes these variables, underlining a different one each time (figure 23).307

He makes another list for power, weight (g, i.e., gravita), motion and time.308 He develops similar lists in the Codex Atlanticus (1502-1504) adding quantitative values to the symbols: e.g. s2 and t2.309 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.

40

The Windsor Corpus provides further evidence that Leonardo is collecting these ideas in a systematic fashion, with the explicit purpose of writing a book. For instance, he notes (1506-1508): "In this 4th book I have to treat of six things as instruments, that is, the axle, round beam, lever, cord, weight and motor."310 On this same folio he also outlines the elements necessary to study: "the nature of the working parts required for the functioning of the capstan."311 Directly beneath this is another of his charts with six variables (figure 24). On the same folio Leonardo considers a combination of five variables (figure 25). He pursues these problems on elsewhere in the same notebook (1506-1508) where he outlines the elements involved in a screw (figure 26). On the same folio he makes a corresponding list pertaining to pulleys (figure 27). 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.312

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 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 a 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 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."313 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 in the Windsor Manuscripts (c. 1509-1510) describes the contents of the book of which this was to have been a part:

41

On machinesSince 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.314

Elsewhere in the same manuscripts (c. 1508-1510) Leonardo tells us that "the book of the science of machines precedes the book of the movements."315 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. He applied them to his transformational geometry that led to the treatise in Forster I and became a basis for later writings. In the case of 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.

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

7. Plans for Publication

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

His military notes are almost always secretive, although when he writes to Ludovico Sforza, the Duke of Milan he offers to teach him "my secrets."316 In everyday work he is guarded. In the Codex attanticus, for instance, he makes a note to himself to "make this secret."317 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.318

42

Some of Leonardo's notes are personal. Sometimes he is reminding himself that he has done something: “On the first day of August 1499, I wrote here on weight and motion."319

Sometimes he reminds 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."320 Sometimes this idea is put more succinctly as: "Experiment of tomorrow,”321

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."322

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 in the Codex Atlanticus there is a revealing little note (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."323 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 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.324

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." 325

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."326 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."327

On other occasions Leonardo gives instructions to specific readers. When he writes to Diodarius of Soria, the lieutenant of the sacred Sultan of Babylon, "Do not be dismayed, O Diodarius, by the tardiness of my reply to your desirous request,”328 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."329 In BN 2038 Leonardo refers to what the painter must consider, in the third person.330 But on one occasion at least he shifts to the second person: "Hence, since you, o painter, know."331 Similarly he writes "When you, o draughtsman, wish to make a good and useful study."332

Elsewhere this becomes a plural: "When you, o draughtsmen, wish"333 and in like fashion: "If you historians or poets or other mathematicians had not seen things badly with the eye..."334 In the Windsor Corpus there is further direct discourse: "o observer of this

43

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."335 In Manuscript E (1513-1514) he refers again to painters: "remind yourself, O painter, that the shades of shadow are as varied..."336 and on the next folio: "O anatomical painter".337 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."338

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..."339 Sometimes it is in the form of a question: "I ask you."340 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.341

Leonardo designed his own printing presses342 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.343

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.

44

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.”344 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.345

8. Influence

Vasari's claim about Leonardo's notebooks being read by others has an unexpected confirmation. There is physical evidence of actual readers in the notebooks themselves. The notebooks are written in mirror script. But in the Codex Trivulzianus, for instance, we find at least a half dozen instances where someone has written in ordinary script that this is a note346 about architecture347, water348, painting349 or a battle.350 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."351

Forster II contains a similar note in Latin: "Most powerful mechanics beginning at the end352" and five notes instructing one to invert the book353 (i.e. read it in a mirror) followed by another: "N.B. This writing is inverted and is to be read in a mirror354," which phrase is repeated at the beginning of Forster III355 and then repeated in abbreviated form another half dozen times.356 Manuscript B has at least 49 notes in Spanish written right side round identifying the subject matter357

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 Sketchbook358. 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.359 Professor Pedretti has noted copies of mechanical drawings in Florence and Munich360, and has brought attention to various sixteenth century manuscript copies of the Treatise of Painting361. Leonardo's instruments were studied by the clockmaker, Lorenzo della Golpaia,362 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 Leonardo363; 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 Leonardo364.

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

45

was developed into the Mordente compass.365 As already noted above, in the Codex Atlanticus there is also a new kind of adjustable compass which Leonardo designates as a proportional compass (plate 117a)366. Two generations later this compass of proportion found its way to Nürnberg, where it became part of a manuscript on perspective attributed to Lencker367; to Kassell, where Bürgi developed a particular version which became so linked with his name through publications by Hulsius (plate 117b)368 and Bramer369 that this instrument is still frequently assumed to have been his invention370. Probably via Mordente, word about this compass also reached Antwerp where Coignet371

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).372 It is perhaps instructive to note that Galileo made analogous claims about having invented the telescope (plate 123, cf. plates 117-120).373

Galileo's fame is also firmly linked with his inclined plane experiments yet another theme that Leonardo explored a century earlier.

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.374

The rise of universal measuring devices including various kinds of proportional compasses was a second consequence.375 A third involved the way in which astronomy was studied. Leonardo's conviction that each planet is at the centre of its own elements, led him to the study the elements of earth, air, fire and water in relationship to centres of gravity. Astronomy and cosmology became for him problems of both geometrical and physical models. So he developed instruments to observe the heavens and he built orreries to explore the relationships of various planets. In the next generation this idea of model making was taken further by Peter Apian, author of the Astronomicum Caesareum (Ingolstadt, 1540), which used perspective to create three dimensional views of sections

46

of the heavens and employed elaborate movable volvelles to picture relationships of planets and stars. He was, of course, visualizing the Ptolemaic world view and the details of his approach were occasionally "wrong."

But the model making impulse remained alive and Jost Bürgi, a member of the Brahe circle at Kassell who studied the book, made his own physical models of the universe linked up with clock mechanisms376; while Kepler took Plato's abstract theories from the Timaeus and tried to construct physically a model of the universe involving the five regular solids, which he described in detail in his Mysterium Cosmographicum (1596). Ultimately it was the discrepancy between this instrumental model of the universe and the evidence of heavens which he observed by means of instruments that led Kepler away from the Ptolemaic world view. In other words, the so-called paradigm shift of the Copernican revolution was not simply an abstract decision for which we have simply to imagine theoretical philosophical, psychological or sociological explanations. It was inspired by a new confrontation of evidence from instruments of observation, with that of instruments of model making: precisely that nexus that Leonardo brought into focus.

In this context Leonardo's work on compasses and telescopes can no longer be dismissed as amusing toys, or neat gadgets. His emphasis on instruments such as balances, automatons and clocks (cf. plates 28, 81) provoked much more than mechanical metaphors for the natural world. They provided the very framework that made it possible to think of the world in a scientific way. They established a visible standard which permitted one to insist on observation and experiment. And by means of instruments Leonardo set European culture on a quest for laws concerning four powers of nature: weights, which Tartaglia, Benedetti and Guidobaldo del Monte would take much further; motion (e.g. plate 89), on which Galileo would build his reputation; percussion, which became the basis of Huygen's philosophy and force which in some senses had to wait for Newton. In short, Leonardo did have a method in his work and the questions he asked gave basic directions to the research programmes of the centuries that followed. As we have shown elsewhere, there was also a method to his painting.377

9. Limitations

Why then have these contributions never been recognized? Why is Leonardo regularly dismissed as a chaotic amateur? It is mainly because hardly anyone reads the notebooks. In the last generation there were only six persons known to have read the 6,500 extant pages of his notebooks. Artists feel that they can limit themselves to his paintings. Scientists believe that they need not read them because they assume that unpublished works had no effect. The few individuals who have nonetheless read the notebooks have usually taken the surface chaos at face value. Venturi, Solmi and Keele were three notable exceptions mentioned at the outset.378 Their work inspired the present study.

It is easy, however, to overemphasize this structure and method, and necessary to note very important limitations on various fronts. Some are practical. Had paper not been so scarce, had Leonardo had more space on which to write his work, he would not have needed to be so cramped in his writings, sometimes writing in the margins, sometimes

47

having to skip a few, or even a great number of pages, to find the next empty space to pursue his idea. Some limitations are more subtle. He may have a mechanical view of the universe, but the microcosm-macrocosm analogy lingers. His mechanical anthropo-morphism leads him to consider mechanical birds, but not airplanes.

Some are problems of classification. A modern reader who finds a page379 with discussions of images hitting the eye, sounds hitting the ear and hammers hitting the ground may assume this is pure chaos. In Leonardo's mind it is not: for him all three are instances of percussion. Some limitations are technical: Leonardo sets out to make lenses in order to make the moon appear large. But he could not hope to see what Galileo did a century later with better equipment. Some are procedural. While praising the mathematical sciences, Leonardo views mathematics mainly in terms of geometry. He does not treat geometry and arithmetic together in the systematic way that begins in the 1580's. He has effectively no algebra. He has no conception of trigonometry, let alone calculus.

Leonardo’s mediaeval prodecessors were frequently concerned with learning and keeping secrets to themselves. When faced with the equivalents of industrial espionage, Leonardo is also secretive. But for the most part he wants to communicate his ideas. He teaches, noting "If you do not teach you will only be excellent."380 He writes with a view to being read, and also wants to be published in order to be more widely read. If the limitations of printing at the time made infeasible the publication of his many illustrations, his new visual demonstrations, this was not his fault. He lived in a time when war and politics repeatedly forced him to move. So the chaos of his notes is partly the chaos of his times. And for all these reasons he is Leonardo and not Galileo, Newton, Helmholtz or Einstein. To acknowledge these limitations does not threaten his unique place in the history of early modern science. He was one of those giants on the shoulders of whom later scientists stood, whose greatness lay in focussing attention on four fundamental principles, which introduced to the myriad impressions of nature herself a new sense of structure and method.

10. Historiography

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

48

The history of science as pictured by nineteenth century scientists such as Whewell381 was straightforward. There was progress and there were great men. The Middle Ages had basically been a millenium of darkness. Then a genius such as Leonardo brought about a scientific revolution which was carried further by the next genius, Galileo, and so on. This was so obvious that there was no need to document the details of how it had occurred. This view remained largely unquestioned until the autumn of 1903 when Pierre Duhem found irrefutable evidence that Leonardo had studied mediaeval sources such as Jordanus of Nemore.382 Duhem soon stated his case more forcefully: "There is no essential idea in the mechanical works of Leonardo da Vinci which does not derive from the mediaeval geometers."383

Soon, even sceptics accepted that there must be some continuity between Mediaeval and Renaissance science, a gradual evolution rather than some sudden event.384 This evidence was particularly taken up in the United States. Sarton included mediaeval science in his great Introduction,385 while individuals such as Thorndike,386 Benjamin,387 and Clagett388

founded mediaeval science as a proper field of study. Their studies established that the scope of mediaeval science was much larger than had been assumed, that knowledge of mediaeval sources in fifteenth century Italy was much more widespread than had been imagined. So Leonardo was not alone. But while Duhem concluded that Leonardo had acquired all his ideas from this mediaeval tradition, Clagett decided that Leonardo's knowledge of it was incomplete at best. Clagett's students were less generous and became convinced that Leonardo's knowledge of mediaeval sources was scanty at best. Hence even if there was a continuity Leonardo was not really part of the story.

Meanwhile, in Europe, the quest to understand the continuity of early modern science led Olschki to write his great two volume history of early technical-scientific literature.389

This had various effects. It firmly established the concept of artist-engineers, which Gilles390 subsequently took up. Leonardo now became one in a long line of engineers from Guido da Vigevano through Taccola, Francesco di Giorgio, pointing to the work of Agricola, Besson, Ramelli and ultimately individuals such as Leupold. The overall thrust of Olschki's work was to focus attention on technology rather than science and while confirming beyond doubt that there existed a continuity at the level of technology, it implicitly raised doubts whether something different might be the case in terms of science.

Olschki's work was used in very different ways. The marxist Zilsel391, concerned with the sociological roots of science, cited it to support his thesis that there were three distinct strata of intellectual activity from 1300 to 1600: university scholars, humanists and artisans and that the first two strata were uninteresting. Zilsel assumed that science was synonymous with causality and that "craftsmen were the pioneers of causal thinking in the period."392 They used quantitative methods but lacked methodical intellectual thinking.

According to Zilsel, the two components of the scientific method were separated by a social barrier: logical training was reserved for upper-class scholars; experimentation, causal interest and quantitative method were left to more or less plebeian artists. Science

49

was born when, with the progress of technology, the experimental method gradually overcame the social prejudice against manual labour and was adopted by rationally trained scholars. This was accomplished around 1600 (Gilbert, Galileo, Bacon).393

In Zilsel's scheme Leonardo became one of a list of craftsmen which included: Ghiberti, Piero della Francesca, Alberti, Biringuccio, Dürer, William Bourne, Robert Norman, William Borough and Palissy.394 Having assumed the existence of a social barrier until 1600, Zilsel or his followers had no incentive to look for evidence of both strands in Leonardo a century earlier.

Nor were these ideas considered only by Marxists. Drake and Drabkin395, for instance, shared none of Zilsel's ideological assumptions, yet also argued for the importance of this practical tradition. They accepted a continuity of mediaeval mechanical ideas in the universities, but held that this was not the source of Galileo's discoveries: that Galileo owed most to a tradition of men outside the universities which went back to the time when these began publishing. This had begun with Tartaglia. It now appeared that there was a scientific revolution which began in the 1540's. The year 1543, when Tartaglia produced his vernacular edition of Euclid, was also the year that Vesalius and Copernicus published their great works and was also said to be the year that the University of Padua began its botanical garden. Leonardo, having been dismissed from the continuity thesis on the assumption that he was a craftsman, in spite of his 119 books, was now excluded from the craft tradition because he did not publish.

The Marburg school of neo-Kantians, notably Cohen (1889),396 building upon ideas of Herder and von Humboldt, had propounded the idea of artists as scientists (Künstler als Forscher). Cohen (1912)397 developed this idea, stressed links between artistic work (künstlerische Arbeit) and scientific logic (wissenschaftliche Logik) and specifically mentioned perspective and anatomy in this context. These ideas inspired both Cassirer and later Panofsky. Cassirer cited Duhem concerning Leonardo's study of Cusa.398 Just as Cusa had developed a concept of lay piety, Leonardo established a concept of lay knowledge of which key elements were a reliance on experience, proportion, measurement and ultimately mathematics. Cassirer claimed that these scientific insights derived from Leonardo’s art:

The scientific theory of experience, in the version to be given it by Galileo and Kepler, will base itself on the basic concept and on the basic requirement of exactness as formulated and established by the theory of art. And both the theory of art and the theory of exact scientific knowledge run through exactly the same phases of thought.399

Cassirer cited Panofsky,400 who had claimed that Renaissance artists such as Dürer had discovered principles of descriptive geometry long before the mathematicians. These very specific neo-Kantian claims about artists as precursors of science were soon translated into general notions the imaginative Italian philosopher and historian, De Santillana, at MIT,401 who used Brunelleschi as a key example. Unfortunately, nothwithstanding practical technical skills in building the Cathedral of Florence, there

50

was no serious evidence to link Brunelleschi with theoretical developments in science. This made nonsense of Cassirer’s original claims. It meant that scholars once again ignored Leonardo. Instead they looked to Brunelleschi402 for things that could not be found there. An enormous body of over 120 articles on Brunelleschi’s perspective as a key to early modern science was the result.

One important thrust of Cassirer's claims was that Renaissance science involved more than Galileo's discovery of laws of motion, that there were basic shifts in philosophical framework involved. Cassirer himself explored these in general terms in Substance and Function.403 This possibility excited Burtt (1924),404 who made his own claims: that the Platonic and Pythagorean traditions involved metaphysical speculations asserting a cosmological status of mathematics which provided both a foundation and justification for science. In so doing, he drew attention to the importance of the new astronomy, and associated early modern science with Copernicus Kepler, Galileo, Descartes, Gilbert, Boyle and Newton. He was challenged by Strong (1936) who claimed that:

The meaning of concepts employed by mathematicians and scientists in their work was found to be established in the limited operations and subject matter constituting the science. The conclusion finally driven home was the conviction that the achievements of Galileo and his predecessors were in spite of rather than because of prior and contemporary metaphysical theories of mathematics.405

In terms of sources, Strong focussed on mathematics from Tartaglia through Cataneo, Clavius and Veglia to Galileo. But there was more to this debate than the interpretation of specific texts. Strong was insisting that the history of science had special problems of its own which could, indeed should be, studied in isolation. This set the stage for what would later be termed an internalist approach. On the other hand, what Burtt was arguing, and what Cassirer had assumed, was that history of science was but one manifestation of a larger cultural framework. This would later become the externalist approach.406

The quest to understand the history of Renaissance science in a larger context led in other directions also. At Oxford, Alistair Crombie407 set out to demonstrate that some of the key terms of philosophy necessary for experimental method which were used by Galileo and other seventeenth century thinkers already existed in the thirteenth century. As he presented the evidence it appeared as if Grosseteste had effectively articulated all the key terms of early modern science. That the actual context within which these terms were used might have changed entirely in the meantime was not discussed. Scholars such as Boas-Hall408 were more careful, and while eager to acknowledge some continuity from the mediaeval period, insisted that there was something fundamentally different about the period 1450-1630. Although focussing on astronomy, Boas-Hall emphasized the significance of other fields such as cartography, botany, biology, medicine and mathematics.

Meanwhile, Burtt's claims about the importance of Platonism were taken up by Koyré‚ who became convinced that there were special connections between metaphysics and measurement.409 He linked these concepts firmly with the Copernican revolution.410

51

DATES FIGURES FIELDS SCHOLARS1. 1200-1250 Grosseteste Philosophy Crombie2. 1300-1350 Ockham, Buridan Philosophy, Theology Duhem3. 1400-1425 Brunelleschi Art Santillana4. 1450-1630 Cusa, Leonardo Philosophy Cassirer5. 1450-1630 Cusa, Copernicus, Astronomy Boas-Hall

Brahe, Porta, Digges, CartographyKepler, Galileo Biology, Botany

6. 1540-1700 Copernicus, Kepler, Metaphysics BurttGalileo, Descartes,Gilbert, Boyle, Newton

7. 1540-1630 Tartaglia, Cataneo, Mathematics StrongClavius, Galileo

8. 1540-1630 Tartaglia, Guidobaldo Mechanics DrakeBenedetti, Galileo

9. 1540- Copernicus Astronomy Koyré, Kuhn10. 1600-1630 Galileo, Gilbert, Bacon Craftsmen Zilsel

MechanicsSurveying

Figure 28. Ten claims about when the Renaissance in science began.

Gradually the Copernican revolution in astronomy could be seen as synonymous with the scientific revolution.411 If we stop for a moment to consider these various theories, we find at least ten different claims about when the Renaissance began (figure 28). Such a list suggests how concerns with specific kinds of problems inevitably led most scholars to overlook the possible role of Leonardo. Indeed, the only exception was Cassirer and his assumptions did not require him to make a detailed study.

Leonardo was one of the most visual thinkers in history with somewhere approaching 100,000 skteches and drawings in the 6,500 extant pages of his notebooks. His work introduced a more scientific approach to visualization through the conscious use of techniques such as transparency, cutaways and exploded views. In the nineteenth century this visual dimension was seen as one of the hallmarks of science412 and one of the reasons for interest in Leonardo. This interest in the importance of visualisation continued into the early 20th century with Hilbert and Cohn-Vossen’s classic Anschauliche Geometrie (1932), which appeared in English as Geometry and the Imagination.413

The rise of relativity theory, quantum physics and the indeterminacy principle raised unexpected questions about the limits of visualization. These developments suggested that there were areas of physics and reality which could neither be seen nor rendered visible. This led Arthur I. Miller to re-examine the role of Imagery in Scientific Thought (1984),414 in order to claim that (non-visual) mental imagery was more important than

52

visual imagery. Building on the work of Hadamard (1954),415 Werthheimer (1959),416

Kuhn (1962),417 Medawar (1969),418 Piaget (1970)419 and Holton (1973),420 Miller claimed that he was using cognitive psychology “to shed further light on the history of science.”421

This approach made the history of visualization seem an irrelevant aspect of science. Leonardo seemed no longer to have intellectual relevance.

The historian, Martin Jay (1993)422 has traced the rise of anti-ocular approaches in 20th

century thought, especially in philosophy, anthropology and sociology. According to Jay the anti-visual approach of thinkers such as Husserl, Heidegger, Derrida, and Lyotard reflected a deeper clash between the Hebraic (aural) and the Greek (visual) cultural traditions. If one accepts this interpretation then the attack on visualization in science is linked with Hebraic cultural traditions, and the tendency to emphasize Einstein as a genius and especially Einstein as a conceptual, abstract thinker reflects trends in Hebriac historiography. In any case, the trend towards a non-visual, mental, cognitive approach emphasizes a constructive model of reality.

The rise of relativity theory, quantum physics amf the indeterminacy principle raised many other unexpected questions. They revealed that the Renaissance distinction between subject and object was too simplistic; that the person as observer was more inextricably linked in the process than the rhetoric of objectivity pretended. A number of responses emerged: cybernetics (Wiener, Ashby, von Förster), systems theory (von Bertanlaffy), objective knowledge (Popper), personal knowledge (Polanyi), radical constructivism (von Glaserfeld, Marturana, Varela). These contributions made it clear that the further study of objects needed to include the role of subjects, of persons and their contexts.

In Russia, especially during the reign of Stalin, political consequences of scientific ideas frequently became matters of life and death. In 1931, a delegation of Russian scientists to London in 1931 inspired John D. Bernal423 to focus on social dimensions of science and later on the importance of institutional contexts. This evolved into a Marxist approach to science.424 In the course of the next half century the ideological basis faded, and evolved into a more general claim that the study of science and knowledge needed to include the role of persons, their communities, institutions and their political contexts.

By the 1960s, such ideas led W. O. Hagström to speak of The Scientific Community425; influenced sociologists such as Edward Shils (1961)426 and the physicist turned historian of science, Thomas Kuhn (1962).427 This prompted debates about the ontological status of science through philososhers such as Popper, Lakatos and Feyerabend. They also prompted the historian and philosopher of science, Stephen Toulmin (1968)428 to speak of conceptual revolution in science.

In the 1970s and 1980s, the history of science, which had traditionally been the domain of scientists and historians of science, became increasingly a subject area for philosophers (Hacking, 1975);429 anthropologists and sociologists (Shapin, Schaffer, 1985430). In the process historical dimensions of science became a backdrop for debates about the nature of science. As Bourdieu put it, these studies on

53

the invention of the experimental method enable one to form an idea of what a structural history of the genesis of the scientific field could be: as a universe in which a special form of accumulation takes place, a principle of methodical reinterpretation of all the external demands and pressures that come, as in the case of probability theory, from the legal field or from the economic field or even from ordinary experience.431

This considerable widening of the history of science led in several directions. First, what had begun as an emphasis on conceptual aspects in the philosophy of science soon turned to a cognitive turn which included both sociological and psychological dimensions.432

Second, there has been increasing attention to the role of institutions (e.g. Lenoir,

431 Pierre Bourdieu, “The Peculiar History of Scientific Reason,” Sociological Forum, Volume 6, Issue 1, Mar. 1991, 3-26.See: http://www.compilerpress.atfreeweb.com/ Anno%20Bourdieu%20The%20Peculiar%20History%20of%20Scientific%20Reason%2017 Baldesar Castiglione, The Book of the courtier, trans. George Bull, Harmondsworth: Penguin Books, 1967, p. 149: "Another, one of the world's finest painters despises the art for which he has so rare a talent and has set himself to study philosophy; and in this he has strange notions and fanciful revelations that, if he tried to paint them, for all his skill he couldn't."8 Sebastiano Serlio, Il Primo [-secondo] libro d'architettura, Venice: Giovanni Battista et Marchio Sessa, 1544-1568, Bk. II.3. The following is a translation from the first English edition (London: Printed for Robert Peake, 1611), fol. 8v: Therefore the most notable paynter Leonardus Vinci, was never pleased nor satisfied with anything that he made, bringing but little worke to perfection, saying, the cause thereof was that his hand could not effect the understanding of his mind. And for my part, if I should do as he did, I should not, neither would I suffer any of my works to come forth: for (to say the truth) whatsoever I make or wryte, it pleaseth me not: but (as I sayd in the beginning of my worke) that I had rather exercise in worke that small talent, which it hath pleased God to bestow upon me, then suffer it to lye and rot under the earth without any fruit.11

2. SOURCES AND CONTACTS? Mad. I 12v:

"Dice Giulio aver visto nella Magni[a] una di queste rote essere consumate dal polo m."

12 CA 370va (1033v, c. 1497-1500): "Dello scriver lettere da un paese a un altro Parleransi li omini di rimotissimi paesi d'uno all'altra, e risponderansi."

13 CA 260ra (697r, c. 1508-1510): "Scrivi a Bartolomeo Turco del frusso e refrusso del mar di Ponto, e che intenda se tal frusso e refrusso e nel mar Ircano, over nel mare Caspio."

14 CA 97va (266v, c. 1515-1516): "Vedi Aristotile de cielo e mondo."

Cf. CA 289vc (785bv, c. 1487-1490): "Aristotile nel terzo dell'Etica."

54

1982433). Third, both historians of technology (Bijker, 1985, 1987434) and sociologists (Latour, Woolgar435) devoted increasing attention to the role of scientific practice. These new schools linked science in varying degrees with social constructivism.436 This has led some to argue that science is merely a matter of consensus making.437 On the positive side this has led to an emerging field of Science Studies438 sometimes called Science, Technology and Society (STS).439 It has also inspired what Richard Rorty has called the “Phony Science Wars.”440 Scholars such as Hacking (2000)441 and Longino (2001)442

represent different attempts to bridge these rhetorical chasms.

In general terms, these studies focused attention away from the studies of individuals. Indeed they tended to dismiss what they saw as a cult of genius in the 19 th and early

Estimates concerning the extent of Leonardo's reading have varied enormously. One of the most thorough studies of possible sources remains Solmi as in note 3 above.15 CA 183va (503r, c. 1517-1518):

"e provato nelli elementi d'Euclide."16 CA 83vb (226v, c. 1508):

"Omnis motu mensuratur tempore...in 2 de 8o fisicae."17 CA 221va (596r, c. 1517-1518):

"e questa regola e nata dalla 14a e ultima del 2o delli elementi d'Euclide." A list of specific references to Euclid's Elements found in the Codex Atlanticus follows:c. 1500CA 169rb (462ar), 285vc (776dr) 5 Postulates, 11 Definitionsc. 1514CA 90va (244v) I.1c. 1515-16 CA 44va (122v)I.1c. 1517-18CA 174v (476v)I.1c. 1500-05CA 184va (506br) I.5c. 1500, CA 177vd (483bv) I.7c. 1500 CA 169vb (462bv) I.12c. 1500 CA 169vc (462av), (483br)I.13c. 1500, CA 169rc (462br)CA 169rc CA 169rc

CA 169rc I.17I.18I.19I.20c. 1500, CA(776dv)CA (776dv)I.21I.22c. 1503-05,CA203rb (544r) CA 203rb I.34I.43c. 1500,CA184va (506ar) CA184va CA184va CA 221va I.46I.47I.48II.14, II.15c. 1508-10, CA269vb, 726r

CA 269vbV.6V.8Cf. CA 259vb (696v, 1515), where he refers to "una d'Euclide" but cites the second of the common notions, or CA 96va (264v, c. 1500) where he states simply "E qui disse Euclide."18 B 8r: "catapulta come dice nonio e plinio," Cf. G 48v: "Dicie Plinio nel secondo suolibro a 103 capitoli."19 BN 2037 7v (formerly part of Ms. B): "virgilio dicie."20 BN 2037 8v: "Lucretio nel terzo delle cose naturale."21 B 8v: "rhomphea...secondo aulo gelio."22 B 9r:

55

twentieth centuries. They focussed attention increasingly on the role of groups, communities, institutions; organisations within which persons worked and political contexts of actions. They also focussed attention on contemporary issues, away from historical traditions. This led to paradoxical conclusions such as Steven Shapin’s (1996) claim: “There was no such thing as the Scientific Revolution and this is a book about it.”443 In this context, Leonardo was not only too long ago to be relevant. There was no revolution in which he could have made a difference.

Our brief survey of these historiographical trends thus helps us to understand a gradual shift from asking questions about when and where the scientific revolution occurred to

"dicie livio nel settimo delaguera cartaginese secondo trovo inuna comedia di plauto...dice flavio."

23 B 30v: "dicie lucano ciesare che..." Cf. 41v: "ciesere ne fa mentione nel secondo delli sua comentarii."

24 B 41r: "sicome vuole quintiliano...plinio nel VI libro de naturale istorie...ecome vole varone...secondo lucano nel nono."

25 B 41v: "delaqual fa mentione Plutarcho nella vita di graccho."26 B 43r: "ecquesto erma varone festo ponpeo ne testimonia diciendo."27 B 45v: "sechondo amiano marciellino."28 E.g. F 27r: "A Platone si risponde," which begins a commentary on his treatment of regular solids in the Timaeus.29 E.g. L 53v: "Dice vetruvio" or G 54v: "Vetruvio ne pone uno nella sua opera darchitectura."30 M 8r: "Suisset."31 M 11r: "Tebit."32 On CA 203ra (543r, c. 1489-1490) Leonardo translates the opening passage of Peckham's Perspectiva communis. For an English translation see the author's Leonardo Studies, Vol. I, Linear Perspective and the Visual Dimensions of Science and Art, Munich: Deutscher Kunstverlag, 1986, p. 56. CA 277ra (750r, c. 1513):

"E questo solu e bastante alla detta pr[o]va la quale a data nella prespettiva comune eccetera."

33 E.g. Mad. I 171v: "Le lettere dell'abbaco da'pesi in su non parlano del peso che ttira le corde anzi di quello che ssi sscarica sopra le gifelle." Cf. BN 2038 8r: "Onde conquesto che ffacto cholla sperienza. Subito poi legiere la lettera dellabaco che tocha dal filo essai la verita delpeso."

34 E.g. BN 2038 2v: "dicie il pelachane." Cf. Mad. I 133v: "Pruova contro al Pellacane."35 E.g. Arundel 263 31v:

"Dice batista alberti nuna sua opera mandata al signore malatesta da rimjni come quando la bilanca" or Arundel 66r: "Dice battista albertj nuna sua opera titolata exludis rerum mathematicarum..."

36 See Ladislao Reti, in Burlington Magazine, London, vol. CX, 1968, pp. 81-89 with supplements in vol. CX, 1968, pp. 406-410, vol. CXI, 1969, p. 91. Cf. Leonardo's list on

56

problems of how and why. In the past, there was a quest to discover how a systematic approach to undertstanding the universe evolved. Increasingly the focus is on how one systematic explanation was replaced by another, or, as Kuhn has termed it, how one paradigm replaced another.444 It is bemusing to note how the scholars of earlier generations who worked so hard at establishing “hard facts” in terms of documented evidence, notably the Duhems, Sartons and Thorndike’s are dismissed by these new schools as being “mere positivists.”445

From this have emerged assumptions that only changes in the structures of thought count: that it is really a question of shifts in mentality. The versions of these assumptions are

CA 210ra, concerning which see Leonardo da Vinci, Scritti letterari, ed. A. Marinoni, Milan: Rizzoli, 1974, pp. 279-257.37 CA 225r. There are other references to Witelo on B 58r and CA 247r.38 L 2r. There are a number of other notes of this kind scattered throughout the manuscripts. On BM132v he refers to [Amerigo] Vespucci giving him a book on geometry. On Forster III 2v he mentions that "Maestro Stefano Caponi, a physician lives at the piscina and has Euclid's De ponderibus or on Forster III 86r he notes that "the heirs of Maestro Ghiringello [a professor at Padua] have the works of Pelacano".39 Florence, Cod. Ashburnham 361.40 Luca Pacioli, as in note 1, p.33.

3.TREATISES 41 These dimensions are taken from Richter, as in note 13, vol. 1, pp. 108-109.42 Cf. CA 199v.43 On this topic see Nando de Toni, "I rilievi cartografici per Cesena ed Urbino nel mano-scritto `L' dell'Istituto di Francia," Lettura Vinciana, No. 5, 15 April 1974, Florence: Giunti 1974.44 Carlo Pedretti, Fragments at Windsor Castle, London: Phaidon, 1957.45 W 19070v (K/P 113r): "fa legare li tua libri dj noa."46 Arundel 190v: "legare il mil libro."47 A more detailed discussion of his approach is found in the epilogue of the authors’s Leonardo da Vinci Studies II.48 Such as CA 206ra (549r, c. 1497).49 Leonardo Studies I, as in note 32, pp. 57-60.50 Forster II 64r: "Mechanica potissimum in fine incipiendum."51 F 94v:

"Libro mio sastende a mostrare come locean colli altri mari fa mediante ilsole splendere il nostro mondo a modo di luna e a piu remoti pare stella ecquesto provo."

52 F 95v: "Nonsi puo difinire qui per carestia di carta ma va inverso il principio dellibro ha carta 40 cheli edifinita."

53 See the author's Leonardo Studies II-III, Continuity and discovery in optics and astronomy, (Munich: Deutscher Kunstverlag) (awaiting funds for publication).54 F 13v: "Volta carta ennota il retrosa accidentale essua govamento."55 F 26v: "Qui si segue la prova di quel che detto nella opposita facca."

57

several. In mild cases, it is simply a matter of mentioning Foucault or Derrida. Others have subtle formulations about seeking to discover how the scientific mind works; that one must use new conceptual tools, above all that of the thematic content of science. Thus Holton can write about Thematic Origins of Scientific Thoughts and write a book of Case Studies on the Scientific Imagination. So history becomes a history of problem solving: a series of themes, case studies, quick probes. And a generation later chairs on the early history of science disappear at Harvard and this becomes a model for other departments. According to this approach, there is no longer a cumulative picture to be understood, and in this context the Renaissance and Leonardo are too early to be relevant.

56 F 51r: "va acarte 59."58 G 44v: "Ecquesto edisegnato in margine della quarta carta dapiedi."59 G 46r: "Qui seghue quelche mancha disocto arri scontro."60 G 46v: "Leggi in carta 45."61 G 51v: "Va alle 44 charte di quessto."62 G 67r: "Per quel che errischontro dappie e choncluso."63 G 75r: "Qui seguita quelche nella contrapposta carta."64 G 75r : "El subbio e fighurato nella contrapposta faccia."57 E 75r: "Qui sifiniscice quel che mancha nella terza charta innanti a quessta."65 G 80r:

"Qui si finisscie quel che mancha qui dirieto acquesto lato della charta cioe dappie in margine."

66 For an analysis of these passages see Leonardo da Vinci Studies I, as in note 32, pp. 257-268.67 Forster I 3r:

"Libro titolato de strasformatione, cioŠ d'un corpo `n un altro sanza diminuizione o acresscimento di materia."

68 Forster I 3v: "Principiato da me Leonardo da Vinci addi 12 luglio 1505."69 As I plan to present this some day as an independent study I do not document the foregoing paragraphs in detail.70 CA 384ra: "Io dissi nella 7a conclusione come la percussione."71 CA 155vb: "Guarda nel 7o del quinto de polo e rota."72 CA 2ra (10r, c. 1515):

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

73 CA 287ra (780r, c. 1514-1515): "Messer Battista dall `Aquila, camerier segreto del papa, ha il mio libro nelle mani."74 Arundel 12r: "Qui per la 5a del 7o, il peso..."75 Arundel 25r: "Come e provato nel 4 della mja prosspectiva."76 Arundel 25v: "Libro 9e dellacqua de poci permanente e dellacqua fugitiva."77 K 30 [29]r: "Sesto libro."78 F 5r, F 24v, F 72v and F 88r.

79 F 4v and F 24v.

58

There are also more radical versions which argue more strongly. Since these are universal problems, what individuals wrote on given pages of specific documents is too trivial. Hence one is saved the trouble of learning old languages. Indeed one can, for the most part do without texts, and can certainly spare oneself the bother of looking at manuscripts, troublesome archives and other outdated, irrelevant modes of communication. In extreme cases, there is a conviction that history, if approached properly, should stop worrying about the past (which is over anyway) and concentrate instead on philosophy, which explains the logic of basic ideas; psychology, which enables us to see structures of the mind and sociology, which provides a social context for changes in those structures.

80 F 35r: "Libro 42. Delle Pioggie."81 F 37r: "Libro 43. Del moto dellaria inclusa sotto lacqua."82 F 66v: "Principio dellibro."83 W 19023v (K/P 65v).84 W 19037r (K/P 81r).85 W 19023v (K/P 65v).86 W 19061r (K/P 154v).87 W 19037r (K/P 81r).88 I 72 [24v]: "Principio dellibro dellacque." 89 E 59v: "Principio di questo libro de pesi."90 E 27v: "Pruovasi per la nona deperchusione cheddicie."91 W 19064r (K/P 157r): "e per la 5a de forza e provato quel che dj sopra sicontiene."92 W 19009r (K/P 143r):

"Fa chellibro delli elementi machinalicolla sua praticha vada inantj ala djmostratione del moto e forza dellomo e altrj anjmali e medjante quelli tu potrai provare ognj tua propositione."

93 W 19061r (K/P 154r):"ordine del libro... Adunque quj con 15 figure intere ti sara mosstro la cosmografia del mjnor mondo colmedesimo ordjne che inanzi ame fufatto dattolomeo nella sua cosmografia."

94 E.g. K/P 58v, K/P 81r, K/P 113v, K/P 139v, K/P 154r.95 This combination is admittedly hypothetical but consists solely of aspects that Leonardo himself considered.96 W 19061r (K/P 154r).97 Leonardo’s anatomical method is discussed in greater detail in the author’s: Leonardo da Vinci Untersuchungen zum menschlichen Körper in: Gepeinigt, begehrt, vergessen. Symbolik und Sozialbezug des Körpers im späten Mittelalter und in der frühen Neuzeit , ed. Klaus Schreiner, Bad Homburg: Werner Reimers Stiftung, 1992, pp. 287-308.

984. PLANS FOR BOOKS? Mad. I 173v: "Farai tutto il testo insieme e poi piu oltre lo dividi col suo commento."99 CA 117rc (324r,c. 1495):

"Tratterai prima del peso, poi del moto che partorisce la forza e po'd'essa forza e in ultimo del colpo."

100 CA 149rb (403r, c. 1493-1495):

59

In this extreme view, history is about universal ideas, not individual opinions; about objective manifestations of truth, not subjective samples of biased creatures who do not reflect the norm. This extreme view is one that is overheard in conversation with graduate students or found expressed in their essays. If such views seem so outlandish that they are felt to have no place here, then it is sobering to recall the case of a not unknown Harvard scholar, John Hermann Randall, Jr. who, while openly admitting that he had never read the notebooks of Leonardo da Vinci, felt that he could safely "lay down" three propositions: 1) Leonardo was not a scientist, 2) the notebooks did not contain "a single theoretical scientific idea that is essentially new or that was unknown in the organized scientific schools of his day," 3) even if Leonardo had had original ideas in scientific

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

101 K 3r: Dividi il trattato delli uccelli in 4 libri il primo sia dellor volare per battimento dalie il secondo del volo sanza batterali e perfavol divento il terzo del volare incomune come ducelli, pipistrelli, pessci, animali, insetti ultimi ultimo dellmoto strumentale.

102 (moto e colpo). on Mad. I 69v

103 (moto e forza) on 94r

104 (teorica) on Mad. I 140v

105 There is also a sixth book of Encounters of water with objects that fall with circular motion e.g. wheels of aquatic instruments.CA 79ra (III 214br,c. 1505-1506):

Libro delle percussione dell`acqua in diversi obbiettiScontri dell'acqua inobbietti permanenti di diverse figure che superanol'acquaScontri dell'acqua inobbietti immobili coperti dell'acquaScontri dell'acqua inobbietti mobili coperti dell'acquaScontri dell'acqua inobbietti permanenti che suprano l'acquaScontri dell'acqua inobbietti piegabili superati dell'acquaScontri dell'acqua nelliobbietti piegabili che superano l'acquaScontri dell'acqua nelliobbietti che cadono con moto circulare, come sono le rote dellistrumenti acquatici.

Cf. CA 74vb (III 201v) for a related list.106 Mad. I 147v

107 On Mad. I 71v

108 On Mad. I 87r

109 On Mad. I 140v

110 A list of specific books and propositions cited in Madrid Codex I follows: 1.4Mad I 73v1.5Mad I 87r, Mad I 114v1.7Mad I 127v2.5Mad I 65r, Mad I 122r, Mad I 141r2.9Mad I 144r2.20Mad I 19v, Mad I 97r2.28Mad I 153v2.penultimateMad I 26r, Mad I 148v3.5Mad I 102v, Mad I 114v3.penultimateMad I 121r4.3Mad I 144r4.7Mad I 69v4.penultimateMad I 153v5.3Mad I 74r5.penultimate172v6.5Mad I 26r, Mad I 47v, Mad I 70r, Mad I 118r7.5Mad I 73v, Mad I 127v, Mad I 139v7.60Mad I 43r9.5Mad I 443r, Mad I 73v, Mad I 139v9.7Mad I 73v, Mad I 143r

111 F 41v: Aparlare dital materia ti bisognia nel primo libro difinire lanatura della resistentia dellaria nel 2o lanotomia dello uccello e delle sua penne nel terzo la operation dital

60

theory, "they remained unknown until the Paris Codici were published 1881-1891 and the Codice Atlantico in 1894."446

Looking back we can see that the 20th century brought fundamental changes not only in how science is done, but also the historiography of the past five centuries. As Cassirer showed, Renaissance science introduced a basic distinction between a person as a scientist (subject) and the visible, physical world (object). The quest to study nature in terms of concrete physical objects, “objectively”, inspired a whole spectrum of methods including physical/visual models, perspectival images, geometrical schema and abstract algebraic formulae. In the 16th and 17th centuries science was about the whole spectrum.

penne per diverse moti dasse nel quarto la valitudine de lalie e coda sanza battimento dalie confavore divento aversi a guidare perdiversi moti.

112 F 90v: "Ordine dellibro... Poni nel principio cochepofare un fiume." F 45v: "Ordine del libro." E 12r (1513-1514): "Ordine del primo libro delle aque."113 F 87v:

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

114

Eddies whichare superficial“ “which rise from the bottom to the surface“ “go from the surface to the bottom“ “move with the course of the stream“ “change direction“ “are lateral and continuous“ “are lateral and discontinuous“ “are wide above and narrow below“ “are narrow above and wide below“ “are straight from bottom to top“ “are oblique from bottom to top“ “are very large“ “are small“ “have gurgles“ “are pipe-like“ “are screw-like“ “are hollow and filled with air“ “are not hollow.

CA 74va (III 201v, c. 1505-1506): Delle cose percosse dalle acqueDelle cose che percotano l'acqueRetrosi superfizialiRetrosi che si leva<n> dal fondo alla superfizieRetrosi dalla superfizie al fondoRetrosi che si movan col corso del fiumeRetrosi scambievoli nellilor raggiramenti, come son quelli de'refrussi e frussi de'fi<u>meRetrosi laterali continuiRetrosi laterali discontinuiRetrosi larghi di sopra e stretti in fondoRetrosi stretti di sopra e larghi in fondoRetrosi diritti dal fondo al disopraRetrosi obbliqui dal fondo al bisopraRetrosi grandissimi

61

In the course of the 18th and 19th centuries science gradually became associated primarily with the abstract side of this spectrum, to the extent that with formulae seemed to define its essence.

Retrosi brieveRetrosi de'bolloriRetrosi a canneRetrosi a viteRetrosi vacui e ripien d'ariaRetrosi non vacui

At least some of this material found its way into a manuscript attributed to Leonardo in the Biblioteca Barberini in Rome, entitled Trattato della natura, peso e moto delle acque, e osservazioni sul corso de' fiumi which was published as Del moto e misura dell'acqua, ed. Francesco Cardinali, Bologna: a spese di F. Cardinali, 1826 (Raccolta d'autori italiani che trattano del moto dell'acque, tom. 4).115 See, for instance, CA 74ra, CA 74va, CA 74vb (201rv, 1505-1506).E.g. Arundel 35rv, 45r, 122r.E.g. Leicester 5r, Leicester 9r, Leicester 15v. These and a number of the above passages have been translated by Richter, as in note 13, vol. 2, pp. 141-167.116 See: Carlo Pedretti, Commentary: The Literary works of Leonardo da Vinci, London: Phaidon, 1977, vol. 2, pp. 140-145.

5. THEMES

117 F 94v as in note 51.118 I have reconstructed his treatise on cosmology showing where his work on optics, instruments and astronomy fit into this in Leonardo Studies II-III, as in note 53.119 CA 361va (1007v, c. 1490):

"Il vento a similitudine col movimento dell'acqua."120 M 83v: "el notare mostra il modo del volare."121 CA 66rb (186r, c. 1505):

"Il notare sopra dell'acqua insegna alli omini come fanno li uccelli sopra dell'aria." Cf. CA 214rd (571ar, c. 1507-1508): "Scrivi del notare sotto l'acqua e arai il volare dell'uccello per l'aria."

122 E 54r: Per dare vera scientia del moto delli uccielli infrallaria e neciessario dare prime lasscientia deventi laqual proverren mediante di moti dellacqua insemedesima he equesta tale isscientia sensibile fara di se scala aper venire alla chognitione de volatili infrallaria elvento.

123 A 55v: "Adunque se'l corpo della terra non avesse similitudine coll'omo."

62

See Richter, as in note 13, no. 917 for the complete passage.124 A 56r:

"Dico che siccome il naturale calore tiene il sangue nelle vene alla sommità dell'omo... similmente le vene che vanno ramificando per il corpo della terra..." See Richter, as in note 13, no. 969 for the complete passage. Cf. his no. 965 where he cites Arundel 263v.

125 CA 260ra (697r, c. 1508-1510): è l'attrazione e respirazione dell'aria nelpolmon dell'omo. Ora s'è l'attrazione dell'acqua, che farebbe la terra in 12 ore col frusso e refrusso, ci poterebbe mostrare la grandezza del polmon della terra inquesto modo.

126 W 19029r (K/P 71r): "“instrumento mirabile inventionato dalsomo, maesstro."

127 W 19037r (K/P 81v): "Questa figura strumentale dellomo djmonsterremo in <24> figure."

128 CA 161ra (434r, c. 1505): "L'uccello e strumento oprante per legge matematica, il quale strumento è in potesta dell'omo poterlo fare con tutti li sua moti, ma non con tanta potentia."

129 W 19147-8r (K/P 22r): "della terresste macchina al suo cientro."

130 CA 269va (727r, c. 1490): "le vari opinioni... della grandezza de l'orbiculare macchina terreste... o preso ardire creare, over comporre un istrumento, il quale adoperai in questa forma."

131 CA 252rb (681r, c. 1490-1492): "Questa terrestre e mundiale macchina."132 A 59v: "universal macchina della terra." For examples of Leonardo's attempts to treat the four elements in quantitative ways so that they too can fit into his mechanical model see, for instance, CA 79vb (214ar, 1505-1506) and CA 72ra (197r, 1508-1510).133 See Leonardo Studies II, as in note 53.134 See Leonardo Studies I, as in note 32.135 Cf. William A. Emboden, Leonardo da Vinci on plants and gardens, Portland: Dioscorides Press, 1987.136 The manuscripts in which arithmetic is discussed are mainly: Forster II, Manuscript L, CA, and Mad. II.137 See, for instance, CA 69ra (189r, c. 1498).138 The main manuscripts with respect to language are: Trivulzianus, H, I and Windsor.139 The main manuscripts for literature are H, I, CA and Windsor.

63

Figure 29. New ways of looking at science as a building of bridges between abstract and concrete, as linked with persons and their contexts: personal, social and societal.

140 Geometry is found in Forster I, II, Manuscripts A, B, E, G, I, K, M, Arundel, CA and Mad. II.141 K 49 [48 et 15]r:

"La proportione no solamente nelle numeri emisure fia ritrovata ma etiam nelli suoni, pesi, tempi essiti ecqualunche potentia sicia."

142 The extent to which Euclid and indeed mathematics generally was important to Leonardo is a matter of considerable debate. For another side see: A. Marinoni, La matematica di Leonardo da Vinci, Milan: Arcadia Edizioni (Philips S.P.A.), 1982. 143 We have already mentioned his treatise on this subject in Forster I. A note on CA 45va

(II 124v, c. 1515-1516) confirms that he was going to write anew on this topic over a decade after the Forster I:

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

145 Cf. Treatise on Painting (Codex Urbinas 1270), trans. A. Philip McMahon, Princeton: Princeton University Press, 1956, vol. II. Facsimile. Quinta Parte. D'ombra e lume.146 Treatise on Painting, Terza parte. Comincia di vari acidenti et movimenti dell'huomo e prima delle mutationi delle misure dell'huomo, pel movimento delle membre a diversi aspetti. This section is as much inspired by geometry and anatomy as light and shade, however.147 Treatise on Painting, Quarta parte, De panni.148 Treatise on Painting , Sesta parte, D'alberi et verdure.149 The chief manuscripts for military architecture are CA, B, L, Mad. I, II. Civil and ecclesiastical architecture are chiefly in CA.150 The standard work on this topic is Pietro Marani, Architettura fortificata negli studi di Leonardo da Vinci, Florence: Olschki, 1984.151 Most of his weapons are in Manuscript B and CA.152 Leonardo wrote on the margins of 7 pages of Francesco di Giorgio's Cod. Ash. 361 (Florence, Laurenziana).153 Most of Leonardo's machines are in Forst. I (hydraulic), CA and Mad. I.154 See note 5.155 This is discussed mainly in Manuscript B with drafts in CA and the Turin Codex.156 Representation in this sense is much more than an idle pastime. It is man's key to understanding God's action as a creator: a re-creation with glimpses into God's mysteries. Hence painting and science combine to provide Leonardo with a new version of natural

64

Context

Personal Social Societal Psychology Anthropology Political Science Psychiatry Ethnology Economics, Law

Two developments of the past decades have brought renewed attention to visualization. First, the rise of computer graphics, which is effectively recapitulating the Renaissance discovery of perspective in digital form,447 is leading to new fields such as scientific visualization and information/knowledge visualization. Second, the advent of new scientific imaging techniques448 at both the micro-and the macro-level has extended the scale of scientific research from 10-15 to 10.25

These trends, which could readily entail further books, are leading to a reassessment of the scientific process as a whole, and bringing new relevance to the work of Leonardo. The expansion of imaging over an immense range of scales from 10-15 to 1025 means that science is much more than solving problems with the right abstract formulae. High

theology: a fusion of science and art which shines as an underlying unity through the chaotic surface of his notebooks.157 CA 269va (727r, c. 1490):

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

158 The chief notebooks on mirrors are CA and Arundel.159 Clocks are considered in Mad. and CA.160 The chief notebooks on balances are Forster II, CA, E, L, and Arundel.161 Compasses are considered mainly in B and CA.162 CA 321ra (X 882r, c. 1493-1495):

"Come tutte le rote sono di natura di bilancia."163 CA 396rd (XII 1102r, c. 1495):

"Fa menzione e regola generale sopra il contatto de' poli e di tutti i pesi."164 CA 155vb (421v, c. 1495-1497):

"Parla prima del moto, poi del peso, perch‚ nasce dal moto; poi della forza che nasce dal peso e moto, po' della percussione che nasce da peso, moto e spesso dalla forza."

165 CA 267ra (721r, c. 1495): "Regola generale del colpo. Regola generale della forza. In queste 2 regole, cioè di colpe e di forza, si può adoperare la proporzione che Pittagora usò nella sua musica."

Leonardo also compares perspective and music in terms of proportions on A 103r (BN 2038 23r, TPL 31).166 CA 20va (66r, c. 1493-1495):

"Per fare regola generale della differenzia ch'è da peso semplice a peso col colpo di diversi moti e forze."

167 CA 120vc (330av, c. 1497-1498): "Si come tu trovi qui regola a diminuire il peso al motore, ancora troverai la regola di crescere il tempo al moto."

168 Mad. I 152r. See note 300 below.169 CA 151ra (407r, c. 1500):

65

energy physics at the sub-atomic level remains important as do scientific experiments at many other scales. There is an emerging insight that science in a deeper sense is about creating a systematic spectrum of links and bridges between physical reality and its various layers of abstractions.449 In this approach, the abstraction school which emphasizes formulae and the visualization school which emphasizes the role of models, figures, diagrams, and schemas emerge as part of a single spectrum. A future history of science must therefore trace and explain the evolution of this spectrum.

This new history is independent of the ontological status of individual components of this spectrum, and even independent of whether one calls this a scientific revolution. This

Tutte le potenzie naturali hanno, ovver sono da essere dette piramidali, con ciò sia che esse abbino gradi in continua proporzione inverso il suo diminuire, come inverso il loro accrescimento. Vedi il peso, che'n ogni grado del suo discenso, essendo libero, acquista gradi in continua proporzione geometrica. El simile fa la forza nelle lieve.

170 CA 335vd (915br, c. 1503): Molti strumenti si potrebbono allegare i quali in simili cavamenti si sono paragonati colle carette... come si mostra in un mio trattato di moto locale e di forza e peso... e a questa ho ordinato il figurato strumento, del quale quanto sia la sua utilit… e valitudine, le ragioni di quelle assegnate non daranno la sentenzia, le quali sempre fien conferme dalla sperienzia.

171 CA 271re (732cr, c. 1508): "De moto locale... E per la quinta del nono che dice."

172 CA 81vb (220v,c. 1508-1510): Elementi macchinali: Del peso proportionato alla potenzia che'l move s'ha a considerare della resistenzia dove tal peso e mosso, e di questo si farà un trattato.

173 CA 298rb (818r, c. 1495): "Se tu tirarai per la linia bf, il tirare si rende difficile per la quinta del settimo ma per la nona del decimo, tirando per la linia Mb..."

174 CA 283vb (771v, c. 1517-1518): "la qual obbliquità è difinità nel'ottavo libro del peso, dove si divide la gravità accidentale dalla gravità naturale."

175 CA 384ra (1062r, c. 1493-1495): "Io dissi nella settima conclusione come la percussione."

176 CA 241ra (657ar, c. 1513): Dividesi la percussione in libri de quali nel primo si dimostra la percussione de due corpi, de' quali l'uno move il percussore al percosso immobile; l'altro muove li percussori e li percossi scambievolmente l'un contro all'altro; terzo e delle materie liquide; quarto delli corpi piegabili; quinto...

177 CA 241vb (657v, c. 1513): "El libro dell'impeto va inanti a questo e inanti all'impeto va il moto."

Cf. Mad. I 103r:“Quess[t]a 8a e allegata nellibro dell'impeto. Adunque quesste figure vanno in esso libro."

178 CA 374ra (1043r, c. 1515):

66

new history has the challenge of explaining how what began as a matter of personal intuition of pragmatic engineers and the whims of individual thinkers evolved to a general method that was accepted by scientists around the world. This shift is also a shift from unrecorded and/or secret efforts of individuals to recorded efforts of a scientific community. In terms of this wider view of science and art as a systematic programme to create a spectrum of methods from visualization to abstract versions (formulae), Leonardo da Vinci was one of the great pioneers. His precise role in this history remains to be told and for this reason his work deserves new study and is destined to gain new recognition.

"Come e la concavità che in se riceve il polo de 'circumvolubili, si debbe provvedere contro alla sua dilatazione. E di questo si tratterà nel libro de confregazione."

There is a further reference to this work on Mad. I 121r under the heading: Della vite e confregazione: Ho mostrato nella penultima del 3o come tutte le varie quantita delle confregationi che po fare un medesimo peso sopra una medesima natura di resistentia over sostentaculo, essere tutti d'equal fatica al suo motore."

179 CA 58ra (151r, c. 1503-1505): "De 'due cubi i quali son doppi uno all'altro come si prova nel quarto delli Elementi Machinali da me composto."

180 This optical tradition has been considered in the author's Leonardo Studies II, as in note 53.181 Witelo, Vitelonis Thuringopoloni opticae libri decem, Basel: Episcopios 1572, Liber Quintus, 57:

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

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

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

183 For a discussion of these connections see the author's Sources of perspective.184 The relevant passages have all been discussed in the author's Leonardo Studies I, as in note 32.185 Cf. note 17.186 CA 174v (476v, c. 1517-1518):

"Io voglio della mezza porzione di circolo abcd fare una transmutatione in abd coll'aiuto d'una d'Euclide ne'sua Elementi."

67

11. Conclusions

From all this there emerges a picture very different than that of “an ingenious empiricist working in an intellectual vacuum.”450 The evidence of Leonardo's notebooks confirms that he was widely read and had many contacts. His extant treatises reveal much more structure than has generally been assumed. Moreover, they evidence a number of clear plans for books. Examination of his entire extant corpus brings to light another unexpected feature: for all their universality the notebooks are focussed on a surprisingly small number of basic themes: crucial among these are his studies of transformational geometry and a mechanical approach to nature, which uses as a point of departure his

187 CA 183rb (502r, c. 1500): "Io voglio fare d'un circolo infinite varietà di figure curvilinee d'equale capacit…."

188 CA 141ra (386r, c. 1500-1505): "Diminuente equalmente aritmetice l'uno e l'altro extremo di ciascuna proporzione, sempre cresce la proporzione geometrica che prima era."

189 CA 174va (475v, c. 1517-1518): "Ma questa calculazione vole essere geometrica, perchè se la volessi fare per arismetrico, sarebbe impossibile."

190 CA 120rd (331r, c. 1504): "Impara la multiplicazione delle radice de maestro Luca."

191 CA 159ra-va (428v, c. 1508): "Col cerchio br farai una regola di radice quadrate insino in venti, e poi con un altro cerchio farai un' altra regola di radice cube insino in venti, e vedrai la differenzia che è dall'una regola all'altra."

192 CA 102va (281r, c. 1517-1518): "Se voli qualunche radice di numero si voglia, questa e la regola."

193 Trigonometria planorum mechanica, 1617194 CA 231ra (629ar, c. 1505):

"Fanne una grande e vedrai con piu certezza se questa regola e vera."195CA 231ra (629ar, c. 1505):

"La pruova meccanica è vera, benchè con fatica si trovi essa verità..."196 CA 130va (360r, c. 1517-1518):

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

197 CA 218va (587v, c. 1503-1505): "Qui accade la prova meccanica."

198 E 8v: "La mechanicha e il paradiso delle scientie matematiche."

199 CA 220vb (593v, c. 1508): "Geometrica regola."

200 CA 239vba (627r, c. 1516): "La prima che è trovata in questa regola. Addi 3 di marzo 1516."

68

concept of four powers (weight, force, motion and percussion), and serves ultimately to integrate both his study of the microcosm (anatomy) and the macrocosm (astronomy) within a single grand plan.

It was shown that these studies were guided by a distinct method of listing variables systematically and playing with them experimentally. It was claimed that the results of this enterprise inspired him to write treatises and led him to make serious plans for publication. While these plans were unsuccessful, there is nevertheless evidence, mainly indirect, that Leonardo, who was at the centre of action in the major cities of the high Renaissance (notably Florence, Milan and Rome), was not without influence in the

Cf. Mad. II 112r for another example.201 CA 103va (285r, c. 1515-1516):

"E questa regola va in infinito."202 E.g. CA 81vb (220v, c. 1508-1510):

"e cosi ti bisognierebbe procedere inverso lo infinito. Adunque sta bene la regola di prima."

203 CA 107va (297v, c. 1517-1518): "E sopra questa tal diminuzione o accrescimento si darà regola generale, che con precisione verreno a chiara notizia del vero."

204 E.g. CA 221vab (596r, c. 1517-1518): "In questa dimostrazione,"

referring to a figure.205 See note 124. For a discussion of this passage see Ernst Cassirer, The individual and the cosmos in renaissance philosophy, tr. Mario Domandi, New York: Harper and Row, 1963, p. 50 (originally published 1927).206 CA 177rb (483ar, c. 1508-1510):

"E questo si prova nell'ottavo de proportione."207 CA 166vb (454r, c. 1515):

"Regola da sapere la valuta e proporzione di molte parte curvilinie di vari cerchi."208 CA 157vb (425v,c. 1515):

"Sesto proporzionale." CA 385ra (1064r, c. 1513-1514):

"Sesto della proporzione."209 CA 248ra (672r, c. 1513):

"Sesto de proporzionalità in profilo. Sesto di proporzionalità in faccia; e il suo pole e mobile... Questo vale nelle proporzionalità inrazionale."

210 CA 83va (225v, c. 1515): "Con questa delli Elementi si può dare qualunche proporzione di circolo, così inrazionali come razionale."

211 CA 128ra (353r, c. 1508): "Libro d'equazione."

Cf. Mad. II 112r: "Scientia di equiparantia."

212 E.g. CA 157rb (425r, c. 1515): "De trasmutazione di superfizie rettilatere in superfizie curvilinie e de converso."

Cf. CA 160rb (432r,1515-1516):

69

century that followed. To explain why Leonardo was subsequently forgotten, the limitations of his approach were considered, as were the limitations of historiography in our own century.

All this was guided by two further purposes: a) to establish that Leonardo and the sixteenth century context leading to Galileo require more careful study, and b) to suggest that new combinations of earlier approaches are required. It is possible, for instance, to accept that there was some continuity between the thirteenth and the seventeenth centuries without assuming that nothing happened between the time of Grosseteste and Galileo. Leonardo's non-exhaustive knowledge of mediaeval and classical sources had its

"De trasmutazione" and CA 167rb-a (455r, c. 1515):

"De trasmutazione d'equali superfizi rettilinie in varie figure curvilinie, e così de converso"

or CA 242rb (660r, c. 1517-1518): "De trasmutazione delle superfizie."

213 E.g. CA 184vc (505v, c. 1516): "De ludo geometrico"

or CA 174v (476v, c. 1517-1518): "De ludo geometrico."

For a full discussion of this topic see the dissertation by McCabe, as in note 158.214 CA 45va (124v, c. 1515-1516):

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

215 CA 167r (455r, c. 1515): "Trattato de quantità continua." Cf. Mad. I 0v: "Libro titolato de quantita e potentia."216 CA 139rab (381r): "Elementi geometrici curvilini."217 E.g. CA 303rb (828r, 1513-1514): "per una delli Elementi segnasta in margine." Also CA 378vab (1053r, c. 1513): "una delli Elementi geometrici" which, as Marinoni notes in his new edition of CA corresponds to Euclid I.37.218 E.g. CA 374va (1043v, c. 1513): "Per la quinta delli Elementi geometrici."219 CA 249rba (673r, 24 June 1518): "43 del primo delli Elementi geometrici."220 CA 242rb (660r, c. 1517-1518): "E la regola di questo si fa coll'aiuto della ultima del secondo delli Elementi geometrici. Per la ultima del secondo delli Elementi geometrici."221 CA 170ra (463v, c. 1516): Di queste mia superfizie curvilinie molte ne son quadrabile in se medesime colla trasmutazione delle sue propie parte nel suo tutto, e molto ne son che colle sue propie parte sono in quadrabile, ma si da quadrati equali loro, tolti d'altre superfizie. E con queste si compone l'ultima mia opera di cento 13 libri da me composti nella quale è 33 modi variati di dare quadrati rettilini equali a circoli, cioè equali in quantità.222 Cf. Reti as in note 5. See Franz Reuleaux, Theoretische Kinematik. Grundzüge einer Theorie des Maschinenwesens, Braunschweig: Vieweg, 1875.223 See note 82.224 See note 152.

70

reasons. He decided that they could not provide him with an adequate picture. So he studied these sources and challenged them also.

This process continued into the seventeenth century. In 1625, for example, Accolti,451

whom one might have expected to cite Kepler, still acknowledged the thirteenth century writer, Witelo, as the most important authority on optics. At the same time, Accolti treated Witelo critically and challenged him. The continuity question offers a means of documenting how specific experiences and experiments were gradually seen as legitimate tools to challenge first individual passages and finally the very authority of traditional sources. This is of interest, but in our view there are more pressing issues.

225 See note 83.226 CA 154ra-va (416rv). Cf. Pierre Duhem, Les origines de la statique, Paris 1905-1906. E.A. Moody and M. Clagett, The medieval science of weights, Madison: University of Wisconsin Press, 1960.227 CA 124ra (342r, c. 1508): "Elli e provato nel primo De ponderibus."228 Cf. CA 118va (325v, c. 1508-1510): "per la regola d'Archimede." On CA 153rbc (413r, n.d.) he has copied out a passage from a manuscript of De insidentibus in humido, concerning which see: Marshall Clagett, "Leonardo da Vinci and the medieval Archimedes," Physics, 1969, p. 141. Cf. also Mad. II 105rv: "piu vicino al vero che Archimede."229 E.g. CA 176rd (481v, 1505-1508) where he copies definitions from both Euclid and Theodosius concerning the sphere.230 E.g. CA 196ra (528r): e similmente e quadrabile la lunola b per Zenofonte or CA 201vb (540v, c. 1509): "aro fatto il bisogno di Zenofonte."231

6. METHOD

? CA 125ra (345r, c. 1490-1492): "Io trovo per isperienzia che..."232 A 47r: "La ssperientia farai inquesto mo[do]."233 CA 117va (1495) and D 4v (1508).234 CA 338va (924v, c. 1490): "Sperimento del moto causa del colpo."235 CA 151va (407v, c. 1500): "Per isperimentare la proporzione delli intervalli del discenso."236 D 3v: "Per fare sperientia come la virtu visiva ricie[v]a le spetie delli obbietti dallochio suo strumento esara fatto una palla di vitro di cinque ottavi dibraccio perdiamitro."237 See for instance the following articles by Enzo Maccano: "Analogies in Leonardo's studies of flow phenomenon", Studi Vinciani in onore di Nando di Toni, Brescia: Ateneo di scienze, lettere ed arti, 1986, pp.19-49."Multichannel tabulation in the notes on flow in the French manuscripts on Leonardo da Vinci", Raccolta Vinciana, Milan, vol. 22, 1987, pp.213-237."La nocion de presion en la mecanica de fluidos vinciana", Raccolta Vinciana, Milan, vol. 22. 1987, pp.239-263.In addition Enzo Maccagno has made a series of careful studies analysing the major codices of Leonardo, demonstrating their experimental basis. In chronological order these are:"Mechanics of fluids in the Madrid Codices", Scientia, Milan, 1982, 99.333-374.

71

While accepting some continuity, we have suggested that there was a breakthrough in early modern science during the period 1490-1510. Interestingly enough this period coincides with the high renaissance in art and thus confirms in an unexpected way Cassirer's claims concerning links between science and art. But whereas he focussed on philosophical context, we have shown that the practical context of machines and instruments played a central role. That the history of technology offers a key to understanding developments in the history of science is perhaps the most important issue raised by Leonardo's work. It means that Marxists such as Zilsel were right about the importance of craftsmen. It also means that Zilsel's claims need revision on two counts. First, Leonardo's life confirms that there was no invisible social barrier preventing

"Hidrostatica vinciana en el Codice Hammer", Anales de la universidade de Chile, Santiago de Chile, 5ta ser. n.8, agosto 1985, pp71-76.Leonardian fluid mechanics in the Codex Atlanticus, Iowa: University of Iowa, 1986. (Iowa Institute for Hydraulic Research, Monograph no. 100).Leonardian fluid mechanics: what remains to be investigated in the Codex Hammer, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 101). Leonardian fluid mechanics: unexplored flow studies in the Codices Forster, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 102).Leonardian fluid mechanics in the Manuscript C, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 104).Leonardian fluid mechanics in the Codex Atlanticus, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 105).Leonardian fluid mechanics in the Manuscript H, Iowa: University of Iowa, 1988. (Iowa Institute for Hydraulic Research, Monograph no. 106).Leonardian fluid mechanics: unexplored flow studies in the Codex Arundel, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 107). Leonardian fluid mechanics in the Manuscript L, Iowa: University of Iowa, 1989. (Iowa Institute for Hydraulic Research, Monograph no. 108).Cf. Matilde Macagno, Geometry in motion in the manuscripts of Leonardo da Vinci, Iowa: University of Iowa, Division of Mathematical Sciences, Department of Mathematics, 1987.238 CA 109va (303v, c. 1490):

"Fa le proposizione semplice e poi la dimostrazione configure e lettere."239 A 31r (1492) :

"Io ti richordo chettu facci le tue propositioni echettu alleghi lessopra scritte cose peresenpli e non per propositioni chesarebe tropo senplice edirai chosi."

240 A 11r: "questa propositione si pruova perrisperientia." A 57r:

"Questa propositione si pruova chiaramente... per ragione chonferma dalla issperienza."

A 45v: "Quessto si pruova perissperientia."Cf. Mad. I 87v:

"Pruova e conclusione ultima di questo tale moto, provato per lo sperimento della bilancia di sopra."

A 13v: "Propositione... Pruova." Cf. A 15r.

72

different classes from meeting prior to 1600. In theory, as an illegitimate son Leonardo had little social standing, yet he was as a brother with Jacopo Andrea da Ferrara452, a leading Vitruvian commentator, he was friends with dukes, popes and the King of France. Secondly, as we have shown, Leonardo combined both of Zilsel's ingredients of science, causal thinking plus systematic organization, in the period 1490-1510, over a century before Gilbert, Galileo and Bacon.

Indeed, as we have noted, Leonardo was engaged in writing this in the form of a treatise in his Elements of machines and was close enough to completion that Pacioli could mention it in his publication of 1509. Hence, whereas Zilsel and everyone since has

A 45v: "La ragione della propositione."A 46v: "Questa ragione si vede manifestamente."A 47r: "Questa si dimostra chiaramente."241 BN 2038 12v:

"la propositione di sopra e molto bene dimostrata e chonferma dalla sperienza"; BN 2038 15r:

"la ragione promessa di sopra chiara mente apare per isperienza";BN 2038 15v:

"Questa...propositione chiara mente apare essiconferma dala esperienza."242 Mad. I 77r, Mad. I 78r. Arundel 32v, Arundel 67v. Forster II 73r, 95r, 97v, 98r, 99r, 99v, 100r, 104r, 120v, 135r, 146r, 147v, 148r, 153v, 154r.243 CA 126va (348r, c. 1490-1492):

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

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

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

On rare occasions he states clearly that he is skeptical as regards a solution as on CA 75va (205r, 1506-1508):

"Ma di questa non veggo nello umano ingegno modo di darne scienzia."244 CA 86ra (234r, c. 1490-1492):

"La sperienza interpetre della artefiziosa natura ne dimostra questa figura essere per necessità constretta a non altre menti oprare che qui figurata sia." This idea of interpreters of nature recurs on CA 117rb (323r, c. 1490): "E da essere giudicati e non altrimenti stimati li omini, inventori e'nterpreti tra la natura e gli omini."

245 CA 274vb (739v, c. 1495): "Fa che questa figura ritorni nella sperienza, inanzi tu giudichi altro di lei."246 F 91v: "Tutti queste figure anno ausscire dalla ssperientia."247 Arundel 19r: "O fatto pruova io medesimo disegnandole."248 Leonardo Studies I, as in note 32, pp. 202-239.

73

assumed that the sixteenth century could provide only isolated examples of causal thinking, we now have firm evidence of both causal thinking and systematic method in the first decade of the century. In the midst of the high renaissance it is highly unlikely that insights of such magnitude would simply be forgotten. So we need to look afresh, and much more closely453 at the sixteenth century if we are to see in a proper framework the contributions of Tartaglia, Benedetti, Guidobaldo del Monte, Galileo and later thinkers; to establish a chronology of when, which discovery or invention was made where, in order that individual scientists and craftsmen are given due credit.

249 CA 203va (543v, c. 1495-1497): "Ma direno solamente i moti essere di 2 nature, delle quali l'uno e materiale e l'altro spirituale, perche non e compreso dal senso del vedere, overo direno d'uno essere visibile e l'altro invisibile."

250 CA 93vb (257r, c. 1513): "Dove la scienza de'pesi è ingannata dalla pratica."O trovato essi antichi essersi ingannati in esso giudizio de'pesi, e questo inganno è natoperche in gran parte della loro scienza anno usati poli corporei, e in gran parte polimatematici, cioè mentali, o vero incorporei, le quali inganni pongo qui di socto.

251 CA 200r (537r, c. 1515): "Dal meccanico punto al matematico e varietà infinità perchè esso meccanico è visibile e per conseguenza quantita continua e ogni quantità continua e divisibile in infinito."

252 E 8v, as in note 198 above.253 These passages have been translated and discussed in the author's Leonardo Studies, vol. 1, as in note 32, p. 223.254 CA 221vd (597br, c. 1490):

"Queste regole sono da usare solamente per ripruova delle figure."255 CA 274rd (738r, c. 1495):

"Io fo molte figure perchŠ tu conosca tutti i casi, che son sotto posti a una sola regola."

256 CA 149rb (403r, c. 1493-1495): "La regola del tuo libro proceder… in questa forma: prima l'aste semplice, poi sostenute di sotto, poi sospese in parte, poi tutte, poi esse aste fieno sostenitori d'altri pesi."

257 CA 86va (234v, c. 1490-1492): "Regola de'sostentaculi traversi."258 CA 119va (327v, c. 1490):

Molti mi crederanno ragionevolmente potere (mi) riprendere, allegando le mie prove esser contro all'alturità d'alquanti omini di gran reverenza apresso de'loro inesperti judizi, non considerando le mie cose esser nate sotto la semplice e mera sperienza, la quale e maestra vera.Queste regole son cagione di farti conoscere il vero del falso la qual casa fa che li omini si promettano le cose possibili e con piu moderanza, e che tu non ti veli di ignoranza che farebbe che, non avendo effetto, tu t'abbi con disperazione a darti malincolia.

259 CA 337rb (922r, c. 1493-1495):

74

It is important to recognize that Leonardo's example also raises deeper questions of method. For it suggests that questions of why and how the scientific revolution took place are not simply abstract problems of philosophy, psychology or sociology, but intimately connected with the historical evidence itself. If Leonardo had only an astrolabe, a quadrant and a few isolated gadgets, he could not have dared to make his claims about machines in universal terms. The great number of machines and instruments with which he dealt was a vital ingredient in making the universality of his claims possible and credible. Hence, aside from individual characteristics of given machines and instruments, upon which historians of technology have traditionally focussed their attention, there is a

"Effetto delle mie regole... elle tengon la briglia all'ingegneri e investigatori a non lasciare promettere a se medesimo, o ad altri, cose impossibili, e farsi mattu o giuntatore."

Cf. CA 368rc: "Queste regole fanno le operatori solleciti perch‚ scoprano e loro errori."

260 CA 154rb (417r, c. 1508-1510): "La sperienza non falla mai, ma sol fallano i vostri giudizi prommettendosi di quella effetto tale che in e nostri experimenti causati non sono."

261 CA 221vd (597br, c. 1490):"e questa regola e nata dalla 14a e ultima del 2o delli elementi d'Euclide."

262 G 42v : "colla reghola della penultima di pittagora."

263 CA 153vd (415r, 1493-1495): "Pruova e fa regola della differenzia ché da colpo dato coll'acqua sopra l'acqua, a l'acqua che cade sopra cosa dura."

264 CA 337rb (922r, c. 1493-1495): "Ancora farai regola de'diversi viaggi che fa la ballotta."

265 Mad. I 51r: "Fanne sperientia e poi regola."

266 Mad. I 148v: "Quella regola che ttu usi a trovare la natura del tirare ancora userai nella natura dello spingere."

267 CA 271vb (732ar, c. 1508): "Regola."

268 CA 130va (360r, c. 1517-1518), as in note 196 above. Cf. another version of this passage on CA 108rb (IV 300r, 1517-1518):

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

269 Mad. I 129r: "Quando una regola fia conferma da 2 varie ragione e ssperientie quella regola fia allora detta generale."

270 CA 20va (66r, c. 1493-1495):

75

cumulative dimension to their development which makes science possible and which requires study.

Needed is a history of how, what were originally seen as a great variety of individual procedures, techniques and methods, were gradually recognized as part of a single, cumulative programme of mathematical sciences, how trigonometry gradually became an integrating tool. This goes beyond Strong's arguments about procedures and requires more than internalism. Thus far we have encyclopaedias of techniques in terms of first occurrences and latest developments. We need a record of what came in between if we are to make some map of this cumulative interplay between technology and science that is unique to the West. This is not to deny the significance of specific references in the

"Per fare regola generale della differenzia ch'è da peso semplice a peso col colpo di diversi moti e forze."

271 CA 82rb (222r, c. 1493-1494): "Ricordo come tu debbi fare sperienza del reggere, a vero quanto peso po sostenere uno filo di ferro; alla quale sperienza terrai in questo modo... e fa di ciascuna cosa regola generale."

272 CA 253va (682v, c. 1493-1495): "Regola generale: a sapere una trave legate nello stremo da una corda, che sia da uno solo loco tirata e levata in più, e saper dire, in tutti li gradi del suo levarsi, di quanto peso sia al suo motore."

273 CA 268va (723v, c. 1493-1495): "Regola generale come ogni trave sospesa per li sua stremi da perpendiculari corde, darà di sè equale peso a ciascuna corda."

CA 155vb (421v, c. 1495-1497): "Fammi regola generale di dare occhetto che move ogni rota."

274 Mad. I 60r: "Se volle fare regola gienerale infra 2 mobili." Mad. I 77r: "Sperimentate ed e regola gienerale." Mad. I 170v: "Regola gienerale de'pesi sopra li stremi braci delle bilancie." Mad. I 171v: "Questa e gienerale regola."275 Mad. I 164v: "Questa si dimanda pratica, ma ricordati di mettere la teorica dinanzi."276 CA 147va: (398v, c. 1500):

"Nessuno effetto ‚ in natura sanza ragione; intendi la ragione, e non ti bisogna sperienza."

277 Mad. I 152v: "Vedi che mirabil cosa e a considerare [come] questa natura adopera in tutte sue cose e con che legie ell'ha terminato li effetti di tutte le cause i quali e impossibile mutare inalcuna minima parte."

278 Leonardo Studies I, as in note 32, pp. 56-86.279 Kenneth Keele, as in note 6, pp. 43-92.280 Leonardo Studies II, as in note 53.281 See, for instance, H 3v-4r (1494) where he declines the verb "amo" systematically. The roots of this approach can be traced back to his long lists of verbs in the Codex Trivulzianus, which appear to be connected with his efforts to learn courtly language. A thorough study of these lists has been made by Augusto Marinoni, Gli appunti grammaticali e lessicali di Leonardo da Vinci, Milan: Castello Sforzesco, 1944, p. 46 etc.

76

manuscripts and printed texts of the period to Archimedianism, Platonism, Pythagoreanism, Vitruvianism, hermeticism, mysticism and other metaphysical influences. Nor is it to question the value of philosophy, psychology and sociology as tools for the history of science. What the discovery of structure and method in Leonardo's notebooks suggests rather, is that we have not looked closely enough even at the basic facts and that a deeper understanding of the history of science will require that we begin focussing on its history.

In the 1490s, an important development in the use and study of machines and instruments provided one of the key shifts that made possible what is now remembered as the

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

PercotereDivellereDimenarsiFregareCadereDiradicareLevarsiSaltarePrecipitareAbassarsiGittareRuinarePiegarsiBattereIsbalzireDirizarsiUrtareDisbattereTorciersiSpingiereSdracciolareDivincolarsiTirareScorrereTreittareStracinareFugireCrollareRotolare o burlareSfuggireRistringiersiRivolgiereDeripareAllargarsi(Bulare)DimostrareSgambettareMovereSpanarsiCalcitrareCorrereScotersiSollevarsiSbattere

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

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

I am suggesting that his study of language and particularly the systematic grammatical treatment of conjugations was one of the early incentives in provoking Leonardo to think in terms of listing variables and their possible combinations. Yet the idea of a fully systematic approach emerges in his perspectival studies as has been shown in Leonardo Studies I and is first evident in tabular form as in a list of weights in CA 152vb (410v, c. 1490-1495), where balances are shown corresponding to each of the fractions 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2, 0/0. In short, it was his conviction of some underlying mathematical regularity that changed his habit of exhaustive list making into the beginnings of a systematic programme for understanding nature, now frequently thought of as Baconian science.282 Leon Battista Alberti, Ludi matematici, ed. S. Timpanaro, Florence: 1926. Cf. Luigi Vagnetti, "Considerazioni sui Ludi Matematici," Studi e documenti di architettura, Florence: Teorema Edizioni, n. 1, 1972, pp. 175-259.283 Arundel 66r:

"Dice batissta albertj nuna sua opera titolata ex ludis rerum mathematicarum che quando la bilancia."

284 CA 116rb (320r,c. 1495-1498):

77

scientific revolution. It was accompanied and partly inspired by a fresh examination of historical sources which printing had recently made newly accessible. In the 1990s an analogous digital revolution may have begun in the spread of electronic devices. The latest developments in computer memory make feasible for the first time a history of scientific instruments and techniques, which will include not only the first and the last but also stages in between.

Paradoxically, without the help of computers and related modern instruments (CD-ROMs, videos, Internet) a systematic history of these developments in instrumentation connected with Leonardo is not feasible; the enormity of the evidence connected with the

Piu lumi con un corpoUn lume con piu corpiPiu lumi e piu corpiPiu lumi sopra un corpo

285 CA 147va (398v, c. 1500): Le nature regulari de'contrapesi, che premiano i bottini sono 9 cioePiu larghichel bottino e piu greviPiu larghi e piu lievePiu larghi ed equaliPiu stretti e piu greviPiu stretti e piu leviPiu stretti ed equaliEquali e piu greviEquali e piu lieviEquali ed equali

Leonardo returns to this problem on F 96 [48]r (1508) under the heading: Contrapesi regulari dando di loro gravita comparazione allacqua:piu larghee di materia piu gravepiu larghae piu lievepiu larghaed equalepiu strettae piu gravepiu strettae piu dievepiu strettaed equaleequale e piu graveequale e piu lieveequale ed equale.

286 CA 74vb (III 201v, c. 1505-1506): Scontri d'acquaequali in potenzia e in quantitaScontri d'acquaequali in potenzia e none in quantitaScontri d'acquaequali in quantita e none in potenziaScontri d'acquainequali in potenziae in quantita

287 Arundel 154r: la piramide astesa a una data lunghezzala piramide acortata a una data bassezzadella piramidesi facci il cubodel cubo si facci la piramide sua

78

mechanical-instrumental revolution of the 1490s is only becoming possible with the new mechanical means of the 1990s. Perhaps our new scientific revolution will again be accompanied by a parallel revolution in access to historical knowledge, and possibly this more systematic approach to the past will lead to a new appreciation of Leonardo and his method. It is hoped that this first Complete Works of Leonardo da Vinci in electronic form may provide a useful step in that direction.

del cubo si facci la piramide a una data altezzaduna data altezza di piramide se ne facci il cuboduna piramidesi facci una tavola duna data grossezzaduna piramidesi facci la tavola di data larghezzaduna piramidesi facci la tavola di data larghezza e lunghezza

288 CA 99vb (273br, c. 1515):"De ludo geometrico, nel quale si dà il processo l'infinite varietà [di] quadrature di superfizie dilati curvi."

289 CA 246vb (667vb, c. 1505-1506): "Accortare, Allungare, Ingrossare, Assottigliare, Allargare, Restringere."

290 CA 246vb (VIII 667v, 1505-1506): Trasmutatione semplice

racortisia unadatadistantiasanza mutationedi larghezza quanto singrossaracortisia unadatalunghezzasanza mutationedi grossezza quanto salargheraallunghisiadatalunghezzasanza diminutionedi larghezza quanto sasottiglieraallunghisiadatalunghezzasanza diminutionedi lunghezza quanto sallargeraingrossisiadatagrossezzasanza mutationedi larghezza quanto s'accorteraingrossisiadatagrossezzasanza diminutionedi lunghezza quanto si ristrigneraassottiglisiadatagrossezzasanza crescerdi larghezza quanto si distenderaassottiglisiadatagrossezzasanza astensione di lunghezza quanto s'allargeraallarghisiadatalarghezzasanza mutationedi grossezza quanto racorteraallarghisiadatalarghezzasanza mutationedi lunghezza quanto ingrossirarisstringhisiadatalarghezzasanza mutationedi grossezza quanto s'alungherarisstringhisiAdatalarghezzasanza mutationedi lunghezza quanto s'ingrossera291 CA 246vb (VIII 667v, c. 1505-1506):

De Trasmutazion composite Raccortisie ristringasia dati termini: quanto s'ingrosseraRaccortisie allarghisia dati termini: quanto ingrossera o assotiglieraRistringasie assottiglisia dati termino: quanto s'allungher…Ristringasie ingrossisia dati termini: che far… la lunghezzaAllunghisie allarghisia dati termini: quanto s'assottiglieràAllunghisie restringasia dati termini: quanto s'assottiglierà oingrosseraAllarghisie assottiglisia dati termini: che far… da sua lunghezzaAllarghisie ingrossisia dati termini: quanto si raccorterà<Ingrossisi>e raccortisi a dati termini: quanto <s'>allarghera o ristrignera<Ingrossisie allar>ghisia dati termini: quanto s'accorterà<Ingrossisie allar>ghisia dati termini: che mutazione ara la lunghezza

292 When Leonardo crosses something out, it means he has developed it somewhere else, frequently in a different manuscript.

79

293 See Ladislao Reti, as in note 5, p.294 See Keele, pp. 93-130 and the author's Leonardo Studies II as in note 50.295 Ibid.296 CA 136ra:

Saette e corde equali anno equali archiSaette e archi equali anno equali cordeCorde e saette equali anno equali archiCorde e archi equali anno equali saetteArchi e saette equali anno equali cordeArchi e corde equali anno equali saette

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

sa corsa ar-----------cor sacor ar-----------ar saar cor

297 Forster I 12r-11v: 1. accortae non mutare grosseza quanto s'allarga2. accortae non mutare largheza quanto s'ingrossa3. allungae non mutare grosseza quanto si restrigne4. allungae non mutare largheza quanto s'asottiglia5. ingrossa e non mutare lungheza quanto si sstrigne6. ingrossa e non mutare largheza quanto si racorta7. assottiglia e non mutare lungheza quanto s'allarga8. assottiglia e non mutare largehza quanto s'allunga9. allargae non mutare lungheza quanto s'asottiglia10. allargae non mutare grosseza quanto si racorta11. resstrigni e non mutare lungheza quanto s'ingrossa12. restrigni e non mutare grosseza per quanto s'alunga13. acortisi e ingrossisi quanto s'allarga14. acortisi e assottiglisi quanto allarga15. acortisi e allarghisi qual fia la grossezza

80

List of Manuscripts, Abbreviations, Dates

Accademia, Sheets 1511 - 

Ambrosiana, Sheets [1508] - 

Archivio Palatino, Sheet 1507 - 

16. acortisi e restringasi quanto s'ingrossa17. allunghisi e ingrossisi quanto si restrigne18. allunghisi e assottiglisi quanto s'allarga19. allunghisi e allarghisi quanto s'assottiglia20. allunghisi e resstringasi quanto s'ingrossa21. ingrossisi e allarghisi quanto s'acorta22. ingrossisi e restringasi quanto s'alunga23. assottiglisi e alarghisi quanto s'alunga24. assottiglisi e restringasi quanto s'allunga25. ingrossisi e alunghi quanto restrigne26. ingrossisi e acortisi quanto s'allarga27. assottiglisi e alunghisi quanto restrigne28. assottiglisi e acortisi quanto s'alarga

298 CA 65va (III 183v, c. 1508): Della percussione del raro nel raroDella percussione del raro nel densoDella percussione del denso nel raroDella percussione del denso nel denso

299 CA 165va (V 449r, c. 1500-1503): De semplici e composiRetto, curvo e rettoCurvo, retto e curvoCurvo e retto, rettoRetto e curvo, curvoDe compostiCurvo e retto, retto e curvoCurvo e curvo, retto e rettoRetto e retto, curvo e curvoCurvo e curvo, curvo e curvoRetto e retto, retto e retto

For a related list see: CA 98rb (III 269r, c. 1500):E moti composti sono:retto curvo e rettocurvo retto e curvocurvo e retto rettoretto e curvo curv<o>

81

Ashmolean, Sheet  - 

Basel, Sheet [1517] - [1518]

Bonnat Bequest, Sheets [1479] - [1494]

British Library, Print Room, Sheets [1506] - [1508]

Christ Church, Sheets [1485] - [1504]

Cf. Madrid I 131v (1499-1500): "Divisione del moto naturale"SenplicenaturaleSenpliceacidentaleSenpliceneutraleSenplicecirculare d'equidiacente motoSenplicecirculare d'equidiacente poloSenplicecirculare neutraleFlessuoso rettilineoFlessuoso curvilinioAngulare rettilinioAngulare curvilinioMoticomposstiNaturale circulare Accidentale circulareNeutrale circulareNaturale flessuosorettilinio e curvilinioNaturale flessuosocurvilinio e rettilinioNaturale fressuosorettilinio e curvilinioCirculare flessuoso curvilinio d'equidiacente motoCirculare flessuoso rettilinio d'equidiacente poloCirculare flessuoso neutraleCirculare naturale e fressuoso rettilinioCirculare acidentalee fflesuosoAngolorettilinio flessuosoAngolocurvilinio e fflessuoso

300 Mad I 152r : 2 di pesoe 4 di forza e 4 di moto vole 2 di tempo2 di pesoe 2 di forza e 4 di moto vol 4 di tempo2 di pesoe 2 di forza e 2 di moto vol 2 di tempo2 di forzae 4 di peso e 4 di moto vol 8 di tempo2 di forzae 2 di peso e 4 di moto vol 4 di tempo2 di forzae 2 di peso e 2 di moto vol 2 di tempo2 di motoe 4 di forza e 4 di peso vol 2 di tempo2 di motoe 2 di forza e 4 di peso vol 4 di tempo2 di motoe 2 di forza e 2 di peso vol 2 di tempo

82

Codex Arundel 263 Arundel

[1504] - [1516]

Codex Ashburnham 361 Ash. 361

(1480) - 

Codex Atlanticus CA

[1483] - [1518]

Codex Forster I Forster I

[1487] - [1505]

Codex Forster II Forster II

[1495] - [1497]

2 di tempoe 4 di forza e 4 di peso vol 2 di moto2 di tempoe 2 di forza e 4 di peso vol 1 di moto2 di tempoe 2 di forza e 2 di peso vol 2 di moto1 di forzae 4 di peso e 4 di moto vol 16 di tempo1 di tempoe 4 di moto e 4 di peso vol 16 di forza1 di motoe 4 di peso e 4 di forza vol 1 di tempo1 di pesoe 4 di moto e 4 di forza vol 1 di tempo

301 Mad I 152r: dato la lieva e contralieva polo el peso si ricercha il motoredato la contralieva polo peso e motore la lievadato il polo peso motore e llieva ________________ la contralievadato il peso motore lieva e contralieva ___________ il polodato il motore lieva contralieva el polo ___________ il peso

Leonardo had, of course, begun examining these problems much earlier. See, for instance, his comments on A 62r (1492).302 W 19143r (K/P 101r):

Discretion de membri della vite ellor travagliamentiDato la vite dente numero lieva peso si cerche il motoreDato il dente numero lieva peso elmotore la viteDato il numero lieva peso motore e vite denteDato la lieva peso motore vite e dente numeroDato el peso motore vite dente numero lievaDato el motore vite dente numero lieva peso

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

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

303 W 19143r (K/P 101r): dato il polosubbio lieva corda e peso si ricercha il motoredato il subbio lieva corda peso e motore il polodato il lievacorda peso motore e polo subbiodato il cordapeso motore polo e subbio lievadato il pesomotore polo subbio e lieva corda

83

Codex Forster III Forster III

[1493] - 

Codex Huygens 1560 - 1580

Codex Leicester (Hammer, Gates) Leicester

[1504] - [1506]

Codex Madrid I Mad I

[1493]? - [1500]?

Codex Madrid II Mad II

[1503] - [1505]

dato il motore polo subbio lieva e corda peso304 W 19143r (K/P 101r):

Esercitatione e natura membri delle taglie ellor circunstantie - libro 4o Dato il diametro numero polo peso e corda si cercha il motoreDato il numero polo peso corda e motore________ il diamitroDato il polo peso corda motore e diamjtro_______ il numeroDato il peso corda motore diametro e numero____ il poloDato la corda motore diametro numero e polo_____il pesoDato il motore diamjtro numero polo e peso ______la corda

As Kenneth Keele has noted in his edition of the Corpus of Anatomical Drawings in the Collection of Her Majesty the Queen (1977-1980) there are related passages on CA 381r, Mad. I 4v, Forster II and B 70v.305 Mad. I 152r:

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

306 CA 212vb (565v, c. 1502-1504): Se una potenzia move un grave una quantità di spazio in tanto tempo, la medesima potenzia moverà la metà di quel grave 2 tanti quello spazio nel medesimo tempo tutto.Ovvero essa intera potenzia tutto quel grave nel mezzo tempo la metà ditale spazio; ovvero nella metà dello spazio, della intera potenzia e tempo dui tanti quel grave; overo l'intera potenzia nella metà del temp<o>, tutto quel peso nella metà di quello spazio.

307 Other aspects of this history of quantification have been studied by E. J. Dijksterhuis in De mechanisering van het wereldbeeld (Amsterdam, 1950). English translation. The Mechanisation of the World Picture, Princeton: Princeton University Press, 1986. 308 CA 212vb (565v, c. 1502-1504):

pg t spg t spg t spg t s2tg p

309 CA 355va (982v, c. 1503-1505): p g t s2 p g t sp g t s p g t2 sp g t s p g t s

84

Codex Resta [1510] - [1511]

Codex Trivulzianus Triv.

[1487] - [1490]

Codex Urbinas 1270 (Trattato della pittura) TP

[1480] - [1516]

Codex Vallardi [1483] - 

Codice sul Volo Degli Uccelli Volo

1505 - 

p g t s p g t310 W 19141v (K/P 99v):

"Io o per istrumenti dj questo 4ø libro a mannegiare 6 cose coe polo, subbio, lieva, corda, peso e motore."

311W 19141v (K/P 99v): "Travagliamento e natura de membri apartenenti allusito dellargano."

312 W 19141v (K/P 99v): Lequalj cose dette sono 6 ora esenda 5 ericerchasi la natura della sessta la quale invero essottile invesstichatione e nonsi fara maj sanza la sua teoricha coe della difinjtione della 4 potentie come peso forza moto e percussione.

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

rotti per rotti sani e rotti per rottisani e rotti per sane rottisani e rotti per rottirotti per sani e rottisani e rotti per sanisani per sani e rottisani per rottirotti per sani

On CA 165va (449r, 1500-1505) Leonardo makes a list involving moving objects:Retto il moto del motore e circulare il mobileCirculare il motore e retto il mobileCirculare il motore e l'mobileRetto il motore e'l mobile: quandoil caval tira da barca.

On CA 241vb (657v, c. 1513) there is a list involving another of Leonardo's four powers:Del percosso immobileDel percosso veloce quanto il percussoreDel percosso men veloce che'l percussoreDella percussion del circunvolubile colla parte incidenteDella percussion del circunvolubile colla parte refressa

85

Collection, Grand Ducal [1499] - [1500]

Collection, Henry, Prince of Netherlands, Sheet  - 

Collection, Stefan Zweig, Sheet [1487] - [1490]

Ecole des Beaux Arts, Sheet 1483 - 1485

Libro delle ombre, e dei lumi 149_? - 1519?

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

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

In addition to the above there are also lists such as those on CA 29rb (82r, 1499-1500), CA 202vb (542r, c. 1500), CA 105ra (290r, c. 1500), CA 79va (214av, 1505-1506) which are not entirely systematic. There are also systematic listings of multiplication tables as on CA 110va (307v, 1517-1518) and of geometrical proportion, which I have discussed in Leonardo Studies I, as in note 32, pp. 240-277.313 See note 302.314 W 19060r (K/P 153r):

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

315 W 19070v (K/P 113r, 1508-1510): "il libro della scientia delle machine va inanzi al libro degovamenti. fa legare litua libri dj novo."

7. PLANS FOR PUBLICATION

316 CA 391ra, (1082r, c. 1482): "farmi intender da vostra excellenzia, aprendo a quella di secreti mei."

317 CA 102ra (279v, c. 1503-1505): "Fa la segreta."

318 Cf. the drafts of a letter to Giuliano de'Medici on CA 247vb (671r, c. 1514-1515); CA 283ra (768r, c. 1515); CA 182vc (500r, c. 1515) and CA 92rb (252r, c. 1514-1515). These are translated in The Literary Works of Leonardo da Vinci, ed. Jean Paul Richter, London: Phaidon, 1970, vol. 2, pp. 336-339.319 CA 104rb (289r, c. 1493-1495):

"A di primo d'Agosto 1499 scrissi qui de moto e peso."320 CA 375rc (1047r, c. 1513-1514):

86

Literary works of Leonardo da Vinci 1883 - 

Louvre, Sheets 1480 - 1500

Manuscript A A

1492 - 

Manuscript B B

1488 - 1489

Manuscript C C

1490 - 

"Fa domane figure discendenti infrall'aria, di varie forme di cartone, cadenti dal nostro pontile; e poi disegna le figure e li moti, che fanno li discensi di ciascuno in varie parte del discenso."

321 M 38r: "sperientia di domane."

Cf. CA 220ra (VII 591r, 1506-1508): "Vedi domattina se l'uccello che gira venendo conto al vento."

322 Mad. II 157v: "Qui si fara ricordo di tucte quelle cose le quali fieno al proposito del cavallo de bronzo del quale al presente sono inn opera."

323 CA 214rd (571ar, c. 1507-1508): "Vedi doman tutti questi casi e li copia e po'cancella li originali e lasciali a Firenze, acci• che se si perdesse quelli che tu parti con teco che non si perda la invenzione."

324 Mad. I 6r: Legimi lettore setti diletti di me, perch‚ son rarissime volte rinata al mondo. Perchè la patientia di tale professione si trova in pochi che vogliono di novo riconpore simile cose di novo. E venite e omini e vedere i miracoli che per questi tali studi si scopre nella natura.

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

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

326 CA 119va (327v, c. 1490): Or guarda, lettore, quello che noi potremo credere ai nostri antichi i quali anno voluto difinire che cosa s[ia a]nima e vita, cose inprovabili q[uando] quelle che con isperienzia ognora si possono chiaramente conoscere e provare sono per tanti seculi ignorate e falsa mente credute.

327 CA 268va (723v, c. 1493-1495): "Io richiedo a te, lettore, quando io nomino trave, che tu intenda io volere dire perso d'equale lunghezza e d'equale peso, cioŠ corpo, che … lunghezza d'equale peso e grossezza."

328 CA 145va-vb (393v, c. 1500): "Al Diodario di Soria, loco tenente del sacro Soltano di Babilonia....Non ti dolere, o Diodario, del mio tardare a dar risposta alla tua desiderosa richiesta."

87

Manuscript D D

1508 - 

Manuscript E E

1513 - 1514

Manuscript F F

1508 - 1509

Manuscript G G

1510 - 1516

Manuscript H H

1493 - 1494

329 Mad. I 152v: "Io ricordo a te conponitore di strumenti."330 BN 2038 10r: "Il pittore debbe prima....sempre il pittore debbe chonsiderare."331 BN 2038 2r: "Adunque chonosciendo tu pittore."332 BN 2038 27v: "quando tu disegniatore vorai fare bono e utile studio."333 BN 2038 21v: "Quando vorete ovoi disegnatiori."334 BN 2038 19v:

"Se voi storiografi opoeti oaltri matematici nonnavessi visto le cose male."335 W 19075r (K/P 179r):

"ho sspechulatore dj questa nosstra machina non ti contristara perche collaltrui morte tu ne dja notitia ma rallegrati che il nostro altore abbia fermo lo intelletto a ttale ecellentja djstrumento."

336 E 18v: "Richordati opictore chettanto son varie le osscurita dellonbre."

337 E 19v:"O pictore natomista."

338 G 47r: "Ho sspechulatore del le chose nonti laldaro di conossciere le cose che ordinariamen te perse medesima la natura... chontucie. Ma rallegrati dicho nossciere il fine di quellelle chose che son disegniate dalla mente tua."

339 CA 384ra (1062r, c. 1493-1495): "Io dissi nella la conclusione come la percussione...Ora tu da te sperimenta il bastone."

340 Mad. I 149r: "Io ti dimando."

On occasion we find a more complex relationship where Leonardo uses a dialogue form with the reader as on CA 211rb-vc (VII 562r, 1495-1497):

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

In addition he has a dialogue in indirect speech in the form "the adversary says...to which it is replied," which derives from mediaeval and ultimately classical sources. There is also the phrase "Tell me" (Dimmj) which occurs on numerous occasions as listed by Carlo Pedretti in his Commentary on Richter, London: Phaidon, 1977, vol. 2, pp. 310-311.341 W 19007v (K/P 139v):

88

Manuscript I I

1497 - 1503

Manuscript K K

1504 - 1509

Manuscript L L

1497 - 1503

Manuscript M M

1500 - 

Metropolitan Museum, Sheets 1495 - 

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

342 Cf. CA 357rf (991v, 1480-1482), CA 358rb (995r, 1480-1482), and CA 372rb (1038r, c. 1497) which shows a moveable printing press.343 Mad. II 119r:

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

See Ladislao Reti, "Leonardo da Vinci and the graphic arts: the early invention of relief-etching," Burlington Magazine, London, vol. CXIII, n. 817, April 1971, pp. 188-195.Another clear reference to Leonardo's acquaintance with printing methods is found in CA 263va (710br, 1510-1515):

"E se tu li volessi gittare di piombo ovver della materia che si gittano le lettere da stampa, li pezzi sarebbono il medesima di sopra."

Cf. CA 83vb (226v, c. 1508): "Stampala a corpo premanente, a parte a parte facendo tale lista d'un pezzo."

344 Vasari, as in note 72, vol. 2, p. 157.345 Leonardo da Vinci, Traité de la peinture, trans. Roland Fréart de Chambray, Paris: Impr. de I. Langlois, 1651.

8. INFLUENCE

346 Triv. 28r: "notta." Cf. Triv. 36r: "del sono del martello con la incudine."347 Triv. 8r: "architettura notta."348 Triv. 37r: "notto de laacqua."349 Triv. 38v: "notta di pittura" and Triv. 39r: "pittura notta."350 Triv. 53v: "di bataglia nota."

89

Nantes, Fragment 1504 - 

Pinakothek, Sheet [1490] - [1492]

Turin, SheetsTurin

[1505] - 

Uffizi, Sheets 1473 - 1478

Un libro scritto ....discorso della prospettiva [1480]? - [1519]?

351 Forster I 3r: "hoc est libro intitolato de trasformatione coe d'un corpo n'un altro senza diminutione o accrescimento di materia."

352 Forster II 64r as in note 50. 353 Forster II 2r, 3v, 7v, 24v, 30v.354 Forster II 160v:

"N.B. Haec scriptum inversa et in speculo legenda est."355 Forster III 1r.356 Forster III 2v, 13v, 32r, 36v-37r, 40v, 48r-47v.357 See: B 5v, 6r, 7v, 8r, 10r, 10v, 11r, 11v, 15v, 20r, 21r, 23v, 26v, 29v, 34r, 36v, 38r, 47v, 49v, 50v, 51v, 52v, 53r, 53v, 54r, 54v, 55v, 56r, 57r, 57v, 58v, 62r, 62v, 64r, 64v, 65r, 65v, 66r, 68v, 71r, 72v, 75v, 76r, 81r, 81v, 88v, 89r, 90v. This adds a new context to the remarks of Giorgio Vasari, The Lives of the painters sculptors and architects, trans. A.B. Hinds, London: Dent, (1927) 1963, vol. 2, p. 157: "Many designs for these notions [of engineering] are scattered about and I have seen numbers of them" and p. 163

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

358 On the problem of Dürer copying from Leonardo's W12613 in the Dresden Sketchbook, fol. 130v and 133v, see: A. Weixgärtner in Mitteilungen der Gesellschaft für vervielfältigende Kunst, vol. II, 1906, pp. 25-26 and Lord Kenneth Clark's edition of The Drawings at Windsor Castle, London: Phaidon, 1968, vol.1, pp. 126-127.359 See: Marilene Putscher, "Ein Totentanz von Titian: die 17 grossen Holzschnitte zur Fabrica Vesals (1538-152)", Metanocite, 1984, 23-40. This dicusses a work entitled: Notomie di Titiano. The National Union Catalogue, vol. 65, p.378. lists an edition of c.1650 edited by Domenico Maria Bonavera noting that:

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

360 Carlo Pedretti, Commentary [on Jean Paul Richter, The Notebooks...], London: Phaidon, 1980, vol. 2, plates 45-48.361 Ibid., volume 1, pp.12-47.362 Cf. Carlo Pedretti, Documenti e memorie riguardanti Leonardo da Vinci a Bologna e in Emilia, Bologna: Editoriale Fiammenghi, 1953.

90

Weimar, Sheet Weimar

1506 - 1508

Windsor Castle, Sheets

363 Claude de Boissière Delphinois, L'art d'arithmétique contentant toute dimension tant pour l'art militaire que par la géométrie et autres calculations, revue et augmentée par Lucas Tremblay, Paris: Guillaume Cavellat, 1561, particularly pp. 53r, 72v.364 Abel Foullon, Usaige et description de l'holometre pour scavoir mesurer toutes choses qui sont soubs l'estendue de l'oeil, Paris: P. Béguin, 1555.365 Fabrizio e Gaspare Mordente, Il compasso, Antwerp: Plantin, 1584. These developments will be explored at greater length in the author's Mastery of Quantity.366 CA 157vb (425v), 358ra (1064r), 248ra (672r). Cf. notes 190-191 above. For further evidence of the proportional compass in a manuscript dated 1509 cf. Anonymous, Breve corso di matematica, Modena, Biblioteca estense, Ms. It. 211=a.W.6.22.h.367 Hans Lencker [attributed to], Perspectiva, Chicago, Newberry Library, Ms.B 128, particularly fol. 55v. Cf. the Landgraf of Hesse's instrument in: Wilhelm IV, Landgraf Hessen, (attributed to), Circini proportionalis descriptio, Biblioteca apostolica Vaticana, Reg. lat. 1149, fol. 2r:

1. Datam rectam lineam iuxta datam proportionem dividere2. Datam lineam circularem in propositas partes secare3. Datam superficiem in similem superficiem multiplicare aut minuere4. Datam corpus in simile corpus multiplicare aut minuere5. Rationem cuiuslibet diametri ad suam circumferentiam invenire6. Superficiem circularem aut quadratam in aliam transferre7. Datum globum et quinque corpora regularia in sese invicem transferre.

368 Levinus Hulsius, Dritter Tractat der mechanischen Instrumenten Levini Hulsii, Beschreibung und Unterricht dess Jobst Burgi Proportional Circkels... Frankfurt: Levini Hulsii, 1604; Philip Horcher, Libri tres in quibus primo constructio circini proportionum edocetur, Mainz: Apud Balthasarum Lippium, 1605.369 Benjamin Bramer, Bericht und Gebrauch eines Proportional Linials neben kurtzem Underricht eines Parallel Instruments, Marburg: P. Egenolff, 1617.370 Cf. Ivo Schneider, Der Proporzionalzirckel. Ein universelles Analogrecheninstrument der Vergangenheit, München: Oldenbourg Verlag, 1970 (Deutsches-Museum. Abhandlungen und Berichte. 38 Jg., 1970, Heft 2).371 Concerning Coignet see Paul Lawrence Rose, "Origins of the Proportional Compass from Mordente to Galileo", Physis, Florence, vol. 10, no. 1, 1968, pp. 53-69.372 Galileo Galilei, Le operazione del compasso geometrico e militare, Padua: Casa dell'Autore, Per Pietro Marinelli 1606. On Galileo see: Stillman Drake, "Galileo and the First Mechanical Computing Device," Scientific American, New York, vol. 234, no. 4, April 1976, pp. 104-133.

91

List of Illustrations1a-b. Leonardo, Turin. Self-Portrait, Biblioteca Reale, Turin

2a. Vinci Photograph See: www.kausal.com/leonardo/vincig.html

2b. Castello Sforzesco, Milan PhotographCf.www.discountmilano.com/tour/ Rinascimento/Castello.

373 This evidence has been considered in detail in the author's Leonardo Studies II, as in note 50. For a standard view on the history of the telescope see: Albert van Helden, The invention of the telescope, Philadelphia: American Philosophical Society, 1977. (Transactions of the American Philosophical Society , vol. 67, n.4, 1977).374 See, for instance, Giacomo Barozzi il Vignola, Le due regole della prospettiva pratica, ed. E. Danti, Rome: Zanetti, 1583, Prob. XI, Prop. XL, p.49: Perche oltre alle descrittione delle figure rettilinee, apporta gran commodita al prospettivo saperle trasmutare d'una nell'altra, ho voluto in queste tre seguenti propositioni mostrare il modo secondo la via commune non solamente di trasmutare il circolo e qual si voglia figura rettilinea in un altra, ma ancho di accrescerle e diminuirle in qual si voglia certa proportione, accio in questo libro di prospettiva habbia tutto quello, che a cosi nobil pratica si fa mestiere.It is instructive to note that Leonardo's verb for transformation (trasmutare) is also used by Danti.375 For examples see Edmond R. Kiely, Surveying instruments, their history and classroom use, New York: Bureau of Publications, Teachers College, Columbia University, 1947 (National Council of Teachers of Mathematics, Yearbook 19). For the underlying philosophy see the author's "Mesure, quantification et science," in: L'Epoque de la Renaissance, 1400-1600, Budapest: Akademiai Kiado, vol. 4, 2000, pp. 401-415.376 For an introduction to the context at Kassell see: L. von Mackensen, Die erste Sternwarte Europas mit ihren Instrumenten und Uhren, 400 Jahre Jost Bürgi in Kassel, Munchen: Callwey Verlag, 1982, particularly pp. 89-114. For two standard works see Klaus Maurice, Die Deutsche Räderuhr, München: Beck, 1976; Klaus Maurice und Otto Mayr, Die Welt als Uhr: Deutsche Uhren und Automaten 1550-1650, München: Deutscher Kunstverlag, 1980. For more specialized studies see: Hans von Bertele-Grenadenberg, "Eine mechanische Mondanomaliendarstellung auf Basis der copernicanischen Sekundären Epizyklen", Der Globusfreund, Nr.21-23, 1973, S.162-168; John H. Leupold, Die grosse astronomische Tischuhr des Johann Reinhold, Augsburg 1581 bis 1592, Luzern, 1974.377 See the author’s Leoanrdo Studies, volume 1.378

9. LIMITATIONS? See notes 2, 3 and 6.379 E.g. A 1v

380 CA 333v (909r, c. 1485-1487): "Non insegnare e sarai solo eccellente."

92

3a-b. Facsimiles of notebooks Ranging from 16o to large Folio size.

4a. Cesenatico, Fort Photograph b.Leonardo, Manuscript L, xx. Drawing

5. Vaprio d’Adda, Crossing Photograph Leonardo, W 12400 Drawing and detail.

7. F. di Giorgio Martini,454 fol. 64v Dredging device Leonardo, Manuscript E

38110. HISTORIOGRAPHY? William Whewell, History of the inductive sciences from the earliest to the present time, London: J.W. Parker, 1837.382 On this and the complex religious, political and social context of the time see R.N.D. Martin, "The genesis of a mediaeval historian: Pierre Duhem and the origin of statics," Annals of science, London, vol. 33, 1976, pp. 119-129.383 Pierre Duhem, Les origines de la statique, Paris: A. Hermann, 1905-1906, vol. 1, p. 192:

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

Duhem subsequently published Études sur Leonard de Vinci. Ceux qu'ils a lus et ceux qui l'ont lu. Paris: A. Hermann, 1906-1913, 3 vol. His most comprehensive treatment was in the Système du monde, Paris: Hermann, 1913-1959, 10 vol.384 Important in this context was Anneliese Maier, Studien zur Naturphilosophie der Spätscholastik, Rome: Edizioni di storia e litteratura, 1951-1955, 3 vol.385 George Sarton, Introduction to the history of science, Baltimore: Pub. for the Carnegie Institute by Williams and Wilkins, 1927-1931 [i.e. 1950] 3 vol. in 5. Particularly II, pt. 1-2, III, pt. 1-2.386 Lynn Thorndike, A history of magic and experimental science, New York: Macmillan, 1923-1958. 8 vol. particularly vol. 2-3.387 Francis S. Benjamin Jr.'s most important contribution was establishing the Benjamin Data Bank of medieval scientific manuscripts now directed by Professor Wesley Stevens (Winnipeg).388 Marshall Clagett, The Science of mechanics in the middle ages, Madison: University of Wisconsin Press, 1959.389 Leonard Olschki, Geschichte der neusprachlichen-wissenschaftlichen Literatur, Heidelberg: Winter, 1919-1922, 2 vol.390 Betrand Gilles, Les ingénieurs de la renaissance, Paris: Hermann, 1964.391 Edgar Zilsel, "The sociological roots of science," American journal of sociology, Chicago, vol 47-48, no. 4, 1942, pp. 544-562.392 Ibid., p. 544.393 Ibid.394 Ibid., p. 561.

93

8. Brunelleschi Crane for Cathedral of Florence Leonardo, CA

9. Frontinus Hoisting machines. Leonardo, W 12627

10. F. di Giorgio Martini, Fol. 50v Dragging Machines and principles Leonardo, Madrid

11. Leonardo, Madrid Exploded view of spinning wheel mechanism

395 Stillman Drake and I.E. Drabkin, Mechanics in sixteenth century Italy, Madison: University of Wisconsin Press, 1969.396 Hermann Cohen, Kants Begründung der Ästhetik, Berlin, 1889, pp. 228-29.397 Hermann Cohen, Aesthetik des reinen Gefühls, Berlin, 1912, pp. 26-30. 398 Cassirer, as in note 387, pp. 48-51. Cassirer did but not mention Leonardo’d study of Jordanus of Nemore, partly because he wished to emphasize Leonardo's break with the past, his challenge against tradition and authority. 399 Ibid., p. 163.400 Erwin Panofsky, Idea, Ein Beitrag zur Begriffsgeschichte der Älteren Kunsttheorie, Studien der Bibliothek Warburg, V, ed. Fritz Saxl, Leipzig: B. G. Teubner, 1924.401 Giorgio De Santillana, "The role of art in the scientific Renaissance," Critical problems in the history of science, ed. M. Clagett, Madison: University of Wisconsin Press, 1969, pp. 33-65.402 There has been an incredible explosion of literature on Brunelleschi, particularly in terms of the science-art connection. For a recent bibliography see: Corrado Bozzoni, Giovanni Carbonara, Filippo Brunelleschi, Saggio di una bibliografia, Rome: Università degli studi, Istituto di fondamenti dell'architettura, 1977.403 Ernst Cassirer, Substance and Function, trans. William Curtis Swabey, Marie Collins Swabey, Chicago: Open Court Publishing Company, 1923 (New York: Dover Publications, 1953).404 Edwin Arthur Burtt, The Metaphysical foundations of modern science, New York: Doubleday and Co., 1924. Cf. particularly p. 28 re: Cassirer.405 Edward W. Strong, Procedures and metaphysics, Berkeley: University of California Press, 1936, pp. 10-11.406 For an introduction to some leading exponents of the two factions see: George Basalla, ed., The Rise of modern science, external or internal factors, Lexington (Mass.): D.C. Heath and Co., 1968.407 Alistair Crombie, Robert Grosseteste and the origins of experimental science, 1100-1700, Oxford: Clarendon Press, 1953.408 Marie Boas, The Scientific renaissance, 1450-1630, New York: Harper and Row, 1962 (Rise of modern science, vol. II).409 Alexandre Koyré, Metaphysics and measurement. Essays in the scientific revolution, trans. R.E.W. Maddison, London: Chapman and Hall, 1968.410 Alexandre Koyré, Etudes galiléennes, Paris: A. Hermann, 1939, 3 vol.; Ibid., The astronomical revolution, trans. R.E.W. Maddison, Paris: A. Hermann, 1973.

94

12. Leonardo, Madrid Spinning wheels

13. Leonardo, Madrid Exploded view of detail and Milling device

14. Leonardo, W 12669v Stage 1. Multiple themes: seeming chaos K/P 127v

15. Leonardo, CA 37 va Stage 2. Three themes in no apparent order.

16. Leonardo, CA 144va, Stage 3. One theme no apparent order

411 Thomas S. Kuhn, The Copernican revolution: planetary astronomy in the development of western thought, Cambridge, Mass: Harvard University Press, 1957.412 E. J. Marey, La méthode graphique dans les sciences expérimentales, Paris: G. Masson, 1878.413 David Hilbert, Stefan Cohn-Vossen, Anschauliche Geometrie, Berlin: Springer, 1932. English Edition: Geometry and the Imagination, New York: Chelsea Publishing Company, 1952 etc. 414 Arthur I. Miller, Imagery in Scientific Thought. Creating 20th Century Physics, Boston: Birkhäuser, 1984; Cambridge Mass.: MIT Press, 1986. 415 Jacques Hadamard, The Psychology of Invention in the Mathematical Field, Princeton: Princeton University Press, 1945.416 Max Wertheimer, “Relativity and Gestalt: A Note on Albert Einstein and Max Wertheimer,” Journal of the History of the Behavioral Sciences, 1965, i, pp.86-87. 417 Thomas S. Kuhn, Structure of Scientific Revolutions, Chicago:University of Chicago Press, 1962. 418 Peter B. Medawar, Induction and Intuition in Scientific Thought, Philadelphia: American Philosophical Society, 1969.419 Jean Piaget, Genetic Epistemology, New York: Columbia University Press, 1970420 Gerald Holton, Thematic Origins of Scientific Thought: Kepler to Einstein, Cambridge, Mass.: Harvard University Press, 1973.421 Miller, as in note 412, p. 4.422 Martin Jay, Downcast Eyes, The Denigration of Vision in Twentieth-Century French Thought, Berkeley: University of California Press, 1993. For a review of some the main ideas see the present author’s Literature on Perspective, chapter 7, section 5. Anti-ocularism. 423 Helena Sheehan, Marxism and the Philosophy of Science: A Critical History,   Atlantic Highlands, NJ: Humanities Press International, 1985 and 1993.See: http://www.comms.dcu.ie/sheehanh/bernal.htm424 Cf. Robert M. Young, Marxism and the History of Science. See: http://human-nature.com/rmyoung/papers/pap104h.html425 W. O. Hagstrom, The Scientific Community. New York: Basic Books, 1965.426 Edward Shils,“Center and periphery.” In The Logic of Personal Knowledge: Essays Presented to Michael Polanyi on His Seventieth Birthday. London: Routledge & Kegan Paul, 1961. 427 Chicago: University of Chicago Press, Structure of Scientific Revolutions, 1962. See: http://www.emory.edu/EDUCATION/mfp/Kuhnsnap.html. Kuhn has since inspired

95

17. Leonardo, W 19148v-19147v Stage 4a. One theme. Beginnings of orderK/P 22v

18. Leonardo, W 19150r-19149r Stage 4b. One theme, columns, figures in margins K/P 118r

19. Leonardo, D 5v Stage 4c. One theme, columns, text or figures neatly

20. Leonardo, F 90 and F , Stage 4c. One theme, columns, text or figures neatly

21. Leonardo, Leicester 14r Stage 4. Text with images in margin

a considerable literature. Cf. Paul Hoyningen-Huene, Alexander J. Levine, Reconstructing Scientific Revolutions: Thomas S. Kuhn's Philosophy of Science, Chicago: University of Chicago Press, 1994; H. Floris Cohen, The Scientific Revolution: A Historiographical Inquiry, Chicago: University of Chicago Press, 1994. This literature frequently has little to do with Kuhn and even less with the details of history of science. Cf. Steve Fuller, Thomas Kuhn, A Philosophical History for our Times, Chicago: University of Chicago Press, 2000; Ziauddin Sardar, Thomas Kuhn and the Science Wars, Blue Ridge Summit, PA: Totem Books, 2000.428 Stephen Toulmin, “Conceptual revolution in science.” In R. S. Cohen and M. W. Wartofsky (eds.), Boston Studies in the Philosophy of Science, Vol. III: 331-337. New York: Humanities Press, 1968; Ibid., 1972 Human Understanding, Vol. I. Oxford: Oxford University Press. 429 Ian Hacking, The Emergence of Probability. Cambridge: Cambridge University Press, 1975. 430 Steve Shapin, Simon Shaffer, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life, Princeton: Princeton University Press, 1986; A Social History of Truth: Civility and Science in Seventeenth-Century England, Chicago: University of Chicago Press, 1994; Simon Shaffer, "Glass Works: Newton's Prisms and the Uses of Experiment." In The Uses of Experiment, D. Gooding, T. Pinch and S. Shaffer (eds.), Cambridge University Press., 1989. 432 Steve Fuller, Marc De Mey, Terry Shinn, Steve Woolgar, ed., The Cognitive Turn: Sociological and Psychological Perspectives on Science, Dordrecht: Kluwer Academic Publishers, 1989 (Sociology of the Sciences/a Yearbook).Steve Fuller, James H. Collier, Philosophy, Rhetoric and the End of Knowledge: A New Beginning for Science and Technology Studies, University of Wisconsin Press, 1993; 2nd

ed.: Lawrence Erlbaum, 2003. Francis Remedios, Legitimizing Scientific Knowledge: An Introduction to Steve Fuller's Social Epistemology, Lanham, MD: Lexington Books, 2003.433 Timothy Lenoir, The Strategy of Life: Teleology and Mechanics in Nineteenth Century German Biology, Dordrecht and Boston: D. Reidel, 1982; paperback edition by the University of Chicago Press, 1989; Ibid., Inscribing Science: Scientific Texts and the Materiality of Communication, Stanford: Stanford University Press, 1998Cf. http://www.stanford.edu/dept/HPST/TimLenoir/Mario Biagioli, Galileo, Courtier: The Practice of Science in the Culture of Absolutism, Chicago: University of Chicago Press, 1993.

96

22. Leonardo, L, Stage 4: Numbered paragraphs

23. Leonardo, F 46v and F . Stage 4: Numbered paragraphs

24. Leonardo, CA 353va Three versions of a seige machine.CA 15raCA 391va (1084)

25. Leonardo, CA 366ra (1021r). Preliminary Draft for a Clock Mechanism (cf. CA 366ra (1021r, 1495-1497).

Lorraine Daston, Katherine Park, Wonders and the Order of Nature, 1150-1750, New York: Zone Books, 1998. 434 Wiebe E. Bijker, Of Bicycles, Bakelites, and Bulbs: Toward a Theory of Socio-technical Change,Cambridge Mass; MIT Press, 1985; Wiebe E. Bijker, Thomas P. Hughes, Trevor J. Pinch (Editors), The Social Construction of Technological Systems: Ibid., New Directions in the Sociology and History of Technology, Cambridge Mass; MIT Press, 1987.435 Bruno Latour, Steve Woolgar, Laboratory Life: The Construction of Scientific Facts, Princeton: Princeton University Press, 1986. Bruno Latour, We have never been Modern, Harvard: Harvard University Press, 1993Bruno Latour, Pandora's Hope: Essays on the Reality of Science Studies, Harvard: Harvard University Press, 1999Bruno Latour, Politics of Nature: How to bring the Sciences into Democracy, Harvard: Harvard University Press, 2004.http://www.ensmp.fr/~latour/biographie.html436 Cf. Philip Brey, “Philosophy of technology meets Social Consructivism,” Techne, Spring-Summer, vol. 2, no. 3-4, 1997. See: scholar.lib.vt.edu/ejournals/SPT/v2_n3n4html/brey.html. 437 Martin Kusch, Knowledge by Agreement: The Programme of Communitarian Epistemology, Oxford: Oxford University Press, 2002. Karin Knorr Cetina, Epistemic Cultures: How the Sciences Make Knowledge, Harvard: Harvard University Press, 2004.438 David J. Hess, Science Studies, An Advanced Introduction, New York University Press, 1997; Mario Biagoli, The Science Studies Reader, London: Routledge, 1999; James C. Peterson, Sheila Jasanoff, Trevor Pinch, Gerald E. Markle (Editors), Handbook of Science and Technology Studies, Thousand Oaks, CA: Sage Publications, 2001. Society of Social Studies of Science (4S) See: http://www.lsu.edu/ssss/439 For a useful site on Science and Society See: www.ncsu.edu/chass/mds/stslinks.html. An early effort to capture some of these trends is reflected in the name: Institute for the history and philosophy of science and technology at the University of Toronto.440 Richard Rorty, “Phony Science Wars, Atlnatic online, November 1999. See: http://www.theatlantic.com/issues/99nov/9911sciencewars.htm.441 Ian Hacking, The Social Construction of What, Harvard: Harvard University Press, 2000; Ibid.: Historical Ontology, Harvard: Harvard University Press, 2000.

97

26. Leonardo, CA 27va (77v). Draft for a Clock Mechanism

27. Leonardo, CA 27va (77v) Draft copied as detail. Madrid I 111v.

28. Leonardo, Madrid I 111v. Clock mechanism.

29. Leonardo, CA 6ra (23r) Studies for a Vice.CA 375ra (1046r)

442 Cf. Helen Longino, The Fate of Knowledge, Princeton: Princeton University Press, 2001. 443 Steven Shapin, The Scientific Revolution, Chicago: University of Chicago Press, 1996.444 Thomas S. Kuhn, The Structure of Scientific Revolutions, Chicago: University of Chicago Press, 1962 (2nd ed. 1970).445 On this complex problem cf. W.F. Boeselager, The Soviet Critique of NeopositivismThe History and Structure of the Critique of Logical Positivism and Related Doctrines by Soviet Philosophers in the Years 1947-1967. Translated from the German by T.J. Blakeley, Dordrecht: Kluwer, 1975; András Gedö, Crisis Consciousness in Contemporary Philosophy. Translated by Salomea Genin; edited by Doris Grieser Marquit. Minneapolis: Marxist Educational Press, 1982. (Studies in Marxism; v. 11) [Original German edition: Philosophie der Krise. Berlin: Akademie-Verlag, 1978.] Chapter Two, part one, pp. 20-34, plus endnotes (208-215): “Neopositivism: Linguistic Philosophy and Critical Rationalism.”See: http://www.autodidactproject.org/other/gedoco2a.htmlCf. Richard Rorty, Consequences of Pragmatism, Minneapolis: University of Minnesota Press, 1982. Introduction. See: http://www.marxists.org/reference/subject/philosophy/works/us/rorty.htmCf. K. Brad Wray, “Defending Longino’s Social epistemology.”See: http://www.umkc.edu/scistud/psa98/papers/wray.pdf.Cf. Larry Laudan, Beyond Positivism And Relativism. Theory, Method, and Evidence, University of Hawaii, 1996. 446 John Hermann Randall, Jr., "The place of Leonardo da Vinci in the emergence of early modern science" in: Roots of scientific thought, ed. P. Wiener, A. Noland, New York: Basic Books, 1957, pp. 207-218.

11. CONCLUSIONS 447 Cf. Richard Mark Friedhoff, William Benzon, The Second Computer Revolution. Visualization, New York: Harry Abrams, 1989. 448 This fascination was introduced partly by a film by Charles and Ray Eames called Powers of Ten. Cf. The Invisible World. Sights Too Fast, Too Slow, Too Far, Too Small for the Naked Eye to See, London: Secker and Warburg; John Darius, Beyond Vision, Oxford: Oxford University Press, 1984. For a more recent publication see: Ivan Amato, Super Vision. A New View of Nature, New York: Harry Abrams, 2003. 449 Cf. The author’s “Visualisation and Perspective,” Leonardo e l’eta della ragione, Milan: Scientia, 1982, pp. 185-210

98

30. Leonardo, CA . Studies for a Vice

31. Leonardo, CA 16rb (54r) Presentation Drawings of a Vice.CA 16vb (56r).

32. Leonardo, W 12424 Presentation Drawing of Star of Bethlehem Plant.

33. Leonardo, CA 1 vb (4r). Presentation Drawing of a Crane.

34. Leonardo, W 12284 Presentation Map of Imola.

35. Leonardo, W 12278 Presentation Map of Tuscany with Lake Trasimeno.

36. Leonardo, W 19058v-19057r Frontal View and Cross section of a cranium. K/P 43v, 42r.

37. Leonardo, W 19057r-19058r Cutaway View of Cranium and entries of chiasmaK/P 42r, 43r

38. Leonardo, K/P 113r Optic chiasma and links to ventricles

39. Leonardo, K/P 103r Optic chiasma and links to ventricles

40. Leonardo, Weimar Optic Chiasma and Ventricles in Exploded view

41. Leonardo, W 19009v Skeletal hands from 2 viewpointsK/P 143v

42. Leonardo, W 19009v Skeletal hands from 2 viewpointsK/P 143v

450 Cf. note 10 above.451 Pietro Accolti, Lo inganno de gl'occhi, Florence: Pietro Cecconcelli 1625, p. 116 where Witelo is called "unico, & principal capo della scuola de perspettivi."452 Luca Pacioli, as in note 1, p. 33.453 It is striking that notwithstanding the monumental work of Ernst Zinner, Deutsche und niederländische astronomische Instrumente des 11-18 Jahrhunderts, München: Beck, 1972, which dealt mainly with astronomy, but touched on manuscripts on surveying, gauging and related fields, we still have no proper census even of the source materials. The same is true with respect to instruments although a great contribution has been made through the team led by Helmut Minow, Historische Vermessungsinstrumente. Ein Verzeichnis der Sammlungen in Europa, Wiesbaden: Chmielorz Gmbh, 1982. For an excellent survey of gauging literature another of the important themes see: Menso Folkerts, "Die Entwicklung und Bedeutung der Visierkunst als Beispiel der praktischen Mathematik der fruhen Neuzeit," Humanismus und Technik, Berlin, Bd. 18, Heft 1, 1974, pp. 1-41.454 Francesco di Giorgio Martini, Trattati, ed. Corrado Maltese, Milan:Il Polifilo, 1967.

99

43. Leonardo, W 19009r Hands in two different layersK/P 143r

44. Leonardo, W 19009r Hands in two different layersK/P 143r

45. Leonardo, W 19009r Hands in two different layersK/P 143r

46. Leonardo, W 19000v Outstretched arms at different layers of detail. K/P

47. Leonardo, W 19000v Outstretched arms.K/P

48. Leonardo, W Studies of front legs of horse in motion

49. Leonardo, W 12333 Battle of Anghiari and study for rear of horse.

50. Leonardo, Volo Weights and balances as basis for flight of a bird

51. Leonardo, Volo Weights and balances as basis for flight of a bird

52. Leonardo, Weights and balances as basis for flight of a bird

53. Leonardo, Flight of Bird as geometrical spirals

54. Leonardo, Visualisation of air flow of a Bird

55. Leonardo, . Sketch of a Bat

56. Leonardo, B . Plan for a Mechanical wing (cf. fig 55).

57. Leonardo, . Plan for a Mechanical Wing.

58. Leonardo, B . Plan for a Mechanical wing

59. Leonardo, B . Plan for a Mechanical wing

60. Leonardo, CA 116b. Systematic variation of light sources and bodies.

61. Leonardo, C , C . Systematic treatment light and different apertures.

62. Leonardo, C , C . Systematic treatment of light and shade.

100

63. Leonardo, A 90v , . Light and shade with two and three objects.

64. Leonardo, A , . Gradations of shade

65. Leonardo, . Gradations of shade

66. Leonardo, C 2r, C 1v. One light source and one object

67. Leonardo, C 1v, C14r. Light sources and objects

68. Leonardo, C 14r, C 19r. Light sources and objects

69. Leonardo, . Square and Octagonal Ground plans

70. Leonardo, CA37ra (104r) Octagonal Ground plans developed.B30r.

71. Leonardo, B 56v, B 22v Systematic Play with Ground Plans

72. Leonardo, , B 25r Systematic Play with Ground Plans

73. Leonardo, B 19r, BN 2038 5v Systematic Play with Ground Plans

74. Eric R. Dobbs. Reconstructions of Leonardo’s ground-plans

75. Eric R. Dobbs. Reconstructions of Leonardo’s ground-plans

76. Leonardo, Declension of Latin verb

77. Leonardo, H 3v-4r, Declension of the verb: amo, amas, amat

78. Leonardo, CA 136va Systematic variables with vowels

79. Leonardo, W 19114r. Draft: variables in transformational geometry

80. Leonardo, CA Systematic variables in transformational geometry

81. Leonardo, Madrid, . Drawing of Clock Mechanism

82. Leonardo, Madrid I 13r. Systematic play with elements of Clock Mechanism

83. Leonardo, Madrid I 13r. Systematic play with elements of Clock Mechanism

84. Leonardo, Madrid I Systematic play with elements of Clock Mechanism

85. Leonardo, Madrid I Systematic play with elements of Clock Mechanism

101

86. Leonardo, Madrid I Systematic play with elements of Clock Mechanism

87. Leonardo, Madrid I 86r. Two examples of rotary motion

88. Leonardo, Madrid I 86r. Two more complex examples of rotary motion

89. Leonardo, Madrid I 64r. Experiments with inclined planes.

90. Leonardo, CA 193 rb. Systematic play of variables with respect to motion. Text reversed for readability.

91. Leonardo, . Systematic examples of percussion.

92. Leonardo, Systematic treatment of weights.

93. Leonardo, Madrid I 137r. Systematic treatment of weights.

94. Leonardo, Madrid I 137r. Systematic treatment of weights.

95. Leonardo, Madrid I 136-137. Systematic treatment of weights.

96. Leonardo, Systematic treatment of weights.

97. Leonardo, Systematic treatment of weights.

98. Leonardo, CA 274va Systematic treatment of weights and pulleys.

99. Leonardo, CA 274va Detail

100. Leonardo, CA 274va, CA Detail and development of same.

101. Leonardo CA Systematic treatment of weights and pulleys.

102. Leonardo, CA , CA Detail and development of same.

103. Leonardo, CA , CA Detail and development of same.

104. Leonardo, CA . Systematic treatment of weights and pulleys.

105. Leonardo, CA 149rb. Systematic treatment of weights.

106. Leonardo, . List of Books (chapters) on Water.

107. Leoanardo, D xx, Illustrations from draft treatise on vision.

108. Leonardo, D xx. Detail of same. Object in front of Eye

102

109. Leonardo, Leicester, xx Planets

110. Leonardo, Leicester xx Eclipse of the Moon

111. Leonardo, Leicester xx Sun’s reflection in waves of oceans of moon

112. Leonardo, Arundel xx Sun’s reflection in convex mirror

113. Leoanardo, CA xx Sun’s reflection in waves of water.

114. Leonardo, Arundel 25r. Sun’s reflection in waves of water.

115. Leonardo, CA xx. Phases of the moon.

116. Leonardo CA 385ra (1064r) Reduction Compass

117. Leonardo, Arundel 47v Reduction CompassCA 248ra (672r)

Levinus Hulsius, Detail from Titlepage Beschreibung und Unterricht dess Jobst Bürgi Proportional-Circkels, 1604.

118. Leonardo, CA xx, Mordente Compass and proportional compass.

119. Galileo Galilei Sector, Museum History of Science, Florence.

120. Leonardo, CA 252vb (681v) Examples of terrestrial telescopes. CA 252va (680v), CA 251rb (679r).

121. Leonardo, Arundel 104r Details showing surface of the moon

122. Leonardo, CA 674 Detail showing surface of the moon.123. Galileo Galilei, 1623 Frontispeice Il Saggiatore: telescope and sector.

103

104