STUDIES ON LEONARDO DA VINCI II

Continuity and Discovery

in

Optics and Astronomy

Kim H. Veltman

in consultation with

KENNETH D. KEELE

1

To

Kenneth and Mary Keele

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CONTENTS

Preface and AcknowledgementsIntroduction

PART ONE. HISTORICAL CONTEXT1. The Optical Tradition2. The Simile of Percussion3. Basic Definitions4. "Images all in all, and all every part"

PART TWO. PHYSICS OF LIGHT AND SHADE1. Definitions and Categories2. Seven Books on Light and Shade3. The Camera Obscura

PART THREE. THE EYE AND VISION1. Anatomy and Physiology of the Eye2. The Visual Process3. Appearance and Illusion4. Optimal and Minimal Conditions of Vision

PART FOUR. ASTRONOMY AS GOAL OF OPTICS1. The Structure of Manuscripts F and D2. The Sun's Image in Water3. The Fourth Book: Of the Earth and its Waters

PART FIVE. CONCLUSIONS AND EPILOGUE1. Optical Instruments2. Mirrors3. Meteorology

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Preface

In February 1973, under the auspices of the Wellcome Institute in London, Dr. K.D. Keele, M.D., F.R.C.P., and the author set out to answer a straightforward question: whether Leonardo da Vinci's writings on linear perspective had an experimental basis. A single experiment was first repeated successfully. This led to further experiments and, in turn, to a complete search of Leonardo's perspectival notes (1975-1976). Dr. Keele, who had in the meantime become engaged with the new edition of the Corpus of Anatomical Manuscripts in the Collection of Her Majesty, the Queen, at Windsor, acted as mentor, advisor and friend. The perspectival writings revealed many connections with optics. A complete search of the optical notes thus followed (1976-1977). Two volumes were now projected: a first on linear perspective; a second on optics. Subsequently, it was decided to add a third, which would serve as a concordance.

In Wolfenbuttel, a draft for volume one was written (1977-1978). The draft for volume two proved more difficult (1978-1980). In the case of linear perspective there had been only a very short tradition, consisting essentially of four fifteenth century authors, Alberti, Filarete, Francesco di Giorgio Martini and Piero della Francesca. The optical tradition, by contrast, extended over nearly two millennia and included such important thinkers as Euclid, Ptolemy, Alhazen, Witelo and Pecham.

Leonardo had modestly described himself as a man without letters (omo sanza lettere). But then, so too had Cicero. Leonardo's position with respect to ancient and mediaeval optical traditions was therefore examined. This revealed a much greater debt than had been expected. It was found, for instance, that Leonardo's concept of percussion, underlying his physics of light, was based on specific similes that could be traced directly to Aristotle. More important: Leonardo's treatment of these ancient similes followed a distinct pattern. That which his predecessors had been content to employ as a verbal image, Leonardo insisted on exploring visually. A traditionally vague image was now put to the test and challenged by experiment. In other words the rise of visualization and the development of experiment in the Renaissance were directly linked.

This visualization was also related to a new interest in taking verbal images, words, literally. Seneca had found it sufficient to note that sound is propagated like the waves produced by a pebble in water. Leonardo, on the other hand, could not be content until he had thrown various pebbles into water and recorded the waves they produced. In the epilogue to volume one it was noted that this new approach to literalism, which some see as a direct product of fourteenth century nominalism, also had profound religious consequences. Hence the scientific revolution and the religious revolts in the sixteenth century had common roots in a nexus between visualization and literalism explored by Leonardo. The attempts by Ong1 and Foucault2 to pinpoint a basic shift in approach to the word and language thus involved a shift initiated by visual images.

Leonardo's visualization of Ancient and Mediaeval similes offers new insights into the relations between tradition and innovation, continuity and discovery; serves, in fact, as a dramatic illustration of Whitehead's claim that Western civilization is essentially a series of footnotes on Plato and Aristotle. And the Renaissance, to which Leonardo was so central, now emerges as a distinct shift in approach to traditional knowledge rather than an actual break. Albertus Magnus,

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Leonardo and Galileo did not use dramatically different sources. But their attitude to these sources marks the difference between mediaeval interest in the natural world and early modern science.

As the draft for volume two progressed it became clear that Leonardo's optical studies are not just concerned with understanding sight and light per se. They have an ulterior motive: astronomy, but then in a special sense, namely, optical phenomena relating to the heavenly bodies: Why does the full moon shine? Why do the stars twinkle? Why are there eclipses? and so on. Here again Leonardo is building on a tradition and his contributions to it, in turn, help explain why Kepler, a century later, should have devoted his classic work (1604) to the astronomical part of optics.

By the time that the draft for volume two had been completed (January 1980) the consequences of visualization loomed anew. The notebooks contain approximately 100,000 diagrams. No author before Leonardo, nor practically anyone since, had produced this number. Leonardo had, moreover, explicitly emphasized the primacy of visual images over verbal ones, of pictures over words. It seemed likely, therefore, that these diagrams would offer insights into Leonardo's method. A four months scissors and paste project ensued (February-May 1980). The diagrams were organized in sequence. Systematic themes were now found to underlie the seemingly chaotic notes. In light of this both volume one (June 1980-March 1981) and volume two (September 1981-July 1983) were written afresh.

This work would have been unthinkable but for the generous and continued support of foundations. It had begun in 1973 as a hobby while the author was preparing a doctoral thesis at the Warburg Institute with support from the Canada Council. From August 1975 through July 1977 a Research Fellowship from the Wellcome Trust enabled the author to pursue these studies on a full time basis. For the period, August 1977 through July 1979 a similar grant from the Volkswagen Foundation made it possible to continue at the Herzog August Bibliothek in Wolfenbuttel. There, work proceeded for the next two years with a Sonderforschungsstipendium from the Alexander von Humboldt Foundation and then, for another six months, still under their auspices, but with funds from the Fritz Thyssen Foundation. With the generous support of the Gerda Henkel Foundation (July 1982-July 1983) the optical sutides were completed.

I am very grateful for the moral support provided by various members of these institutions, and in particular Dr. Marie-Luise Zarnitz (Wolkswagen), Dr. Thomas Berberich (Alexander von Humboldt) and Frau Lisa Maskell and Dr. Ulbrich (Gerda Henkel).

In the preface to volume one a list was given of the many persons who contributed both directly and indirectly to this project. As the same list applies to volume two it will not be repeated here. A few individuals require special mention. Professor R.H. Weale (London) and Professor A.I. Sabra (Harvard) kindly read and criticized parts of the text. In Wolfenbuttel, I wish to thank in particular Dr. Sabine Solf, and Professor Paul Raabe. Among the many friends who provided moral support I thank especially Udo Jauernig. My great debt to Dr. Kenneth Keele and his wife Mary I cannot frame in words. Without him, this project could simply not have been carried out. I am very grateful to Ms. Shirley Fulford for typing the manuscript and to Dr. Michael Meier of the Deutscher Kunstverlag for his personal help in seeing this work through the press.

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Introduction

1. Definitions of Optics 2. Images 3. Survey of Literature 4. The Problem of Sources 5. Scope of the Present Study

Definitions of Optics

The modern term, optics, closely resembles the Greek optike. However, that is about as far as the resemblance goes. In Antiquity optics was chiefly a study of sight: today it is primarily concerned with the physics of light. In Euclid's time optics was chiefly subjective: today it ranks among the most objective and quantitative of the physical sciences. In today's terms Euclid's work would be classified as psychological optics which is but one of a spectrum of contemporary branches ranging from geometrical, physical/physiological, and meteorological to what might be termed metaphysical optics.

The present study will focus on a single chapter in the complex story of how and why the scope of optics has changed so fundamentally in the twenty-four centuries since Euclid. We shall examine in detail Leonardo da Vinci's writings on optics and suggest where he stands in relation to his seventeenth century successors Kepler, Huygens and Newton.

1. Images

In the story of how optics developed and changed, Kepler's Ad Vitellionem Paralipomena (1604) marks an obvious milestone. Here Kepler made a basic distinction between two kinds of images: one which would be seen in the air but not measured (imago); the other which could be focussed on walls and other surfaces and be measured (pictura). The imago of sight was subjective: the pictura of light was objective. The study of vision now posed itself as an obstacle to objective science. Kepler's successors followed his advice to concentrate on the pictura and within two generations Huygens and Newton had formulated laws of optics strictly in terms of the science of light. Kepler's distinction between imago and pictura thus played a critical role in shifting the definition of optics from a study of vision to a science of light. What then were the factors that made Kepler's distinctions possible?

In Antiquity the distinction would simply have been unthinkable. Plato, for instance, in his Timaeus speaks about the images of vision and the images of dreams indiscriminately.2 For him the eidola of nature and the eidola of the mind are interchangeable. He makes no distinction between a physical eye and the mind's eye. Lucretious' attitude is the same.3 The ambivalent treatment of images, whereby mental and physical images are interchangeable, continues in the late Antique writings of Ptolemy and throughout the Arabic tradition. Alhazen appears to come close to making a distinction4 but does not. Nor do his successors in the Latin West: Witelo, John Pecham or Piagio Pelacani da Parma. For these thinkers optics is primarily an intellectual pursuit, is given a certain flair by geometrical diagrams, but remains ultimately a philosophical topic with metaphysical overtones. Experiments one might perform provide interesting themes for discussion. But there is little concern with actually carrying them out.

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Leonardo works fully within this tradition. There is evidence that he consulted Witelo's optical compendium.5 We know that he copied out the beginning of Pecham's Perspectiva communis.6 Yet, where his predecessors had been content to discuss possibilities, Leonardo appears to have made experiments. In volume one, we showed how Leonardo's experimental approach led him to find perspectival laws the existence of which his predecessors and even his elder contemporaries had denied. In the realm of optics he did not actually discover fundamental laws.

His experimental approach was, nonetheless, of enormous significance: it brought to what had traditionally been a domain of mental speculation, a new criterion of physical demonstration. Thereby he set the stage for precisely that distinction between physical and mental image made famous by Kepler. In short, we would claim that Leonardo's work on optics and astronomy provides a basic context for Kepler's treatise on optical astronomy (astronomia pars optica). So dramatic a claim, one might expect, would long since have been thoroughly examined in the secondary literature. In fact it has not.

2. Survey of Secondary Literature

As noted earlier, (vol. 1, p. ) Leonardo's scientific work was certainly known in the sixteenth and seventeenth centuries. In the eighteenth century Robert Smith mentioned Leonardo's contributions to binocular vision in his Complete System of Opticks (1737).7 Venturi (1797) basing his comments on a direct study of the notebooks, claimed that it had been Leonardo and not Maestlin or Kepler who had discovered that the moon's light was caused by reflection from the earth. According to Venturi, Leonardo had also discovered that the twinkling of the stars was actually an illusion originating in the human eye. Venturi went on to note that Leonardo had described the camera obscura and had compared its function with the eye long before Porta or Kepler. Venturi also held that Leonardo had explored the possibility of a telescope.

In the Philosophical Transactions (1838) Wheatstone published a significant essay "On some remarkable and hitherto unobserved Phenomena of Binocular Vision." Wheatstone had observed that when an "object is placed near the eyes that to view it the optic axes must converge: under these conditions," a different projection is seen by each eye. This led him to conclude that it was impossible for artists to give a faithful representation of any near solid object.... When the painting and the object are seen with both eyes, in the case of painting, two similar objects are projected on the retina, in the case of solid object the pictures are dissimilar. Wheatstone searched for precedents and found only the comments in the Treatise of Painting (cf. Smith, 1737 above) and he therefore praised Leonardo as a pioneer in the study of binocular vision.

Even so Leonardo's fame was still far from being universal. When Emil Wilde set out to discuss all writers on optics from the thirteenth to the seventeenth centuries in his standard Geschichte der Optic (1838) he included A. Thylesius and B. Telesius but omitted Leonardo altogether. Libri in his Histoire des sciences mathematiques en Italie (1840) included some fifty pages on Leonardo's science in general, but only a single page to optics. Here Libri based his remarks primarily on Venturi, but also added two discoveries to Leonardo's name: capillary action and diffraction.

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Gilberto Govi (1872) had more to say on this theme. He noted Leonardo's work on the camera obscura and his treatment of the eye as an instrument. According to Govi, Leonardo had anticipated Maurolycus with respect to the shape of images on passing through an aperture, and also Bouguer with respect to comparison of different intensities of light. Govi acknowledged the significance of Leonardo's work with concave and convex mirrors. He denied, however, that Leonardo had invented the telescope. In Leonardo's notes on coloured shadows, Govi found an anticipation of von Guericks, Buffon and Scherffer.

Brun (1879) touched on Leonardo's optics, crediting him with invention of the camera obscura and anatomical dissection of the eye. Guardasoni (1880) mentioned Leonardo's interest in stereoscopic vision without citing any specific texts. At the same time he queried whether Leonardo had ever been read by his successors. Raab's Leonardo da Vinci als Naturforscher (1880) appeared that same year. According to Raab the Ancients had explained vision in terms of an extromission process with rays emanating from the eyes and Leonardo had corrected this erroneous view by demonstrating the reverse. To do this, claimed Raab, Leonardo had used a camera obscura. Moreover, to convince his students, he had used a model eye. In Raab's opinion Leonardo had anticipated Kepler by a century.

In 1881 Ravaisson-Mollien published the Manuscript A of the Institut de France. Within a decade he had edited Leonardo's most important notebooks on light and vision (Manuscripts C, D and F) complete with transcription and French translation. Meanwhile, Ludwig (1882) had published the Treatise of Painting with a German translation and Richter his Literary Works of Leonardo da Vinci (1883). Under such headings as "Six Books on Light and Shade" and "Theory of Colours" Richter made a first attempt at organizing Leonardo's optical notes.

Leonardo had claimed that stars seen through a small aperture appear smaller than those seen with a naked eye. He had also held that two flames positioned close to one another, would appear joined together if seen from afar with a naked eye, whereas they would be distinguished as two separate flames if viewed through a small aperture. These claims convinced Theodor v. Frimmel (1892), himself myopic, that Leonardo must also have been myopic. Venturi (1797) and Raab (1880) had credited Leonardo with the invention of the camera obscura. Nonetheless, Muntz (1898) devoted an entire article to the theme and this served, in turn, as starting point for Elsasser (1900) who turned to how the eye functioned. Elsasser claimed that Leonardo had not been aware of the inverted position of retinal images and had instead believed that images in the eye were inverted twice. Nor, according to Elsasser, had Leonardo been aware that the eye functions as a camera obscura. Indeed, he claimed that Leonardo's anatomical ideas of the eye were effectively identical with those of Alhazen. Elsasser was thus led to conclude, that even if Leonardo had described two experiments later repeated by Scheiner, he had not brought about any dramatic advance in the theory of vision.

In this period the other major notebooks containing optical work were being published. Piumati edited and transcribed the Codex Atlanticus (1894-1904) and in the meantime, with Sabachnikoff, produced an edition with accompanying French translation of the Windsor Anatomical Folies A and B (1898-1901). At the same time Rouvèyre was also publishing facsimiles of the Windsor collection (1901). A decade later, Vangenstein, Fonahn and Hopstock began producing a more thorough edition (1911-1916) complete with English and German

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translation. The optical material in the Codex Arundel (1923-1930) and the Forster Codices (1930-1936) was only published later.

Meanwhile, Colombo (1903) a teacher of optics at Bologna, had studied the existing literature and set out to determine precisely what contribution Leonardo had made in this field. Colombo's point-form conclusions provided a succinct account of the state of scholarship at the time:

Leonardo da Vinci not only knew the principle of the camera obscura but also described the optical instrument that goes under this name (excluding the lens). He recognized that the same principle of the camera obscura should apply to the eye. As a logical consequence he admitted the crossing, within the eye, of luminous rays coming from the objects seen and thus the re-inversion of their images. He observed the movement of the rainbow under the stimulus of light and understood the reason through experiment. He observed that the retina was dazzled through the excess of light and understood the disastrous consequences for the eye of staring at the sun. He noted some phenomena that are characterized by asthenopia, (and) remarked on the existence of the near point of distinct vision and the impossibility of simultaneously seeing nearby and far [things] distinctly. He described retinal adaptation, the persistence of retinal images, [and] consecutive images. He noted that the field of indirect monocular vision exceeded the right angle along the horizontal meridian. He described the phenomenon of the fosteni of pressure. Meanwhile, Baratta (1903) in an appendix to a book on Leonardo's science, published an important discovery: a passage on CA 203ra had been copied directly the Pecham's Perspective communis. This established that Leonardo had first hand knowledge of the most popular mediaeval textbook on optics.

Edmondo Solmi's Nuovi studi sulla filosofia naturale di Leonardo da Vinci (1905) was the first major study of Leonardo's optical writings. Solmi took it for granted that Leonardo had diligently studied earlier authors and that Leonardo was in the mainstream of Western science and philosophy. Solmi divided his study into three parts. The first section dealt with experimental method and had a brief chapter on the role of observation. The second section on astronomy discussed why the apparent rise of the sun, stars and planets was different at the horizon than when seen directly above; why objects in the mist appear larger; what shape is assumed by a light source after passing through an aperture of a different shape; that the twinkling of the stars was an optical illusion, which idea Leonardo probably adopted from Cecco d'Ascoli or via Ristoro d'Arezzo; questions of moonlight and spots on the moon; Leonardo's conviction that the stars have no independent light and his awareness that if the light from individual stars could somehow be combined, this would produce far more light than the moon.

Solmi devoted the third part of his study specifically to Leonardo's theory of vision, beginning with his idea that not only light but also sound, odour, magnetism and even thought are all propagated by wave motion. Solmi went on to show that although Leonardo considered both extromission and intromission theories of vision he clearly favoured the latter. Basic to Leonardo's optics, claimed Solmi, was the concept that every point in the air carries with it infinite images of the things opposite. To test this concept, Solmi claimed, Leonardo had made experiments with the play of light on objects of different colours, with camera obscuras and various mirrors placed in different positions.

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Solmi then examined Leonardo's concepts of light: his notion that it is a spiritual force (virtu spirituale) and not material; that light is propagated rectilinearly and spherically; that this spherical propagation is well illustrated through the analogy of a stone thrown into water that produces waves, an analogy that applies equally to sound. Finally he examined Leonardo's studies of mirrors to show that the angle of incidence equals the angle of refraction.

In a second chapter Solmi considered Leonardo's writings on the structure and function of the eye: how he compared the eye with a camera obscura; how he rejected the traditional notion that vision occurs at a point, appealing to both experience and experiments to confirm this and to arrive at the idea that images are formed on the surface of the retina; how and why he conceived of images as "all in all and all in every part"; how he concluded that the visual image was twice inverted within the eye and why he believed that the visual power was located at the extremity of the visual nerves.

In his third chapter "On Sensations and Visual Perception" Solmi considered Leonardo's work on after-images, the effects of pushing the eyeball with a finger; his experiments to determine how pupil size affects the perception of light and shade, optical illusions, binocular vision and its consequences for artists wishing to draw in relief. In his final chapter Solmi examined Leonardo's concepts of colour, his notion that the amount of colour varies with light, his definition of the simple colours, his concept that a mixture of colours could produce infinite variety, his ideas on the rainbow, on the effects of background on both the apparent intensity and size of coloured objects in the foreground and how the apparent size of some lights increase with distance.

Solmi's book was a major contribution towards an understanding of Leonardo's theories of vision and light. It was the first survey to indicate the scope of Leonardo's optical researches. At the same time it made clear that these studies were not to be understood in isolation, that Leonardo's questions and concepts were both a development from earlier authors such as Aristotle and Witelo and a preview of the later work of persons such as Huygens and Helmholtz. To understand Leonardo one thus had to study the entire optical tradition.

Perrod (1907) wrote an article on Leonardo's studies of dioptrics of the eye, which was essentially a collection of quotations from the notebooks to explain how the eye functioned, how it could be likened to a camera obscura, what were the limits of visual acuity, how retinal images persisted and about spectacles. Perrod pointed out that Leonardo had concentrated on healthy vision and had avoided almost entirely the pathology of vision. According to Perrod, he had done little or nothing by way of actual dissection of the eye, and the resulting gaps in his anatomical knowledge had led to inevitable errors. Nonetheless, Perrod acknowledged the profundity of Leonardo's experiments and claimed that his methods were essentially those codified by Helmholtz, Claude Bernard and others.

In 1910, Otto Werner under the supervision of Wiedemann, published a doctoral dissertation at Erlangen, which remains the best overall survey of Leonardo's theories of vision and light to date. The dissertation set out to understand Leonardo's physics and opened with a brief outline of his life followed by a list of sources that Leonardo may have used. Werner provided an outline of earlier theories of vision and turned then to consider Leonardo's concepts; his definitions of the visual species, how these were propagated, how they intersected without interference, his

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conviction that light did not move instantaneously, his concept of the visual pyramid, its strength and its function.

Werner proceeded to consider Leonardo's ideas on the eye and the visual process. Using a series of diagrams, mainly from the Manuscript D and the Codex Atlanticus, Werner traced in details how Leonardo kept modifying his ideas of the double inversion of images within the eye. This reliance on Leonardo's diagrams marked an important step. For Werner was the first scholar to realize intuitively that the diagrams in Leonardo's notes were not simply illustrations of claims made in the text, but actually functioned as independent visual statements. Werner believed that Leonardo's theory of vision owed much to the Arabic tradition, but had also been modified as a result of his own experiments with the camera obscura. Werner went on to consider Leonardo's work on other optical problems: binocular and stereoscopic vision, optical illusions, his experiments to demonstrate the rectilinear propagation of light. These, claimed Werner, derived mainly from Alkindi and Alhazen. (Werner's supervisor Wiedemann was an expert an Arabic optics and science.) In the case of the camera obscura and images projected through irregular apertures Werner claimed to find precedents in Kamal ad Din al-Farissi and Levi ben Gerson.

Werner next examined Leonardo's statements on reflection in mirrors of various shapes, and proceeded to discuss his notes on dioptrics: his refraction experiments using both glass spheres filled with water and half spheres of crystal. Werner concluded the section with Leonardo's comments on spectacles and rainbows. In the remainder of his thesis, Werner considered briefly Leonardo's acoustics, - the laws of which, as he pointed out, Leonardo himself had compared to optics - and finally heat and magnetism. In spirit Werner's thesis furthered Solmi's approach. Both authors assumed that Leonardo's problems were only to be understood in light of the Western-Latin and Arabic - tradition. In retrospect, however, Werner's search for sources appears somewhat over zealous.

A period of lesser contributions ensued. That same year there appeared also a book by Seidlitz on Leonardo (1910), with a chapter on light and shade, but this scarcely got beyond basic definitions. Feldhaus (1914) produced a short paper on Leonardo's diving and other spectacles. An article by Angelucci (1919) entitled "La maniera in pittura e le legge ottiche di luci e colori" offered less precise analysis than its title promised. Vetri (1926) returned to the theme of binocular vision and stereoscopy. Marcolongo (1929) provided a likely reconstruction of Leonardo's instrument to deal with Alhazen's problem. Moller (1930) published the Weimar sheet (our fig. ) and that same year, Nicodemi 91930) wrote briefly on Leonardo's light. McMurrich (1939) entered upon the problem of the ocular nervous system. In the Corriere della Sera appeared an article which cited Leonardo's perspectival drawing of an armillary sphere (cf. vol. 1, fig. 272) as proof that he had invented the telescope.

The great exhibition in Milan in 1939 led to the construction of 275 models of Leonardo's inventions, plus a two-volume work with chapters on individual aspects of his studies. Among these was an essay on optics by Argientieri, which remains the most enthusiastic assessment of Leonardo's contributions to this field. Argientieri saw in Leonardo's description of the wave motion of light not only a precedent but even a possible source for Huygen's theories. The Ancients, believed Argientieri, had maintained that light was propagated instantaneously and, according to him, Leonardo first recognized the finite speed of light a century and a half before Roemer (1676).

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Leonardo, claimed Argientieri, had given a clear explanation of the intromission theory of vision. Moreover, Argientieri suggested, he may well have been aware of the Doppler-Fizeau principle. Leonardo, in his discussion of the propagation of rays cites Aristotle to claim that Nature always acts in the shortest way possible. Argientieri forgot Aristotle and saw in Leonardo's claim an early statement of Fermat's law.

Argientieri did acknowledge that Leonardo had been aware of earlier authors, that he was, for instance interested in Alhazen's problem and stimulated by Witelo's refraction studies. But Argientieri's prime concern was to prove Leonardo's modernity: how the fifteenth century genius had discovered properties of colours by mixing light three centuries before Fernel; the laws of photometry long before Bouguer and Rumford; how he experimented with diffraction before Grimaldi; how he invented a projector and even the telescope.

Argientieri pointed out that Leonardo had made model eyes and claimed that his notion about images being twice inverted in the eye had nothing to do with psychological prejudices. It was, according to Argientieri, a result of trying to account for the role of the crystalline lens. Leonardo had also experimented with pupil size, had explored the limits of the visual field and, if he made a few wrong conclusions, Argientieri noted, Leonardo nonetheless anticipated later experiments by Czermak and Scheiner concerning visual perception and anticipated Kepler and Wheatstone in their study of stereoscopy. If Argientieri's study was too generous in its assessment of Leonardo's originality, it nevertheless marked a valuable contribution. Its 104 illustrations, most of them photographs taken directly from the notebooks, provided an important visual survey of Leonardo's optical researches.

The most controversial element in Argientieri's article was his reconstruction of what he believed to have been Leonardo's telescope described on Ms.F 27v at the Institut de France. In July of the same year (1939), Vasco Ronchi, then director of the National Institute of Optics, wrote a three page typescript essay in which he rejected outright the possibility that Leonardo had invented such a telescope. By way of a reply, Argientieri immediately published a 37 page paper explaining a full detail his assumptions and method in arriving at the reconstruction.

Meanwhile the Milan exhibition was attracting interest abroad. In Britain the ophthalmologist Walter Gasson (1939) was prompted to write an article on Leonardo's optical contributions in which he mentioned themes such as binocular vision, rectilinear propagation of light, the twinkling of stars, optical illusions and the laws of perspective. Gasson claimed that Leonardo appeared not to have been acquainted with Euclid's Optics. Gasson believed that Leonardo had been aware of the possibility of telescopes, but found no clear evidence of his ever having constructed one.

The following year the models from the Milan exhibition travelled to the United States where they inspired further interest. Ingalls (1940), in an enthusiastic Scientific American article reported how he believed he had been the first to guess that Leonardo was acquainted with telescopes until, on consulting Nicodemi, he learned of Argientieri's work. Maestro (1941) mentioned another optical invention of Leonardo: the diploscope. The following year, Boring (1942) brought Leonardo's work on binocular stereoscopic vision to the attention of psychologists, describing it as "Leonardo's paradox."

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At the instigation of Elmer Belt, Nino Ferrero (1951) produced "an original new translation" of the Manuscript D, but with the exception of fol. 9v, he omitted the diagrams. Ferrero offered no analysis, only a note to indicate that Leonardo had discussed optics elsewhere in his writings, and a bibliographical survey listing the studies of Angelucci, Argientieri, Elsasser, McMurrich, Perrod and Venturi. O'Malley and Saunder's (1952) publication served to render more accessible some of the pertinent anatomical drawings but did not significantly advance understanding of Leonardo's optics. Pirenne (1952) in an important article established that linear perspective corresponded with optical reality.

Hofstetter and Graham (1953) claimed that Leonardo had invented contact lenses. Sergescu (1953) returned to Leonardo's instrument for dealing with Alhazen's problem. Abetti (1953), made general claims that Leonardo's optical work was based primarily on Witelo who had, in turn, drawn upon "the rudimentary theories of the Arabic scientist Alhazen." According to Abetti, Leonardo had made accurate studies of shadows, had explored the camera obscura, was interested in the principles underlying the telescope but there was no evidence that he ever constructed one. Senaldi (1953) mentioned Leonardo's anatomical studies of the eye.

Zubov (1954) in an important article in Russian, identified several passages with striking parallels between Leonardo and his mediaeval predecessors Alhazen and Witelo. At the same time, Zubov noted differences in Leonardo's approach: where Witelo had been content with a general statement, Leonardo had turned to experiments. Where Witelo had been satisfied with certain isolated examples, Leonardo had insisted on every possible combination, claimed Zubov. Indeed, he concluded, Leonardo's studies could be seen as a concrete version of Witelo's abstract themes.

Brizio (1954) showed that there were clear connections between some optical notes in the Codex Atlanticus, the Folio B at Windsor and the Manuscripts A and C at the Institut de France. Ronchi (1954) believed that Leonardo's philosophical doubts had ultimately led him to mistrust the sense of sight. Ronchi claimed that Leonardo had learned about the camera obscura and various other optical phenomena from Alhazen and Witelo, but that he had never actually experimented with the camera obscura. Ronchi was, moreover, of the opinion that Leonardo had probably been slightly myopic. As evidence he cited a passage in the Treatise of Painting in which Leonardo claimed that the white hat of a woman dressed in black would appear larger than it actually was. Ronchi turned to what he considered to be Leonardo's entirely confused notions of how bodies produce rays that are propagated spherically. Ronchi was, moreover, convinced that there was no serious evidence that Leonardo had ever constructed a telescope.

Keele (1955) assessed Leonardo's work on vision from a medical standpoint, acknowledging his debt to earlier authors but adding that the experiment described in Manuscript K marked "one of the earliest examples of the technique of imbedding tissue for section-cutting." Leonardo's belief in the double inversion of images in the eye was, claimed Keele, based on observation and experiments. According to Keele, Leonardo had also experimented with the camera obscura, had made models of the eye, and had examined the effect of adjustments in the pupil's size. Keele went on to suggest that Leonardo had worn glasses for presbyopia and that this would explain why Leonardo discussed presbyopia while making no mention of myopia. Keele

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noted that Leonardo had also studied the optic nerves, had arrived at a masterful diagram of the optic chiasma, explored eye movements and stereoscopic vision.

Garin (1956), found in Ronchi's assessment of Leonardo's optics a "most serious and balanced contribution," but felt, nonetheless, that Ronchi had overemphasized Leonardo's intromission theories. Garin pointed out that the notebooks also evidenced extromission theories. Garin saw certain similarities between Leonardo and Bacon; was of the opinion that Leonardo must have known Alhazen and Witelo either second-hand or through a compendium; disagreed with Abetti's judgment that Alhazen was rudimentary. In conclusion, Garin called for a more detailed comparison between Leonardo's notes and to his optical treatises.

Giovannini (1957) offered another summary of Leonardo's optical work, alluding to his anatomical researches, his experiments with the camera obscura, his study of binocular vision, the principles of photometry and the laws of contrast. In Giovannini's opinion Leonardo was far ahead of his time. Keele (1961) returned to develop themes raised in his earlier article. He examined the development of Leonardo's concepts of the optic chiasma and the cranial nerves, analysed the Weiman sheet and emphasized the significance of his injecting the cerebral ventricals with wax. Thereby, Leonardo had revealed a cerebral central mechanism of vision. That same year, Garin (1961) broached afresh the problem of Leonardo's sources, challenging the approach of Duhem and Solmi who had assumed Leonardo could read. Garin believed that Leonardo was illiterate (omo sanza lettere) and that he had acquired his knowledge either second-hand through compendia or even third-hand through conversations with learned friends and patrons. By modern standards, claimed Garin, Leonardo's scientific achievements had been "very modest." In support of this opinion he cited Ronchi's negative conclusions.

A study by Pedretti (1962) of a manuscript, (H 227 Inf.) in the Ambrosiana provided important evidence concerning lost chapters from Leonardo's book on light and shade. The manuscript, compiled by Father Antonio Gallo (1639-1640) appears to have marked a first anthology of Leonardo's notes. Pedretti's study ended with a useful concordance between the manuscript and the notebooks. A general article on Nature and vision in Leonardo's work by Bottari (1963) may simply be mentioned in passing.

Brizio (1963) basically accepted Ronchi's negative assessment of Leonardo's optical work. She questioned, however, whether Leonardo was best understood through comparison with modern concepts. Her alternative was to compare him with earlier authors such as Witelo on the question of reflection in convex and concave mirrors. The close parallels she found between Leonardo and Witelo led Brizio to challenge Ronchi and Garin's claim that Leonardo had been unaware of the mediaeval optical tradition.

A doctoral dissertation by Strong (1967), supervised by Pedretti, marked the most detailed study to date of Leonardo's optical researches. Strong centred his work around the Manuscript D, which he translated into English before reassembling its contents and assessing its significance for the chronology of the other manuscripts. Strong compared statements in the Manuscript D with analogous comments in other notebooks. In a chapter on Leonardo's method Strong emphasized how his ideas had evolved with time. After 1505, claimed Strong, Leonardo became increasingly interested in determining the causes of things and establishing the mathematical certitude of

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phenomena, a trend which Strong attributed to Pacioli's influence. Strong then turned to assess the position of the Manuscript D in the late mediaeval optical tradition. He characterized the change from thirteenth to fourteenth century optics as a shift from metaphysical to physical interests. Strong noted some problems of definitions of terms and how Leonardo had alternatively considered both intromission and extromission theories of vision. In reply to Garin's claim that Leonardo had acquired his knowledge primarily from discussions and vernacular texts, Strong insisted that the evidence is overwhelming that Leonardo sought out and confronted directly classical and Mediaeval treatises pertinent to his interests in the libraries of Florence, Milan, Pavia, Urbino and Rome.

Strong proceeded to mention some of the texts most likely to have influenced Leonardo: the anonymous Della prospettiva, Euclid's Optics, Giorgio Valla's De fugiendis et expetendis rebus, the anonymous De visione stellarum, Alhazen and Witelo's optical writings, Roger Bacon's Opus Maius, which offered direct comparisons with Leonardo's ideas, and John Pecham's Perspectiva communis. Study of this mediaeval tradition revealed the Leonardo had not been the inventor of the camera obscura nor the first to discuss eye movements. In a final chapter, Strong considered the impact of Leonardo's optics in his art; how his optical studies of the mobile eye could account for Leonardo's interests in anamorphosis, and that these studies on the mobile eye had, in any case, aimed to eliminate Alberti's "recalcitrant space" and his principle of the immobile eye. Strong went on to claim that Leonardo's modelling techniques were an attempt "to exploit the mobile eye and its contact with the form to intensify the visual experience." Strong saw the forma serpentina of the Leda as an illustration of this.

Lindberg (1975), in an otherwise important book on the history of visual theories came to a very negative assessment of the "man without letters," which began with a brief summary of Leonardo's opinions on radiant pyramids and intromission/extromission theories. The standard interpretation claimed that Leonardo's approach to vision was mechanistic. Lindberg rejected this interpretation, claiming that it resulted from a misunderstanding of Leonardo's analogy between the spherical diffusion of light and the circles produced by stones thrown into water.

In Lindberg's estimation Leonardo's anatomical work was "exceedingly primitive" but had, nonetheless, led to two contributions: study of the variable diameter of the pupil and the analogy between eye and camera obscura. Lindberg believed that if Leonardo owed a "very substantial debt to the past," his writings confirmed that he had no understanding at all of the central issue of traditional optics - the problem of a multiplicity of rays from every point in the visual field influencing all parts of the eye. According to this view Leonardo is thoroughly confused concerning the entire mediaeval tradition and his writings reveal that "the problem of sight was not to be solved through a fresh start by an ingenious empiricist working in an intellectual vacuum."

Borsellino and Maltese (1976) analysed Leonardo's experiments in the Manuscript D using a needle viewed through a pin-hole in a piece of paper and concluded that Leonardo had thereby discovered "some curious deceptions of the senses," but had not arrived at the modern explanation.

In his important Richter Commentary, Pedretti (1977) drew attention to new manuscripts, hitherto untranslated passages on light, shade and vision as well as providing a host of suggestions concerning the chronology of individual notes. Kemp (1977), cf. 1981) felt that scholars had not

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appreciated sufficiently the traditional roots of Leonardo's ideas and set out to show that Leonardo's optical studies "centre on essentially mediaeval themes." As examples he cited the concept of the visual pyramid and the idea that vision is "all in all and all in every part." Ackerman (1978) also analysed this idea in an article which surveyed Leonardo's optics in relation to the mediaeval tradition and mentioned some consequences of his optics on painting. While introducing little that had not been discussed previously, Ackerman provided a useful survey for non-specialists. An article by Ehrich (1979) provided a useful summary of literature concerning Leonardo's supposed invention of contact lenses and provided an experimental reconstruction of the same.

The new edition of the Windsor Corpus by Keele and Pedretti (1979-1981) has provided a valuable new transcription, a first complete English translation of the anatomical notes, and has radically altered our picture of the chronology of these notes. Keele (1983), who describes knowing "how to see" as Leonardo's gateway to science, provides a lucid survey of his studies on optics and vision. Keele emphasizes links between vision and perspective in Leonardo's approach, outlines his definitions of light and shade; his concepts of the pyramidal nature of light and connections with the four powers of Nature, especially percussion; his experiments with the camera obscura, how light spreads in circular waves and how all the powers involve movement. In the second part of the chapter Keele examines Leonardo's writings on the physiology of the eye, the optic chiasma, the ventricles, their location, their link with the soul; Leonardo's theories of extramission and intramission, the experiments he carried out concerning these theories, his concepts of the pupil, cornea, optic nerve and imprensiva.

The above survey, in which we have deliberately limited ourselves to the most important articles and books on Leonardo's optics, reveals that a great deal has been written on the subject. It also confirms that much remains to be done. Most of the contributions have been piecemeal, concentrating on whether Leonardo invented or mentioned this or that. How often he mentioned a problem: whether it occurred merely in passing or whether it was a recurrent theme in his notebooks has usually been overlooked. And notwithstanding the admirable contributions of Werner and Strong, there exists as yet no comprehensive study of Leonardo's optical researches. This will be an aim of the present study.

3. The Problem of Sources

On the question of Leonardo's literacy, whether he stood fully within the optical tradition or whether he did not, there are two radically opposed schools of interpretation. One, championed by figures such as Santillana, Sarton, Garin and Lindberg believes that Leonardo was truly a man without letters8 (omo sanza lettere) and that the great wealth of classical and mediaeval learning was literally a closed book for him. According to this school his awareness of the optical tradition was at best incidental and gained second-hand through vernacular compendia and by word of mouth with learned friends. It bears noting that no member of this school has read the complete works of Leonardo carefully.

A second school has insisted that Leonardo's optical - and other - studies cannot be understood without a detailed knowledge of classical and mediaeval sources. The champions of this approach have tended to be specialists on Leonardo: Solmi, Werner, Brizio, Keele, Pedretti, Strong and Kemp9 to mention only some. Considering that Leonardo had a personal library of at

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least 116 books10, many of them in Latin, there is, in our opinion, little doubt that he must have been aware of the optical tradition and in the course of this work we shall present evidence to establish this viewpoint.

Indeed, at one point in the research it seemed a reasonable ideal to read through all the major optical sources such as Euclid, Ptolemy, Alkindi, Alhazen, Witelo and Pecham in an attempt to solve these debates once and for all. On reflection however it became clear that this attempt really involved two quite independent questions: first, whether Leonardo was aware of a problem in the tradition and secondly, what was the particular source of his knowledge of that problem. The first of these questions can be answered assuming one knows sufficiently the whole of Leonardo and the whole of the optical tradition. The second question often cannot be answered.

The second question concerning direct sources remains difficult. Euclid's Optics was available in various mediaeval manuscript. Witelo had also included almost all its propositions in the fourth book of his great optical compendium to which, incidentally, Leonardo refers at least five times. A number of the propositions in Euclid's Optics recur in Bacon, Pecham, Biagio Pelacani da Parma, the anonymous author of Della prospettiva and even in Alberti's Della pittura.

A serious search for sources would therefore require confronting each of Leonardo's phrases with each of these authors and then with all their manuscript variants. For this reason we have limited ourselves to pointing out parallels between the tradition and Leonardo's work. It remains for a future student, possibly with the aid of a computer, to establish whether one can identify specific manuscripts from which he copied verbatim. For the present only Pecham's Optics and Alberti's Elementa picturae (cf. below p. ) are known.

4. Scope of the Present Study

Nonetheless, the general question of Leonardo's relation to the optical tradition will be one of the basic themes of our study, which will open with an introductory chapter on how the meaning of optics gradually shifted from sight to light, how the qualitative concerns that dominated Euclid's Optics were gradually replaced by a quest for quantitative analysis. In short we shall outline the tradition that prepared the way for Kepler's distinction between subjective and objective images, between the imago and the pictura. Within this context we shall examine in detail Leonardo's optical researches and contributions.

In the first volume of our studies it was shown that the whole of Leonardo's science sprang from a combination of his perspectival studies with his concept of the four powers. This second volume offers a case study of the general proposition explored in the first. There we were concerned with the whole of his science. Here we are concerned only with his optics. There we were concerned with all four powers: percussion, force, impetus and movement. Here we are concerned chiefly with percussion, since this power underlies the whole of his physics of light and shade. The link between percussion and optics, we shall find, is an Ancient one: it can be traced back at least to Aristotle, and we shall follow its development through Ptolemy, Alhazen and Pecham, examining how Leonardo inherits it and transforms it.

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We shall turn then to Leonardo's definitions of basic optical concepts such as point, line, rectilinear propagation, pyramid and pyramidal diffusion, again note precedents, and go on to examine the physical and metaphysical roots of Leonardo's concept that images are "all in all and all in every part."

With this understanding of the historical context of Leonardo's ideas we shall, in part two, enter into the details of his physics on light and shade. Following a list of basic definitions and distinctions we shall reconstruct the contents of his proposed seven "books" - chapters in our terms - on light and shade, plus a further book on the movement of shadows. This will reveal various aspects of his systematic approach. Parallel with these demonstrations in the open air, there are a number of others involving the use of a camera obscura. That Leonardo was familiar with the camera obscura. That Leonardo was familiar with the camera obscura is well known: that he devotes over two hundred and thirty figures to this problem has never before been suspected. Among these diagrams will be found further evidence of his systematic experimental approach, which ultimately served to confront metaphysical speculation with the realities of physical demonstration.

Part three will focus on Leonardo's studies of the eye and vision. By way of introduction his notes on the eye as the noblest of the senses and as window of the soul will be cited. His demonstrations concerning the anatomy and physiology of individual parts of the eye such as eyelids, cornea and optic nerves will then lead to his theories of the visual process: his opinions on extromission versus intromission theories; his study of the idea that images converge to a point; his demonstrations to refute this, his interest in both single and double inversion of images in the eye and his experiments in which he substitutes a camera obscura for the pupil and a glass sphere filled with water for the crystalline lens.

Leonardo is also very interested in the conditions under which the eye is deceived. It will be shown that the situations which he considers have numerous parallels with Euclid's Optics, a difference being that were Euclid remains abstract and geometrical, Leonardo provides concrete examples from everyday experience. This interest in subjective visual appearances leads him, in turn, to explore optimal and minimal conditions for vision, including the role of the central ray, the visual field, distance, size, light as well as monocular and binocular vision.

Part four will open with Leonardo's demonstrations and comparative studies of pupils in animals and humans and lead to consideration of a connection that he saw between pupil size and apparent size, a connection that reveals why optics and astronomy are so closely linked in his mind and indeed why he felt that his studies of the eye were ultimately a prelude for his studies of the heavens. Leonardo was preoccupied with the optical part of astronomy as was Kepler a century later. To support this claim the structure of the Manuscript D will be reconsidered. An analysis of the Manuscript F will reveal another treatise of 21 consecutive pages on this theme. A further series of notes from the Manuscript F and the Codex Arundel will serve in the reconstruction of Leonardo's projected Fourth Book: On the Earth and Its Waters which reveals why he became so fascinated with reflections of the sun in water. If the surface of the water is completely smooth, the reflection is small. When there are waves, each of these waves functions as an individual mirror and reflects the sun, such that the whole surface becomes aglow with reflected light. Leonardo takes this principle and proceeds to ask what would happen if one looked at the ocean from a

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position high above the earth. He imagines that the entire ocean would reflect the sun's light. His imagination now takes over. If a planet were entirely covered with oceans, he reasons, then the whole planet would reflect light. Once a month the full moon reflects light over its entire surface. Ergo the moon must be covered with oceans.

Very much aware that there are competing explanations for the full moon Leonardo sets out to explore their viability: one is that the whole surface of the moon is like a convex mirror. He therefore studies the nature of images in convex mirrors and in the end demonstrates why the mirror hypothesis cannot explain why we see the full moon. And having accounted for this Leonardo turns finally to eclipses. Thus the whole of his study of optics as it is on earth is ultimately in order to understand astronomy as it is in heaven.

In the appendices will be collected Leonardo's notes on optical instruments such as spectacles, his much debated telescope, lens grinding instruments, his work on mirrors plane, convex and concave and finally on meteorology celestial - haloes - and terrestial - rainbows. The challenge of presenting this material in a comprehensible form has led to a threefold approach. First, there is a problem of understanding Leonardo in his own terms. This requires that we imagine ourselves in his world, (what the Germans call sitz im Leben and some term horizontal history), that we enter into his way of thinking and re-construct his arguments, as he himself might have done had there been more time, or if he had a good secretary. This approach is the main focus of this study.

A second approach is useful in order to establish the validity of his claims using a positivistic standard (verticle history). Ideally, such an approach would include an experimental reconstruction of all his demonstrations and claims: a task for a future project. Inherent in this approach, however, is the temptation of a holier than thou attitude, as if history could be reduced to the question: how many marks would Leonardo get on a modern physics test? To understand an individual also requires insight into why a person stops asking questions; to discern how a man with a critical mind can accept a wrong answer as the correct solution; to perceive how a way of explanation can become so convincing that it remains a close system. In becoming sensitive to why a Leonardo is often blind by our own standards, we become aware that even the most modern among us today is, in turn, equally blind when judged by standards of tomorrow. Hence blind alleys of the past may prompt new avenues for the future. But these are questions of philosophy.

Although an historian of ideas may point to such questions his main task lies elsewhere. In creating an intellectual biography of Leonardo it is not enough to end with an elegant list of the man's knowledge and opinions. Insight is required into why Leonardo wrote the notebooks the way he did. Here a third approach is necessary to comprehend why an individual with such coherent arguments should present them in what strikes us at first as an incoherent manner (eg. figs. ).

Such a threefold approach may be eminently logical and easy to describe in theory but in practice this does not eliminate the problem of comprehension. In Leonardo's notebooks a paragraph ostensibly devoted to one problem inevitably slides into two or three others. If, for instance, on W19150r (KP 118r, 1508-1510), he begins by speaking about the rectilinear propagation of light, he may well demonstrate this by means of a camera obscura and then, carried

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away by his camera obscura-eye analogy, discuss the nature of rays in the eye. Hence a paragraph that begins as physics of light ends as physiology of vision.

Leonardo's train of thought is easy to follow. The problem comes when we wish to analyse his work from a positivistic point of view and need to break things down into separate problems: physics of light, camera obscura, anatomy of the pupil, physiology of the eye, etc. For we then find ourselves with a choice of either repeating each paragraph three or four times or creating what easily becomes a labyrinth of cross-references.

There are further problems. When he feels something is important he repeats it to the point of boredom and beyond. The idea that the surface of a body participates in the colour of the surrounding objects is a case in point. There are some 75 passages on this idea alone (see below pp. ). On closer inspection it is found that these passages involve various kinds of proofs. One involves concrete objects in the open air (fig. ), another involves a camera obscura (fig. ), and a third involves a purely mathematical demonstration (fig. ). Each of these demonstrations is in turn related to others of its kind, for instance, the mathematical demonstration to show that colour participates is one of a coherent set of abstract mathematical demonstrations. In this case it is desirable to refer to a proof once under the idea it supports and a second time under the type of proof.

The result is a web of ideas which is often repetitive and which, on first encounter, prompts the question: "why didn't he...?" As we enter further into the labyrinthine logic of his mind, however, we find ourselves thinking afresh about physics of light and shade, the visual process, problems of perception and connections between optics and astronomy. The quality of some minds is measured in terms of the rightness of wrongness of their answers. There are rare minds, however, where the assessment of quality involves other criteria: where the actual answers are less important than the process of asking questions; where questions become windows into new landscapes of experience which provoke one to look afresh at oneself and the world around one with a new sense of wonder. This study is a journey into the landscapes of such a mind, replete with fields of unfinished thoughts.

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Part One Chapter One The Optical Tradition

1. Introduction 2. Ancient Philosophy and Optics 3. Ancient Medicine and Optics4. Euclid and Optics 5. From Illusion to the Correction of Deception 6. Distance as a Requirement for Sight 7. Distance in Alhazen and Witelo 8. Surveying and Optics 9. Astronomy and Optics 10. Changes in the Scope of Optics 11. Summary

1. Introduction

Much of Ancient optics is not to be found in optical treatises such as those of Euclid or Ptolemy. Although Aristotle classed optics under geometry, it was partly a philosophical problem. Hence it was discussed in Plato's Timaeus, Aristotle's De Anima and Lucretius' De rerum natura. It was also partly a medical problem. Hence its appearance in Galen's On the Usefulness of the Parts. Moreover, optics was linked with surveying and astronomy. These connections between optics and other disciplines are significant because in the course of time they evolved and gradually transformed the whole scope and content of optics itself. For this reason an outline sketch of their role in the optical tradition will be desireable if we wish to understand the context of Leonardo's optical researches.

2. Ancient Philosophy and Optics

Aristotle considers the relation of philosophy to optics in the Analytica Posteriora Bk.I.13 (79a 10-15):

As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.1

In the following paragraph (79a 17-20) Aristotle refers to optics as one of the "sciences that investigate causes."2 Hence, while the natural philosopher records optical phaenomena, the "optician" is expected to explain why these Lucretius were all, in this sense, opticians and in the optical sections of their philosophical texts they concentrate on three causal problems: (1) how and why the process of vision relates to concepts of matter; (2) why the process occurs through extromission or intromission and (3) how and why the veracity of vision, or its absence, is determined.

Plato's discussion of optics in the Timaeus is quite short. He begins with analogies between the (cold) fire of sight and the fire of light; between the images of sight by day and the images in dreams by night. Plato distinguishes between different kinds of fire and proceeds to explain reflection in a mirror in terms of a combination of internal and external fire. He believes that sight involves accessory causes not true ones but nonetheless praises it. In a later section he considers colours which, he claims, are different kinds of flame composed of various particles. These in turn yield sensations of different colours.3

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Aristotle's discussion in De Anima opens with the question of the objects of sight. This leads to consideration of basic concepts such as the visible, colour, the transparent, light, the medium and the common sensibles of sight.4 In De sensu Aristotle again begins5 with the objects of sight and common sensibles before broaching the nature of vision; possible links between the five senses and the four elements and hypotheses concerning the ratios of colours.

Theophrastus devotes much more attention to these questions in his work On the Senses.6 In book one he concentrates on the sensory process outlining a school which holds that vision takes place by similarity between the eye and object (Parmenides, Empedocles, and Plato) and then the opposed school which believes that vision occurs by contrast (Anaxagoras, Heraclitus). In his second book, Theophrastus considers the objects of sight, chiefly colour. He outlines possible links between the four elements and the senses, concluding with a discussion of heavy and light qualities. Lucretius in his On the Nature of Things7 prefaces the optical section with a consideration of the existence and character of both subjective and objective images, the rapidity of their formation and their velocity. He then turns to vision and mental pictures, deceptions of sight and the criteria for their correction with comments on the veracity of the senses.

This link between philosophy and optics is of great importance because it means that cosmological speculations on the structure of matter are bound up with theories of vision. As a result theories of vision remain a subject for intellectual debate rather than careful observation. This leads Galen to attack the Sophists' notions of vision who care "not for truth but only for glory." Nonetheless, the tradition continues and even in the seventeenth century the habit lingers of making lists of conflicting theories of vision in the manner that one lists conflicting philosophical positions.9 In this context, Leonardo's protracted notes on the pros and cons of both extromision and intromission become more comprehensible.

3. Ancient Medicine and Optics

While Galen is critical of the habits of some "practical physicians who call themselves oculists," he himself clearly represents a practical school also interested in theory. The section on the eye in On the Usefulness of the Parts10 begins with the question why the eyes are where they are and leads to a description of various parts of the eye such as the crystalline lens, the choroid membrane and the iris. Akin to Plato, Galen believes that the essence of the faculty of sight is of the nature of light. He gives examples to show that excessive light hurts the eye, that a lesser light is overcome by a greater one and that one eye increases when the other is closed - all themes that are discussed by various mediaeval authors and by Leonardo who, studied Galen, according to Vasari.

Following an excursus on the motions of the eye and eyelids, Galen considers geometrical aspects of the visual process itself: the visual rays, the cone of vision, monocular vision, binocular vision, plus an explanation why the crystalline lens is round. It is striking that Galen is very reticent to use geometrical explanations on the grounds that most people pretending to some education not only are ignorant of this but also avoid those who do understand it and are annoyed with them.11 Indeed, Galen insists that it is "only in obedience to the command of a divinity12 that he has used geometry at all. This is important because it suggests that medical, mathematical and philosophical explanations of the eye traditionally existed independently of one another, with no

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attempt at synthesis. This habit continued throughout the Mediaeval period and helps us to understand how even Leonardo, who aimed at synthesis, could alternately use medical, mathematical and philosophical approaches to optics and often not compare the results.

4. Euclid and Optics

Euclid's Optics represents a tradition quite different from that of the philosophers and physicians mentioned above. This Euclidean tradition is presumably what Aristotle had in mind when he wrote in the Physica Bk.II.2 (194a 6-11):

Similar evidence is supplied by the more physical of the branches of mathematics, such as optics, harmonics, and astronomy. These are in a way the reverse of geometry. While geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical not qua mathematical.13

Although couched in geometrical terms Euclid's Optics deals, however, with what would in our day be termed psychological optics: its prime concern being subjective appearances and optical illusions. Debates whether vision occurs through intromission or extromission do not interest Euclid. It is likely that these debates were the domain strictly of the philosophers and that those writing on mathematical optics, such as Euclid, had other concerns, as we learn from a passage in Hero's Definitions:

Optics does not deal with physical questions and does not study whether given rays flowing out from the eyes go forth to the boundaries of objects or whether images that are detached go forth from corporeal objects [and] enter the eye along a rectilinear path or whether the intervening air is stretched or contracted by the ray-like pneuma from the eye. It is only concerned whether, at each reception (of an image) the right direction of movement or tension is maintained as well as the requirement that the convergence to a point occurs at an angle when objects are seen that are larger or smaller than the eye.14

Upon reflection it becomes clear that Ancient optics was not one discipline but at least four which tended to appear in various combinations (Chart 1).

Profession/Discipline Type of Optical Problem/Question1. Natural Philosophers ‘What’ of optical phenomena, illusions2. Philosophical Opticians ‘How/why’ in terms of (meta-physical and cosmological structure3. Mathematical Opticians ‘How’ in terms of mathematics4. Medical Opticians/Oculists ‘How’ in terms of physical structure

Chart 1. List of different disciplines and their respective optical interests.

There was often interplay between some of these, e.g. Aristotle, Lucretius (1,2); Galen (3,4); Ptolemy (1,2,3,). Alhazen in the eleventh century is among the first to study all four disciplines together. If we examine the structure of Euclid's treatise more closely (see charts 2, 4) we discover that of the 57 theorems there are four that deal with surveying and stand apart from the rest. As Theisen has rightly noted: “ Their inclusion in the work is . . . most significant, since these

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propositions add a quantitative dimension to what is otherwise a purely qualitative work on vision.”15 In Euclid's text the surveying propositions appear simply to be interjections without theoretical justification. This changed, and the seeming accidental link between optics and surveying gradually assumed great significance. To understand this will require an excursus on two basic changes within optics generally: one involving the role of illusions; the second, objects of sight. We shall consider each of these in turn.

5. From Illusion to the Correction of Deception

Plato's famous attacks on vision/optics due to the deceptiveness of sight introduced a tradition that emphasized the fallibility of optics in particular and the senses in general. There is evidence, however, that the Platonists may have been more concerned with the problem of how one gets beyond the deceptions of vision, than with the deceptions as such. Sextus Empiricus in his Against the Logicians claims to "set forth the Academic tradition from Plato down"16 and refers to careful distinctions made in the school of Carneades between different kinds of vision, in terms of three categories.

Quantitative Problem Qualitative Deceptions, Illusions Proposition

Distinctness, visibility, invisibility 1-3Size 4-8Position 9Relative size with movement 10-14

Size, depth, length 18-21Amount of shape at a given position 22-23Shape with movement 24Size relative to position 25-36Sizes moving relative to one another or one object moving relative to eye 50-55

Changes in size and illusion of movement 56Changes in position 5

Chart 2. Survey of quantitative, qualitative themes in Euclid's Optics.

Logicians claims to "set forth the Academic tradition from Plato down"16 and refers to careful distinctions made in the school of Carneades between different kinds of vision, in terms of three categories. A first category involves things seen that are evidently false: A second category, those which are apparently true.17 This second category is subdivided into three groups: (1) the probable presentation, (2) the probable and irreversible presentation and (3) the presentation that is "at once probable and irreversible and test."18 This final group requires the certification of all the factors in the visual process, namely:

the subject that judges the object and the object that is being judged and the medium through which judgment is effected and distance and interval, place, time, mood, disposition, activity.19

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Hence within the very school famous for its attacks on the deceptiveness of vision/optics emerged a set of criteria for overcoming such deceptions and certifying the veracity of sight. Perhaps such a quest is also implicit in Euclid's analyses of illusions in the Optics. If so the Optics was ultimately a manual for getting beyond deception. In Lucretius' On the Nature of Things this ideal is more apparent. Lucretius acknowledges the existence of deceptions but, nonetheless, refuses to impugn the veracity of sight:

And yet in this we don't at all concedeThat eyes be cheatedTis after all the reasoning of the mindThat must decide, nor can our eyeballs knowThe nature of reality. And soAttack thou not this fault of mind to eyesNor lightly think our senses everywhereAre tottering.20

In his list of stock deceptions that follows there is the implicit assumption that experience allows one to see through a deception as when he mentions that

to gazers ignorant of the seaVessels in port seem, as with broken poopsTo lean upon the water, quite agog.21

Here the illusion is only caused by lack of familiarity. Through experience one can get beyond deception. That the aim of optics is to give an explanation for illusions is stated specifically by Geminus as reported by Proclus: "optics . . . explains the illusory appearances presented by objects at a distance, such as the converging of parallel lines or the rounded appearance of square towers".22 The passage goes on to discuss two further branches of optics: catoptrics, concerned with the reflection of light and scenography, concerned with assuring that drawings of objects will not be seen as disproportionate or shapeless when seen at a distance.

A similar classification is found in Hero of Alexandria's Definitions. He too describes optics as a discipline implicitly concerned with illusions; catoptrics, concerned with reflection, mirrors, rainbows, shadows and then scenography, concerned with the painting of buildings. Since things are not what they appear, claims Hero, one must not draw things as they are but as they appear.23 Aulus Gellius, in the Attic Nights provides us with yet another source which confirms that the aim of optics was to explain optical illusions.24

In Ptolemy's Optics the theme of the basic veracity of vision under proper conditions is continued. Ptolemy is very much aware, however, that the eye can be deceived and hence devotes practically the whole of Book two to these questions of deception. 25 But Ptolemy's underlying concern is to know why the deceptions occur. He also wishes to know at precisely what point in the visual process a deception occurs: be it due to the object's distance or position; to manipulation by the actual eye or due to the mind.26

25

Ptolemy also cites examples from everyday experience to illustrate his categories of deception. But he then proceeds to introduce experiments designed to determine when, for instance, we see an image as double and when we see it as normal.27 Hereby, the process of getting behind the deceptions becomes testable. In so doing, Ptolemy has expanded the scope of optics. For whereas Euclid had restricted his treatise to the mathematics of subjective visual appearances, Ptolemy goes beyond an actual description of appearances and seeks to identify the conditions in/by which vision can inform us about objective elements of the measured world.

Nemesius, carries Ptolemy's approach further in spirit if not in detail. Nemesius emphasizes the role of memory and thought in vision and uses this to defend the veracity of vision: "When then we suppose a wax apple to be a real apple, it is not sight that errs, but thought.28 He proceeds to give examples of deceptions owing to a lack of proper conditions such:

as when someone sets out to meet a friend, meets him and walks right past him, because his thoughts are on other matters. But this is not really a failure of sight as much as mind. For sight saw and gave notice, but mind would not attend to the notice given.29

This idea, which derives from Aristotle's De sensu 30, recurs in developed form in Macrobius' Saturnalia:

The organ of sight would therefore be of little help without the faculty of reason; this is so

true that an oar seen in the water appears broken to us and a polygonal tower appears round to us at a distance. But if reason rectifies these errors, the tower becomes angular again and the oar takes on a straight line. It is by these that we redress so many false impressions which have led the Academics to calumniate the senses, since the senses aided by reason should be counted among the most certain of things, albeit a single sense sometimes does not suffice in distinguishing a species.31

To illustrate this Macrobius returns to the example of the imitation apple that Nemesius had also used, before concluding that the senses owe their efficacy to reason.

Alhazen, the great tenth century natural philosopher and mathematician, develops these ideas very considerably. In an approach reminiscent of Ptolemy, Alhazen32 gives detailed attention to the requirements for vision and its objects. He is particularly interested in the criteria needed for the certification of what is seen, sometimes describing experiments akin to those described by Ptolemy33, often providing additional vivid examples, which could have been tested. By Alhazen's time optics is clearly devoted to explaining "what is there," to getting beyond deceptions. Whereas Plato had usually assumed that optics is concerned with describing subjective aspects of what the eye sees, Alhazen is convinced that optics must inform us about objective elements of the physical world of Nature. The great challenge of optics is now increasingly: what are the characteristics of the objects seen, in spite of how they appear? Through the contributions of Ptolemy and Alhazen in particular the realm of optics begins to shift from the debates of philosophical theory to testable predictability of experimental demonstration. What had begun as an objection to illusion has now become a commitment to getting beyond deception. As the aims of

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vision were being redefined, the requirements and objects of vision were being reconsidered also. Particularly interesting in this regard is the concept of distance.

6. Distance as a Requirement for Sight

Distance as a requirement for vision can be traced back to Aristotle's De anima:

If what has colour is placed in immediate contact with the eye, it cannot be seen . . . . Hence it is indispensable that there be something in between - if there were nothing, so far from seeing with greater distinctness, we should see nothing at all.34

Here distance means primarily "lack of contact." The Ancients were also concerned that distance should not be excessive, as Carneades35 noted and as Lucretius illustrated by means of a vivid example:

And when from far away we do beholdThe squared towers of a city, oftRounded they seem, - on this account becauseEach distant Angle is perceived obtuse,Or rather it is not perceived at all;And perishes the blow nor to our gazeArrive tis stroke, since through such length of airAre borne along the idols that the airMakes blunt the idol of the angle's pointBy numerous collidings.36

In such passages distance remains a qualitative prerequisite: a compromise between not in contact and not too far away. By late Antiquity this begins to change. Ptolemy lists distance as one of his distinguishing characteristics37 and he emphasizes the importance of a moderate distance between eye and object. To this end he requires that there be a perceptible proportion between the size of the object and the distance involved:

The eye perceives size accurately when the diameters of the base, which is above the object

seen, have a perceptible proportion to our distance from the object, which is the case when the rays containing it are disposed at a perceptible angle at the tip of the pyramid.38

Unlike Euclid, who had relied solely on a concept of angular size, Ptolemy is convinced that visual angles alone are not sufficient to determine the apparent size and distance of objects and points out that other factors such as the position of the object may also play a significant role.39 Ptolemy does not abandon altogether the Euclidean notion of angular size and yet, in his insistence on the importance of the central ray40, he implicitly establishes a relation between the (measured) size of objects and their (level) distance from the eye.

Nemesius, in the fourth century, is more extreme in emphasizing the importance of distance: he makes it one of the basic requirements of vision: "Sight needs four chief conditions for

27

clear discernment, unimpaired organs, measured motion, moderate distance and the air clear and light."41 Alhazen, in the eleventh century, makes distance one of six prerequisites for vision, along with position of the object, light magnitude, transparency and density or solidity.42 Later in his treatise Alhazen makes distance one of eight prerequisites of vision, adding time and health of vision to his former list.43 He also repeats the Aristotelian notion that distance in the sense of "lack of contact" is a prerequisite for sight.44 Just how much more distance meant to Alhazen than it did to Aristotle becomes clear, however, when we consider the role of distance as an object of sight.

Distance as an Object of Sight

In the Ross edition of Aristotle's De anima we read that: "The object of sight is the visible and what is visible is (a) colour...."45 Two words in this translation bear closer attention: "colour" and "object." In today's terminology, colour simply connotes black, white, red, blue, etc. Aristotle defines colour quite differently: "Every colour has in it the power to set in movement what is actually transparent: that power constitutes its very nature".46 Hence colour is, for Aristotle, not just a thing that is passively seen: it is an active agent that is vital to the visual process: colour sets the process of vision in action. The term "object" (of sight) is equally problematic. The original Green To c aTOV** literally means "the seen." In other words, Aristotle is saying: the seen is the visible and what is visible is that which sets the process of vision in action, namely colour. Hence colour is essential for Aristotle because it activates the visual process.

In Aristotle's student, Theophrastus, a shift in interpretation is evident when he writes concerning the things seen: motion, distance and size are visual objects and yet produce no image."47 This idea that distance is among the things seen, and is an object of sight is not mentioned by Ptolemy who lists instead: "body, magnitude, colour, shape, position, movement and rest."48 Whereas, Ptolemy cites seven things seen, Nemesius in the fourth century goes on to mention twenty-one, including distance:

Vision operates along straight lines and in the first place perceives colours. Along with the colour, it recognizes the body so coloured, its size, its shape, relative position and distance away, together with the number of its parts, whether it is in motion or still, whether it is rough or smooth, even or uneven, sharp or blunt; as well as its constitution, whether, say, it is watery or earthy, moist or dry.49

Nemesius acknowledges the Aristotelian view that the seen is colour but proceeds to qualify this claim: “But hard upon colour follows perception of the body possessing the colour, the position in which the thing seen may chance to be and the space or distance between the person seeing and the object seen.”50 Hence body, position and distance are now objects of sight. Each of these he describes, ending with distance:

Sight, on the other hand, can operate also from a distance. And since it receives its characteristic impression across an intervening space, it necessarily follows that sight by itself can recognize the distance of its object, and, likewise, the size of its object, provided that the object can be apprehended in a single glance.51

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If the starting point of Nemesius' approach is clearly Aristotelian, his interpretation of the concepts is basically different. Colour is, for him, no longer something that stimulates the visual process: it is merely something such as black, white, red or blue. Colour's special role in the visual process is thereby lost, and Nemesius is, therefore, led to the obvious conclusion that other things such as distance should be included among the objects of sight. Hereby the way is set for surveying, which measures distance, to assume a central role in optics. That which Nemesius mentions in the fourth century, Alhazen explores in detail in the eleventh.

7. Distance in Alhazen and Witelo

A study of Ptolemy's Optics prompted Alhazen to write his Doubts on Ptolemy, in which he notes that instead of seven, there are twenty-two objects of sight52, which he then includes in his Optics: light, colour, distance, position, body, figure, magnitude, continuity, discreteness, separateness, number, motion, rest, roughness, lightness, transparency, thickness, shadow, obscurity, beauty, ugliness, similarity.53 In this list, distance comes directly after light and colour. How important this concept is for him becomes clear when we turn to the second book of his great optical treatise: De aspectibus. Here, he begins with a careful distinction between distance, which can be quantitatively measured, and mere "lack of contact," which is qualitative.54 Alhazen proceeds to discuss the mind's role in vision: it is, he claims of great importance in the certification of distance55 (II:24). If there are a continuous number of ordered bodies the mind begins by determining the distance of one that is fairly close and on the basis of this moves onto the next one, thereby certifying distance as it goes along56 (II:25). If the distance be moderate then the position (situs) of the object can also be determined (II:26)57 and its location (locus) can, in turn, be deduced from its position, provided that moderate distances are involved (II:27).58

How one discriminates between two kinds of position is now mentioned. A direct position is indicated when the distance from the eye to the extremities on either side is equal. An oblique position is indicated whenever the distance from the eye to the two extremities is not equal (II:28).59 Alhazen points out that if the distance of the objects be extreme, then the eye does not certify their position, with the result that even obliquely positioned things seem as if they were facing the viewer (II:29).60 He claims that the various parts and boundaries of the objects seen, as well as the position of the separate objects, all depend on whether the lines leading to the extremities are equal or unequal in distance (II:30).61

Alhazen now mentions the different requirements for perceiving a body accurately: sometimes the eye alone is sufficient, sometimes it requires judgment. If the body be too far one cannot be certain at all (II:31).62 Beginning with circles, Alhazen discusses how various shapes are perceived (II:32).63 He then studies how perception of a convex surface is determined (II:33)64 and how this differs from a flat one (II:34)65 which leads in turn to the problem of how we perceive a plane surface at a moderate distance. There follows a consideration of how we perceive the size (magnitude) of objects. This, Alhazen admits, is a matter of debate. There are some who think it depends on the visual angle. Others say it depends on a comparison of this visual angle with the actual distance involved. But neither of these explanations will do66, he claims, and he proceeds to show why they are inadequate, including amidst his arguments a convincing test (fig. 1):

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(figure)

Fig. 1 Diagram from Alhazen's optical treatise to refute the Euclidean theorem of visual angles.

Now if the object seen were one cubit away from the eye and it were then moved until it were two cubits away (i.e. ab. is moved to de) then there will be a great difference between the two angles subtended by the two objects at the eye (i.e. /bca. and /dce.) and yet the eye will not apprehend the object two cubits away as being smaller than the object one cubit away. And similarly if it is moved three or four cubits away it will not appear smaller even though the angles at the eye vary immensely.67

He goes on to compare a direct view from immediately above the square with various oblique views as the distance increases. What is important in Alhazen's description is the way that he relates distance to vision in terms that can readily be tested experimentally. The actual mention of cubits of measurement indicates how close optical theory has come to the problems of surveying practice. Alhazen's conclusion to this particular demonstration is that if vision depended solely on visual angles then we could not see a square shape. In short, he realizes that a strict acceptance of the visual angles theory precludes any possibility of constancy in perceptual images (II:36).68

Having destroyed the "conventional wisdom" he turns to set out his own ideas. He claims that the size an object appears depends on the size of the surface of the eye affected by the image (in quam pervenit forma) as well as the angle of the optical pyramid (II:37).69 This serves, however, to introduce his main point (II:38) that the real (measured) size of an object depends on a comparison of the base of the triangle with the length of the optical pyramid, which we can illustrate in terms of a simple diagram in which apparent size depends on a comparison of ab. (distance) with cd. (measured size70 (fig. 2):

(figure)

Fig. 2 Author's reconstruction of Alhazen's claim (II:38) that apparent size depends on a comparison of distance with measured size (ab. with cd.).

The implications of this claim are profound, for Alhazen has hereby introduced the notion of a simple relation between measured size and level distance into his theory of vision which implies, in turn, that the basic principles of surveying are now at one with those of optics.

Alhazen goes on to present a rough version of the inverse size law. The eye, he claims, will note how the object seen will tend to get smaller as one goes further away and larger as one gets closer. Indeed, experience will show that to the extent an object seen is removed from the eye, to that extent will the location of its form in the eye diminish and the angle which the object seen subtends at the centre of the eye. Alhazen is not content to leave the matter here. Granted he does not go as far as Leonardo who insisted on demonstrating these principles experimentally but he proceeds, nonetheless, to drive home his theoretical concept by making an important appeal to the principle of occlusion which, albeit long-winded, is worth citing at length as an example of his approach to problems:

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... And to the extent the visible object is moved further away, and the eye certifies the

quantity of its remoteness, to that extent is it comprehended to be larger, e.g. when someone looks at a distant wall which is a reasonable distance from the eye and the eye certifies the distance of this wall and its size and it certifies the quantity of its length. If the person then places his hand between one eye and the wall while closing the other eye he will then find that his hand will occlude a great portion of the wall and he will comprehend the quantity of his hand in that situation and he will comprehend that the quantity of the wall occluded by the hand is far greater than the quantity of hand and the eye will simultaneously comprehend the limits (verticationes) of the radial lines and it will comprehend the angle which the radial lines contain.

The eye will then comprehend, therefore, that the angle which the hand and the wall subtend, is the same angle, and then he will also comprehend that the part of the wall occluded by his hand is far greater than his hand. And since this is so, the discriminating faculty (virtus distinctiva) comprehends in this comprehension that the more distant of two visible objects - at different distances, both subtending equal angles - is of greater size. Then if someone averts his eye while he is in that position and he looks upon another wall more remote than this wall and he positions his hand between his eye and that all he will find that what is occluded of the second wall is greater than what is occluded of the first. And if he then looks at the sky he will find that his hand occludes a half of what appears in the sky or a great portion of it. Nevertheless, the viewer will not doubt that his hand is nothing with respect to what is occluded in the sky according to his sense (of sight). From this experience it will, therefore, be determined that the eye does not comprehend the size of an object seen, unless from a comparison of the size of the thing seen with the quantity of its remoteness in comparison to the angle and not just from a comparison of the angle. And if the comprehension of the quantity of the magnitude were simply as a result of the angle then it would have to be that two objects seen at different remotenesses, subtending the same angle at the centre of sight, would appear equal. And this is not so ...71

To put it simply, Alhazen has launched an open attack on classical optics. Euclid's fourth assumption had been that "things seen under a larger angle appear larger, under a smaller angle appear smaller and unequal equal angles appear equal."72 Alhazen rejects this. Whereas Euclid had accepted the convenience of angular size at the expense of a direct relation between (measured) size and (level) distance, Alhazen chooses, instead, to relegate visual angles to an ancillary role. Nor does he stop here. All that remains now, he claims, is to explain how the eye comprehends the distance of continuous ordered bodies (corpora ordinata continuata) and how it determines their size clearly:

These bodies which are ordered and continuous with respect to the distance of the objects

seen are, for the most part, pieces of land and objects which are seen regularly and are always comprehended by the eye. The comprehension of the sizes of these pieces of land stretched out between the viewer and the objects seen can only come from a comparative measurement of the pieces amongst themselves and comparing the remoter parts of the land with closer ones whose size has been certified.

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Alhazen notes the importance of habit and memory in these experiences and explains in greater detail how this comprehension of distance occurs:

Its origin, the extent of which is clearly determined by the organ of vision, is that which is at the feet. This is because this extent which is at the feet is comprehended by the eye and the distinguishing power and the organ of sight determines it clearly through a measurement of the human body. This is because that land which is at the feet is always measured by man, without attention (intentione), by his feet where he walks over it and by his arm when he extends his hand to it. And all land which is close to man is always measured by way of the human body without attention and the eye comprehends that measurement and perceives the same. And the distinguishing power comprehends that measurement and knows it and determines clearly from it the extents of the parts of the land with the human body.73

There is, therefore, an automatic process of judging distance which beings with the ground at one's feet:

Thus if a man were standing up straight and he looked at the earth at his feet there would be

lengths of radial lines in keeping with the extent of the man's height and the distinguishing power would know for certain that the distance between the organ of sight and that piece of land is the size of a man standing up and the length of the locations of the land continuous with the body of the man are known and these perceived distances in the distinguishing power and their shapes come to rest in the mind (anima).74

In volume one we analysed Leonardo's careful attention on Ms.A 37r to the appearance of furrows of land with respect to his perspectival studies.75 This, we now realize, had a mediaeval precedent. In Alhazen's view the judgment of distance is not merely a vague one: the eye perceives distance on level ground in terms of standardized measurement:

... similarly it (the distinguishing power) obtains from a cube and a palm and from some measured quantity a determined quantity. Hence if the viewer apprehends some space and wants to know how many cubits are in it, he compares the form obtained by his imagination of that same space and compares it with the form his imagination had obtained of a cubit and he would comprehend the quantity of that space with respect to a cubit through this comparison.76

This standard of measurement also aids in the judgment of heights:

And similarly the quantities of the heights of objects above the land at some distance (such as walls and mountains) are apprehended by the eye in the way that quantities of pieces of land are apprehended and the distances of these objects are apprehended through a comprehension of their heights.77

He concludes this mini-chapter (II:39) by re-emphasizing that these principles only apply within moderate distances.78 Alhazen's further comments on distance need not concern us here.

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What makes the particular passages we have cited so important is the evidence they provide that Alhazen is very much concerned, at least in theory, with relating vision to distance, not in the vague sense of "interval" or "space between," but specifically in terms of precise standardized measurement.

With Alhazen, what had traditionally been a philosophical discussion on the role of distance in accurate vision, becomes transformed into a more scientific approach, wherein distance has become a quantifiable factor and whereby the aims of optics and surveying become effectively synonymous. The great twelfth century writer on optics, Witelo79, who borrows heavily from Alhazen's ideas, plays a significant role in making these advances of the Arabic tradition available to the Latin West. For example, Witelo gives a long paraphrase of Alhazen's discussion concerning the certification of distance in the case of continuous, ordered bodies which, as he explains, are those placed in a practically straight line and effectively at equal distances from one another as are trees or mountains or high towers and the like80 (IV:10):

Now the body of land which is interposed between these bodies is measured by the eye through the number of feet, since the foot is the minimal measure ordinarily used by men in measuring nearby sections of land and through these nearby sections of land the more distant sections of land are measured by the distinguishing power of the mind on account of the frequency of the comprehension of sections similar to that section of land, the measure of whose parts remains in the mind such that even the mind does not perceive the duration of these parts in itself. Moreover, these measures come to the mind since the quantity of the spaces which are at the feet of men are comprehended by the eye for they are even measured unintentionally through the feet of men as they frequently walk over those spaces as they are also measured with the lengths of arms (brachiorum cf. braccia). The distinguishing faculty comprehends this true measure and from this it certifies the quantities of the parts of the land which are continuous with the body of the man seeing and this remaining in the mind is the principle of measurement of all distances by the estimative (power).81

By comparison Alhazen seems unnecessarily wordy (cf. II:24): Witelo not only transmits his predecessor's ideas but also clarifies them. Throughout this description it is striking that the process of measuring distance remains theoretical and notwithstanding references to "true measure" there is no evidence that Witelo is committed to the practical testing of his ideas. In encyclopaedic fashion Witelo reproduces the greater part of Euclid's Optics - in book four of his massive work - and thereby repeats the entire visual angles theory as if it were his own. In the midst of this discussion, however, he introduces Alhazen's attack on the Euclidean theory as faithfully as he had stated the defence. Witelo's summary amply communicates the spirit of this approach (IV:27):

Thus the distinguishing power in distinguishing the true quantity of the thing seen, will not

consider the angle alone nor the distance alone, since neither of these suffices in themselves but it will consider the angle and remoteness simultaneously. Hence the quantities of the things seen will not be comprehended unless through distinction and comparison: moreover this comparison will be simultaneous and it will be between the base of the radial pyramid (which is the surface of the object seen by Bk.3, theorem 18) and the angle of the pyramid

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and the quantity of the length of the axis of the pyramid, which is the line of remoteness of the object seen from the eye.82

Hence Witelo, like Alhazen, rejects the simple equation between size of the visual angle and apparent size, insisting that this be checked through a comparison with distance. At the same time, as heir to the classical division of optics into four disciplines (Chart 1) Witelo is unconcerned with contradictions arising between the various schools. Thus he can report Alhazen's important philosophical optical comments on distance and then proceed a few folios later to cite Euclid's propositions in terms of geometrical optics without attempting to reconcile the two. Moreover, he can discuss the careful computation of distance in measured cubits, without testing the results. His fourteenth century successor Biagio Pelacani da Parma does test the results but still describes them verbally.83 Francesco di Giorgio Martini (Fig. 5) and Leonardo (Fig. 6) illustrate the results visually.

In volume one we noted a connection between these diagrams and others in Witelo (Fig. 4) and Euclid (Fig. 3). We are now in a position to understand why these connections existed. The general orientation of optics had shifted from an interest in qualitative appearances and illusions to a study of quantitative elements with a view to getting beyond deception. At the same time the concept of distance had been so integrated into optics that theoretical optics and practical surveying now had parallel goals. The quest for veracity of vision, which had begun on a strictly philosophical level in Antiquity could now be pursued on two levels simultaneously: one in terms of theoretical arguments, the other in terms of practical demonstrations.

As late as the 1390's Biagio Pelacani da Parma can still pursue both these strands in his lectures on optics. In the course of the fifteenth century, however, these practical demonstrations evolve to such an extent that they inspire independent treatises or parts of treatises by authors such as Alberti, Filarete, Francesco di Giorgio Martini, Piero della Francesca and Leonardo. In the minds of the fifteenth century authors the theoretical and practical demonstrations are interdependent. Hence they refer to both as perspectiva, the mediaeval term for optics. Modern historians, to avoid ambiguity, usually refer to the theoretical demonstrations as optical treatises and those with practical demonstrations as treatises on linear perspective.

8. Surveying and Optics

In Euclid's Optics, the inclusion of four theorems on surveying appeared fortuitous. Our excuses on how distance became a concept central to optical theory explains why this link between optics and surveying becomes ever more important: both disciplines are concerned with the same concept of distance. Hence as early as the second century A.D. we find evidence in Hero of Alexandria84 that surveying is a branch of optics. By the tenth century, Al-Farabi goes much further. Concerning optics (sic) he writes:

This art makes it possible for one to know the measurement of that which is far distant, for example, the height of tall trees and walls, the width of valleys and rivers, the height of mountains and the depth of valleys, rivers... then the distances of the celestial bodies and their measurements.85

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The twelfth century Toledan philosopher and translator, Gundissalinus, effectively copies this passage verbatim in his De divisione philosophiae.86 His work, as Crombie87 has noted, in turn influence Grosseteste and the Oxford school of the thirteenth century.

(figure)

Figs 3-6 Surveying principles in the optical tradition: Fig. 3, Euclid, Optics Theorem X; Fig. 4, Witelo, Opticae, IV:22; Fig. 5, Francesco di Giorgio Martini, Codice Torinese Salazziano 148, fol. 33; Fig. 6, Leonardo, CA 36vb.

Indeed this ideal of distance common to optics and surveying explains why the two disciplines so often appear together in mediaeval manuscripts as, for example, in Grazia de Castellani's De visu, 88 which includes various propositions on surveying. Moreover, this common ideal of distance explains why the practical demonstrations of optics that evolved into the independent science of linear perspective, should have had their start in practical surveying treatises. In short we now have a reason for the links between perspective and surveying described in volume one.

9. Astronomy and Optics

Aristotle, in the Analytica Posteriora, specifically claims that optics is linked with geometry whereas the data of observation, as he calls them, are linked with astronomy.89 Nonetheless, there also appear to have been direct links between optics and astronomy. Certain propositions in Euclid's Optics may have been written to account for astronomical phenomena.90 Plato in his Timaeus is unequivocal about this link between optics and astronomy:

For I reckon that the supreme benefit for which sight is responsible is that not a word of all that we have said about the universe could have been said if we had not seen stars and sun and heaven. As it is, the sight of day and night, the months and returning years, the equinoxes and solstices, has caused us the invention of number, given us the notion of time, and made us inquire into the nature of the universe .... Let us rather say that the cause and purpose of God's invention and gift to us of sight was that we should see the revolutions of intelligence in the heavens and use their untroubled course to guide the troubled revolutions in our own understanding.91

This connection would explain why Ptolemy, the greatest writer on optics in Antiquity should also have been the author of the Almagest. St. Anatolius in his Fragments of the Books of Arithmetic provides further evidence of this connection when he writes of the mathematician that:

he ought to be cognisant of the course of the stars and their velocity, and their magnitudes, and forms and distances. And besides, he ought to investigate their dispositions to vision, examining into the cause, why they are not seen as of the same form and of the same size from every distance, retaining, indeed, as we know them to do, their dispositions relative to each other, but producing, at the same time, deceptive appearances both in respect of order and position.92

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(Chart)

Chart 3. Summary of different systems of classifying optics and astronomy

As the mediaeval period progressed the Aristotelian scheme of classification was gradually replaced by others in which optics and astronomy were more closely linked (Chart 3). Among them was that of A1-Farabi who, as we have just cited, claimed optics was concerned with "the distances of the celestial bodies and their measurements."93 These connections between optics and astronomy are reflected in the optical treatises themselves. In the case of Alhazen's great treatise it becomes customary to add a short treatise De crepusculis as an appendix.94 In Witelo's optical compendium astronomical considerations are incorporated within the test itself. In Pecham's treatise astronomy plays a greater role. This interplay between optics and astronomy also explains why Biagio Pelacani da Parma should have been studying Ptolemy's Planisphere in a course on optics.95 It also explains why Dante, in his Convivio (II-III-6), speaking of the moving heavens should write that:

The position of these is manifest and determined by an art, which is called optics (perspettiva) and by arithmetic and geometry, is sensibly and reasonably seen, and by other sense experiences.96

In this context, Leonardo's preoccupation with optics as a means of understanding phenomena in the heavens can be seen as a natural outgrowth of a tradition. And Kepler's decision, a century later, to focus his optical studies on astronomy, becomes a logical next step. Optics which had begun as a philosophical problem, had now acquired a problem solving function.

10. Changes in the Scope of Optics

This interplay between optics and disciplines such as philosophy, medicine, surveying and astronomy, dramatically expanded the scope of optics itself as becomes evident from even a cursory survey of the contents of the chief treatises. Euclid, as we have seen, devotes his Optics primarily to deceptions of vision (Chart 4). In a separate work he deals with catoptrics. Ptolemy's Optics, by contrast, is divided into five books. His first book, now lost, dealt with vision and light, how they were imparted, how they were comparable and how they differed. His second book considers those things properly perceptible by means of sight, which topic leads him to pay considerable attention to illusions. In book three, Ptolemy examines basic problems of plane and convex mirrors. In his fourth book, he concentrates on concave mirrors, and his fifth, on refraction.

Alhazen's great optical treatise, in the version that became famous in the West, has seven books. Book one97 begins with the claim that light has an effect on the eye and then considers colours, composition of the eye, quality of vision, the use of sight and the prerequisites for healthy vision. Book two opens with a description of how vision takes place, examines the twenty-two objects of sight and the diversity of things seen by the eye. In book three he considers deceptions of sight and the various causes thereof. In a fourth book, Alhazen discusses problems of reflection in general, before turning, in book five, to discuss the position of images in plane, convex, concave and cylindrical mirrors. In book six he is concerned with explaining errors brought about be reflection in concave and convex mirrors. In book seven, he deals with refraction, its properties,

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and the illusions it occasions. Often appended to Alhazen's treatise is a short work entitled De crepusculis, which considers optical phenomena relating to the sun, shadow and clouds.

Witelo, in the thirteenth century effectively compiles an encyclopaedia of optical knowledge at the time. In a first book he concentrates on mathematical principles gleaned from Euclid, Archimedes, Apollonius, etc.98 In his second book he examines the projection of light and shade. In book three Witelo considers simple vision, disposition of the organ of sight, conditions of sight, and properties of the first two objects of sight: light and colour. In book four he considers the remaining twenty objects of sight and the deceptions that these involved. Here he integrates virtually the whole of Euclid's Optics.

(chart)Chart 4. Summary of themes of the other optical treatises from Euclid to Kepler. Evident to a diversification of topics and increasing attention to astronomy.

In book five Witelo turns to problems of reflection common to all mirrors and the properties of plane mirrors.99 In book six he examines convex, columnar and pyramidal mirrors. These latter types he discusses further in book seven. In book eight, Witelo studies concave mirrors. In his ninth book, he returns to consider some special properties of columnar and pyramidal (conic) mirrors and parabolic burning mirrors. In his tenth book, Witelo concentrates on refraction, which leads him, in the final section, to consider optics with respect to astronomy.

Pecham's Perspectiva communis100 is most probably an abridged version of Witelo's compendium. Pecham's work is divided into three books and opens with a consideration of light and its properties of propagation. The eye and the visual process are discussed, then visual perception, the conditions for sight and the objects of sight as well as deceptions and the problem of stars on the horizon. Book two turns to investigate mirrors: their nature, differences between them, the manner and positions of reflection, how one locates images, as well as various errors of reflection, ending with a note on the twinkling of stars. In book three, Pecham opens with a discussion of refraction and then devotes no less than eleven propositions to astronomical and meteorological questions ranging from twinkling stars to rainbows.

Kepler, in his Ad Vitellionem Paralipomena quibus Astronomiee pars optica traditur 101 (1604) focusses on this astronomical dimension of optics. His work is divided into eleven chapters and opens with a discourse on the nature of light, which leads to a consideration of the shape of light (chapter two), a study of mirrors and the position of their images (chapter three), the measure of refraction (chapter four), and in turn to the visual process in chapter five where he makes his distinction between the imago and the pictura. The remaining six chapters are devoted to astronomy, beginning with a consideration of the light of the stars, then the shadow of the earth (chapter seven), the shadow of the moon and diurnal shadows (chapter eight); parallax (chapter nine), the optical basis of the movement of the stars (chapter ten) and how one can determine the diameters of the sun and moon (chapter eleven).

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9. Summary

On the one hand this continuity of the optical tradition during the 1900 years from Euclid to Kepler kelps explain many parallels which we shall find in Leonardo's optical writings. On the other hand, within this continuity the whole nature of the approach had changed. What had begun as an interest in qualitative, subjective aspects of vision has become transformed into a concern for a quantitative, objective physics of light and shade. As our brief survey of the historical context has shown the theoretical roots of this shift were gradually established in the course of the Mediaeval period. Leonardo's optical researches provide us, in turn, with an important chapter in the story of how that theoretical shift became a practical one, in short how a tradition of speculative metaphysical questions was translated into a set of problems in physics.

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Part One Chapter 2 Percussion: Ancient Simile: Modern Concept

1. Introduction 2. Ancient and Mediaeval Precedents 3. Percussion in Leonardo's Optics and Acoustics 4. Percussion and Leonardo's Physics in General 5. Percussion and the Four Powers6. Percussion in Solid and Fluid Particles 7. Percussion in Water and Air: Mediaeval Precedents8. Leonardo's Wave Theory 9. Conclusions

1. Introduction

The starting point of Leonardo's physics of light and sound is the principle that they consist of impeded movement, producing the equivalent of a blow in the context of the four elements: earth, air, fire and water. This impact Leonardo terms either blow (colpo) or percussion (percussione). "Blow," writes Leonardo on A27v (c.1492) "I say to be the terminus of speedy motion made in bodies of resisting objects." This he restates on A32r: "Blow is motion interrupted by a resisting object." On BM138v (c.1505) he drafts an other formulation which he presents on BM Arundel 90r (c.1505-1508): “Percussion is the terminus of incident motion and the beginning of reflected motion achieved in an indivisible speed, time and position.” For Leonardo percussion constitutes a major problem. As early as c.1497 we find him on CA384ra alluding to a seventh conclusion concerning percussion. On CA29rb (c.1500) he notes that:

In percussion you need to consider four things, namely, the power that moves the percussor, the nature of this percussor, (and) the nature of the percussed object and the object which sustains this object that has been struck.

On Mad II 137v (c.1503-1505) he distinguishes between simple and compound percussion. In a note on CA354va (c.1505) he asks himself: why percussion on water makes more waves? Drafts for percussion of heavy bodies on CA252ra (c.1505) and heavy spherical bodies on CA354va (c. 1505) follow. Shortly afterwards as on CA74vb (c.1505-1508) he is making lists of different kinds of percussion:

Of the percussion of waters of various sizesOf the percussion of waters wide and smoothOf the percussion of waters narrow and roughOf percussion in dense objectsOf the percussion in water of objects with apertures

Expanded versions of this list recur on CA79ra, 79vb and 65va (c.1505-1508). On CA241ra (c.1508-1510) Leonardo outlines further plans to organize his material:

You will divide percussion into books of which the first will be of two bodies of which the one percussor moves the immobile percussed object; in the second the percussor and the percussed move one another reciprocally; the third is of liquid materials; the fourth is of flexible bodies; fifth...

A further note on CA241vb (c.1508-1510) makes it clear that these books on percussion

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formed part of a larger work on the four powers of nature: "The book of impetus will go before this and before impetus goes motion." In Leonardo's mind, weight, force, motion and percussion are all interdependent. By way of introduction we shall, nonetheless, concentrate on the historical roots of percussion, before explaining its function in both Leonardo's physics and optics.

2. Percussion: Ancient and Mediaeval Precedents

The concept of percussion in optics can be traced back to Democritus (c.460-370 B.C.) who, according to Theophrastus held that:

the air between the eye and the object of sight is compressed by the object and the visual organ and thus becomes imprinted; since there is always an effluence of some kind arising from everything. Thereupon this imprinted air, because it is solid and is of a hue contrasting < with the pupil > is reflected in the eyes which are moist.1

According to this Democritean theory of imprints "the air is moulded like wax that is squeezed and pressed."2 Aristotle developed this theory with respect to his concept of the senses in De anima:

By a 'sense' is meant what has the power of receiving into itself the sensible forms of things

without the matter. This must be conceived of as taking place in the way in which a piece of wax takes on the impress of a signet-ring without the iron or gold...3

Elsewhere in the De anima when describing acoustics Aristotle expressed this imprint theory in terms of a blow:

What is required for the production of sound is an impact of two solids against one another and against the air. The latter condition is satisfied when the air impinged upon does not retreat before the blow, i.e. is not dissipated by it.

That is why it must be struck by a sudden sharp blow if it is to sound...An echo occurs, when, a mass of air, having been unified, bounded and prevented from dissipation by the containing walls of a vessel, the air originally struck by an impinging body and set in motion by it rebounds from this mass of air like a ball from a wall.4

This analogy between an echo and a ball bouncing from a wall is taken up by Leonardo (cf. figs. 7-10). Aristotle went on to compare these effects of sound with these of light:

What happens here must be analogous to what happens in the case of light; light is always

reflected - otherwise it would not be diffused and outside what was directly illuminated by the sun there would be a blank darkness; but this reflected light is not always strong enough, as it is when reflected from water, bronze, and other smooth bodies to cast a shadow, which is the distinguishing characteristic by which we recognize light.5

Implicit in Aristotle's analogy is a comparison between sound which is bounced from a wall

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like a ball and light which is reflected/bounced from a wall like an image in a mirror: a comparison that the author of the Problemata pursued, beginning with a question: “Why is it that objects which fall to the earth and rebound describe similar angles to the earth's surface on either side of the point at which they touch the earth's surface?”6 As if unsatisfied with his first answer he posed the question anew this time concluding with an analogy that made explicit Aristotle's implicit comparison in De anima:

As then, in a mirror, the image appears at the end of the line along which the sight travels,

so the opposite occurs in moving objects, for they are repelled at an angle of the same magnitude as the angle at the apex (for it must be observed that both the angle and the impetus are changed) and in these circumstances it is clear that moving objects must rebound at similar angles.7

Hence if Aristotle borrowed the image of a bouncing ball from kinematics to explain principles of optics and acoustics, the author of the Problemata in turn borrowed the principle that the angle of incidence equals the angle of reflection from catoptrics to illustrate properties of kinematics. Lucretius in his On the Nature of Things continued this analogy between light and the blow caused by an object.

The peacock's tail, filled with copious light,Changes its colour likewise, when it turns,Wherefore, since by some blow of light begot,Without such blow these colours can't become.

And since the pupil of the eye receivesWithin itself one kind of blow, when saidTo feel a white hue, then another kind,When feeling a black or any other hue...8

To describe this blow caused by light Lucretius used the verb percutior, to percuss. Pliny in his Natural History also described reflection in mirrors in terms of percussion: “Still, the property of reflecting images is marvelous; it is generally agreed that it takes its place owing to the repercussion of the air which is thrown back into the eyes.”9 Alexander Aphrodisias, in his commentary on Aristotle's Meteorology10 also referred to percussion in a similar context. The comparison made in the Problemata between the reflection of images in a mirror and objects bouncing from polished surfaces was repeated by Ptolemy11 and Hero of Alexandria.12 In the eleventh century this comparison was taken up afresh by the great Arabic optical writer Alhazen:

And since it [the image in a mirror] preserves in itself the force and nature of its prior motion, it is reflected in the direction from which it came, and along lines having the same position [i.e. slope] with the prior ones. Moreover, we can see something similar to this in natural motions and also in accidental ones. If we permit a heavy spherical body to descend from some altitude perpendicularly onto a polished surface we shall see it reflected along the [same] perpendicular [as the one] by which it descended.13

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Elsewhere in his text Alhazen briefly compared sight and hearing14, again describing hearing in terms of percussion and suggesting that this was analogous to the action of images reflected in a mirror. For Alhazen, however, these were images in passing. Percussion did not play an important role in his optical explanations. This was also the case for Witelo and Pecham. For Roger Bacon, an elder contemporary of these two, percussion played a more significant role as we learn from his Optics:

In addition it can be expressed otherwise that there is a different ratio for light [than for]

both sound and odour, for light is carried more swiftly by far through the air than these, as [when] we see [a person] somewhere from afar striking (percutiente) with a hammer or a staff [and] we see him strike before we hear the sound generated. For we perceive the second percussion with the eye before the sound of the first percussion comes to the ear.15

Implicit here was the notion that hearing and vision both involve some corporeal action, which the anonymous author of Della prospettiva, in early fifteenth century made a starting point to his treatise:

The first thing to be noted in the first part, is proposed in the form of a conclusion - It is not

possible that any incorporeal thing, that is, non corporeal, be seen, speaking of human vision and of other animals...16

3. Percussion in Leonardo's Optics and Acoustics

Leonardo stands clearly within this tradition. He believes that objects can only be heard and seen by means of corporeal instruments, but that sound and light themselves involve incorporeal energy, residing in space without occupying it. For this reason the mathematical point (cf. below pp. ) and not the atom is the basis of his physics of light and shade. From the outset percussion and the other powers of nature play a significant role in Leonardo's conception, as is evidenced by a passage concerning acoustics on B4v (c.1487-1490):

There can be no voice where there is not movement and percussion of the air; there can be

no percussion of this air where there is not an instrument. There can be no incorporeal instrument. This being so a spirit can have neither voice, nor form, nor force and if you take it to be a body, it cannot penetrate nor enter where the exits are closed. Beware the precepts of those speculators whose reasons are not confirmed by experience.

And if someone were to say: by [means of] air congregated and restricted together, the spirit produces the bodies of various forms and by this instrument it speaks and moves with force, to this side I reply that, where there is not nerve and bone, there cannot be force operated in any movement made by the imagined spirits.

The need for a corporeal instrument in perceiving this incorporeal energy Leonardo discusses further on CA345rb (C.1505-1508):

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one cannot see a spirit in the countryside which the other cannot see also.

Hence no spiritual or transparent object can see anything positioned opposite it, because a dense and opaque object is necessary in it and if it be thus it does not require a spirit.

This serves, in turn, as a:

Proof that no object can be seen except through an aperture through which passes the air filled with the species of the objects which intersect in the dense and opaque sides of the

aforesaid apertures. And for this [reason] no object which does not have body can see either the figure nor the colour of any object. Whence it is necessary that it is a dense and opaque instrument through the aperture of which the species of objects impress their colours and figures.

In other words the incorporeal energy of sound cannot speak, nor that of light shine without a corporeal instrument of percussion. This theme Leonardo takes up again on W19048v (KP 49v, c.1508-1509), continues throughout W19048r (KP 49r) and into W19047vz (KP48v, c.1481-1510) in a passage entitled:

Whether the Spirit can Speak or Not.Wishing to show, whether a spirit can speak or not it is necessary to define first what thing a voice is and how it generates itself and we shall say in this way: voice is [a] movement of the air rubbed together in a dense body, or a dense body rubbed together in the air which is the same, which rubbing of dense with rare condenses the rare and makes resistance and again the swift rare in the slow rare condense one another on contact and make sound and very great uproar. And the sound or murmur made by the rare which moves in the rare with mediocre movement, like the great flame generating the sound in the air and the greatest uproar made by the rare with the rare and when the swift rare penetrates the immobile rare, like the flame of the fire issuing from the cannon and is percussed in the air and again the [like] the flame issuing from a cloud percusses the air in the generation of (the) bolts. Hence we shall say that the spirit cannot generate noises without movement of air and air in it is not, nor can it emit from itself what it has not and if it wishes to move that in which it is infused it is necessary that the spirit multiplies and multiply it cannot if it have not quantity. And by the 4th which says: no rare [body] moves if it does not have a stable spot whence ittakes its motion and maximally having to move the element in the element which does not move of itself, if not by uniform evaporation at the centre of the evaporated thing, as happens in the sponge squeezed in the hand which stands under water, from which the water flees every which way with equal movement through the fissures interposed between the fingers of the hand in which it is squeezed.

The passage ends with a volley of questions:

Whether the spirit has an articulated voice and whether the spirit can be heard and what thing is hearing and seeing and how the wave of the voice goes through the air and how the species of objects pass to the eye?

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If Leonardo clearly has difficulties in formulating his concepts of incorporeal energy there can be little doubt that his ideas on percussion build directly on a well established tradition. On Manuscript B 90v (c.1487-1490), for example, he sets out to give an explanation relating to acoustics:

The voice, having parted from the man and having repercussed on the wall, will escape

upwards. If you have an overhang above this wall at right angles [to it] the upper face will send the voice to its source, as the voice of the echo must do which, for everything that you say, will be replied to you in many voices.

A diagram (fig. 7) follows which is then described:

[There are] 150 braccia from one wall to the next. The voice that issues from the horn is formed on the opposite wall and from there it bounces to the second and from the second [back] to the first, as a ball which bounces between 2 walls which diminishes its bounces and so too diminishes the voice.

Leonardo's analogy between echoing sound and bouncing ball is the same as that found in Aristotle's De anima II 8 (419b 20ff.). The difference between the Stagirite and Leonardo is one of emphasis. What had functioned as an image in passing for Aristotle becomes, for Leonardo, a recurrent motif that involves a systematic concept.

(figure)

Figs. 7-10: Visual demonstrations of sound involving percussion or a blow. Note the hammer striking the ear, in fig. 10. Fig. 7, B90v; fig. 8, C6v; fig. 9. C 6v; fig. 10, C16r.

(figure)

Figs. 11-15: Analogies between percussion, reflection, sound, light and sight. Figs. 11-13, A19r; fig. 14, A19v; fig. 15, A113r.

Having mentioned the image of bouncing balls on B90v, Leonardo illustrates it on C6v (c.1490, fig. 8), following which he makes a note which recalls Bacon's idea cited earlier:

On the sound made by percussionSound cannot be heard so close to the ear that the eye has not first seen the contact of the blow.

Accompanying this text is a diagram showing a hammer striking a bell (fig. 9). Alongside this diagram is an X. Near the top of 16r in the same treatise there are two other X's and lower down on this folio there is another diagram of a hammer striking a bell (fig. 10). Immediately following this is a text entitled:

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Of corporeal movementsI say that the voice of the echo is reflected by (the) percussion at the ear as at the eye (the) percussions are made in mirrors by minds of objects. In the same way that the resemblances fall from the thing to the mirror and from the mirror to the eye under equal angles, so too will fall and rebound, at equal angles, the voice in the concavity of the first percussion at the ear.

Here Aristotle's loose analogy between acoustics and optics has become an explicit comparison between the percussion of sound in echoes and the percussion of images in mirrors. Some two years later on Manuscript A 19r (c.14592) Leonardo develops this comparison, beginning with a general statement: "It is possible with the ear to know the distance of thunder seeing its flash of lightning, through the similitude of the voice of the echo". Below this he draws a bouncing ball (fig. 11) which he explains in a caption alongside: "The line of percussion and that of the ball are set in the middle at equal angles." This he reformulates: "Every blow hitting the object rebounds back at an angle similar to that of the percussion." This is effectively a paraphrase of the analogy made in the Problemata, a text that Leonardo possessed under the title Problema D'Arisstotile.17 But whereas his predecessor had been content to use the analogy in passing, Leonardo explores it in detail, turning first to the case of sound, and again using Aristotle's image of the bouncing ball as a starting point for his explanation:

This proposition appears clearly for if you strike a ball on a wall it will rebound back at an

angle equal to that of its percussion, that is, if the ball b., is thrown to c., it will return (back) by the line bc. because it is constrained to leave equal angles on the wall fg.; and if you throw it along the line bd., it will return back along the line de. and thus the line of percussion and the line of the ball will make one angle on the wall fg. situated in the middle of two equal angles as appears in the middle of fmn.

Hence, if one stands at b. and shouts, his voice is all in all the line fg. and all in (each) part. Hence, whoever stands, as was said, at b. and shouts, it will appear to him that he hears his voice in c. and comes to his ear by the line bc. And if, at the same time, one were at e., it would appear to him that he hears the voice b. in the place d., and [that] it comes by the line de.

Leonardo now formulates a general principle:

The voice is all in all and all in [each] part of the wall where it percusses, and that part which is formed in such a way that it is apt to send back its percussion, renders the voice in so many various particles of itself as are various the positions of the hearers.

This principle that the voice is "all in all and all in each part" is, as shall be seen presently, of great importance both for his acoustics and optics.

Immediately following this statement of a general principle Leonardo offers another specific example in diagram form (fig. 12) below which he writes another caption:The voice made at n. will percuss at the angles a,b,c. [and] d. and for every voice made at

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n., a,b,c,d. will send back the one [as] four.

Leonardo draws another diagram (fig. 13) this time with a longer caption:

If the person who stands at m. (and) shouts, his voice will be returned to him from r. and he who stands at n. will hear the echo from r., so near the percussion that he will confuse the one with the other, and will not be able to discern the original from the echo.

He ends with another brief note on acoustics: "The ear receives the species of sounds (voce) by straight, curved and bent lines and yet no bending can interrupt its operation." At the top of A19v Leonardo considers colours reflected in mirrors and provides a diagram of images in a mirror (fig. 14). That this is not a digression becomes clear from the proposition which follows in which he develops the comparison between images in a mirror and percussion of sound:

The voice percussed at the object will [re] turn to the air by a line of such obliquity as is the line of incidence, that is, the line which carries the voice from its source to the place where this voice is apt to be reformed and this voice acts like the similitude of a thing seen in a mirror which is all in all the mirror and all in [each] part. That is, let us say that the mirror is ab. and that the object mirrored is at c., [then] just as c. sees all the parts of the mirror, so too (do) all the parts of the mirror see c. Hence c. is all in all the mirror, since it is all in its parts and it is all in [each] part, because it is seen in as many various parts as are various the positions of the viewers. Hence if the object c. be in n. it appears as far inside as it is outside. Hence c. will be seen at d. and that which is at f., seeing the object d., will see it along a straight line i. hence, [he will see] the object d. at the part of the mirror e. and he who is at m. will see the object d. at t.

Thus both verbal and visual images are percussed; both are reflected and possess the property that they are "all in all and all in each part." On A94v (BN 2038 14v, TPL156, C.1492), this time without a diagram, Leonardo returns to his comparison between a bouncing ball and the percussion of light:

On reverberationReverberations are caused by bodies of a clear quality, of flat and semidense

surfaces that are percussed by the light which, like the bounce of a ball, repercusses it at the first object.

Nineteen folios later on A113r (BN 2038 32r, c.1492) Leonardo takes up once again his comparison between percussing light and bouncing ball in a passage entitled (fig. 15):

How one is to understand which part of the body must be more or less luminous than the other.

If f. be the light and the head be the body illuminated by it, (and) that part of this head which receives above it the ray between more equal angles will be more illuminated and that part which will receive the rays under less equal angles will be less luminous. And this light in its function is like a blow, insomuch that a blow that falls under equal angles will be

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in a first degree of power and when it falls under unequal [angles] it will be that much less powerful than the first, by the extent to which the angles are more disform.

For example: if you throw a ball at a wall, the extremities of which are equidistant from you, the blow will fall between equal angles and if you throw the ball at that wall, standing at one of its extremities, the ball will fall under unequal angles and the blow will not remain.

Once again it is instructive to compare Leonardo's passages with tradition. Aristotle, had considered the blow of sound with a bouncing ball and had implied that this was comparable with the action of light. The author of the Problemata had, in turn, compared the blow of light on a mirror with a projectile rebounding from a polished surface. But each of these comparisons had served as isolated examples. By contrast, in the Manuscript A passages just cited (19rv, 94v and 113r), Leonardo has made explicit the connections between these comparisons and his integrated them into a coherent framework such that the percussion/blow of sound and light are equivalent to the percussion/blow of a bouncing ball or similar projectile. At the same time he has stressed how all percussion obeys the fundamental law of catoptrics, whereby the angle of incidence equals the angle of reflection.

4. Percussion and Leonardo’s Physics in General

In order to appreciate how these examples of percussion in light and sound fit into Leonardo's concept of physics a digression is useful (a) to consider other examples of percussion and b) to examine how percussion fits into a larger mechanistic scheme. The author of the Problemata had been content to suggest in passing that projectiles, like images in mirrors, had their angle of incidence equal to angle of reflection. Leonardo, by contrast, examines this comparison in detail. On C28r (c.1490) he draws three diagrams showing balls in percussion (fig. 17), two of which illustrate the reflection principle. The caption accompanying one diagram, entitled "Blow," is in general terms. It is followed by a specific description:

The angle [of incidence] caused by the percussion of equal spherical bodies is always equal

to that of reflection.

If ef. were a wall, the ball s., departing from b., and knocked against this wall, would rebound to a. and likewise, the ball t., departing from fc., will rebound to d., after its percussion in the ball s.

A series of fourteen diagram ons A 8r (fig. 16, c.1492) illustrates Leonardo's determination to explore all the variants of this principle. On A22r he draws five further sketches of reflected projectiles (fig. 18). On A24r he compares the distance covered by a bouncing ball with that of one hurled directly through the air (fig. 19) a problem to which he returns on 161[13]r (c.1497-1499, fig. 20) this time in answer to a question.

(figure)

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Fig.15: Visual list of fourteen types of percussion, A8r

(figure)

Figs. 17-18: Further examples of reflected percussion. Fig. 17, C28r; Fig. 18, A22r.

(figure)

Figs. 19-21: Comparison between direct and reflected percussion Fig. 19, A24r; fig. 20, 161[13]r; fig. 21, 122r.

“I ask if the movement made by the stone in a continual line is equal to movement which will be in a reflected line, that is to say before the bound and after the bound?” Another diagram (fig. 21) on 122r answers a related question:

I ask in [the case of] an equal quantity of motion made by 2 equally heavy bodies which will give the greater percussion to its object, the direct motion ab. or the reflected, or genuflected motion ac?

On I128[80]r (c.1497-1499) he poses a related question and outlines an experiment:

About the bounce

If the first bounce is ten braccia, tell me how much will the second be? Hold the ball in such a way that it marks the place where it strikes the marble or other hard surface and follow this rule for all successive rules and thus make the general rule.

Leonardo pursues this problem on I14v, this time using the example of a man on stairs, in a passage headed:

On movement and percussion

If one descends from step to step making a jump from one to the other, (then) if you joined together all the powers of the percussions and weight which such a man would give when he fell by a straight line perpendicularly from the head to the foot of those stairs from the height.

Again if this man fell from a height hitting from degree to degree objects which bent like a

spring in such a way that the percussion from one to the other is small, you would find that such a man at the last part of his descent would have diminished his percussion as much had he fallen by a free and perpendicular line as in adding up all the percussions that were made at each degree of the said descent on these said springs.

(figure)

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Figs. 22-24: Visual demonstrations that the angle of incidence equals the angle of reflection in the percussion of objects. Fig. 22, L42r; fig. 23, K1v; fig. 24, E28v.

On Forster II 45v (c.1495) this problem recurs. This phenomenon of a problem being raised in one passage and then answered elsewhere, perhaps in a different manuscript, may strike us as irregular, but with Leonardo it is almost a norm. Hence the basic problem of projectiles being reflected like images in a mirror, described on C28r (fig. 17, c.1490-1491) is repeated on L42r (fig. 22, c.1497-1503), K1v (fig. 23, c.1503-1050), F22v (fig. 25, c.1508) and E28v (fig. 24, c.1513-1514). Similarly, the question of what occurs when two unequally sized balls collide, mentioned in passing on A8r (c.1492), is posed again in quantitative terms on I76[28](r) (fig. 26, c.1497-1499 cf. fig. 53). The question of percussion produced by a falling ball, broached on A22r, is posed afresh on I41(v) (c.1497) as a quantitative experiment. And the percussion of cannon-balls, discussed in isolated examples on A26r, 43v, 44r (c.1492) is considered more systematically on L43v (fig. 27, c.1497-1503).

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Figs. 25-27: Reflected percussion of projectiles. Fig. 25, F22v; fig. 26, I76[28]r; fig. 27, L43v.

(figure)

Figs. 28-31: Simple examples of percussion with cannon-balls. Fig. 28, BM192v; fig. 29, BM85r; figs. 30-31, BM91r.

(figure)

Figs. 32-39: Systematic demonstrations that the angle of incidence equals the angle of reflection. Fig. 32, BM90r; fig. 33, BM83v; figs. 34-35, BM82v; fig. 36, BM81v; fig. 37, BM93r; fig. 38, Mad I 147r; fig. 39, BM18r.

(figure)

Figs. 40-44 Systematic studies of percussion with cannon-balls. Figs. 45-49 Systematic studies of percussion with water. Fig. 40, BM36r; fig. 41, BM92v; figs. 42-43, BM128r; fig. 44, BM228v; fig. 45, CA81ra; fig. 46, Mad I 134v; fig. 47, Mad I 151r; fig. 48, F53v; fig. 49, F16v.

In the Codex Arundel this problem of the percussion of balls becomes an independent theme. Here we find a whole spectrum ranging from simple sketches on Arundel 192v (fig. 28, c.1505-1508) and 85r (fig. 29, c.1505, in which the percussion of a weight is compared directly with percussion in a mirror) or elementary geometrical diagrams on Arundel 95r (figs. 30-31, c.1505), to a series in which he approaches the problem more systematically: Arundel 90r (fig. 32, C.1505); 83v (fig. 33, c.1505), 82v (figs. 34-35, c.1505); 81v (fig. 36, c.1505), 93r (fig. 37, c.1505) in which the results are quantified (cf. Mad I147r, fig. 38, c.1499-1500) and BM Arundel 18r (fig. 39, c. 1508). This series culminates in five comparative diagrams: 36r (fig. 40, c.1505); 128r (fig. 41, c.1505); 92v (figs. 42-43, c.1505) and 226v (fig. 44, c.1500-1505). Leonardo is intent on

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applying his bouncing ball image wherever possible. On A63v (c.1492), for example, he uses it to explain how the percussion of water erodes the banks of rivers with its blows:

And this is seen clearly and it is understood that the waters which strike the banks of rivers act like balls percussed from walls which depart from these at angles similar to those of (the) percussion and are going to batter the side of the wall opposite.

On I115[67](v) he proceeds to illustrate graphically this comparison between a bouncing ball and bouncing water (fig. 50) a theme to which he returns on K1v (fig. 23, c.1503-1505) in the form of a question.

(figure)

Figs. 50-51: Percussion of bouncing balls and bouncing water. Fig. 50, I115[67](v); fig. 51, K99[19]r.

(figure)

Figs. 52-53: Reflected percussion in water and wind. Fig. 52, I114[66]v; fig. 53, BM276v.

Whether the stone or water struck by the incident mobile object follows (the) reflected motion in the way the incident mobile [object] would follow by itself after its percussion or not [?]

On K99[19]r (c.1503-1505) he makes further sketches concerning the reflection of bouncing water (fig. 51). This leads to comparative studies (figs. 45-49) analogous to those involving cannon balls (figs. 40-44). On BM Arundel 276 this principle of reflected percussion is applied also to wind (fig. 53 cf. fig. 52). Thus an image that had traditionally been mentioned in passing, Leonardo uses systematically with respect to light, sound, projectiles, water and wind. In his De sensu Aristotle had used the image of the bell with respect to the production of sound.18 For Leonardo the image of the bell and hammer also serves to illustrate his concept of percussion. Two early examples have already been cited (figs. 9-10). On A22v (c.1492) he returns to this image under the heading:

Of the blowThe blow on a bell leaves behind it its similtude impressed [on it], like the sun in the eye or odour in the air. But it is to be seen if the similitude of this blow remains in the bell or in the air and this you will learn [by] placing your ear on the surface of the bell after this blow.

Immediately following is another passage in which the resonance accompanying sound is discussed again under the heading:

Of the blowThe blow given to the bell will [make] respond and move somewhat another bell similar to it and the cord of a lute which is sounded will [make] respond and move another similar cord of like mouth in another lute and this you will see by putting a piece of straw on the

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cord similar to the one that is sounded.

In the late writings Leonardo returns to the bell and hammer comparison on G73r (c.1510-1515) under the heading:

Every impression tends to permanence or desires permanence.

One proves it with the impression made by the sun in the eye of the spectator and in the impression of sound made by the hammer that strikes the bell.

Every impression desires permanence ... (and the eye which looks at the sun proves it) as is shown to us [by] the image of movement impressed on the mobile [object].

Many other examples illustrating the physical nature of percussion could be cited. On A31r (c.1492), for example, he writes a draft, crosses it out, then adds an interjection concerning method:

I remind you that you make your propositions and that you adduce the aforementioned things with examples and not by propositions, which would be too simple and you will say as follows:

A general proposition is now given under the heading:

Experience

A blow given in some dense and heavy body passes naturally through this body and injures whatever finds itself in the surrounding dense or rare bodies (that there are).

This he illustrates with a specific example

(and) let there be many fish in (a) water, which enter(s) under a rock and if you give a big blow to this rock, all the fish which find themselves below or to the side of this rock will come, as if dead, to the surface of this water....

Why Leonardo is so concerned with percussion is explained, in part, by his belief that this concept can be quantitatively tested. In the Manuscript A (c.1492) he outlines several demonstrations/experiments on this theme. On A32v, for instance, he notes that 100 blows on a glass vase with a needle would not break the vase, yet one blow with a needle 100 times as heavy would break it. On A4r Leonardo proposes to compare the effects of a one pound hammer falling 100 times from a height of one braccio, with a 100 pound hammer falling one time from a height of one braccio. A related experiment is described on A23r:

On the blowWhether 10 blows of one pound by a blow that has fallen on a place, falling from one braccio [in] height fixing a nail of one braccio a given amount [will fix it] as much as would a combined weight of 10 pounds? This shows [that they do] not, for if you wished to fix a nail with the weight of another similar nail, this would be impossible, for [even] if you beat

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on this ten thousand similar blows, all would be nothing. And if you took a weight 20 times as much, its blow will be in proportion to the nail that you wish to fix.

(figure)

Figs. 54-55: Quantitative experiments concerning percussion of hammers and of light. Fig. 54, M83v; fig. 55, Forst III 58v.

In the next paragraph he compares combined percussion in voices. But he does not forget the problem. On Forster II.2 74r (c.1495-1497) he notes that "100 pounds given in one blow gives a greater percusssion that a million [blows] of one pound given one after the other." The problem is pursued on M83v (c.1499-1500) in the form of an experiment (fig. 54):

If the hammer of 10 pounds drives a nail into a [piece of] wood in one blow, a hammer of one pound will not drive then such a nail entirely into the said wood in 10 blows. And a nail less by a tenth part will not be the more driven in by the said hammer of one pound in a single blow even if it is in equal proportion to the first given, because what is lacking is that the hardness of the wood does not diminish the properties of its resistance, that is to say, that it is hard as before.

If you wish to treat of the proportions of movement of things which penetrate wood driven

by the power of a blow, you need to consider the nature of the weight which strikes and the place where the percussed thing is fixed.

A passage on A3v (c.1492) suggests that such experiments were intended to test a more general hypothesis: "The powers (potenzie) separated will not have, all at one time and in one operation that power (virtu) and effect (alturita) as when they are united." On A3v, Leonardo offers further test cases involving separate and combined voices, forces, supports and, to return to the theme of optics, lights:

Many little luminous bodies joined together will be of greater power in themselves than they would be being separate. The proof you will see if you take many lights in a straight line and you stand at a certain distance facing the centre of this line and you note the quality of the light made by these lights and then join them together. You will see that the place where you stood is more luminous than before. Again it is known that the stars are of an equal light to that of the moon and if it were possible to join them together, which would compose a body much larger than that of the moon, and nonetheless, even though it be calm and all [the stars are] shining, if the moon be not in our hemisphere our part of the world remains dark.

Leonardo pursues this theme of separate and combined lights on Forster III 58v (fig. 55, c.1493) under the heading:

On the Duplication of Lights.

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If one light is of 4 ounces (and) it appears that joined together two of these lights make 8 ounces.

Again, the problem whether the combined light of two candles is more intense than that of one candle on its own, is not new with Leonardo. It is raised in a contemporary commentary on Aristotle's de Generatione under the quaestio: "Whether a like can act on a like."19 Nonetheless, the context is fundamentally different. The Aristotelian commentator had been concerned with the example as an isolated instance of an abstract philosophical concept, Leonardo is interested in the example to test a principle in his physics of percussion. For Leonardo percussion represents a basic law of nature by means of which he can account for the physics of light, sound, wind, water and various solid projectiles.

Percussion, as we have already mentioned earlier, is a complex concept which involves all four of his natural powers. This is implicit in a brief definition that he drafts on A27r (c.1492): "Blow (colpo) is terminus of the motion caused by force and operated by bodies in resisting objects in indivisible time". On A27v (c.1492) this relationship with the other powers becomes clearer in the course of three longer drafts, headed:

BlowBlow I say to be the terminus of speedy motion made by bodies in resisting objects. This same is the cause of all sounds, breaker and transmuter of various things, recauser of second motion. No thing is shorter, nor of greater power and it goes diversifying itself by means of the causes.

Blow is the terminus of speedy motion caused by force and operated by weight in an object, caused by sounds, transmuter of its effect. And no thing is of shorter operation or of greater power. Its result is of greatest velocity and penetration in every counterposed variety of object and departing from its source through circular movement and it is all in all and all in [each] part.

Blow is the terminus of speedy motion, caused by force and operated by bodies in the resisting objects. From this derive the sounds, from this the breakings, and no thing is of shorter operation, nor of greater power. Its result is of the greatest velocity and penetration in every counterposed variety of object.

In attempting to define blow, Leonardo refers to the related powers of force, movement and weight. These four powers dominate the succeeding twelve folios as Leonardo explores various definitions and examples. Then it becomes clear that these four powers are fully interdependent and belong to a larger system.

5. Percussion and the Four Powers

The terms violence, weight, force, motion and blow are all familiar from he Aristotelian tradition. But on A35r (c.1492) Leonardo combines them in a fresh way:

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Violence is composed of 4 things, that is, of weight, force, motion and blow (colpo); and some say that violence is composed of 3 passions, that is, force, motion and blow and that which is more powerful has less life, that is, (the) blow; second, is (the) force; third in weakness would be motion and if weight be accepted in this number, it is the weakest and most eternal of any of the above mentioned.

In the paragraph following Leonardo turns to weight and on the verso of the folio he continues the discussion under the heading: “Of weight, force, motion and blow.” The interdependence of the powers he notes in passing on CA173vb (c.1490-1495): "Motion is not without percussion. Percussion is not without body and weight." As Leonardo's thought matures the "four things" which began as components of violence, assume an independent identity as accidental powers. These powers are "spiritual" and "incorporeal" (B63r, c.1490). That which he had at first termed blow, he now terms percussion. On BM Arundel 181r (c.1497-1500) he drafts a revised definition:

Gravity and force (are) together with percussion are 3 accidental powers which are nonetheless to be called generators of motion (which [is] created) generated by this.Gravity, (the) force along with motion are nonetheless to be said to be generators of motion which is daughter of this. (For from the motion is again caused motion. This motion is generated by it. Whence it is concluded that the one without the other without the one [does] not.) For these are again caused by motion; motion without these cannot be...

These versions he crosses out. The next version he finds acceptable:

Gravity and force which are interchangeably daughters and mothers of motion and sisters of impetus and of percussion always combat their cause which life and (if force and gravity lose their being) conquer one another and kill [themselves].

On BM Arundel 151v (c.1495-1497) he pursues the theme:

Material motion along with gravity, force and percussion are the four (powers) accidental powers with which all the works of mortals have their being and their death.

On the recto of the same fol. 151 he again modifies the definition: "Force, along with material motion, weight along with percussion are the four accidental powers with which all the works of mortals have their being and their death." This he repeats almost verbatim on Forster II2 116v (c.1495-1497): "Gravity, force, accidental motion along with percussion are the four accidental powers with which all the evident works of mortals have their being and their death." In his anatomical writings Leonardo develops this concept of the four powers into a basic principle of organization. On W19060r, (KP153r, c.1509-1510), for example, he notes:

Why nature cannot give motion to animals without mechanical instruments as is shown byme in this book on the motive works of this nature made in animals and for this [reason] I have composed the laws (reghole) in the 4 powers of nature, without which nothing can by itself, give local motion to these animals.

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Hence I shall first describe local motion and how it gives birth to and is born from each of the other three powers. Then I shall describe natural weight, even if no weight can be said to be other than accidental, but it is preferred to name it thus in order to separate it from force which is of the nature of weight in all its operations and for this [reason] it is named accidental weight and this force is placed as the 3rd power of nature or naturated (because) the fourth and ultimate power is said [to be] percussion, that is, terminus or impediment of motion. And I shall say first of all that every insensible local motion is generated by a sensible motor, as in a clock, the counter-weight, lifted on high by man, its motor.

Here the four powers have become a crucial theme. As Keele,20 has so lucidly shown, the powers are a key to the whole of his science. For our purposes it is important that we remain conscious of this framework, as we focus on the concept of percussion which, as is clear from a late passage on G62v (c.1510-1515), Leonardo came to regard as the most significant of the powers:

Among the accidental powers of nature, percussion exceeds by a great excess each of the others which are produced by the motors of heavy bodies in equal time with various movement, weight and force. This percussion is divided into simple and composed. Simple is that [in] which the motor is joined with the mobile percussor at the junction of the percussed place. Composed is that for which the mobile which strikes does not terminate its movement at the place of its impression, as (is) the hammer which strikes the coin which impresses coins. And this composed percussion is much feebler than simple percussionbecause if the mouth of the hammer had attached [itself] to the coin which it is to press, which it had struck on the die of the impression and [such] that in this head of the hammer there had been engraved the concavity opposite the money, then the impression would be more expedite and clean-cut on its side percussed by simple motion than on the side of composed percussion, as is the money which remains struck in the corner where the descent of the hammer has struck it and the percussion is reflected and reverberated against the front of the hammer.

Whether he terms it colpo in the early period or percussione in the late period, his concept of percussion remains consistent, inviting a mechanistic interpretation.

6. Percussion in Solid and Fluid Particles

Thus far we have concentrated on instances of percussion involving solid objects. But Leonardo was also interested in the percussion of both solid and fluid particles. For example, on C22v (fig. 56, c.1490) he considers what happens when water falls on water:

Water or another thing which falls on water makes that this water which receives the blow enlarges under this blow and surrounds it and the cause of this blow being overcome it passes above this in pyramidal form:

The reason for this is that, [with] a drop of water, falling from a roof on other water, that

part of the water which receives this blow, cannot have place, nor flee behind other water

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with that velocity with which it was assailed because it would need be that it raises too much weight in order to enter beneath such a quantity of water. Hence, needing to obey with its own flight being chased by the thing which chases it from its site and finding the nearby water which does not receive the blow (which) is not prepared to make a similar flight, this first cannot penetrate it and hence it seeks the shortest way and runs to the thing that produces the least resistance for it, that is, the air. And this first circle which surrounds the percussed place, closing it furiously, because it stood raised outside the common surface of the water, reduces the water which flees upwards in a pyramidal form. And if you believed that the water which falls was that which leaps up, make the water fall on a pebble and you will see the water likewise leaping up and not the pebble.

(figure)

Figs. 59-61: Connections between percussion, pyramids and quadrature of the circle. Figs. 59-60, Mad I 126v; fig. 61, BM188r.

Leonardo develops a similar conclusion regarding the pyramidal qualities of small dust particles. In a note on A32v (c.1492) this idea is implicit when he mentions that "if you beat on a flat surface you will see the dust that is there reduce itself to small heaps." On Forster II1 16r (c.1495-1497) he proposes a general rule for the pyramids thus formed by percussion (fig. 57): "Every hillock of sand either in a plane or on a slope will be twice as wide in its base as its axis is long." On Forster II2 68v (c.1495-1497) he refers to experiments with circular motion and then adds: "I want to do the same with dust that is beaten." The accompanying diagram shows a hammer and a little pyramid of dust (fig. 58). On Mad I 126v he describes this phenomenon in greater detail (fig. 59-60):

(figure)

Figs. 62-65: Pyramidal and inverted pyramidal percussion. Fig. 62, I 60[12]v; fig. 63, L64v; figs. 64-65, L1r.

If you beat the board [with] dust, this dust will reduce [itself] to various little heaps, which little heaps will always pour such pulver through the point of their pyramid and descend to its base. Then having re-entered below it will pass through its centre and will fall again through the summit of this hillock. And so it will go passing again through the orthogonal triangle amn., as many times as such percussion follows.

Alongside is another passage describing the pyramidal hillock that has been formed (fig. 59):

The hillock made by many particles, of some solid material, which falls from a single aperture always has its width twice its height.

Let h. be the aperture where grains or dust are poured; let an. be the height of this "mountain, om. its width. You thus see that om. enters twice into an.

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The geometrical form of the accompanying diagram (fig. 60 cf. 61) brings to light links with quadrature of the circle problems. On L64v (c.1497-1504) Leonardo pursues this theme asking: "What difference is there between the percussion of the thing that is united and the thing that is disunited?" His reply is in the form of a sketch (fig. 63 cf.62) and a brief text: "Grain thrown in the air with a sieve leaps pyramidally". On L1r (c.1497-1504) he considers the related problem of water striking a flat surface which he again illustrates (figs. 64-65):

The water which falls pyramidally by a perpendicular line will rebound up and will finish the apex towards the base of such a pyramid and will then intersect itself and pass beyond and fall below.

On F61r (c.1508) he returns to the example of hillocks produced by striking a table and suggests that this principle accounts for the sand dunes along the Po and in Libya. From such examples a connection between percussion and pyramidal shapes becomes clear which, as we shall see presently, explains aspects of his physics of light.

7. Percussion in Water and Air: Classical and Mediaeval Precedents

Leonardo's studies of percussion extended equally to water and air. Here again he was working within a well-established tradition, which it will be well to consider by way of introduction. Aristotle, in his De anima had mentioned analogies between water and air.21 He had also explained that the effect of any blow or contact was dependent on the medium involved:

Thus if an object is dipped into wax the movement goes on until submersion has taken place, and in stone it goes no distance at all, while in water the disturbance goes far beyond the object dipped: in air the disturbance is propagated furthest of all...22

The Roman architect Vitruvius developed this comparison between different mediums. Speaking of the voice he wrote:

It moves in an endless number of circular rounds, like the innumerably increasing circular waves which appear when a stone is thrown into smooth water, and which keep on spreading indefinitely from the centre unless interrupted by narrow limits, or by some obstruction which prevents such waves from reaching their end in due formation.23

The Stoic philosopher Seneca in his Natural Questions, in trying to account for haloes around the sun, took this analogy of the stone in water and applied it to light:

When a stone is thrown into a pond, the water is observed to part in numerous circles, which, very narrow at first, gradually widen out more and more until the impulse disappears, lost in the surface of the smooth water beyond. Let us suppose something of the same kind to occur in the atmosphere. When condensed, it is capable of receiving an impact: the light of the sun, moon, or any heavenly body encountering it forces it to recede in the form of circles. Moisture, be it observed, and air, and everything else flat takes shape from a blow, is driven into the same form as that possessed by the object that strikes it. Now every kind of light is round. Therefore, the air when struck by light will assume this

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form. Accordingly the Greeks gave the name Threshing-floor (i.e. Halo) to a brightness of this kind, because spaces set aside for threshing corn were, as a rule, round.24

For our purposes this passage is significant for at least two reasons. First, it introduces the analogy between the circular waves caused by a stone thrown into water and waves of light in the atmosphere. Second, it relates this analogy to the concept of a blow. Leonardo, as will be shown, developed both of these ideas. In the third century Diogenes Laertius referred to the analogy between waves in water and waves in air while discussing Zeno in his Lives of the Eminent Philosophers (VII:158):

We hear when the air between the sonant body and the organ of hearing suffers concussion, a vibration which spreads spherically and then forms waves and strikes upon the ears, just as the water in a reservoir forms wavy circles when a stone is thrown into it.25

In the thirteenth century, John Pecham, the Archbishop of Canterbury, used this same analogy in his standard textbook on optics, the Perspectiva communis:

Although, as should be known, all pyramids in a single body of illumination constitute essentially one light, they nevertheless differ virtually, that is, in efficacy. In the same way, when a stone is thrown into water, different circles are generated which, nonetheless, do not divide the water, so that it is in some way not all separate. From any (given) point of a luminous body a ray of light departs powerfully and the more direct it is, the more strength it has.26

8. Percussion in Water and Air. Leonardo's Contributions

On Mad II 2v (c.1494) there is a "record of the books that I keep locked up in the chest." Among these in Pecham's Prospettiva commune.27 As early as 1490, and probably while he was staying in Pavia with Fazio Cardan, editor of the first printed version of Pecham's work, Leonardo translated, or at least copied out, on CA203ra a translation of the opening paragraph of Pecham's treatise. It is therefore very likely that Pecham's comparison between waves in water and waves in air served as a direct source for Leonardo. On CA373rb(c.1492-1497) Leonardo drafts a passage, on circular diffusion of images in the air:

Every (opaque) body, placed in the luminous air fills circularly (this air) the infinite parts of this (air) with its similitudes, and is all in all and all in the part and goes diminishing itsspecies by the equidistant space surrounding it like the...

Of the 4 elements and 2...

Immediately following this he explores the traditional analogy between waves in water and waves in air in a series of eight more drafts which reveal how laboriously he reformulates an idea:

1. (Just as) The stone thrown in the water fills this (with waves) of circles makes itself the centre of various circles, which have for their centre the percussed place.

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And the air similarly fills itself with circles, of which (they make themselves the centre of the vo[ice] their centres are sounds and voices made in this.

As the water with various circles surrounds the place percussed by the stone.

(As ) The stone, where the summit of the water percusses, causes circles around it, which go expanding (themselves) to such an extent that they disappear, and also (and) the air, percussed by a voice or by a noise, (by) similarly departing circularly, is gradually lost, such that the nearest is better heard and the far away less heard.

a[s].

Just as the stone thrown in the water makes a centre (in various circles) is cause of various

circles, so too, the sound made in the air, sends to the equidistant ears equal[lly], circularly spreads its (sound) voice.

Just as (that) the air (which is re) percussed by the voice (is that) the water (which is percussed) by the stone, goes in a circular movement (spreading out and fleeing from their source) showing their source. Which circles make themselves centre of the percussed place and the more they become distant.

Just as the water and the air [are] percussed, the one by the voice and the other by the stone,

you will see at the water by various circles demonstrate the percussed place, thus you will feel at an equal distance the sound of the voice made in the air

(As)As (in bodies)As circularly the species of bodies are spread beneath the infinite parts of the nearby

air....

To the modern reader Leonardo's almost obsessive reformulation of ideas is often boring. Nonetheless, it offers a clue why there are so few instances where he merely copies: his transformative mind is too active. These drafts on CA373rb (c.1492) lead to a clear formulation on A9v (c.1492):

Just as the stone thrown in the water produces a centre and causes various circles, sound

made in the air, spreads circularly.

(figure)

Figs. 66-69: Percussion, circular propagation and non-interference of waves. Fig. 66, A61r; fig. 67, Forst III 76r; fig. 68, H69[21]v; fig. 69, Leic. 23r.

(figure)

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Figs. 70-73: Circles representing waves of water or air inter-secting without interference on CA300rb.

(figure)

Figs. 74-75: Non-interference of circular wave patterns in Jan van Eyck's Ghent Altapiece and in Leonardo's treatise BM135r.

Thus every body placed between the luminous air spreads circularly and fills the surrounding parts with its infinite similitudes and appears all in all and all in every part.There follows a diagram (fig. 133) and a brief text: "This is proved by experience, because if you close off a window facing the West and make an aperture," which then breaks off. Here he has linked the concept of circular wave motion in water and air with (a) the principle that images are "all in all and all in every part" and (b) the camera obscura. (We shall return to this remarkable nexus of ideas, central to his physics of light, cf below pp. ). The topic of circular wave motion in water, which just broached on A9v (c.1492), is developed on A61r (fig. 66, c.1492):

Even though voices which penetrate this air part with circular movements from their sources, nevertheless, the circles springing from different origins meet together without any interference and penetrate and pass [through] one another always maintaining their causes through [their] centre.

There follows an assumption: "Since in all cases of motion, water has great conformity with the air I shall add to the above (mentioned) proposition with an example." This assumption is because it explains why Leonardo uses the example of a stone thrown in to water to illustrate effects of sound and light in the air, as for instance on A73 (c.1492), where he compares turbulent water and turbulent air. It also explains why he sometimes draws intersecting circles in contexts where they could equally represent waves of water or waves of air as on CA300rb (figs. 70-73, 1508-1510). In addition it accounts for a basic aspect of structure in Leonardo's notes: why, for instance, in the Manuscript C which is devoted primarily to light and shade, Leonardo should discuss water together with optics (cf. C6v, 22r-28v; figs. ); a combination of themes that dominates the Manuscript F and recurs throughout his writings.

(figure)

Figs. 76-77: Demonstrations that circular waves can contract as well as expand. Fig. 76, CA126vb; fig. 77, Mad I 126v.

In his example on A61r (C1492) Leonardo examines how two spreading waves in water, - and by implication, in air - can meet and cross without interference (fig. 66, cf. figs. 67-75):

I say: if at the same time you throw 2 small stones somewhat distant from one another on an expanse of water without motion, you will see caused around these two said percussions. 2 separate quantities of circles, which quantities growing, will come to meet one another and then incorporate one another, the one circle intersecting the other, always maintaining as centre the places percussed by the stones. And the reason is that even if there appears some

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demonstration of movement, the water does not depart from its site, because the aperture made by the stones immediately closes itself again and this motion, made by the sudden opening and closing of the water makes in it a certain stir which can sooner be called a tremor than a movement and, in order that what I say be made more manifest to you, call to mind those fescue grasses which through their lightness stand above the water, which by the waves made beneath them by the advent of the circles will nevertheless not part from their first site. Now this stir of the water being a tremor, rather than a movement, they [the waves] cannot, in meeting, break one another, since the water, having all its parts of a same quality, it is necessary that the parts convey (apichino) this tremor from one to the other without moving from their place, for the water, staying in its place, one can readily take this tremor from nearby parts and send it to neighbouring parts, always diminishing its power until the end.

(figure)

Figs. 78-81: Circular waves in a bowl and transformation from a triangular waves to circular waves. Fig. 78, Leic. 12v; Figs. 79-81, Mad I 95v.

(figure)

Figs. 82-85: Transformations from triangular waves to circular waves. Fig. 82, CA173rb; fig. 83, CA199vb; figs. 84-85, Leic. 14v.

Leonardo wishes to demonstrate that these circular waves can contract as well as expand, and hence describes an experiment on CA126vb (fig. 76, c.1490-1492):

Here one gives an exit to the water close to the surface and one demands what part of the surface of the water will take more motion, speedily or more slowly in taking the water to such an exit. And in order to make a rule you will put [into the water] particles of things which remain noticeable, which are equal, as are some minute seeds of herbs, and place them in a circle equidistant from the exit at rmf and note how that which first reaches the mouth closes the water and retains the circle.

On Mad I 126v (fig. 77, c.1493-1495) he describes another demonstration of the same principle:

If a bowl, full of water, is beaten on one of its sides, it is certain that the water will begin

circular motions which will diminish at the centre of its surface. But if the water which is contained in the bowl be struck at its centre, then circular movement will be caused which, going from minimal circles, will terminate in very large circles.

He pursues this problem on Mad I 95v (figs. 79-81, c.1499-1500), under the heading:

Of percussion and motion in waterWater, which is contained in some bowl, if it be percussed in its centre, will cause circles,

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which will make a common centre from that percussed place. But if the bowl be percussed from the outside, then the circles will be caused by the contact which the water has with the bowl and will flee diminishing towards the centre of this bowl. And the percussion having been made at this centre, they will return back, increasing until the contact of the bowl, and then they will redo this to the centre of the circle so many times that they lose themselves.

If the percussion made on the water in the middle of its bowl, be of a triangular shape, this,

in long movement, will make itself nearly circular and will return, after the percussion made at the banks, towards the centre and then to the banks. In such a way that at its end, such motion of the water will be of a nearly perfect rotundity. And this motion will only be in that part which borders with the air.

This problem of a triangular shape mentioned in the last paragraph is again one which can be traced with some precision. On CA173rb (c.1500) Leonardo makes a quick sketch of a circle within which he inscribes three simple dots (fig. 82). In the caption alongside he asks: "Why the triangular percussion makes a round wave in the water, if the circular waves penetrate one another in their encounters?" In a further note on CA199vb (also c.1500) he restates the problem: "Test if the triangle thrown in still water in the end makes its wave of perfect circularity." The diagram alongside shows a triangle clearly inscribed within a circle (fig. 83). The diagrams (figs. 84-85) and text on mad I 95v (c.1499-1500) clearly represent the next step.

In his camera obscura studies Leonardo also experiments with triangular apertures (figs. ; cf. below pp. ) while exploring how light rays take on a rounded shape with distance. This parallel is no coincidence. He considers water as a medium that helps render visible corresponding effects in air and as such, water is his equivalent to a modern slow motion camera, allowing him to study more closely the transformations of waves from one shape to another.

In volume one (pp. ) it was noted that Leonardo's concern with transformational geometry was linked with his studies of the movement of the heart. In these transformations of waves of water and/or light it can be seen that Leonardo's fascination with De ludo geometrico had other links with physical reality (figs. ). For him it is far more than an abstract geometrical game; transformational geometry is a key to natural phenomena. Whence he claims that the quadrature of curvilinear surfaces is the goal of the geometrical sciences, and equates science with the ability of transforming one shape into another and back again. Meanwhile he has become preoccupied with another problem related to circular wave motion, which he posed as a question on Forster II 76r (fig. 67, c.1493): "Why do the circles of water not break when they intersect?" His preliminary answer is an experiment on H31v (fig. 87, March 1494):

The stone thrown into dead water will make equal circulation of motion, the water being of equal profundity.

If two stones be thrown [into water] the one near the other by an interval of one braccia, the circles of the water will grow equally the one body into the body of the other without the one causing the other interference (sanza guastamento lunu dellaltro)But if the base be not equal the circulation [i.e. production of circular waves] will not be

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equal except at the surface.

On H 69[21]v (c.1494) he again draws two non-interfering wave patterns (fig. 67). Below this he draws four diagrams in which he compares the non-interference of streams of water and wine or tinted water. Accompanying this is a brief text: "[Streams of] water of equal course and descent which move(s) - counter to one another pass(es) through one another without changing natural course." On H 67[19] (c.1494) he compares waves in water, with waves in other mediums:

Water percussed by water makes circles with regard to the place percussed. Over a long distance the voice in air [does the same], [over a] longer [distance] in fire. More so the mind in the universe. But why does the finite not extend to the infinite?

In another passage on CA175ra (c.1494) he again considers the circular waves of vision:

... each eye in itself (has a cen[tre]) causes infinite visual lines which are in the eye of so much greater power to the extent that they are closer to the centric line which is in the first degree of visual power. Now these lines spread out circularly from this centric line and are adopted in the powers of the species and similitudes of the things which are placed in front of the eyes.

(figure)

Figs. 86-87: Circular seats in the Roman theatre at Aspendos and circular waves of light/sight in Caesariano's commentary on Vitruvius (1521).

(figure)

(Figs. 88-91: Intersecting wave fronts on Ca83vb.

(figure)

Figs. 92-100: Waves in agitated water. Fig. 92, H31v; fig. 93, Leic. 14v; fig. 94, I87[39]r; fig. 95, I86[38]v; figs. 96-97, I87[39]r; figs. 98-100, Leic. 14v.

This passage bears comparison with a diagram in Caesariano's commentary on Vitruvius23 (fig. 87). On CA83vb (c.1508) Leonardo devotes four more sketches to the problem (figs. 88-91) this time with only a brief caption: "Every part of the wave which percusses on the other waves is reflected towards the centres of their centres." Meanwhile he has also been reconsidering the possible interference of expanding circular waves on M I86[38](v) (fig. 95, c.1497):

I ask whether a circle, which has its growth, which meets with the growth of another enters with its wave penetrating the wave of the other, as n. passes in c. at the same time that n. passes in d. or, truly, if at their percussion they bounce back under equal angles as when c.

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entering n. jumps to d. and likewise d., hitting n., bouncing back on c.

This is a beautiful and subtle test.

Not content with a static situation, on the following folio, I87[39](r), he examines what happens in the case of flowing water (fig.s 96-97):

If the stone is thrown in motionless water, its circles are equidistant from its centre. But if the stream moves the figures will form long figures, nearly ovoid and will go with their centre, from the place where it was created, following the course of the [stream].

Leonardo considers further aspects of the physics of wave motion on Leic. 14v (figs. 84-85, 98-100, c.1500-1516):

More speedy (is the wave) is the motion of the wave than the motion of the water which generates it. This is seen in throwing a stone in a dead [pool of] water which generates around the place of percussion circular motion, which is speedy, (which) and the water, which (it) produces such circular inundation, does not move from its situation, nor even the things which sustain themselves above the water. For what reason are the reverse sides of the waves of the water more circular than tided with various faces? This arises because the water which moves itself in a curved way at the beginning cannot direct its course in a straight line if it does not find water of less resistance than the first and, not finding this, it needs be that it observes in the middle and the end of motion, that which from the beginning was given it.

Elsewhere is the same treatise on Leic. 23v he again compares (cf. A61v above) the equivalence between waves in various elements

The wave of the air takes the same role under the element of fire as the wave of water under air or the wave of sand, that is, earth under water and their motions are in proportion as their motive forces are amongst themselves.

The problem continues to fascinate him. On CA77vb (c.1505, cf. CA264rh, C.1500-1505 where he speaks of the sphere of water seeking perfect rotundity) Leonardo notes that the voice "acts like a circular wave in the sea." On BM Arundel 204r(c.1505-1508) he develops his air/water analogy:

(figure)

Figs. 101-105: Percussion and circular waves beyond an aperture-like channel. Figs. 101-103, Leic. 14v; Figs. 104-105, CA281ra.

(figure)

Figs. 106-109 CA281ra Lunule studies on CA281ra. Cf. figs. 104-105.

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More speedy (is the wave) is the motion of the wave than the motion of the water which generates it. This is seen in throwing a stone in a dead [pool of] water which generates around the place of percussion circular motion, which is speedy, (which) and the water, which (it) produces such circular inundation, does not move from its situation, nor even the things which sustain themselves above the water. For what reason are the reverse sides of the waves of the water more circular than sided with various faces? This arises because the water which moves itself in a curved way at the beginning cannot direct its course in a straight line if it does not find water of less resistance than the first and, not finding this, it needs be that it observes in the middle and the end of motion, that which from the beginning was given it.

On this same folio (Leic. 14v) he draws three sketches of water in a very narrow canal, with waves extending beyond this in a half circle (figs. 101-103). He draws related diagrams on CA281ra (figs. 104-105, 1516-1517) where they appear in connection with lunule studies (figs. 106-109). In Leonardo's mind there is a connection between the (theoretical) intersections and transformations of circles in geometry and the (practical) intersections and transformations of circles in water. His associative mind compares waves in various elements as on A61v cited earlier, or on Leic. 23r:

The wave of the air takes the same role under the element of fire as the wave of water under air or the wave of sand, that is, earth under water and their motions are in proportion as their motive forces are amongst themselves.

(figure)

Fig. 110 Circular sound waves in the open air on A43r;

Fig. 111 Experiment with circular waves in water passing through an aperture (cf. Newton).

Such comparison fascinates him. On CA77vb (c.1505) he notes that the voice "acts like a circular wave in the sea." On BM204r (c.1505-1508) he develops his analogy between air and water:

The air is a liquid body invested with a spherical surface and is penetrated by solar rays which restrict themselves in their concourse and the more they are heated the more they are restricted at the point of their concourse.

He repeats the comparison between circular expansion in water and air on CA112vb (c.1508-1510) and CA251rb (c.1510-1515). On CA84va (c.1510-1515) he returns once more to the non-interference of waves:

If you throw a stone into a body of water (pelago) with different sides, all the waves that strike these sides reflect towards the percussion, and in encountering other incident [waves] they never impede the course of one another.... All the impressions of percussions made on the water can penetrate one another without their destruction.

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Meanwhile, on CA77vb (c.1507) Leonardo had considered a special case in the analogy between waves of air and water, namely, when the waves are broken or interrupted:

...The note of the echo is either continuous or intermittent

...The note of the echo is intermittent when the place which produces it is broken and interrupted. It is single when it is produced in one place only. Accompanied is when it is generated in several briefly. Long is when it turns round in the percussed bell or in [a] cistern or other concavity, or clouds in which the voice extends itself in degrees of space with degrees of time and always diminishing, the medium being uniform and it acts like the circular wave in a pond m.

(figure)

Figs. 112-114: Experiments with circular waves of water passing through apertures on W19106v (K/P 126r).

On W19106v (K/P 126r, c.1510) he describes an experiment concerning such interrupted waves (figs. 111-114):

Go in a boat and make the enclosed place nmope and inside it put two pieces of board sr and tr and give percussion to a and see if the interrupted wave passes with its convenient part to bc and that which you experiment with the wave cut from the circular wave ... of the water, such you will conceive as having experimented on the part of the wave of the air which passes through the tiny aperture where the human voice, enclosed in a box, passes [through],m as I heard at Campi with someone enclosed in a cask open at the bunghole.

More than a century and a half later Newton repeated this experiment in his Principia.28 Leonardo has crossed out this passage on W19106v (K/P126r, c.1510). Immediately to the left he draws a diagram (fig. 113) without text: it shows a rectangular enclosure, three sides of which contain an aperture centrally positioned. An object has been dropped into the centre of this enclosure and has produced circular waves. These continue beyond the apertures and maintain their characteristic circularity. To the North and South of this rectangular enclosure two other objects have been dropped into the water and the circles which emanate from these objects, having passed through the apertures, move towards the centre of the enclosure, crossing the outgoing waves without interference. To the left is a tiny sketch showing two circles of waves meeting without interference (fig. 112).

(figure)

Figs. 115-118: Four examples of percussion. In Leonardo's mind the blow of an axe, falling water, the blow of a hammer and light striking the eye are all examples of a single concept. Percussion, motion, gravity, and force constitute his four powers by means of which he believes that he can account for the whole of reality. Figs. 115-116, C22v; figs. 117-118, A1v.

On this far left and slightly lower down is another passage that has been crossed out: "test of throwing the object in an expanse of water and you see the wave, where it is interrupted, which

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is what it does in fo." Below this Leonardo has drawn a final diagram (fig. 114) showing three sides of a right-angled container, the side mn of which has an aperture in the middle. From a central point within the container emanate circular waves. These pass through the aperture and go beyond it to form the arc mban. Meanwhile, waves coming from outside are coming through the aperture in the opposite direction as they move or, as Leonardo might say, their tremors go towards the centre of the container. A one line caption accompanies the sketch: "ab. is the voice from the aperture cd." This confirms that he is again concerned with waves of sound akin to those produced by the man in the cask referred to in the first passage cited from this folio.

9. Conclusion

In this chapter we have stressed the continuity between classical ideas and Leonardo's approach. We have shown that his concept of percussion can be traced back at least to Aristotle, and that his simile comparing sight with waves of a stone thrown into water has precedents in both Vitruvius and Seneca. Leonardo's late experiments to determine what happens when circular waves pass of water or sound pass through small apertures (W19106v) could even be seen as demonstrations of a Vitruvian passage23 cited earlier (p. , cf. figs. 86-87).

(figure)

Figs. 119-120: Percussion of circular waves of water on Mad II 126r.

Figs. 121-122: Percussion of circular waves of light in Christiaan Huygens Treatise of Light (1690).

While Leonardo's debts to tradition cannot be denied, his innovative qualities must not be overlooked. Similes, which his ancient and mediaeval predecessors had mentioned in passing, Leonardo uses systematically. That which they had used in a loose figurative sense, he adopts literally. This taking literal of the figurative again stands in a tradition - it is an outgrowth of nominalism - and it points in turn to the increased emphasis on literal interpretation and the veracity of verbal images that was to take place in the next generation with Luther and Calvin.

Distinctive in Leonardo's approach is how he renders visual the process of taking the figurative literally. He is not just interpreting words, he is trying to illustrate them as pictures. Breughel builds on this principle when he paints the Netherlandish Proverbs (Berlin, Dahlen). It is easy at this point to slip into a Hegelian mentality which insists on a complete harmony linking new ideas in science, religion and art. The details of history are always richer than such generalizations, varying from town to town, and ultimately from one person to the next. But if we keep these differences in mind it is of interest to see that underlying the contrasts between Netherlandish and Italian art; between science in the North and science in Italy and even between Protestantism and Catholicism there was a new relationship between pictures and words, between visual images and verbal images to which the whole of Europe was (re)acting.

Leonardo's visualisation of traditional verbal images concerning circular wave motion is of particular interest (cf. Argentieri, 1939) because it points to the work of Christiaan Huygens. Parallels between the two thinkers are striking. Huygens specifically mentions that light "spreads by spherical waves, like the movement of Sound."29 In describing the foundations of light and

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sound Huygens refers to "Laws of Percussion,"30 describes propagation in terms of "agitation"31 and speaks of waves as being "struck by blows,"32 images familiar from Leonardo's theories of percussion and the four powers. Leonardo's principle of new waves propagated from points along a wave front, recurs in Huygens treatise33 (figs. 121-122, cf. figs. 119-120). From such examples an extraordinary continuity comes into focus. Certain, concrete, observable situations are mentioned by Aristotle by way of similes. Leonardo takes these literally, tests them experimentally and demonstrates them visually. This prepares the way for the radically mechanistic approach of Huygens who claims:

It is inconceivable to doubt that light consists in the motion of some sort of matter.... This isassuredly the mark of motion, at least in the true Philosophy, in which one conceives the causes of all natural effects in terms of mechanical motions. This, in my opinion, we must necessarily do, or else renounce all hopes of ever comprehending anything in Physics.34

Hence there is a direct continuity between Aristotle's similes, Leonardo's visualization and Huygen's mechanism. Why then does the standard view hold that the mechanistic world picture came about through a rejection of Aristotelianism? Historically speaking there were effectively two Aristotles. Aristotle, number one, has an organic framework of knowledge, based on concepts of growth (generation) and decay (corruption) which are basically qualitative. Aristotle, number two uses specific similes are potentially quantifiable. Leonardo concentrates on Aristotle number two and begins to express his similes visually and quantitatively. Leonardo's successors explore the implications of this quantitative approach only to find that it contradicts the organic, qualitative framework of Aristotle number one. Galileo accordingly finds it convenient to make Aristotle number one, along with Ptolemy and the Church, into straw men, symbolic of everything qualitative and opposed to everything quantitative, modern and scientific.

Galileo's rhetorical picture of Aristotle overshadows and effectively eliminates the historical role of Aristotle number two. The course of the continuity is thereby forgotten and the way is prepared for a historiography that can interpret the mechanistic science of the seventeenth century strictly as a rejection of the past. This trend continues today; witness the view of Aristotle found in Dyksterhuis' Mechanization of the World Picture:

Aristotelian physics thus has the advantage over classical mechanics in that it deals with concrete, observable situations constantly encountered. But from a scientific point of view this very advantage constitutes its weakness, for those situations are so complicated (the reader needs only think of a vehicle drawn through the air along a rough road, or of a body of any form thrown upwards) that even with the aid of perfected classical mechanics they can be treated mathematically only by approximation and at the expense of comparatively arbitrary suppositions.35

If we are right then it is precisely these concrete, observable situations of Aristotle that hold the key to understanding how seventeenth century mechanism became possible. At the very least, such instances in Aristotle explain how Leonardo's physics of light and shade can be both Aristotelian and open to a mechanistic interpretation. To understand why Leonardo's physics is not mechanistic in the seventeenth century sense will require a more detailed analysis of his basic definitions.

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Part One Chapter Three Basic Definitions

1. Introduction 2. Point 3. Punctiform Propagation 4. Line 5. Recilinear Propagation6. Speed of Light, Vision 7. Cone or Pyramid of Vision 8. Pyramidal Diffusion

1. Introduction

We have shown that Leonardo's concepts of the four powers (and percussion in particular) involve physical analogies, which lend themselves to a purely mechanistic physics of light and shade. Nonetheless, his own conception of light is neither atomistic nor corpuscular. For Leonardo light is ultimately composed of mathematical points, which occupy no space as becomes clear from a detailed examination of his definitions of point, line and pyramid.

2. Point

Leonardo's definitions of a point build directly on previous authors.1 One of his earliest passages devoted to this theme on CA253vd (c.1490-1491) is a direct translation of Alberti's Elementa picturae2, as is evident from the parallel texts below:

Alberti LeonardoPoint, they say, is that which Point (is), they say, that whichcannot be divided in any part. cannot be divided in any part.

(The) line they say is made The line, they say, is made bywith a point drawn in length. drawing the point in length.

Hence the length of the line Hence the length of the linewill be divisible, but its will be divisible, but the widthwidth in its entirety will be in its entirety (is in-) will beindivisible. indivisible.

Surface, they say, is as if you Surface, they say, is as if youextended the width of a line, extended the width of the line,whence it is that its length whence it is that its lengthand its width can be divided. and also its width can be divided.But depth there will not be. But depth there will not be.

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But body, they affirm, is that But body, they affirm, is that,of which length, width and of which length, width and depthdepth is divisible. and is divisible.

This, in short, is what the .............ancients used to say. We addthe following.

Body, I call, that which is Body, I call, that which iscovered with a surface in covered with a surface infront of the eye/(s)/ and front of the eye/(s)/ andwith light can be seen. with light can be seen.

Surface I call the outer skin Surface, I call, the outer skinof a body which defines the of a body which defines theborder. form of the body and its borders.

Border,I call, the extreme Border, I call, the extremecircuit of each surface seen, circuit of each surface seen,of which the terminus is the of which the terminus is thedivision. division.

Alberti also wrote an Italian version of this treatise. A careful analysis of the texts confirms, however, that Leonardo's passage is a precise translation from the Latin and not simply a copy from the Italian, another confirmation that the omo sanza lettere had direct access to and understood the learned traditions. Leonardo returns to this definition when he is drafting basic concepts of linear perspective on A3r (1492): "Point they say, is that which in no part can be divided."

Alberti, both in his Elementa picturae and De punctis et lineis apud pictores,3 had made a fundamental distinction between the theoretical point of mathematicians and the practical point of painters. Leonardo also makes this basic distinction and makes various attempts to define both kinds of point. With respect to the theoretical point of mathematicians Leonardo relies largely on Aristotle and Euclid. "A line," Aristotle had claimed in the Physics,4 "cannot be composed of points, the line being continuous, and the point indivisible." "A point," claims Leonardo, on Triv.34r (1487-1490), "is not part of a line." "A point," wrote Euclid at the beginning of the Elements,5 "is that which has no part." "A point," writes Luca Pacioli,6 citing Fibonacci (Leonardus Pisanus) as his source, "is that which has no part." "A point," echoes Leonardo in a draft on BM173v (1500-1505), "has no part." This statement he redrafts in several versions on BM173r and 176v (1500-1505) only to cross them out again. On BM173v (1500-1505) he arrives at a more polished version which he later crosses out.

In the course of these drafts on BH173v,r and 176v,r Leonardo explores an analogy between an instant, which has no time, and a point, which has no part. Aristotle had explored the same analogy in the Physics? Leonardo restates the idea that a point has no part no BM190v (1500-

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1505). On CA68rb he drafts another definition of a point as having no centre and being indivisible, which he restates on BM160r (1505-1508; cf. BM266r, 1505-1508; 267v, 1505-1508). On CA289ra (1505-1508) he devotes an entire folio to this theme. In the right-hand margin he writes related terms: terminus, contact, separation, conjunction which he crosses out. Next come draft phrases concerning line and point and then, not crossed out: “Nothing is the lack of the being or of the thing. The point is terminus of the being or of...the thing.” These marginal notes come to an end with a final draft phrase, again crossed out. The main body of the text on cA289ra opens with a series of arguments raised by an adversary which he subsequently crosses out:

1st. The adversary says: either the point is in a site, or it is not in a site. 2nd. And if it is in a site, it is in being. 3rd. And if it is not in site, it does not exist in nature. 4th. And if the point exists in nature, either it is one, or there are many. 5th. And if it is a single one, either it is mobile, or it is immobile. 6th. And if it is mobile, motion describes a line composed of as many points, as are the changes of site made by the point.

Not crossed out is Leonardo's rebuttal in terms of a reductio ad absurdum:

Hence, the site, (of the point) occupied by the point, being equal to this point, the line is composed of sites left by the motion of the point, the continuation of which compose such a line.... And if there are more points, and the juncture of two points (occupies) in itself is divisible in two parts, which are two points. Hence the point is a part and this is contrary to the definition of a point.

Leonardo next replies to five of the adversary's objections in systematic fashion:

It is replied to the first that the point (has being) is in a site, without occupation of the site; to the 2nd, that the point (is is) exists in nature; third, that the points are infinite; 4th that the point is mobile along with the site where it resides; 5th (that it describes) that the motion of the point describes an imperceptible line, which is divisible infinitely, and its points are two.

Another objection is raised and disposed of:

The adversary says that the conjunction of the two points is in itself divisible with a single division, with which the 2 points are separated anew.

It is replied. The two points joined together do not make a divisible line, but a single name,

which is divided by a name, since the two points conclude the line within themselves.

Here Leonardo interjects a quick-thought:

Between the nothing and the thing there is an infinite proportion, because nothing is less than nothing and the thing in itself is divisible to infinity.

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There follow two definitions of a point and one of a line:

The point has no centre and its boundaries is the contact of the front of two lines.

The point has no centre and its boundary is (the) nothing. The line has a middle in length,

but not (in thickness) in width or depth, and the boundaries of its width are two lines. Surface has no....

This folio ends with three further draft definitions of a point. In the case of key terms Leonardo appears to be inexhaustible in his search for an acceptable formulation. On CA176vc (c.1510) the drafting process continues. Again the objections of an adversary are considered and rejected.

The point has no centre and its boundary is nothing.The adversary says that nothing is the vacuum and that the vacuum does not exist in the elements.

It is replied that (the) nothing (...) is that which does not occupy place and (the) vacuum is in place, which has to be, and consequently it is not nothing like the point, which is a place without occupation of place.

The drafting process continues on CA289vb (1505-1508) now under the heading of First Book. Here Leonardo drafts a definition of a point, the objections of an adversary, a reply to these objections and a conclusion; then a second draft of each of these: a third draft of the conclusion, definition, the adversary's objection, and finally arrives at a version which he does not cross out:

The nothing has no middle and its boundaries are the nothing.The adversary says that the nothing and the vacuum is one and the same thing with two names, of which one speaks and they do not exist in nature.

It is replied that if the vacuum existed and if there were a place which surrounded it, and the nothing exists without occupation of place, it follows that the nothing and the vacuum are not similar because the one is infinitely divisible and the nothing does not divide itself, because no thing can be less; and if there existed parts of this, this part would be equal to all and all to the part.

In the last sentence of this passage Leonardo hints at a connection between his definition of a point and the concept of "all in all" which is of central importance for his optics (cf. below pp. ). He develops this connection between a point and things being "all in all" on BM159v (1505-1508):

The point is that than which nothing can be said to be smaller and it is common terminus of the nothing with the line and it neither nothing nor line, nor does it occupy a position between the nothing and the line. Hence the end of the nothing and the beginning of the line are in contact between themselves, but not joined and in this contact is the dividing

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point of the continuation of the nothing with the line. It follows that the point is less than nothing and if all the parts of nothing are equal to one, it could be concluded on the whole that all the points are equal to a single point and one point is equal to all.

This idea he repeats on BM204v (1505-1508) and then reformulates more succinctly on BM205v (1505-1508) under the heading:

The mathematical point is thisThe point has no centre and its termini is nothing: it follows that the point is indivisible, nor is it part of any thing.Hence all the points joined together are equal to one and one to all.

This simple conclusion is of the greatest significance for Leonardo's thought because it provides a theoretical justification for the concept that an image can be "all in all and all in very part" (cf. below Pt.1:4). Alberti in his Elementi di pittura 8 had also provided a more practical definition of a point: "The point we call in paint that little inscription, than which nothing can be smaller." As a painter Leonardo is likewise interested in a more practical definition as he explains on BM159r (1505-1508), where he arrives at a formulation very close to Alberti's:

The point is said not to have a part and by this it follows that it is indivisible and indivisible things have no middle and what has no middle is terminated by nothing. Hence the point is nothing and on nothing no science can be begun. And to flee such a beginning we shall say: the point is that than which no thing can be smaller.

In the Treatise of Painting he goes on to note (TPL 3) that: "the beginning of the science of painting is the point." On folio 27r of Francesco di Giorgio Martini's La praticha di gieometria9 (Ash. 361) where that author makes his own definitions of a point, line, etc., Leonardo writes in the margin:

The natural pointThe smallest natural point is greater than all the mathematical points and this is proved because the natural point is a continuous quantity and every continuous quantity is divisible to infinity and the mathematical point is indivisible, because it is not a quantity.

He pursues this distinction between a mathematical and a natural point on CA200rb (1508-1510):

What thing is a mathematical point?Mathematical point is that which has no middle and there nothing in nature that is less than it and for this [reason] it is indivisible.

What is the natural point?

The natural point is that...impression that the point of some [piece of] iron leaves of itself and this is divisible infinitely.What difference is there between centre and mathematical point:

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The centre is there between centre and mathematical point:The centre and the mathematical point is one and the same thing, but only varies in the place where it is joined: insomuch that the centre is placed in the middle of the quantity or gravity of something, but the point is terminus of a line or some angle.

On BM204v (1505-1508) Leonardo pursues this discussion of natural and mathematical points:

Every continuous quantity is divisible to infinity. Hence the quantity of this (truly) divided will never lead to the print given (by the extre-) by the extremity of the line. It follows that the length and width of the natural line is divisible to infinity.

The times to divide things successively by half are equal insomuch that if, with the mind, you begin to divide the universe, it is the same as dividing the natural point with the mind in the same time.

Hence from the natural point to the mathematical there is an infinite proportion as there would be in dividing the infinite in half would not make two finite parts and thus you will make in the same time from the natural point because in the (the point) one and in the other division made in equal number successively one will never experience the mathematical point. Hence things of near infinite proportion have an equal (divisi-) number of divisions made in equal time with the mind because in actuality it cannot be done and there is no other difference except that the infinite has a greater number of parts than the natural point.

This passage is written around a sketch of an optical pyramid, which has its apex in a point (fig. 123). The pyramid and point are both a mathematical abstraction and something physically real. The significance of the passage written around the pyramidal figure thus becomes clear. Leonardo is trying to reconcile a traditional tension between mathematical (theoretical) and physical (practical) reality. How these opposites are to be reconciled he clarifies on BM132r (1505-1508).

(figure

Fig. 123: Illustration of point, line and pyramid on BM 204v.

Here he begins with definitions of a point and line in terms of limits: "The point is limit of the line and no other thing can be less...." He notes that "All the limits of things are not at all a part of these things because the limit of one thing is the beginning of another." This bears comparison with Aristotle's criticism of Plato in the Topics10 where he objects to his teacher's definition of a point as the extremity of a line. From this premise that a point is not the extremity of a line, Leonardo draws a striking conclusion:

Here, since the limits of things are not parts of these things nor of those things which touch them, these limits do not occupy anything and all things which occupy nothing are equal amongst themselves and all together equal to each of them and each of them [is] equal to all. (In) Whence in this case, it follows that the part will be equal to the all, and all to the

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art, and the divisible to the indivisible, and the finite to the infinite. Hence from what has been said the surface, the line and the point is nothing because it occupies nothing and because they occupy nothing they are each equal to all and all to one as is proved in arithmetic.

Two brief drafts follow before he restates this idea lucidly:

All things which occupy nothing are equal amongst themselves and all joined together will be equal to one and each in itself equal to all. (This demonstrates that the part is equal) to all and all to the part and the divisible to the indivisible, the finite to the infinite.When we examine in detail Leonardo's concept of "all in all and all in every part" in the

next chapter, the full import of the above "demonstration" will become clearer. Here, it is fascinating in itself to see the depth with which Leonardo approaches a problem conceptually.

On BM132r (1505-1508) he makes further preliminary definitions:

The point being indivisible occupies nothing. All things which occupy nothing are nothing. The boundary of a thing is the beginning of another. That which is not part of any thing is said to be nothing and that which is not part of anything occupies nothing...what has no limit has no figure at all.The limits of 2 bodies joined together are in turn the surface one of the other as is water with the air.

He begins a fresh draft: "All the points are equal to all an all to one." This he crosses out and then makes a further demonstration with accompanying sketches (fig. ):

If in a circle there is only one point to which concur infinite lines, within every 2 lines 1 angle is included and each separate angle terminates in a point, hence many angles have many points which, returned in the circle are equal to a single point, centre of this circle, whence it is manifest that many points are equal to one and 1 to many, which thing cannot happen except in nothing. Hence (the point in ) it is said the point is nothing and nothing because it occupies nothing.

On BM131v (1505-1508), the folio opposite he reformulates these ideas in terms of basic tenets and numbers them. The order is not yet final. In the next chapter, when we examine in detail Leonardo's concept of "all in all and all in every part," the full import of the above "demonstration" will become clearer. On BM132r (1505-1508) he makes further preliminary definitions:

The point being indivisible occupies nothing. All things which occupy nothing are nothing. The boundary of a thing is the beginning of another. That which is not part of anything is said to be nothing and that which is not part of anything occupies nothing...what has no limit has no figure at all.

The limits of 2 bodies joined together are in turn the surface of one another as is water with the air.

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He begins a fresh draft: "All points are equal to all and all to one." This he crosses out and then makes a further demonstration with accompanying sketches (fig. 124):

If in a circle there is only one point to which concur infinite lines, within every 2 lines 1 angle is included and each separate angle terminates in a point, hence many angles have many points which, returned in the circle, are equal to a single point, centre of this circle, whence it is manifest that many points are equal to one and 1 to many, which thing cannot happen except in nothing. Hence (the point in) it is said that a point is nothing and nothing because it occupies nothing.

On BM131v (1505-1508), the folio opposite he reformulates these ideas and numbers them, although the order is not yet final.

(figure)

Figs. 124-125: Punctiform propagation of light on BM132r and 131v.

1. The surface is limit of the body.2. The limit of a body is not part of this body.3. That is nothing, which is not part of any thing.4. That is nothing, which occupies nothing.5. The limit of a body is the beginning of another.

Below this he redraws three of the figures (fig. ) sketched on BM132r (1505-1508) and restates his fundamental concept.

If 1 single point placed in a circle can be the beginning of infinite lines and limit of infinite lines by such a point infinite points are separated, equally reduced, returning to one [point] it follows that the part is equal to the all.

This claim is of the greatest importance because it is implicitly a rejection of atomism. It explains why images can be all in "all and all in every part" and is the basis of his concept of punctiform propagation of light.

2. Punctiform Propagation

Punctiform propagation was by no means a new concept. It had been clearly formulated by Alhazen in the eleventh century.11 It was restated by Witelo.12 Pecham in the Perspectiva communis noted that "Any point of a luminous or illuminated object simultaneously illuminates the whole medium."13 The anonymous author of Della prospettiva restated the idea: "from every point of the visible thing infinite rays are multiplied terminating in various parts of the said medium or space."14

(figure)

Fig. 126: Proof how every part of light makes a point on W12604r.

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On MB232r (c.1490) Leonardo notes:

2. Every surface is full of infinite points.3. Every point makes a ray.4. The ray is made up of infinite separating lines.

He restates this principle on CA144va (c.1492): "every point of the luminous body makes itself the cause of infinite luminous pyramids." Behind this principle lie more than traditional explanations, however. As early as 12485 he attempts to visualize the nature of such a punctiform propagation (eg. CA353vb, fig. ). In the course of the next two decades these attempts continue (eg. W19148v, K/P 22v, 1489: CA144va, 1492; CA179rc, C.1505; CA345rb, 1505-1508, figs. ). By 1488 on W12604r (fig. 126) he offers a

Proof how every part of light makes a point.Even though the balls a., b., c. have lights from a window nonetheless if you follow the lines of its shadows you will see that these make an intersection and point at the angle n.

Elsewhere on the same folio he alludes to such a proof as a given:

Because it has been proved that every limited light makes or appears to originate from a single point, that part illuminated by it will have its particles more luminous on which the luminous line will fall between two equal angles.

He explains why light has a single centre in more detail on W19147v (K/P 22v, figs. ,1489):

The reason why light has in itself a single centre is this. We know clearly that a large light is much greater than a small thing. Nonetheless, even if its rays surround it much more than half, the shadow always appears on the first wall and is always seen.Let us posit that cf. is the large light and that n. is the object opposite it which generates shadow on the wall and that ab. is the wall. It appears clear that the large light would not conduct the shadow n. to the wall. But since the light has in it a centre, I prove by experiment, that the shadow conducts itself to the wall as the figure motr.

This experiment he repeats on CA204ra (c.1492) and, elsewhere as will be seen later (cf. pp. figs. ). For the moment, however, we need to examine his definitions concerning lines and pyramids.

3. Line

As noted above ( 2) one of Leonardo's sources for his definition of a line was Leon Battista Alberti whose Elementa picturae he copied. Another source was Aristotle who, in his De anima had written: "They say a moving line generates a surface and a moving point a line."15 "A line," claims Leonardo on BM173v (1500-1505), "is the transit made by a point." In the period 1500-1505 he repeats this definition on BM176v, 176r and again on 190r where it is accompanied by an

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alternative even closer to Aristotle's concept: "The line is made by the movement (moto) of the point." This he reformulates on BM159r (1505-1508): "The line is a length made by the movement of a point," and restates afresh on BM204v (1505-1508): "The line is a length born together with the motion of the point and terminated at the limit of the movement of this point than which line nothing can be said to be thinner".

(figure)

Fig. 127: Demonstration concerning the nature of a line on W19151r (K/P 118r).

On CA289ra (1505-1508) Leonardo drafts a further variant: "The motion of the point describes a line composed of as many points as are the mutations of the sites made by the motion of the point." This he crosses out and defines anew: "The motion of the point describes an imperceptible line, which in itself is divisible infinitely, and its points are two." On W19151r (K/P 118r, 1508-1510), to clarify the nature of a line, he sets out to demonstrate that a (fig. 127):

Line cannot intersect itself.This is proved by the motion of the line af. to ab. and of the line eb. to ef. which are the sides of the surface afeb. But if you move the line ab. and the line ef. with the front ends ae. to the position c. you will have moved the opposite ends fb. towards each other at the point d. And from the two lines you will have made the straight line cd. which resides in the middle of the intersection of these two lines in the point n. without any intersection. For if you imagine two such lines to be corporeal, through the said motion one will necessarily completely cover the other being equal to it without any intersection at the position cd. And this is enough to prove our proposition.

(figure)

Fig. 128: Sketch on A113v relating to rectilinear propagation of light in the mountains.

4. Rectilinear Propagation

In Leonardo's physics of light a straight line defines the path of rays. This concept of rectilinear propagation of light and lines of sight was again a well established one. Euclid, in the first definition of his Optics had posited that "straight lines" emanate from the eye."16 The author of the Problemata had been even clearer in this respect: "the course of sight can take only one direction, namely, a straight line, as is shown by the rays of the sun and the fact that we can only see what is directly opposite us."17 Elsewhere in the same treatise, he wrote "light travels in a straight line only."18 Galen in The Usefulness of the Parts reported that the rectilinearity of light is established by the sun's passage through a narrow opening.19 Similar demonstrations occur in Alkindi20, Alhazen21 and Witelo.22 Like Euclid, Leonardo makes the rectilinearity of light a preliminary axiom (A4v, 1492):

Mention of the things which I demand be conceded me in the proofs of this my perspective.I demand that it be conceded me the affirmation that each ray passing through the air which is of equal subtlety, go along a straight line from their source to the object of percussion.

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On A9v (1492) he restates this basic principle: "The concourse of the pyramidal lines caused by the objects and terminating at the eye must be rectilinear." On A113v (BN2038 32v, TPL747, fig. 128, 1492) he invokes this principle once more under the heading.

Way where the shadows made by objects must terminateIf the object be the mountain here drawn and the light be the point a., I say that from b., d. and similarly from c., f. there is no light, except for reflected light and this occurs because luminous rays do not adapt themselves except along straight lines and the second rays which are reflected will do the same.

On CA222va (1492) he alludes briefly to this principle of rectilinearity: "No species is carried to the eye passing through equal air, that is not along straight lines." On TPL11 (c.1492) he cites this principle as a reason why sight is a more dependable sense:

The eye at moderate distances and in moderate mediums is less deceived in its function than any other sense because it does not see except by straight lines which compose the pyramid which makes itself the base of the pyramid and conducts it to this eye, as I intend to show.

Leonardo paraphrases his earlier statement on A8v (1482) when he returns to this principle of rectilinearity on Mad.I 0r (1494): "I wish that it be conceded me that the line which goes from the object to the eye through the same quality of air be straight."

(figure)

Figs. 129-131: Rectilinearity of light in a convex mirror, model eye and a camera obscura on W19120v (K/P117v); D10v and D10r.

On CA150ra (1500-1505) he writes: "All the luminous rays are straight which pass through an equal space." On W19120v (K/P 117v, c.1508-1510) he illustrates this principle (fig. 129) in connection with reflection in convex mirrors, adding the caption: "Every action of nature is made by the shortest way possible." Almost the same phrase recurs on BM85v (c.1505), this time in connection with concave mirrors: "Every action made by Nature is made in the shortest way." On D10v (1508) he identifies Aristotle23 as the source of this idea (fig. 130):

All vision made in the same quality of air is rectilinear. Therefore since it is possible to draw a straight line from the eye to each part of the air seen by this eye, this vision is rectilinear. And this is proved by that of Aristotle which says: every natural action is made in the briefest possible way etc. Therefore vision will be made through the shortest line, i.e. a straight [one].

On D10r Leonardo goes on to note that (fig. 131) "the dark or luminous particles of any given ray are always rectilinear," an idea which he restates on CU613 (1508-1510): "The boundaries of derivative shadows are rectilinear." These passages are particularly interesting because they point to a close link between his principle of rectilinear propagation of light and camera obscura demonstrations (see below pp. ).

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By 1508 Leonardo refers to rectilinear propagation in terms of propositions. On D10v and r it is "the third proposition." On D4v he lists it as "the ninth of the first, in which it is stated that each act of vision is made by the eye in the same way and that this is accomplished by straight lines." On W19149r (K/P 118r, 1508-1510) he refers to "the second of this...which shows that all rays which convey the images of objects through the air are straight lines." On CU630 (1508-1510) he refers to "the fourth of this which states that all umbrous and luminous rays are rectilinear." On W19150v (K/P 118v, 1508-1510) he lists it as "the seventh of this where it is said: Every form projects images from itself by the shortest line, which is necessarily a straight line." These frequent references confirm that rectilinearity is a basic principle of his physics of light and shade.

5. Speed of Light, Vision

In Antiquity, there had been a debate whether the propagation of light is instantaneous or not. Aristotle, for instance, had argued that light was instantaneous and criticized Empedocles for holding a contrary view.24 This Aristotelian claim was in turn challenged by the Arabic optical thinker, Alhazen, who devoted a long chapter to how "light and colour in themselves are perceived in time."25 Witelo, who borrowed heavily from Alhazen, nonetheless, continued to argue that light is instantaneously propagated.26 Roger Bacon, on the other hand, reviewed conflicting theories on the subject27 and went on to assert that light is temporal, that it is much swifter than sound or smell, and that, even when its motion is imperceptible, light is still not instantaneous.28

Leonardo's position on this question is equivocal. He broaches the problem at least eleven times in the extant writings. In five cases he seems to agree with Aristotle that light is instantaneous. On CA204va (1490-1492), for instance, Leonardo claims that the eye, a ray of the sun and the mind are the most speedy motions which exist. He points out that, the moment the sun appears in the east, it immediately sends its rays to the west. The moment the eye opens it sees all the stars of the hemisphere. The mind, Leonardo adds, goes from east to west in "a wink" and all spiritual things are similar. Aristotle, in his De Anima had made a similar comment about light moving from extreme east to extreme west instantaneously.29 Leonardo makes an unequivocal statement concerning the instantaneous nature of light on A27r (1492):

PerspectiveImmediately the air is illumined, it is filled with infinite species, which are caused by various bodies and colours which are collected in front of it; of which species the eye acts as a target and magnet.

This recalls Pecham's proposition: "Any point of a luminous or illuminated object simultaneously illuminates the whole medium adjacent to it."30 On A81r (BN 2038 1r, 1492), in a discussion why the eye does not send out visual rays by extromission (cf. below pp. ), the simultaneous propagation of light is again implicitly suggested. In the Treatise of Painting (TPL22, c.1492) Leonardo contrasts the slowness of hearing with vision: "But the work of the painter is immediately comprehended by its regarders." On CA179rb (1497-1500) he restates the instantaneous nature of light: "The air is, in itself, capable of producing without time and leaving every species and similitude of anything seen by it."

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In various other passages, however, he suggests that light and vision are not instantaneous. For instance, on C6v (14590), he mentions how the eye is always swifter than hearing, noting that a blow is always seen prior to its being heard, Roger Bacon, in a passage cited earlier (p. ), had made the same point. On A108r (BN 2038, 28r, TPL44, 1492) the temporal nature of sight is broached again: "We clearly know that sight is the most speedy operation that there is and in a point infinite forms are seen. Nonetheless only one things is understood at a time." By way of illustration he cites the example of how the eye cannot read an entire page at once, but must read it bit by bit.31

In three passages Leonardo alludes to the physiological transit of images. On CA90rb (1490-1492) he merely notes in passing that the eye is the swiftest sense "because it is the sense nearest the imprensiva." On CA250ra (1490-1495) he claims that "to the extent that the eye in its function is faster than the ear, to this extent it reserves the similitudes of things impressed in it." The ambiguity of Leonardo's position is most evident on CA203va (1492) where he discusses various branches of motion. He begins with temporal motion which, he claims, "embraces all others." Fourth on his list is the motion "of species of things spread through the air by rectilinear propagation." This he claims "appears not to be defined by time because it occurs in an indivisible time." A few lines later he adds that the motion of similitudes is swiftest and that of mind, second. The passage ends with a physiological remark:

Of the motion of the senses we will not make mention except for hearing because, applied to visible things, it is accomplished in time as demonstrated in...sound, voices....

Finally, in this regard may be mentioned a passage on BM176v (1500-1505). Here he considers and accepts the possibility of a potential infinite: cf. Aristotle in the Physics.32 But he then rejects infinite motion in practice because every medium exerts resistance and a vacuum does not exist or to use his own words "the medium does not concede it." Aristotelian physics thus leads Leonardo to reject Aristotle's idea that light has an infinite speed.

6. Cone or Pyramid of the Eye

The notion of a cone of vision is found in Euclid's Optics.33 Ptolemy used the term at least 46 times in the extant books of his Optics 34 and implicitly connected it with the notion that images are everywhere in the air.35 Ptolemy's work was translated into Arabic and when Eugene of Sicily translated it anew into latin in the twelfth century he chose to translate the Greek, cone, (konos**) with the Latin, pyramid, (piramis) in all but one case.36 Hence from the twelfth century onwards the visual cone becomes the visual pyramid. It is a standard term in Roger Bacon, Witelo, Pecham, Biagio Pelacani da Parma, Alberti and the anonymous author of Della prospettiva. Throughout these texts there is a convention of drawing the pyramid as a triangle. Leonardo often follows this convention. The evidence leaves no doubt, however, that the pyramid is conceived as being three-dimensional (e.g. fig. 123). Because the concept of the visual pyramid is so well established in the optical tradition, Leonardo only rarely defines this term as on BM176v (1500-1505):

A pyramidal body is that of which all the lines parting from the angles of its base concur in a point. And such a body can be invested with infinite angles and sides.

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Euclid had primarily been concerned with a single cone of vision. Alhazen's concept of punctiform propagation had led him to claim that pyramids originated from every point of an object.37 He went on to show that: "between the visible object and a mirror innumerable pyramids are produced with alternating based and vertices."38 Witelo pursued this idea in his optical compendium:

As many as are the points on the surface of a mirror, so many are the pyramids to the entire surface of the form of the terminated body, which surface is the basis of all these pyramids and as many are the points in the whole surface of the body the form of which falls on the mirror, so many are the pyramids to the entire surface of the terminated mirror, which is the base of all these pyramids.39

Pecham repeated this concept40 which gave rise, in turn, to the third quaestio of Biagio Pelacani's treatise: "Whether the entire visible object and each of its parts terminates a radiant pyramid of its light or colour in every part of the medium."41 Notwithstanding doubts Biagio fully accepted the concept of infinite pyramids as is clear from a passage at the end of the seventh argument:

Since on the authority of the quaestio, at any point some part of the visible object terminates its pyramid, therefore in the eye there will be a pyramid of the whole and infinite pyramidsof infinite parts.42

In the third conclusion Biagio claimed that "in every sphere of light there are de facto infinite pyramids.43 That he was basing his ideas on Pecham is confirmed by the first conclusion to the second article:

at any given distance one sees the whole and any of its parts, however small, is seen. This is proved since, as is stated in the fourth conclusion of the perspective [i.e. proposition 4 of Pecham's perspectiva communis] every point of the luminous body terminates a pyramid in every part of the medium.44

Leonardo's concepts of pyramidal diffusion grow out of this tradition as is evidenced by his preliminary drafts on CA256rc (1492):

Every body makes raysIn each part (of) a given ray (is the concourse) concur (all the...) pyramidally the similitudes of that part of a body which faces this line.Every part of a body which faces the ray (line), which results from it with pyramidal concourse is all in all of this line and all in every part of it.

These notes may well have served as a draft for a long compilation on BM232r (1490-1495), which opens with numbered definitions of body, surface and ray (1-4) and is followed by five consecutive propositions concerning pyramids (i.e. 6-11 because 5 is omitted):

6. In each point of the length of any line, lines parting from the points of the surfaces of bodies intersect and they produce pyramids.

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7. Each line occupies the entire point from which it originates.8. In the extremity of each pyramid lines parting from the whole and from the parts of the bodies intersect such that from this extremity one can see the whole and the parts.9. The air that is found between bodies is full of intersections made by the radiating similitudes of these bodies.10. And colours of each body transfer themselves from the one to the other by pyramids.11. Each body fills the surrounding air by means of these rays with its infinite similitudes.

The last of these propositions is restated on A2v (1492): "Each body fills the surrounding air with its similitude, which similitude is all in all and all in every part." Immediately following these propositions on BM232r (1490-1495) he adds another six, which he again numbers:

5. The similitude of each point is caused by this point in the whole and in the part of this line.6. Through its similitudes each point of one object is capable of the entire base of the other.7. Each body makes itself the base of innumerable and infinite pyramids.8. Each pyramid, which originates within more equal angles will give a truer similitude of the body from which it originates.9. A same base is the cause of innumerable and infinite pyramids turned in various directions and of various lengths.10. The point of each pyramid has in it the entire similitude of its base.

Not content he adds eight further unnumbered propositions regarding pyramids:

The centric line of pyramids is filled with infinite points of other pyramids(The) One pyramid passes through the other without its confusion. The quality of the base is in every part of the length of the pyramid.That point of the pyramid which includes within it all those which originate at the same angle will be less demonstrative of the body from which it departs than any other one enclosed within its.The pyramid of [a] more subtle point will demonstrate less the true form and quality of the body from which it originates.That pyramid will be more thinner, which has more disform angles of the base.That pyramid which is shorter will transform in greater variety the similar and equal parts of its base.Infinite lengths of pyramids will originate from the same quality of angles.The pyramid of thickest point will tinge the place percussed by it with the colour of the body whence it derives, more than any other.

Nor does he stop here. On BM232v (1490-1495) he outlines thirteen further propositions:

Pyramids which derive from spherical bodies will always be of equal angles in their origin.If one pyramid finds itself originating on plane bodies of equal angles...That pyramid which falls on bodies under more equal angles will tinge the place percussed by its more with the colour of its base.

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Every pyramid in transit with the friction of the sides of the bodies opposite will produce an intersection after which it will cause opposite it a new and inverted pyramid.In their percussion inverted pyramids render the species upside down.The upside down (pyramids) species will be that much smaller than their origin to the extent that the inverted pyramid is shorter than the right side up [one].The base of the inverted pyramid is infinite.The bases of inverted pyramids if they are in a dark place will show the form and the colour of their origin upside down.By the intersections of pyramids it is possible to know the true size of their bases.If the angles of the intersection of the pyramid be equal to those of its base...each base will be proportionate, the one to the other as is the proportion of the length of the cut pyramid to the whole.If they intersect the bases of inverted pyramids are redoubled in their quality.The eye does not see except by pyramids.The perspective of painters does not occur within pyramids.

These final two propositions confirm that, for Leonardo, the pyramid is essential to both vision and linear perspective. Hence the pyramid plays an important role in drafting his definition of linear perspective as knowing how to draw well the function of the eye (A3r, 1492):

which function extends itself solely in drawing of the forms and colours of all things placed opposite it by means of pyramids. By pyramids, I say, because there is no thing so minimal that it is not larger than the place where these pyramids are conducted in the eye. Hence if you take the lines at the extremities of each body and their concourse leads to one point, it is necessary that these lines are pyramidal.

Two drafts follow and then a definition in the final paragraph of A3r: "Pyramidal lines I intend to be those which depart from the superficial extremities of bodies and by distant concourse conduct themselves to a sole point." As we have shown elsewhere (vol. 1, Pt.I.2) such definitions on A3r are the starting point for his treatise on perspective (A36v-43v). Thus the concept of the pyramid is fundamental for both his linear perspective and his physics of light and shade. To understand the complexities of Leonardo's theories concerning pyramids we need to examine how he relates them to his claim that images are "all in all and all in every part." This is the concern of the next chapter.

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Part One Chapter Four Images All in All and All in Every Part

1. Introduction 2. Classical and Mediaeval Sources 3. Mirror Demonstrations 4. Camera obscura Demonstrations 5. Further Drafts and Statements 6. Philosophical and Theological Dimensions 7. A Systhesis of Circular and Pyramidal Propagation

1. Introduction

The concept that images are "all in all and all in every part" can be traced back to Antiquity and is found in various mediaeval sources. Leonardo is not content merely to cite such sources. He develops demonstrations using both mirrors and camera obscuras. He drafts numerous versions of the concept exploring its philosophical and theological implications. His efforts lead him to a synthesis of circular and pyramidal modes of propagation: a visualization of the ubiquity of images.

2. Classical and Mediaeval Sources

The idea that images are everywhere in the air can be traced back to Lucretius' Of the Nature of Things:

For every outside streams away

From off all objects since discharge they may...From things there must be borne, in many modesTo every quarter round, upon the moment,The many images of things, becauseUnto whatever face of things we turnThe mirror, the things of form and hue the same respond.1

Lucretius' reference to a mirror in this connection is particularly interesting. It recurs in Alhazen2, and Leonardo. Two paragraphs later Lucretius returns to this theme:

To such degree from all things is each thingBorne streamingly along and sent about.To every region round; and nature grantsNor rest nor respite of the onward flow.3

(figure)

Fig. 132: Punctiform propagation of light in a seventeenth century interpretation of Lucretius.

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In the paragraph that follows he pursues the idea:

Now these same films I name are borne aboutAnd tossed and scattered into regions all.But since we do perceive alone through eyesIt follows hence that witherso we turnOur sight, all things do strike against it there with form and hue.4

The significance of these passages in Lucretius becomes apparent through a seventeenth century optical writer such as Zahn5 (fig. 132) who cites the Roman author to illustrate the concept which Leonardo expresses with his description of images as "all in all and all in every part." Lucretius refers to both this concept and to a cone of vision but does not relate the two. Ptolemy's approach in the Optics is comparable. In the eleventh century Alhazen builds on this notion of a cone or pyramid and emphasizes how "light and colour are reflected from every point of a polished surface."8 In the thirteenth century, Witelo goes further to argue that what applies to light and colour applies equally to the forms of objects: "The surfaces of terse polished bodies of whatever shape they may be, reflect the lights, colours and forms of things opposite from each point [of their surface] along straight lines."9 Witelo's version of Alhazen's proposition concerning reflection in mirrors and pyramidal diffusion is therefore somewhat different:

Between the points of forms falling on the surface of any mirror and the surface opposite the mirror, it is necessary to draw infinite pyramids, cones and bases...

...Therefore the entire form of the body will be in any one point of the mirror and the form of any point of the body in the entire surface of the mirror. Hence as many as are the points on the surface of the mirror, as many are the pyramids of the form of the object terminating at the entire surface.10

Whether this form is a mental image or a physical one troubles neither Witelo, nor his younger contemporary, John Pecham, who is more intent on emphasizing that pyramids are propagated from every point of objects:

Any point of a luminous or illuminated object simultaneously illumines the whole medium adjacent to it (Prop.3).... The pyramid of light originating from the whole luminous or illuminated object terminates at any point of the medium (Pro.4).... Pyramids proceed from pyramids at every point of an illuminated medium (Prop.5).... Radiant pyramids proceeding from the same or different surfaces differ in effective strength and weakness at different points in the medium (Prop.6)....11

To illustrate this paradoxical concept that pyramids and images stem from every point of objects and that they are everywhere in the air, Alhazen had used the example of images in a mirror.12 Witelo13 and Pecham14 also use this image in passing. Leonardo da Vinci writing in the 1490's, develops it.

3. Mirror Demonstrations

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One of Leonardo's earliest appeals to a mirror to demonstrate that images are "all in all and all in every part" (`9V, 1492) has already been cited in connection with percussion (cf. above pp. ). This is followed by a second example involving mirrors taken from everyday experience:

Let us take the example of the sun, which, if you walk along the bank of a river and you see the sun mirrored in that river, to the extent that you walk along this river to that extent will it appear that the sun walks with you and this because the sun is all in all and all in every part.

On CA138rb (1492) he considers an example with two mirrors positioned opposite one another:

PerspectiveThe air is filled with infinite similitudes of the things, which are distributed beneath it and all are represented in all and all in one and all in each, whence it happens that if there were 2 mirrors turned in such a way that by a straight line they faced one another, the first would be mirrored in the second and the second in the first. The first which is mirrored in the second carries with it all the similitudes of itself with all the similitudes that are represented in front of it, among which is the species of the second mirror and from similitude to similitude they go into infinity in such a way that each mirror has in front of it mirrors, one smaller than the other and one in front of another.

Following a digression on vision, to be examined later (see pp. ) he gives further examples of mirrors to demonstrate that images are "all in all and all in every part":

That the species of all things are [dis]seminated through the air is seen by the example of many mirrors in a circle, which mirror one another an infinite number of times and joining one another it leaps ahead to its cause and there diminishing it again leaps to the object and then returns and thus it does an infinite number of times.

If you place a night light between two flat mirrors which have an interval of one braccio you will see in each of these mirrors infinite lights, the one less than the other. If at night you place a light beneath the wall of a room, all the parts of this wall will become tinted by the similitudes of this light. And all these which will be seen by the light, the light will see similarly, that is, when there is no opposition betwixt them which [dis]rupts the concourse of the species.

This same example appears better in the concourse of solar rays which, all in all and each in itself carries to its object the similitude of its cause.... That each body by itself fills all the counterposed air with its...similitudes and that this same air is capable at the same time, of receiving in itself the species of infinite other bodies which are before it, is clearly demonstrated by these examples. And each body appears all in all of said air and all in every minimal part of it, all in all and all in every minimal part, each in all and all in the part.

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He again uses two mirrors opposite one another to illustrate the principle that images are "all in all and all in every part" in a rough draft on BM Arundel 186r (c.1492):

All bodies have their similitudes and species infused and mixed being luminousHence all and the similitudes of bodies hasHence all and the part of the species of bodies appears all in all and in the part of the air opposite it and the body of the air appears in all and in the part of the surface of these bodies.Whence we can clearly state that each body is all in part in each part and in all the opposed bodies as is demonstrated by mirrors opposite one another.

This he crosses out and re-writes on CA179vc (C.1500):

All bodies have infused and mixed all their species and similitudes in all the quantity of the air opposite it.The species of every point of corporeal surfaces are in each part of that airAll species of bodies are in each point of the air and part of the similitude of air is in each point of the surface of bodies. Thus part and all of the species of a body appear in all and in part of the air opposite it. And the body of air appears in all and part of the surface. Whence we can clearly say: the similitude of each body is all and in part in each part and in all of the opposite bodies reciprocally, as is seen in mirrors, one opposite the other.

On Leic.7v (1504-1508) he again mentions this mirror demonstration: "The sun is mirrored all in all and all in every part of the water opposite it." As will be shown later (Pt.4:3) this principle has important consequences for his theories of astronomy. In the late period Leonardo mentions the mirror demonstration once more in connection with problems of visual perception on W19076r (K/P 167r, c.1513):

Describe how no object is terminated by itself in a mirror, but the eye which sees it inside such a mirror, terminates it. For if you represent your face on a mirror the part is similar to the whole, such that the part is all in all the mirror and is all in every part of the same mirror and the same occurs with the entire image of the entire object placed opposite this mirror etc.

4. Camera Obscura Demonstrations

In the eleventh century Alhazen had used a pinhole in a darkened room to demonstrate that light and colour do not mix when passing through an aperture.15 From this he concluded that light comes "from each point of each body."16 Witelo used the same demonstration.17

Leonardo builds on this tradition. In the passage on A9v (1492) cited earlier (p. ) he begins by comparing the waves of a stone in water and the waves of sound in air. These he compares to light which, he claims, appears "all in all." Next he draws a sketch (fig. ) and notes: "this is proved by experience because if you close off a window facing the west and make an aperture...." The meaning of this unfinished phrase can be reconstructed with aid of diagrams (figs. 136-138). An object such as a tree (AB) sends forth its images in circular waves (C) which

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pass through a pinhole (D), continuing in the shape of a now inverted pyramid (E) to appear inverted on the wall opposite (B,A).

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Figs. 133-138: Demonstrations of the "all in all" principle using a camera obscura. Fig. 133, A9v; figs. 134-135, CA373rb; figs. 136-138 Author's reconstructions.

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Fig. 139: Illustration how images are all in all and all in every part from Mario Bettini's Apiaria universae, (1646).

Leonardo is fascinated how a pinhole, effectively a single point, conveys the images of an entire object. Indeed this point conveys the images of all surrounding objects, which send out circular waves passing through one another without interference, like the waves produced by pebbles thrown into water. Leonardo's sketch shows a pinhole in a shut off window (A. in fig. 137) through which pass two laterally positioned objects (C. and D. in fig. 136) which then appear inverted on the two side walls (as C., and D. in fig. 137).

With Leonardo, a sketch that is difficult to interpret is usually presented more clearly elsewhere. In this case we need to return to the draft for A9v on CA373rb (1490-1495). Here he draws two sketches. One is a simple line drawing showing the inversion principle in a camera obscura (fig. 134). The second sketch (fig. 135) shows two objects, analogous to these on A0v, intersecting one another without interference in a camera obscura. A seventeenth century illustration from Mario Bettini's Apiaria universae demonstrates the basic principle more dramatically (fig. 139). Leonardo pursues the problem some sixteen years later on W19151r-19152r (K/P 118r, 1508-1510), beginning with a sketch (fig. 140) to which he adds letters, which he subsequently forgets to mention in the text:

How we conclude that a surface reduces itself in a point.The angular surface reduces itself into a point when it terminates itself in its angle, or if the sides of such an angle are produced in a straight line, then, behind that angle another surface will be generated [that is] less or equal [to] or greater than the first.

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Figs. 140-141: Camera obscuras to demonstrate that images are "all in all and all in every part" on W19151r-19152r (K/P 118r).

He then draws a second sketch (fig. 141) which shows two light sources of different colours, yellow and blue, passing through the point of an aperture without mixing. Alongside he explains:

How innumerable rays of innumerable species can reduce themselves to a single point.

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Just as in a point all the lines pass without occupying one another through being incorporeal, so too can all the species of the surface pass through it, and just as every given point sees every opposite object, every object sees the natural point opposite. The diminished rays of such species can also pass through this point, beyond which passage they are re-formed and the size of such species again grows. But their impressions will be reversed as is proved in the first of the above, where it is said that every species is intersected at the entrance of the narrow apertures made in material of minimal size.

A note directs us to: "read opposite in the margin":

In accordance with the inversion of the central line necessity requires that all the species which enter by thin and narrow apertures in dark places are turned upside down along with all the species of bodies which invert them.

Comparison of this late passage on W19152r (K/P 118r, 1508-1510), with the early passage on A9r (1492) points to the continuity of Leonardo's thought. Consideration of further camera obscura demonstrations confirms this. On A93r (BN 2038 13r, 1492), for instance, he begins with a general proposition:

all bodies together and each in itself fills the surrounding air with its infinite similitudes which are all in all and all in the part carrying with them the (essential) quality of the body, the colour and shape of their source.

He then draws (fig. ) three light sources, red, white and yellow respectively which penetrate the aperture of a camera obscura without interference, beneath which he adds:

That bodies with their similitudes are all in all the surrounding air and all in the part in body, shape and colour is clearly demonstrated by the species from many various bodies which are produced to a single perforated point where with intersected lines, through contrary pyramids they carry things inverted to the first dark wall, the reason for this is...

Here the text breaks off and no reason is given. But on CA135vb (c.1492) Leonardo produces two further drafts of this camera obscura demonstration; repeating the phrase "the reason is," and this time providing an answer. The draft opens with a familiar phrase: “Every body is all in part of the air and each in all the opposite air, all in part and all in all.” This he restates: “Every body appears by similitudes all in every part of the (opposed air) counterposed air and ag-(appears)-ain appears.” He goes on to draft a related idea:

But since (the air) each body appears by similitudes (in each part of the) air positioned opposite, that is all the object in all in the part, all the objects in all the air and all in the part, speaking of this air which is capable of receiving in itself the straight (lines of rays ra-) and radiant lines of the species sent by the objects, whence for this [reason] (in) [it] appears (to be) necessary to confess it to be the nature of this air (which attracts like a lodestone to itself the similitudes of the things positioned in front of it.

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Immediately following Leonardo turns to his camera obscura demonstration (which Pedretti dates c.1508) under the heading:

Proof how all the objects positioned in a site are all in all and all in a part.I say that if a face of a building or piazza or landscape (campagna), which is illumined by the sun has opposite it a habitation and in this face [of a wall] which does not see the sun a little round aperture be made, that all of the illumined objects sidestep their similitudes through this aperture and appear inside the habitation on the opposite wall, which should be white, and they will be there exactly (a punto) and inverted and if in many points of this face [of the wall] you made similar apertures, there would be a similar effect in each. Hence the species of the illumined things are all in all of this face [of the wall] and all in every minimal part of it.

Why this should be so he now proceeds to explain

If the object positioned...opposite the eye sends to it from itself the similitude (to it), the eye also sends its similitude to the object. If from the object and through the departed similitudes nothing streams off from any cause neither to the eye nor to the object (the eye does the same), then we can sooner (say) believe (it) to be the nature and power of this luminous air (whi) which attracts and catches in itself the species of things which are in front of it, rather than the nature of the objects which sends species through this air.

This he redrafts in terms of his "all in all" concept:

If an object positioned opposite the eye (the eye) sends to it a similitude from itself, the eye will have to do the same to the object, whence it would be convenient that these...species were spiritual powers (virtu spirituali). Being thus...it would be necessary that each thing would be diminished immediately because (the air) each body appears by similitudes (in each part of the) in the air positioned opposite, that is, all the object in all the air and all in the part; all the objects in all the air and all in the parts, speaking of this air which is capable of receiving in itself the straight (lines of rays ra-) and radiant lines of the species sent by objects. Whence for this [reason]...it appears (to be) necessary to confess it to be the nature of this air...which attracts like a lodestone to itself the similitudes of the things positioned in front of it.

His camera obscura demonstration (which Pedretti dates c.1508) follows:

Proof how all the objects positioned in a site are all in all and all in a part.I say that if a face of a building or piazza or landscape, which is illumined by the sun has opposite it a habitation and in this face [of a wall], which does not see the sun, a little round aperture be made, that all of the illumined objects send their similitudes through this aperture and appear inside the habitation on the opposite wall, which should be white, and they will be there exactly and inverted and if in many points of this face [of the wall] you made similar apertures, there would be a similar effect in each [cf. fig. 139]. Hence the species of the illumined things are all in all of this face [of the wall] and all in every minimal part of it.

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Why this should be so he now explains:

The reason is [that] we know clearly that this aperture must send some light into this habitation and the light which renders it a mediator is caused by one...or many luminous bodies. If these bodies are of various colours and various kinds, the rays of the species will be of various colours and various kinds and the representations on the dark wall will be of various colours and kinds....

This example of various apertures in a room again illustrates how Leonardo adapts and transforms his sources. Both Alhazen18 and Witelo19 had mentioned that there would be as many images as there are candles in front of an aperture. They had also made the proviso that one should have a white background.20 Nonetheless, it is Leonardo's idea to use this as a demonstration that the images of objects are everywhere in the air:

The reason is [that] we know clearly that this aperture must send some light into this habitation and the light, which renders it a mediator is caused by one or many luminous bodies. If these bodies are of various colours and various kinds, of various colours and various kinds will be the rays of the species and of various colours and kinds will be the representations on the dark wall.

The example of various apertures in a room here cited again illustrates how Leonardo adapts and transforms his sources. Both Alhazen18 and Witelo19 had mentioned that there would be as many images as there are candles in front of an aperture. They had also made the proviso that one should have a white background.20 Nonetheless, the idea of using experience to demonstrate that the images of objects are everywhere in the air is Leonardo's. It is typical for him to consider various alternatives. Hence having described how three lights pass through one aperture on A93r (1492), he proceeds on W19150v (K/P 118v, 1508-1510) to consider a case in which three objects pass through two apertures to produce six images (fig. ). This time he letters the diagram and the accompanying text serves partially as a caption:

The species of the bodies are all infused through the air which sees them and all in every part of it. To test it let a, c, [and] e. [be] objects, the species of which penetrate a dark place through the apertures m. [and] p. and impress themselves on the wall f.; opposite these apertures, which impressions will be made in as many places on this wall as will be the number of the above-mentioned apertures.

On W19149r (K/P 118r, 1508-1510) he explores the role of aperture size in this phenomenon:

Principle how the species of bodies intersect themselves at the edges of apertures penetrated by them.What difference is there in the penetration of species that pass through narrow apertures and those that pass through large apertures? On those that pass along the sides of umbrous bodies?

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He interjects a subtitle before answering these questions:

On the movement of the species of immobile objects.

The species of immobile objects are moved, moving the lips of that aperture through which the rays of the species penetrate and this occurs through the 9th which states that the species of any body are all in all and all in every part of the site surrounding them. It follows that moving one of the lips of the aperture, where such species penetrate a dark place, it permits the rays of species which were in contact with it and joined themselves with other rays of these species which were remote etc....

In previous passages he had used the camera obscura principle to confirm that images are "all in all." Here he uses the "all in all" concept to confirm principles of the camera obscura. Beneath this passage he draws a diagram (fig. ) showing four objects passing through an aperture. He adds nine letters to the diagram but these receive no caption. Diagram and text function independently. He now writes a further heading: "on the motion of right or left, superior or inferior edge" of the aperture, which introduces a more detailed analysis of the phenomenon:

If you move the right side of the aperture, then the impression will move left of the object [on the] right which penetrates through this aperture and all the other sides of such an aperture will do the same and this is proved by the second of this which states that all the rays which carry the species of bodies through the air are rectilinear. Hence, with species of very large bodies which need to pass through minimal apertures and after such an aperture to recompose to their maximal dilation, it is necessary to generate the intersection.

Beneath this he in turn draws three diagrams (figs. ) of which only one is lettered and none is explained.

5. Further Drafts and Statements

In addition to these demonstrations involving mirrors and camera obscuras, there are a number of other drafts and statements concerning the idea that images are "all in all and all in every part." These are primarily of interest because they reveal how painstakingly he juggles with various combinations of a key word term. On BM186v (c.1492), for example, he makes a very tentative draft:

the all the quantities of bodies send all the quanti.the pure air is passage is capable of receiving in itself similitudes fromwithout interval of time. Every...corporeal quality which...is percussed by solar rays or by the cause from an illuminated or other luminous body and all the parts of this are capable of all the things seen by this.the entire quantity of the pure air positioned opposite bodies sees the partEntirely and see [n] by the bodies positioned opposite it. Every particle of a point of this body sees all the said air and all this air sees each point and each point of the air is capable of all.

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Immediately following comes a second draft:

Every part of the air receives in itself all the species of bodies which see it and are seen by them at the same time. Every point of the air is capable of all the species of bodies seen by it (and each point of the bodies is capable of receiving in itself the entire similitude of the air positioned opposite it (the entire (air) sum of the air sees eachall the pure air is capable of all...the bodies...and the species of these (and each body is capable of the similitude of the air seen by it.

On CA396rb (1492) a clearer formulation emerges:

The air is all in all and all by similitudes in the parts positioned opposite it.If there is not an opaque body int he air, all of this is capable of all and of the part and the part of the part and of the all.

Whence we shall say that the air is all in all...interwoven and filled with infinite rays of species of bodies which are situated in it and this air is filled with infinite points and every point is indivisible and all the parts of this indivisible and all the species are capable of the parts of the bodies opposite each other and in this point they are entirely united and entirely divided and distinct without interference (confusion) by one another.And the pyramids of species are in all this air without occupation of one another and each in itself is divided all in all and united. And although the species go pyramidally to the eye, the eye does not recognize [them] if it does not...make contrary pyramids facing the thing seen.

These drafts are not always long-winded. For instance, on CA138vb (1492) there is a short version:

On air.The air is (filled with that) all filled with species (equal) caused by the objects which surround them.(Again) I say that the objects fill the air with their similitudes.

On CA120rb (c.1505) there is another short draft, this time interrupted: "The air is all in all and all in every part of this filled with the similitudes of bodies which in themselves include...." The drafting process continues into the late period as witnessed by a series of crossed out statements on CA345rb (1505-1508):

(The ... species of any visible object are all infused in all the counterposed air and are all separate in each part of the same air.)All (the species see) the objects seen from a single point are seen again from the same point.The species of the objects which, confusedly with their mixture fill of themselves the air opposite are all in all this air and are all in every part of it.The species of the object are all in all the air opposite these objects and all in every part of this air.The objects fill their species in the air seen by these objects.

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All that air sees each object, ... which is seen by that object.

For Leonardo drafting is obviously not just a matter of crossing out a troublesome word. It is a process of reformulating an entire phrase, transforming its elements with a thoroughness that recalls the systematic manner in which he plays with perspectival variables. Immediately following these drafts on CA345rb he interjects a passage on light and shade (cf. p. below) and then continues:

No thing is seen which does not send its species through the airSince it is visible...One cannot see a spirit in the countryside that is other than what one sees.Therefore no spiritual or transparent thing can see anything positioned opposite it, because it is necessary [that there is] in it a dense and opaque object; and if it be thus, it is not called spirit.Proof how no thing can be seen except through an aperture through which passes the air full of the species of objects, which intersect one another ... on the dense and opaque sides of the aforesaid apertures; and for this [reason] nothing that does not have a body can see either the shape or the colour of any object, such that it is necessary that there is a dense and opaque object through which the species of objects impress their colours and shapes.

Throughout the mediaeval tradition it had remained an open question whether the images of sight are spiritual (psychological) or corporeal (physical). With statements such as the above Leonardo brings the problem of vision into the domain of physics. We are a significant step closer to Kepler's concern with the pictura rerum which he distinguishes from the imagines rerum. On the reverse of the same folio (i.e. CA345vb, fig. , 1505-1508) he launches into a eulogy of the aperture in the camera obscura:

Since the species of objects are all in all...the air positioned opposite them and are all in a point of it, it is necessary that the species...of our hemisphere...enter and pass with all the celestial bodies through the natural point, in which they are infused and mix in the penetration and intersection of one another, as the other of the one in which the species of the moon in the east and the species of the sun in the west are united and infused together with our hemisphere at such a natural point. O marvelous necessity, thou, who with reason constrain (with supreme reason) all the effects to participate in their causes, and with supreme and irrevocable laws...every natural action obeys thee with the shortest (law) operation. Who would believe that this...very smallest of spaces would be capable of the species of the entire universe? O mighty action, what ingenuity could penetrate such a nature? What tongue would there be who could unravel so great a wonder? Certainly none. This it is that guides human discourse to divine things, etc.

And [they] imprint themselves in the...wall, opposite the above mentioned perforated point in a thing wall, and by this means the eastern part will impress itself on the western part of this wall and the western on the eastern and likewise the northern on the southern and conversely.

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A recent commentator has claimed that the section between the markers...constitutes an "unconnected" digression from "physical phenomena into meditations on philosophical matters."21 That Leonardo himself would not have considered this an "unconnected" digression becomes clear if we turn to the philosophical and theological traditions.

6. Philosophical and Theological Dimensions

The philosophical conotations of the "all in all" concept can be traced back in Antiquity. Lucretius reported it as the view of Anaxagoras and attacked it.22 In the fifteenth century Nicholas of Cusa in his Docta Ignorantia returns to Anaxagoras' notion that "everything is everywhere" and uses it as starting point for his own concept of God who constitutes "all in all and everything in every point" (omnia in omnibus esse constat et quodlibet in quolibet).23 Nicholas of Cusa uses the example of two mirrors positioned opposite one another to demonstrate this.24 An edition of Cusa's Docta ignorantia was published in Milan in 1502. Luca Pacioli, in the introduction to his Divina proportione (15-09) adapts Cusa's concept of God when listing reasons for the title of his book:

the fourth reason is that just as God cannot change and is all in all and all in every part (tutto in tutto e tutto in ogni parte) so too does our proportion present it always in every quantity continuous and discrete.25

Pacioli's words to describe God are precisely those, which Leonardo uses to describe the nature of images. Pacioli and Leonardo were colleagues in Milan (1496-1499) and later in Florence Leonardo lived in Pacioli's house in Florence.26

These links are not restricted to optics and theology. On Forst.II 128r (1495-1497) Leonardo notes that "Gravity is all in all the length of its support and all in every part of this" (cf. M. 40v, 1499-1500). For Leonardo the concept of "all in all" thus involves connections with various branches of physics, and even metaphysics as evidenced on CA385vc (1510-1515) where he credits Anaxagoras with the idea that:

Everything comes from everything and everything becomes everything and everything turns into everything because that which is in the elements is made from these elements.

In this context the passage on CA345vb (1505-1508) emerges as anything but a digression. His physical demonstrations of the "all in all" concept using mirrors and camera obscuras can be seen as ways of visualizing the metaphysical ideas of Anaxagoras. At the same time these physical demonstrations serve as visible experiments mirroring the invisible Creator, a subtle form of natural theology. By shifting from the physical to the metaphysical on CA345vb he thus establishes a synthesis between natural and divine, microcosm and macrocosm.

7. A Synthesis of Circular and Pyramidal Propagation

Writers on optics in Antiquity had been familiar with both circular (cf. above p. ) and pyramid 1 (cf. above p. ) concepts of rays, but did not consider possible contradictions between

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these modes of explanation. This is probably because Ancient authors never attempted to visualize the progress in terms of three-dimensional diagrams. Granted Euclid had used figures throughout his Optics but these remained abstractions, always viewer-centred and usually amounting only to variations on the theme of visual angles. The optical writings of Aristotle as they have come down to us are virtually devoid of illustrations. In Ptolemy and Galen diagrams were minimal and even in Alkindi and Alhazen they remained at a premium.

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(Figs. 142-143: Pyramidal propagation of light in Witelo's optical treatise (II.37-38).

Witelo's optical compendium (c.1270-1280) marked a significant advance in this respect. It contained illustrations of complex optical phenomena independent of a vanishing point at the observer's eye. In book two27, for instance, Witelo described the properties of rays passing through a pinhole aperture in a camera obscura. Accompanying this he attempted to reconcile pyramidal and circular modes of propagation with the help of two diagrams (figs. 142-143).

A fourteenth century manuscript copy of Pecham's Perspectiva communis 28 went one step further (fig. 144). It showed a circle circumscribed by pyramids, some of which in turn are framed by larger pyramids. In the latter fourteenth century Blasius of Parma wrote a commentary on Pecham's treatise with a third quaestio asking "whether the entire visible object and any point in it terminates a radiant pyramid of its light and colour in every part of the medium"29 The third argument of this quaestio turned on the contradiction between pyramidal and spherical modes of explanation:

Any luminous body (and as I shall say briefly) any natural agent acts spherically, or in the mode of a sphere: hence none such acts pyramidally. The argument holds since the pyramid and the sphere are bodies of various kinds (rationum) and the foregoing is evident by experiment, reason and authority....30

Attempts to reconcile spherical and pyramidal explanations continued. A fifteenth century manuscript of Pecham's treatise contains a diagram31 (fig. 145) analogous to the earlier example just cited, but with one significant distinction: although more roughly drawn each of the circumscribing pyramids is here the source of a larger pyramid. In Bacon's Tractatus de speculis 32, there is a related diagram (fig. 146).

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Figs. 144-146: Pyramidal propagation of light from mediaeval works by Pecham and Bacon.

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Figs. 147-150: Pyramidal propagation of light in Leonardo's notebooks. Fig. 147, C20r; figs. 148-149, CA101vb; fig. 150, A86v.

Leonardo builds on this tradition. On A2v (1492) he formulates his "all in all" principle:

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Every body fills the surrounding air with its similitudes which similitude is all in all and all in the part.The air is filled with infinite lines rectilinear and radiant intersecting and joined together without occupation of one another and will present from each object the true form of their cause.

This he reformulates on A27r

PerspectiveImmediately that the air is illumined, it is filled with infinite species which are caused by various bodies and colours which are arranged beneath it, of which species the eye makes itself a target and lodestone.

Pecham, it will be recalled, had written: "Any point of a medium or illuminated object simultaneously illumines the whole medium adjacent to it."33 On A86v (BN2038 6v, 1492) Leonardo pursues this theme, now attempting to picture how pyramidal diffusion takes place. He draws (fig. 150) by drawing a circle around which he subscribes a series of pyramids in all directions in a manner that recalls the manuscript illustrations in Pecham (figs. 144-145). But unlike these, in which each of the pyramids is extended, Leonardo has extended only one of the pyramids and in order to demonstrate how it becomes narrower as it goes further from its source. He adds letters to this extended pyramid, but does not mention them in the text.

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Figs. 151-153: Drafts of pyramidal diagrams on CA101vb. Cf. figs.148-150.

Even so the text offers another instructive glimpse into how his associative mind deals with a problem. In the right-hand column he continues with the theme of pyramids of light:

Every pyramid composed of a long concourse of rays, contains within it infinite pyramids and each has power for all and all for one.The equidistant circuit of the pyramidal concourse will give to its objects equality of angles and the thing received from the object will be of equal size.

In the left-hand column he balances this with a discussion of pyramids of shade (cf. Pt.2.1):

Every umbrous body fills the surrounding air with its infinite similitudes, which through the infinite pyramids infused through it, represent this body all in all and all in every part.

He begins to draft a another phrase: "Every radiant pyramid from afar," stops and crosses it out. Whereupon he decides to return to an idea which he had begun drafting on CA101vb (c.1492). The first of these drafts had been very rough indeed:

Perspective...body of the air is filled with infinite radiant lines, representing (carries) from the objects the form of their cause, in[ter]sected and intermixed by cause of...courses.

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(Every body) which...by straight lines going through the air, with disunited concourse...give to the first object cause to recreate in it the similitude...as a result of the said rays.

A second draft on CA101vb (figs. 148-149, 151-153, cf. 150) directly below this had been considerably more coherent:

The body of the air is filled with infinite radiant pyramids caused by the object positioned in it, which, intersecting and mixing, without occupation of one another, with disunited concourse infuse themselves with equal power and all are able as much as each and each as much as all and by these the similitude of the body is carried all in all and all in every part and each in itself receives in every minimal part it entire cause.

A second draft on CA101vb (figs. 148-149, 151-153, cf. 150) directly below this had been considerably more coherent:

The body of the air is filled with infinite radiant pyramids caused by the object positioned in it, which, intersecting and mixing, without occupation of one another, with disunited concourse infuse themselves with equal power and all are able as much as each and each as much as all and by these in every part and each in itself receives in every minimal part it entire cause.

This draft becomes the starting point for the lower paragraph on A86v (1492):

The body of the air is filled with infinite pyramids composed of radiant and straight lines which are caused by the superficial extremities of umbrous bodies positioned in this air and the more they go further from their cause the more they make themselves acute. And even though their concourse is intersected and intermixed nonetheless they do not interfere with one another (nonsi confondano) and with disunited concourse they go amplifying and infusing themselves through all the surrounding air and among themselves they are equal in power and all as each and each as all and through this the similitude of the body is carried all in all and all in the part and each pyramid receives in every minimal part of it, the entire form of its cause.

This passage clarifies the integral connection between the "all in all" concept and the principle of pyramidal propagation. It offers no hint, however, as to how circular and pyramidal modes are to be reconciled. A first step towards such a reconciliation is taken on CA126ra (c.1487-1490), here Leonardo draws (fig. 154) a candle, which has its pyramidal rays, mn, blocked by a wall. This is to show: "how the lines, or indeed luminous rays, do not pass through other than diaphanous bodies." A second diagram (fig. 155) shows the pyramidal properties of reflected light or, as he puts it:

How the base...xo, being illuminated by the point p, generates a pyramid which finishes at the point c and recauses another base at rs, which receives upside down that which is at xo.Next, he draws a box (fig. 156) with an aperture at both ends in front of which candles have been placed in order to demonstrate "how the point is cause of the base, and [if you] put a coloured glass in front of each light, you will see the base tinted in this." Implicit here is a

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demonstration how pyramids pass through one another without interference.

He now turns to consider sound. He draws (fig. 157) a hammer hitting an anvil, shows how sound spreads pyramidally and "how the lines of the blow pass through every wall," without the pyramid being interrupted. A further diagram (fig. 158) shows "how, finding an aperture, many lines are caused, each weaker than the first ab." Below this he illustrates (fig. 159) how the pyramidal reflection of sound passes through "the mouth of the echo."

(figure)Figs. 154-165: Studies of pyramidal propagation of light and sound, magnetism and odour. Figs. 159-164, CA146ra; fig. 165, C20r.

In the following diagram (fig. 160) he applies his pyramidal principles to magnetism to show "How the lines of the magnet and those of iron pass through the wall, but that which is lighter is attracted by the heavier." In a second example devoted to magnetism he illustrates (fig. 161) how "being of equal weight the magnet and the iron will be drawn in one way." A final example in this series shows (fig. 162) two rooms joined by an aperture to illustrate "how odour does the same as the blow."

This series of diagrams on CA126ra (1492) illustrates how light, sound, magnetism and odour all follow the same pyramidal laws and thus eiptomize Leonardo's potentially mechanistic view of physics (cf. Pt.I.2). Directly beneath these nine diagrams on CA126ra he draws (fig. 163) a base al., on which he draws a series of seven pyramids, each having their apex along the circumference of a semicircle. He marks various intersections with letters and numbers, does not discuss them, however, and only notes that "every base fills the air with infinite pyramids."

He has so simplified his diagram that it takes an effort to appreciate the complexity of the physical model which it implies. The half-circle which he draws (fig. 164) needs to be interpreted three-dimensionally as a sphere, the circumference of which represents a circular wave-front, simultaneously a circular bundle of pyramids, which is constantly expanding.

(figure)

Figs. 166-169: Author's reconstruction of various expanding and contracting pyramids implicit in Leonardo's diagram. Fig. 170: Leonardo's diagram on C20r (fig. 115) doubled to produce a complete circle.

This expansion can be visualized in various ways. The original centre is the apex for a series of pyramids which expand with distance (fig. 166). Then expansion can also be seen as a series of double pyramids (i.e. squares) and circles, which increase with distance (fig. 167). Each of these pyramids and circles can be seen as a base for other pyramids which become successively narrower with distance (fig. 168, cf. fig. 150). The apexes of these pyramids in turn become the bases of other pyramids whose apexes lie along a common circumference (fig. 169, cf. fig. 163 where he notes that "every point causes infinite bases").

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On C20r (fig. 165, cf. 170) Leonardo integrates these contracting and expanding pyramids within a single picture. He draws a line abc. This serves as diameter of an imaginary half-circle with centre b. Along this diameter he marks 41 points some of which he marks with letters. If we begin at a., for instance, then the first point is m., the twelfth is d., the twenty-first is e., the twenty-ninth is n. and the thirty-second is f. He joins each of these 41 points to each of the points a, b and c producing an interplay of pyramids which are at once expanding and converging. For the point b. is the apex of 42 expanding pyramids while the diameter ac. is the base of 41 converging pyramids with their apex at the circumference of the semi-circle which represents half of a three dimensional sphere. Along the line be., which bisects the semi-circle, he marks the ninth intersection as 1. and then numbers each successive intersection until the final one at 13.

(figure)

Figs. 171-173: Punctiform pyramidal propagation of light. Fig. 171, CA144vb; figs. 172-173, CU2.

In the accompanying text only some of these letters and features are explained:

Universally all points causing the extreme points of pyramidal species of things are continually all in all the air together with these and joined without any interval.Necessity makes that nature orders or indeed has ordered [that] in all the points of the air all the species of the things positioned opposite concur by means of a pyramidal concourse of rays which have parted from these things. And if this were not so the eye would not (recognize) see in every point of the air which is found between itself and the object seen, the form and quality of the thing positioned opposite it.

Let us say that ac. is the thing seen. Let e. be the eye seeing this thing. You see that towhatever point the eye moves along the circle adefc. that the eye always finds the intersection where the entire base a.c. can be seen.

Well over a decade later he returns briefly to these ideas on CU2 (1500-1505) under the heading "Beginning of the science of painting" (cf. pp. ). here the diagrams (figs. 172-173) are so simplified that, if seen out of context, one would hardly guess the complex synthesis of circular and pyramidal propagation which they imply.

6. Conclusions

We have shown that Leonardo's physics of light and shade has its roots in Ancient thought; that his concept of percussion and other powers can be traced back to Antiquity as can his basic definitions of line, point and pyramid. What distinguishes Leonardo from this tradition in his use of experiment and diagrams to render visible complex three-dimensional situations. Hence the circles produced by pebbles thrown into water (figs. 79-81; 113-114) visualize in one plane the spherical and pyramidal propagation of light in the air.

The unending reflections of objects positioned between two plane mirrors illustrate the principle that images are all in all and all in every part. Images which intersect without interference

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at the point of the aperture of a camera obscura provide a further demonstration of the same. These experiments so intrigue him that he becomes almost obsessed with his "all in all" phrase. As we have shown, however, there are also deeper reasons for this. Through Anaxugoras the phrase had philosophical implications and through Nicholas of Cusa it had religious connotations.

As we have shown Leonardo's response to this tradition is an ingenious synthesis. He uses Euclidean concepts of mathematical points lines and pyramids as a basis for his physics and thus develops a concept of non-spatial light which avoids later problems of Huygen's mechanistic wave model (see pp. ) and Newton's corpuscular mode.34 We have also shown how Leonardo uses the four powers (percussion, force, gravity and movement) to provide a theoretical framework for this mathematical physics of light. Ironically, his experiments and demonstrations concerning these powers are so vivid and concrete that they invite precisely the mechanistic wave and corpuscular interpretations which he was at pains to avoid. With this understanding of the historical context and the principles underlying Leonardo's approach, we shall examine the details of his physics of light and shade in part two.

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Part Two Physics of Light and Shade

Chapter One Definitions and Categories

1. Introduction 2. Light and Sight 3. Categories of Light 4. Categories of Shade5. Lustre 6. Colour 7. Darkness 8. Conclusions

1. Introduction

Leonardo's fascination with light and shade relates both to his painting practice and his optical theory. As has been shown elsewhere (Vol. One, Part Three), light and shade, or chiaroscuro as he terms it, makes possible the drawing of muscles (CU50, TPL42, 1505-1510), provides a sense of relief or depth in objects (A96v, BN 2038 16v, CU847, TPL667, 1492, CU844, TPL716, 1508-1510) and makes figures appear life-like (A100r, BN 1038 20r, 1492). For this reason he makes chiaroscuro the second principle of his painting practice (CU4, TPL5, 1500-1505), and describes it as the "excellence" of that science (CU840, TPL671, 1508-1510).1

He believes that the properties of light are analogous to those of sight. This is why his physics of light and shade involves many themes that relate directly to his theories of vision. Hence it will be useful to examine these light-sight analogies before considering his definitions and categories of light, shade, lustre and darkness. He also makes reference to seven books on light and shade. Richter, in the Literary Works make a rough attempt at reconstructing six of these books.2 Pedretti has taken this one step further.3 We shall show, however, that Leonardo's outline can be taken more seriously; that all seven books, plus an eighth on movement can be reconstructed and that his studies in this domain involve a much more coherent structure than has hitherto been imagined. Moreover, these studies reveal an unexpected amount of systematic experimental research.

In a subsequent chapter it will be shown that his studies of the camera obscura go hand in hand with these studies on light and shade. For instance, he makes systematic studies of different sizes and shapes of apertures in order to discern more complex properties of light and shade. He demonstrates that the image in a camera obscura sometimes resembles the aperture, and at other times assumes the shape of the original light source. He also discovers that, under certain conditions, the boundaries between light and shade are fully obscured, leaving a spectrum of gradations. In his early writings he claimed that light and shade were more difficult than design (disegno, A81r, BN 2038 1r). His new findings convince him that the boundaries between light and shade are most difficult of all (CU106, TPL121, 1505-1510).

Having considered his studies of light and shade (part two), it will be shown how these studies serve as a basis for his physiology of vision (part three): how he compares the aperture of a

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camera obscura with the aperture of an eye, and how his work on gradations of light and shade influences his claims that the eye cannot perceive boundaries clearly.

2. Light and Sight

Leonardo's conviction that the properties of light and sight are fully analogous explains the close links between his physics of light and shade and physiology of vision. His earliest extant reference to this analogy occurs on W19147-19148v, (K/P 22v, c.1489-1490) where he notes that: "Light does the same, because in the effects of its lines and maximally in the works of perspective it is very similar to the eye." Some two years later he explores this theme at length on CA204vb (c.1492):

(Sight)Light in the function of perspective is no different from the eye. The reason that light is no different from the eye as concerns occluding that which is behind the first object, is this: you know that as concerns speed of motion and the concourse of straight lines, the visual ray and the luminous ray are the same....

Directly following he gives a demonstration:

For example, if you place a coin...near the eye, that...space which occurs between the coin and the extremity of the site, the more that it is capable of a greater interval, the greater will be that part of the extremity of the site that is not seen by the eye and [hence] the closer that the coin is to the eye, the more will the extremity of the site be occluded.

He expands on this in the next paragraph:

On LightThe same will occur in light, because in bringing a coin nearer or further from this light you will see the shadow increasing or decreasing on the opposite wall. And if you wish an example, do it in this form: have placed in a room many bodies of various objects. Then take in your hand a long pole with charcoal at the end and with this mark off on the ground and on the walls...all the [projected] boundaries of the panels which appear beyond the [actual] boundaries of the panels.

In the right-hand margin alongside this he writes the headings: "On the eye," and "On light." In the paragraph that follows his description continues:

Then at this same distance and height take a light and you will observe that the shadows of the said bodies occupy as much of the wall, as is that part which finds itself included within the marks made by the charcoal at the end of the stick.

Next he describes a related demonstration:

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Experiment.If you wish to see clearly a similar experiment place a lamp at the head of a table...then go

backwards somewhat and you will observe all the shadows of the objects which are between the wall and the light stamped by shadow...in the form of the objects...and all the lines of their length directed to the point where the light is.

Then bring your eye nearer to this light, producing such shadow with the edge of a knife so as not to offend the eye and you will see all the objects positioned opposite without shadow and the shadows which were on the panels of the eye will be occluded at the eye by the interposed objects.In short, the area thrown into shadow by candlelight coincides with the area occluded by the same object as a result of the visual angle. He develops this idea on CA241vd (1508-1510):

In all the functions of shade spherical light is as the eye in its visual power, because one opaque sphere interposed between the eye and a panel will occlude as much of the light on that panel as would the eye.

He pursues this analogy on CA195v (c.1510):

A window does the same on each side whether above or below and likewise on one side as one the other where it borders on the dark wall.And the species that enter the pupil do the same and they are thrown to the right and left, intersected behind the interposed object.

This close analogy between geometrical light rays and visual rays explains (a) why various folies with notes on the physics of light and shade also deal with the physiology of vision and (b) why he continues to use "perspective" in its mediaeval sense meaning "vision" as, for example, on W19037v (K/P 81v, 1489-1490), when describing his anatomical programme, he refers to "perspective, through the action of the eye" or on CA119va (c.1492) where he makes the heading: "Premium to perspective: that is, on the function of the eye." Leonardo takes this analogy further. In his mind the geometry of visual rays, the geometry of rays of light and shade and the geometry of rays of linear perspective are all equivalent. Hence his claim in a draft note on BM103v (1490-1495) that "all the functions of lights are similar to those of the eye as concerns the perspective of painters." On A103v(BN 2038 23v) he claims that in "all cases of linear perspective it is similar to light." For this reason he is usually not concerned with distinguishing clearly between perspective (a) in the sense of "optics" and (b) in the sense of "linear perspective," as is evidenced by his definition of perspective on A3r (1492):

as a demonstrative reason by means of which experience confirms that all things send their similitudes to the eye by pyramidal lines.

For the same reason, he deals interchangeably with problems of vision, linear perspective, and light and shade with respect to anamorphosis in his treatise on perspective on A36v-A42v analysed earlier (see Vol. 1, Part I:2 ). Moreover, this light-sight analogy explains why a number of folios dealing with rays of light from the sun also contain notes on vision (see below pp. figs. ). Indeed, once we understand that analogy is a key to Leonardo's organization, much of the apparent confusion of his notebooks disappears.

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3. Categories of Sight

Leonardo has practically no definitions of light. He calls light the chaser of darkness of CA116rb (1492). He notes that light makes objects evident (dimostra), whereas shade hides them (A102r, BN 2038 22v, 1492). He does not dwell on the metaphysics of light. The importance he attaches to the subject is, nonetheless, clear from the opening passage of Pecham's Perspectiva communis that he copies in translation on cA203ra (1492).

Among the study of natural considerations, light delights its contemplators most. Among the great things of mathematics the certainty of the demonstrations raises most sublimely the mind of the investigators. Hence perspective is to be proposed above all the traditions and humane disciplines, in the domain of which the luminous ray, complicated by the modes of demonstrations, finds the glory of both mathematics and physics, decorated by the flowers of each.Leonardo is more interested in various categories of light. In the Mediaeval optical tradition one had distinguished between primary (direct) and secondary (reflected light.4 There had also been a distinction between lux (A102r, BN 2038 22r, 1492). He also makes a distinction between primitive and derived lights which, according to his definition on C16v (1490), would correspond to the mediaeval terms primary and secondary light: “That light is said to be primitive which primarily lights umbrous objects and derived is said to be that which is reflected from those parts.”On CU158 (TPL157, C.1492) he reformulates this definition, now substituting "original" for "primitive":

...lights are of two kinds (nature): one is called original, the other derivative. Original I claim to be that which derives from the flame of a fire or from the light of the sun or the air. Derived light is reflected light.

In another passage on BM171r (c.1492) he reverts to his earlier terms, primary and derived, but now defines them somewhat differently under the heading:

How lights are of two kinds, the one separate, the other joined to bodies.Separate is that which illumines the body: joined is that part of the body illumined by that light. The one light is called primitive, the other, derived. And likewise shadows are of 2 kinds: the one primitive, the other derived shadow. Primitive is that which separates itself from bodies, carrying in itself the form of its cause to the panels of walls. He returns to this distinction between primitive and derivative light on CA116rb (c.1498 or 1495-1497): "Primitive light is that which is the cause of illuminating umbrous bodies. And derived lights are those parts of bodies...illuminated by the first light."

Alberti in his On Painting had distinguished between light from the stars, i.e. sun, moon, enus, etc. and light from fires.6 He had also mentioned reflected light. Leonardo takes these distinctions further. In an early note on BM 170r (1492), he distinguishes simply between free and restricted light (figs. ):

How there are 2 different lights, the one is called free and the other restricted. The free [light] is that which freely illumines bodies, restricted is that which illumines bodies all the same through some aperture or window.

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On A109r (BN 2038 29r, 1492) he develops this into a threefold distinction:

First you will treat of the lights made by windows, to which you will give the name restricted air; then you will deal with lights of the countryside, to which you will give the name free light and then you will deal with the light of luminous bodies.

This evolves into a fourfold distinction on G3v (1515, cf. CU598, TPL663, c. 1510), under the heading:

Of Lights.The lights which illuminate opaque bodies are 4 kinds, that is, universal, as is the air which is within our horizon, and particular, as is that of the sun or of a window or a door or some other space. And the third is reflected light and then there is a 4th, which passes through transparent things, such as cloth or paper or the like, but not transparent like glass or crystal or other diaphanous bodies, which have the same effect as if nothing were interposed between the umbrous body and the light which illuminates it and of these [various kinds of light] we shall speak distinctly in our discourse.

Elsewhere he is content to omit mention of transparent light and limit his discussion to three kinds as on CU866 (TPL754, 1508-1510):

On giving the necessary lights to illuminated objects in accordance with their sites.One needs to have great respect for the lights accommodated to the things illuminated by them since, in a given story, there may occur parts which are in the countryside in the universal light of the air and other which are in porticoes, which are a mixture of particular and universal light and others in particular light, such as in houses which receive light from a single window.Of these three kinds of light, for the first it is necessary to take the lights of large fields by the fourth of the first which states "such is the proportion of size to size on the part of illuminated bodies as is that of the size to size of the bodies illuminating them." And again of those, namely, which require the reflections from one body to another, where the light enters through narrow places among those bodies illuminated by universal light, because with the lights which penetrate between bodies close to one another, the same happens as with the lights which penetrate the windows and doors of houses, which we term particular lights.And which shall also make the necessary remarks about this in its proper place.

A late reference to this threefold distinction is found on E3v (c.1513-1514) under the heading of:

Painting.Of the 3 kinds of lights which illuminate opaque bodies.The first of the lights with which opaque bodies are illuminated is said to be particular, and this is the sun or some other light of a window or a fire. The second is universal as happens

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in times of cloud or fog or the like. The third is compound, that is in the evening or in the morning when the sun is entirely beneath the horizon.

This third variety of light here mentioned he had described in more detail on CU593 (TPL558, 1508-1510) in a passage entitled:

What is the difference between compound light and compound shade?Compound shade is that which participates more in the umbrous body than in the luminous body. Compound light is that which participates more in the luminous than in the umbrous. Hence, we shall state that compound shade and light take their name from that in which it participates most, namely, that if an illuminated object sees more shade than light, that it is said to be invested with compound shade and if it is invested with more from the luminous body than from the umbrous one then, as we said, it is called compound light.

Hence Leonardo considers various categories of light: primitive (separate) and derived (joined); universal (free), particular (restricted), reflected, translucent and compound. As we shall see some of these categories recur in his discussion of shade.

4. Categories of Shade

Pecham had claimed that "shadows are diminished light."7 "Shade, writes Leonardo on C14v (1490-1491), "is a diminution of light." He returns to this definition in a paraphrase on W19152v (K/P118v, C.1508-1510):

Shade is diminished light mediated by the opposition of an opaque body.Shade is the substitution of the luminous ray that has been intersected by an opaque (body].

On W19076r (K/P167r, c.1513) this definition becomes the first in a series of alternatives:

Shade is the diminution of light and of darkness and it is interposed between these darknesses and lights.Shade is of infinite obscurity and of infinite diminution of this obscurity.The beginnings and ends of shade extend themselves between lights and darknesses and [it] is of infinite diminution and infinite augmentation.Shade is a pronouncement by bodies of their shapes.The shapes of bodies will not give knowledge (notitia) of their quality without shade.

He returns to the first of these definitions on E32v (1513-1514, CU580, TPL550a, 1513-1514): "Shade is diminution of light." A variant of this basic definition is found on A102r (BN 2038 22r, CU577, TPL549c, 1492): "Shade is privation of light and only the opposition of dense bodies opposite luminous rays: shade is of the nature of darkness...." One of these definitions he repeats on CA116rb (c.1498 or 1495-1497): "Shade is privation of light." On CU604 (TPL660b, 1508-1510) he combines two of these basic definitions: "Shade is a diminution or a privation of light." At other times as on Mad II 25r (c.1503-1504) he defines darkness as "a privation of light" claiming that "shade is an alleviation of light. And there is no darkness where there is not some exhalation of air." He returns to this formulation on CA207ra (c.1508-1512):

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Darkness is a privation of light.And shade is an alleviation of this light.Shade is a mixture of darkness with light.Shade is an alleviation of darkness and light.

A related version occurs on CU578 (TPL665, 1508-1510): "Darkness is a privation of light and light is a privation of darkness. Shade is a mixture of darkness with light." On A102r (BN 2038 22r, CU577, TPL549, 1492), he describes shade as being: " of greater power than light, because this prohibits and deprives objects entirely of light, and light can never hide the shade of bodies, that is, dense bodies." On CU582 (TPL556, 1508-1510) he provides yet another definition: "shade is said to be that where no part of the luminous body or illuminated body can see." On CA207ra (1508-1510) he drafts two further definitions:

Shadow in opaque bodies is percussion of the species of dark bodies.Shadow is a mixture of bright and dark and of that much more or less darkness to the extent that the bright which mixes itself with it is of greater or less power.

These he crosses out and writes anew: "Shadow is a mixture of bright and dark and is of that much more or less darkness as the bright with which...it mixes itself will be of greater power." As in the case of light Leonardo considers various categories of shade. A straightforward solution is given on CU597 (TPL569, 1508-1510) in a passage entitled:

How many sorts of shade are there?There are three sorts of shade of which one is born from a particular light such as the sun, moon or a flame. The second is that which derives from a door, window or other aperture, from whence one sees a large part of the sky. The third is that which is born from universal light, such as the light of our hemisphere when it is without sun.

In the passage on BM171r (c.1492) cited above it was noted that just as he distinguishes between primitive and derived light, so too does he distinguish between primitive and derived shade (fig. ). In the extant notes this distinction first occurs on Triv 11v, 29r (c.1487-1490). He returns to it on C14v (1490) where he lists a number of basic definitions:

Primitive shade is that which attached to umbrous bodies.Derivative shade is that which is detached from umbrous

bodies and passes through the air.Repercussed shade is that which is surrounded by an illuminated

wall.Simple shade is that which does not see any part of the

light that causes it.Simple shade begins in that line which parts from the

boundaries of the luminous bodies ab [fig. 174).

(figure)

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Fig. 174: Illustration of simple shade on C14v.

In the introduction to his books on light and shade on CA250ra (c.1490) Leonardo drops the term "primitive" and refers instead to: "Original shadows since they are the first shadows which invest the bodies to which they are attached." He then goes on to note that:

from these original shadows umbrous rays result which go spreading through the air and are of as many kinds as are the varieties of the original shadows, whence they derive. And for this reason I call them derived, because they are born from other shadows.

He pursues these distinctions on A102r (BN 2038 22r, 1492) now referring to primitive shade as joined and derived shade as separate (figs. ):

What is the difference between shade joined with bodies and separated shade.Joined shade is that which never parts from the illuminated bodies, as would be [the case with] a ball which, standing in the light, always has one part of itself occupied by shade, which never divides itself through a change in position made by this ball. Separate shade can exist and not be created by the body. Let us suppose that this ball is one braccia distant from a wall and from the opposite side is in light. The said light will send to the given wall just so much dilation of shadow as is that which finds itself on the part of the ball which is facing the wall. That part of the separate shade which does not appear, is when the light is below the ball, [such] that its shade goes towards the sky, and not finding resistance in its path, is lost.

On CA116rb (1498 or 1495-1497) this theme is taken up anew in a draft:

Primitive shade is that part of the bodies which cannot be seen by light.Derived shade...is only the percussion...of umbrous...rays.

Accompanying this passage is a diagram (fig. ) illustrating primitive and derived light and shade. Some years later he makes a passing reference to this distinction on CU585 (TPL570, 1505-1508): "The species of shadows are of two sorts, of which one is called primitive and the other derived." Almost as cursory is a passage on CU584 (TPL552, 1508-1510) entitled:

On shadow and its divisions.The shadows in bodies are generated from dark objects opposite these bodies and they are divided into two parts of which the one is called primitive and the other derived.

In the late period he returns once more to these terms on E32v (1513-1514, cf. TPL553a, c.1508-1510):

Shade is divided into two parts of which the first is said to be primitive shade and the second is derived shade.Primitive shade always serves as a basis of derived shade.The boundaries of derived shade are rectilinear.

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On C14v (1490) he had referred to repercussed shade. On E32r (c.1513-1514) he returns to this concept, now calling it something else: " On corrupt shadows. Corrupt shadows are said to be those which are seen by a white wall or other luminous bodies." On CU759 (TPL573, 1505-1508) he subdivides the category of derived shade, in a passage headed:

What are the varieties of derivative shade?The varieties of derivative shade are of two sorts, of which the first is mixed with the air which stands opposite the primitive shade and the other is that which percusses at the object which meets this derived shade.

As will be seen in the next chapter he further subdivides the first of these categories into shadows that increase with distance, those that remain constant and those that decrease with distance (figs. ). These basic categories are summarized in our Chart. 5. On CU588 (TPL572, 1505-1508) he considers another subdivision of primitive shade (figs. ):

In how many ways does primitive shade vary?Primitive shade varies in two ways, of which the first is simple and the second is compound.Simple is that which faces a dark place and for which such shadow is tenebrous. Compound is that which sees a place illuminated with various colours, such that this shadow will mix itself with the species of the colours of the objects positioned opposite.

He returns to this distinction between simple and compound shade on BM248v (c.1508): "Simple shade is that which does not see any luminous body. Compound shade is that which is illuminated by one or more luminous bodies." On CU590, (TPL553, 1508-1510) he again describes these two kinds of shade in a passage headed:

On two species of shade and in how many parts they are divided.The species of shade are divided into two parts, the one of which is said to be simple, and the other compound. Simple is that which is caused by a single light on a single body. Compound is that which is generated by a number of lights on a same body or by a number of lights on various bodies.

In the passage immediately following on CU591-592 (TPL553a-b, 1508-1510) he divides simple shade further:

Simple shade is divided into two parts, that is, primitive and derivative. Primitive is that which is joined at the surface of the umbrous body. Derivative is that shade, which parts from the aforesaid body and passes through the air, and if it finds resistance it comes to rest at the place where it repercusses, with the shape of its own base.And the same is said of compound shade.

(Chart)

Chart 5: Classifications of kinds of shade in Leonardo's notes.

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In other words, simple and compound shade have here become generic terms to which the categories primitive and derived are now subordinate, (see Chart 5). He returns once more to this basic distinction between simple and compound shade on CU582 (TPL557, 1508-1510):

What difference is there between simple and compound shade?Simple shade is that where no part of the luminous body can see. And compound shade is that where, between the simple shade a part of the derivative light mixes.

In addition to these categories of light and shade he also provides basic definitions of lustre, colour and darkness. Each of these will be considered in turn.

5. Lustre

Leonardo distinguishes between light and lustre in an early note on A113v (BN 2038 32v, 1492):

What difference is there between light and lustre and how lustres are not among the number of colours and [a lustre] is always whitish and it originates on the extremities of bodies basked [in light]. Light is of the colour of the object whence it originates such as gold or silver or a similar thing.

A pithy definition of lustre occurs on CU769 (TPL664, 1508-1510) under the heading:

On illumination and lustre.Illumination is participation of light, and lustre is the mirroring of this light.

(figure)

Fig. 175: Illustration of lustre on G10r.

On CU774 (TPL775, 1508-1510) he develops this distinction between lights and lustre under the heading:

What is the difference between lustre and light?The difference that there is between lustre and light is that lustre is always more powerful than light and light is of a greater quantity than lustre. And lustre moves with the eye and with its cause, and with the one and the other, but a light is fixed at a given place, as long as the cause which generates it is not removed.

He returns to this theme on E31v (CU780, TPL776, 1513-1514):

Of light and lustre.What difference is there between light and the lustre which shows itself on the terse surfaces of opaque bodies?

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The lights which are generated on the smooth (terse) surfaces of opaque bodies will be immobile in immobile bodies even if the eye of these viewers moves. But the lustres will be on the same bodies in as many parts of its surface as are the sites where the eye moves.

In another late passage on G10r (c.1510-1511) he refers to lustre as an "accident" (fig. 175 ):

On the shadow of a leaf.Sometimes a leaf has 3 accidents, that is, shadow, lustre and transparency, as when the light is at n on the folio s and the eye at m which sees a illuminated, b in shadow and c transparent.

Elsewhere in this same treatise, on G24r (CU949, TPL886, c.1510-1511) he again refers to lustre as an "accident" but this time of colours:

Of the accidental colour of trees.The accidental colours of the leaves of trees are 4, that is, shadow, light, lustre and transparency.

6. Colour

In Antiquity the problem of colour had been linked with philosophical debates concerning the visual process. Leon Battista Alberti, in his On Painting (1434), is aware of these philosophical debates but consciously avoids them.8 He alludes to theory, but emphasizes practice.

6.1 Black and White

With respect to black and white, for instance, Alberti notes that "the painter ought to be persuaded that white and black are not true colours but alterations of other colours"9 and at the same time treats them as if they were colours. Leonardo inherits this ambiguous approach which theoretically rejects and practically accepts black and white as colours. On CU783 (TPL692, 1508-1510), for example, he mentions in passing "if it can be said of white that it is a colour. On CU739 (TPL699, 1508-1510), he notes that "neither white nor black are colours." On F75r (CU204, TPL247, 1508), he goes further and sets out to show "why white is not a colour" (see below p. ). This leads him, on CU205 (TPL215, 1505-1510), to discuss white as the most receptive of colours because it is empty:

What is that surface that is most receptive of colours?White is more receptive of the colours than any other surface of any body which is not mirrored.This is proved, saying, that every object which is empty is capable of receiving that which the objects which are not empty cannot receive. For this [reason] let us say that white is empty or, if you wish to say, deprived of any colour, it being illuminated by the colour of some luminous body it will participate more in this luminous body than would black which in its use is like a broken vase which is deprived of every capacity of a given thing.

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On F23r (1508) he notes, in passing, "no white or black is transparent."

6.2 Simple Colours

Nonetheless, in practical terms, he accepts black and white as simple colours, as on CU176 (TPL254, 1508-1510) where he lists six:

On the colours which result from the mixture of other colours, which call themselves secondary species.The simple [colours] are six, of which the first is white, although some philosophers do not accept either white or black among the number of the colours, because the one is the cause of colours and the other is the privation [thereof]. Yet since the painter cannot do without these we shall place it among the number of the others and we shall say that in this order white is the first of the simple, and yellow is the second and green is the third, and azure is the fourth, and red is the fifth and black is the sixth.

These simple colours he links with various elements:

And we shall let white stand for light, without which no colour can be seen, and yellow for earth; green for water; azure for air, and red for fire because there is no material or size where the rays of the sun have to percuss and consequently to illuminate.

His equivalents bear comparison with those of Alberti10, although the two authors disagree on the colour of earth (Chart 6). Leonardo makes a further list of simple colours on CU178 (TPL213, 1505-1510). Instead of six, he now mentions eight:

Simple colours I call those which are not compound, nor can they be composed through a mixture of other colours.Black, [and] white, even if these are not placed among the colours, because the one is darkness and the other is light, that is, the one is privation and the other generates, I do not want to leave them behind for this reason, because in painting they are the principal [colours] since painting is composed of shade and light, that is chiaro-scuro.After black and white follow azure and yellow, then green and leonine, that is, tan, or if you wish ochre, then morello and red. And these are the eight colours, and there are no more in nature.

6.2 Compound Pigment Colours and Compound Coloured Lights

He goes on to describe how these basic colours can be mixed to produce further colours:

(Chart)

Chart 6: Basic elements and equivalent colours according to Alberti and Leonardo

With these I begin mixtures, first white and black, then yellow and black, yellow and red, and since I here lack paper, I will wait to make such distinctions in my work with a long process

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which will be of great use and also most necessary. And such a description will be intermediate between the theory and practice of painting.

This leads to instructions concerning more complex mixtures of pigments (see below pp. ). Meanwhile, by 1492, he is studying how light passing through coloured glass becomes a different colour, as on CA126ra where, in connection with a camera obscura (fig. 156, see above pp. ), he notes: "how the point is cause of the base: ...put a coloured glass in front of a light and you will see the base tinged in this [colour]." On W19151v (K/P118v(B), 1508-1510) he describes how sunlight passing through both an azure and a yellow pane of glass produces a green light (see below pp. ). In such experiences he finds a new demonstration for the production of compound colours, as, for instance on CU176 (TPL254, 1508-1510):

If you wish with brevity to see the variety of all the compound colours, take pieces of coloured glass and through these look at all the colours of the countryside which are seen beyond this and thus you will see all the colours of things which are behind such a window and you will see what is the colour which you thereby improve or spoil.

This leads to a description of what happens in the case of various colours of glass:

When the aforesaid glass is of a yellow colour, I say that the species of the objects which pass to the eye through this colour can either become worse or better. And this deterioration of such colour will occur in azure and black and white more than in any of the others and the amelioration will occur in yellow and green more than in any of the others. And thus you will go glancing with the eye, the mixtures of colours which are infinite and in this way you will make a choice of colours of new inventions of mixtures and compounds. And the same is done with two panes of glass of different colours positioned in front of the eye and thus you can carry on on your own.

As a result of such experiences he revises his list of simple colours. Azure and green, which he had at first described as simple (CU176, 178), he describes as compound on CU177 (TPL255, 1505-1510):

On colours.Azure and green are not simple per se because azure is composed of light and

darkness, as is that of the air, that is, most perfect blackness and most bright white.Green is composed of a simple and a compound colour, that is it is composed of azure and green.

6.3 Light and Colour

Such experiences make him ever more aware of the extent to which colour is determined by the intensity and colour of nearby light and shade. In the early period, as on CU208 (TPL259b, 1482), he believes that a colour can only be true, if it is not seen by other shadow or lustre:

Of true colour.

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The true colour of some body will be shown in that part which is not occupied by any quality of shadow, nor of lustre, if it is a polished body.

He restates this idea on CU725 (TPL694e, 1508-1510):

When a particular light illuminates its object, which object has opposite it some object illuminated by the same light which is of a bright colour, then a counter-light is produced that is, the reflection or true reverberation.That part of the reflected light which partly invests the surfaces of the bodies will be that much less bright than the part illuminated by the air to the extent that this is less bright than the air.

Meanwhile he is also exploring the conditions for true shade as, for example, on CU803 (TPL703, 1508-1510):

What is, in itself, the true shade of colours of bodies.The shade of bodies should not participate in another colour other than that of the body where it touches. Therefore, since black is not included among the number of the colours, from it are taken the shadows of all the colours of bodies with more or less darkness which is more or less required in its place, never losing entirely the colour of this said body except in the shadows included within the boundaries of the opaque body.

On CU197 (TPL203, 1505-1510) he reconsiders the conditions required for true colour:

How nothing shows its true colour if it does not have light from another colour [that is] the same.No colour will ever show its proper colour if the light which illuminates it is not entirely of this colour.That which has been said is shown in the colour of vestments of which the illuminated folds reflect and give light to the folds opposite [and] make them show their true colour. A leaf of gold does the same in giving light to the other and it does the contrary of taking light from another colour.

He pursues this question of true colours on CU799 (TPL816, 1508-1510):

Precept.The true colours of the shades and lights of any body are such that the walls of the habitations where such a body is found are the colour of the body which is enclosed inside them and such that the light of the paper window covering, which illuminates such a habitation, is also the colour of this enclosed body and thus the habitation will generate sun only with its umbrous parts on the enclosed body which will be of a colour proportionate to that of this umbrous body and the illuminated parts of the colour of the window will correspond to the colour of the illuminated body and to the colour of its shades.

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The extent to which changes in light and shade are important depends in part on the original colour of the object. On CU787 (TPL636, 1508-1510), for instance, he points out that these changes are least in the case of black:

Which principle shade on the surface of bodies will have less difference from the luminous parts and which will have more?

The shade of black bodies, being primary, will have less difference from its principle lights than in the surface of any other colour.

On CU739 (TPL699, 1508-1510), he notes that these changes are greatest in the case of white:

What colours produce more variety between lights and shades?Among colours there will be a greater difference between their shades and their lights, which are more similar to whiteness because white has a brighter illumination and darker shade than any other colour, even though neither white nor black are colours.

He compares the effects of light and shade on white, black, green, azure and grey on CU783 (TPL692, 1508-1510):

Where and in what colour shadows lose the natural colour of the umbrous body more?White, which sees neither incident light, nor any kind of reflected light is that which first loses its proper natural colour entirely in its shadow, if it can be said of white that it is a colour. But black augments its colour in the shadows and loses it to the extent that the illuminated part is seen by light of a greater power.Both green and azure augment their colour in median shades. And red and yellow acquire colour in their illuminated parts and white does the same and mixed colours participate in the nature of colours which compose such a mixture, that is, black mixed with white makes grey (berettino), which is not beautiful in the final shades as is simple black and it is not beautiful in its lights, as is simple white, but its supreme beauty is between light and shade.

6.4 Light and Beauty

As early as 1492 he is aware that light also augments the beauty of a given colour and hence, on A112r (BN 20-38 33r, CU188, TPL210, 1492) he notes;

How the beauty of a colour must be in its light.If we see that the quality of colours is understood through light, it can be reckoned that where there is more light there one sees the true quality of the illuminated colour more. And where there is more darkness, the colour tinges itself in the colour of these shadows. Hence, [o] painter, remind yourself to show the truth of colours in the illuminated parts.

He reformulates this idea on CU187 (TPL207, 1505-1510):

How every colour which does not lustre is more beautiful in its luminous parts than in its umbrous parts.

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Every colour is more beautiful in its illuminated part than in its shaded part. And this arises because light vivifies and gives true information of the quality of colours and shade weakens and darkens the same beauty and impedes the information of this colour. And if it is objected that black is more beautiful in the shadows than in the lights, it is replied that black is not a colour, nor is white.

On CU189 (TPL242c, 1505-1510), he restates this connection between light and beauty:

Of colours.The colour which finds itself between the umbrous and the illuminated part of umbrous bodies is of lesser beauty than that which is fully illuminated. Hence the primary beauty of colours is in the principal lights.

He states this idea more forcefully on CU762 (TPL768, 1508-1510):

What part of umbrous bodies will show their colours with a more excellent beauty?The excellent beauty of a given colour, which does not have lustre in itself, is always in the excellent brightness of the most illuminated part of these umbrous bodies.

In the period 1508-1510 he also explores which colours compliment one another as on CU182 (TPL253):

On colours.The colours which belong together are green with red, or purple and violent and yellow and azure.

In the late period he returns once more to the connection between light intensity and beauty in a longer passage on E18r (1513-1514) which opens under the heading:

Colours positioned in the shadows participate that much more or less in their natural beauty to the extent that they are in lesser or greater darkness.But if the colours are situated in a luminous space, then these will show themselves of that much more beauty to the extent that the light is of greater splendour.

He now introduces the objection of an:

Adversary.The variety of the colours of shadows are as many as the variety of the colours that these shaded objects have.

This he counters with a:

ReplyColours positioned in the shadows will show themselves of that much less variety to the extent that the shadows which are situated there are darker. And a testimony of this is when

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from a piazza we look inside the doors of dark temples where paintings covered with various colours appear entirely covered with shadows.Hence at a long distance all the shadows of the various colours appear of a same darkness.Among bodies covered with light and shade the illuminated part shows its true colour.

6.5 Darkness and Colour

If greater light shows the truth of colours, greater darkness removes it. Alberti had expressed this corollary succinctly in his On Painting: "As shadow deepens the colours empty out and as the light increases the colours become more open and clear.11 Leonardo broaches this problem in passing in his "prophecies" on CA370ra (1497-1500):

Of the nights which do not know any colour.You will see so much that the difference between colours is not recognized, on the contrary, they will all be of a black quality.

He again mentions the effects of darkness on colour on CU194 (TPL201, 1505-1510):

Whether various colours can appear of a uniform darkness through a same shadow?It is possible that all the varieties of colour of a given shadow appear transmuted in

the colour of that shadow.This is manifested in the shadows of cloudy nights in which no shape or colour of a body is recognized. And since darkness is nothing more than the privation of incident light, reflection, through which all the shapes and colours of bodies are understood, it is necessary that when the cause of light has been removed entirely, the effect and recognition of the colours and the shapes of the aforesaid objects is lacking.

On CU696 (TPL715, 1508-1510) he pursues the question:

What is that body which with the same colour and distance from the eye has its lights varying less from its shadows?That body will show less difference between its shades and lights which will be in air of greater darkness and conversely in air of greater brightness, as is shown by the things placed in the shadows which cannot be recognized and the things which are placed opposite the brightness of the sun, that the shadows appear tenebrous with respect to the parts percussed by the solar rays.

He reformulates this idea on CU740 (TPL713b, 1508-1510) in terms of black surfaces:

What surface makes less difference between bright and dark?The black surface is again that which will participate more of this blackness, [and] has less difference between its umbrous and luminous parts than any other part, because the

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illuminated part shows itself to be black and the shaded [part] cannot be other than black, but with little variety it acquires some more darkness than the black illuminated part.

He mentions the problem once more on BM169r (c.1510): "Darkness (tenebre) tinges everything with its colour and the more a thing departs from this darkness, the more it is rendered of its true and natural colour." This idea he restates as a general rule on E30v (CU734, TPL592, 1513-1514):

Quality of shades.Among equal diminutions of light such will be the proportion from darkness to darkness of this generated shadow as there is from darkness to darkness of the colours where such shades are conjoined.

Elsewhere in the same treatise, on E17v (1513-1514) he reformulates this in terms of two propositions:

5th All colours positioned in umbrous places appear to beof equal darkness among one another.

6th But all colours positioned in luminous places never varyfrom their essence.

6.7 Colour Conclusions

At the outset Leonardo probably intended to write an independent treatise on colour. By the period 1508-1510 he intended to annex this treatise on colour to his work on the rainbow at the end of his book on painting, as is clear from a note on W19076r (K/P167r):

Do the rainbow in the last book on painting, but first do the book on the colours originating from the mixture of other colours such that you can, through these colours of painters, produce the generation of colours of the rainbow.

This explains the close links between his notes on colours and discussions of the rainbow (see below pp. ), which links, in turn, confirm a shift in his thinking. By 1510 theoretical questions concerning the substance of colour are no longer of interest to him. For all practical purposes colour is now a phenomenon of varying light and shade. This is why most of his notes on colour emerge as two chapters of his Books on Light and Shade (see below pp. ).

7. Darkness

Leonardo defines darkness as a privation of light, sometimes strengthening this with adjectives such as "entire" (CU665, TPL810, 1505-1510). In the extant notes he repeats this basic definition at least ten times (see Chart 7). On most occasions he treats darkness as an absolute category and carefully distinguishes between shadow and darkness, as on G8r (c.1510-1515): "The shadows of plants are never black because where the air penetrates there cannot be darkness." On rare occasions, as on CA371rb (1510-1515), he suggests that there may also be gradations of

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darkness: “The percussion of derived shade in darkness is similar to the darkness of this derived shade, that is, the variety of its darkness.”

(Chart)

Chart 7: Ten passages in which Leonardo defines darkness as a privation of light.

8. Conclusion

We have examined Leonardo's analogies between light and sight in order to explain the close parallels between his physics of light and shade and studies of vision. We have also outlined his definitions and categories of light, shade, lustre, colour and darkness were outlined. In the following chapter we shall show how he builds on these basic concepts in drafting his seven "books" on light and shade.

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Part Two Chapter Two Seven Books on Light and Shade

IntroductionBook One Light and ShadeBook Two Primary ShadeBook Three Derived ShadeBook Four Derived Shade and Interposed ObjectsBook Five Derived Shade and Reflected LightBook Six Reflected ColourBook Seven Reflected Colour and Distance(Book Eight) Movement of Shadows

Conclusions

Introduction

A careful study of Leonardo's notes on light and shade reveals a much more systematic approach than is at first apparent. He himself considers various ways of arranging these notes. On BM171r (c.1490), for instance, he writes: "You must first describe the theory and then the practice. First you will describe the shade and light of dense bodies and then of transparent bodies." Much more extensive is his outline on CA250ra (c.1490). This begins with a draft:

ProemiumHaving treated of the nature of shadows and their percussion, I shall now deal with the places...in which these shadows are touched and of their curvature or obliquity or straightness or of whatever quality...I could find.

This he crosses out and begins anew:

Shade is a privation of light

It seeming to me that shade is of the greatest necessity in perspective, because without these opaque bodies and cubes are only poorly recognized unless they terminate in a background of a different colour from this body, and this I propose...in the first proposition of shade, and I state it in this form, how every human body is surrounded...and superficially clothed with shade and light and on this I build the first book. Besides this, these shadows are of various qualities of darkness, because they are...abandoned by various quantities of rays...,

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and these I call original shadows, because they are the first shadows that invest bodies where they are attached. And on this I shall build the 2nd book,... From these original shadows there result umbrous rays which go spreading out through the air...and they are of as many qualities as are the varieties of original shadows from which they derive and for this [reason] I call these shadows, derivative shadows, because they are born from other shadows, and on this I shall make the 3rd book. Again these derivative shadows, in their percussions, make as many various...effects as are various the locations, where they percuss and of this I shall make the fourth book. And because,...the percussion of the derivative shade is always surrounded by the percussion of luminous rays which, through a reflected concourse, bouncing back towards their source, find the original shadow and mixes itself and converts itself into this, altering its nature somewhat, and on this I shall build the fifth book. Besides this I shall make the sixth book, in which will be treated the various and many diversifications of reflected rays [that are] bouncing back, which alter the original with as many various colours, as are the various locations, whence these reflected luminous rays derive. Moreover, I shall make the seventh division concerning the various distances which exist between the percussion of the reflected ray and the place whence they are born, and the equally various similitudes of colours which this [ray] brings in the percussion of the opaque body.

The themes of these seven books are summarized in Chart 8 and have been used as a starting point for a reconstruction of various chapters these books might have been contained. In this reconstruction, an attempt has been made to indicate not only the chapters that he foresaw in 1490, but also the modifications resulting from sub-sequent researches (Chart 9).

Book Themes 1 Light and Shade 2 Primary Shade 3 Derived Shade 4 Derived Shade and Interposed Objects 5 Derived Shade and Reflected Light 6 Reflected Colour 7 Reflected Colour and Distance

Chart 8: Survey of Themes of the Seven Books on Light and Shade drafted on CA250ra (c.1490).

BOOK ONE: LIGHT AND SHADE1. Punctiform Propagation2. Central Lines3. Opposing Theories

BOOK TWO: PRIMARY SHADE1. Definitions2. Degrees

BOOK THREE: DERIVED SHADE1. Kinds of Light2. Light Source: (a) Equal to Opaque Body

(b) Smaller than Opaque Body

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(c) Larger than Opaque Body (d) Comparative Sizes

3. Object: (a) Comparative Distances (b) Comparative Sizes* (c) Comparative Sizes and Distances*

4. Eye: (a) Comparative PositionsBOOK FOUR: DERIVED SHADE AND INTERPOSED OBJECTS

1. Introduction2. Degree of Light+3. Angle and Intensity of Light*4. Angle and Intensity of Shade5. Position and Intensity of Light Source6. Size/Shape of Light Source/Object7. Position/Shape of Projection Plane8. How/Why One Light Source and One Object Produce

Two Shadows+9. Compound Shade: (a) Preparatory Studies

(b) Multiple Lights and Objects (c) Columns* (d) Experiments with St. Andrew's Cross*

10. ConclusionsBOOK FIVE: DERIVED SHADE AND REFLECTED LIGHT

1. Introduction and Basic Propositions2. Lustre3. Elementary Demonstrations4. Interposed Rods5. Interposed Walls *6. Theoretical Demonstrations

BOOK SIX: REFLECTED COLOUR1. Introduction2. Mirrors *3. Water *4. White Objects *5. Faces *6. Landscape and Verdure *7. Yellow, Azure and Green *8. Walls *9. Light and Pigments *10. Further Demonstrations *11. Precepts *12. General Statements *13. Conclusions

BOOK SEVEN: REFLECTED COLOUR AND DISTANCE1. Introduction2. Demonstrations *3. General Statements *

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4. Links with Perspective *

Chart 9: Reconstruction of chapters of the Seven Books on Light and Shade outlined by Leonardo (CA250ra, 1490). Chapters marked with an + indicate themes which he considered in 1490 but subsequently develops. Chapters marked with an * indicate themes that he does not mention in 1490 but discusses at a later stage, especially after 1505.

Leonardo's researches lead him to consider alternative schemes of organization. On CU841 (TPL673, c.1490-1495), for instance, he considers a fourfold scheme, later adding a fifth, in a passage headed:

Of the four things which one needs to consider primarily in light and shade.The principal parts which one needs to consider in painting are four, namely, quality, quantity, site and shape. By quality is understood which shade or part of shade is more or less dark. Quantity, that is, how large such a shade is with respect to others nearby; site, that is, in what way things need to be situated and to what part of a member that it attaches itself. Shape, that is, what is the shape of this shadow, that is to say, whether it is triangular or participates of the round or the square, etc.The aspect of the shadow is also to be numbered along with the parts of shade, that is, whether the shadow is long, [and] to see in which aspect the sum of such a length is directed, whether the shadow of a brow is directed towards the ear, whether the lower shadow of the eye socket is directed towards the nostril of the nose and likewise with similar encounters of the various aspects to situate these shadows. Hence the aspect is to be placed before the site.

On CU843 (TPL739, 1508-1510) he outlines a further book that he intends to write on the subject:

On the master shadow, which stands between the incident and reflected light.Note the true shape which the master shadow has, which interposes itself between the reflected and the incident light. Such a shadow is not intersected, nor does it have a boundary, except with the object...to which it attaches itself. And its sides are of various distances from its centre and of various boundaries of this incident and reflected light, such that it sometimes shows clear boundaries and sometimes imperceptible boundaries, sometimes it bends from its rectitude, sometimes it observes rectitude, sometimes the boundaries are at unequal distances from the middle of the principal shadow.

And about this you will compose a book.

Book Theme1. On the usefulness of shadows2. On the motion of shadows3. On the shape of shadows4. On quality5. On quantity6. On boundaries7. On simple shade

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8. On compound shade9. On decompounded shade10. On darkness11. On light12. On light penetrating through apertures of different shapes13. On light passing through various numbers of apertures14. On the composition of multiple luminous rays15. Whether it is possible that rays which penetrate one another depart from a same luminous body16. Whether parallel rays can [come] from a single light and penetrate through some apertures

Chart 10: List of themes concerning light and shade outlined on CA277va (c.1513-1514). The book numbers have been added by the author.

On CA277va (c.1513-1514) he composes a completely new list of sixteen chapter headings (see chart 10). At about the same time, on W19076r (K/P 167r, c.1513) he reminds himself to: "Reserve for the last [book] on shade, the figures1 that appear in the study of Gerard the miniaturist at San Marco in Florence." On this same folio Leonardo proposes to include themes relating to light and shade in his treatise of painting:

Do the rainbow in the last book on painting, but first do the book on the colours originating from he mixture of other colours such that, through these colours of painters, you can produce the generation of colours of the rainbow.2

Hence he himself remains undecided about the final arrangement of his notes on light and shade. Any attempt at a reconstruction of his intended treatise must therefore remain tentative. Our concern is to understand the chief themes that preoccupy him and gain insight into the systematic aspects of his approach. As will be shown, the scheme of seven books outlined on CA250ra (c.1492) lends itself admirably to these concerns. In addition we shall examine his notes on the movement of shadows. Almost all these demonstrations involve light and shade in the open air. In a subsequent chapter we shall show that these demonstrations are parallelled by others involving camera obscuras which theme will lead in turn to the physiology of vision.

Light and Shade Book One

Every opaque body is surrounded and its whole surface is enveloped in shadow and light on this I shall build the first book (CA250ra).

The purpose of Leonardo's first book on light and shade is to show that every body has its surface covered with luminous and umbrous rays. Had he actually managed to write it, this book might well have had three chapters beginning with a first on punctiform propagation, a second on the role of central lines and a third to deal with opposing theories.

Book One. 1. Punctiform Propagation

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We have already analysed the earliest of Leonardo's extant notes on W19147v (K/P 22v, figs. 176-177, c.1489-1490) to demonstrate that light originates in a point and that its punctiform propagation spreads everywhere, or, as he puts it, "all in all and all in every part" (see above p. ).These early demonstrations lead him to conclude (W19147v, K/P 22v, 1489-1490):

It will appear clear to experimentors that every luminous body has in itself a hidden centre from which and to which arrive all the lines generated by a luminous surface and from there they return or leap back outwards and unless they are impeded they are dispersed by an equal distance through the air.

(figure)

Figs. 176-182: On the properties of a large light source in front of a small object. Figs. 176-177; W19147v (K/P 22v); fig. 178, CA262v; figs. 178-183, A97r.

On A97r (BN 2038, 17r, 1492) he returns to this theme:

On light which operates in all its quantity in a single luminous centre.If a large sphere illumines another sphere much less than it, it would be fitting that if the luminous rays parted from the surface of the light, that the lesser light would be surrounded and illuminated by more than half. If it were so then the shadow, the further it is from its cause, would be smaller until the end. But experience shows the contrary, because the lights of candles which are long and narrow, when they illuminate a spherical body, the shadow on the wall which ought to be round would be long and low.

He illustrates this with a concrete example (fig. 183 cf. 182):

For this reason, let us suppose that ab, and cd are the height and length of a light. If its surfaces are to function then ab will illuminate that much more than one half of the spherical body to the extent that pn is from ys and the shadow on its wall will appear much smaller than it is on the spherical body. The surfaces of the width of the light, at cd, will illuminate the spherical body at py, that is, in the middle. It being thus, the shadow will go wide like that of the wall. Hence, the shadow on the ball will be wide and low and like this will be that of the wall which, since it is demonstrated by experience that it is of a round form and always larger than its cause, it is convenient to abandon both of the above demonstrations and to confess that the centre of every light is the cause of shadow.

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Figs.184-186: Demonstrations how a large light source in front of a small object produces expanding shade. Fig. 184, A97r; fig. 185, A109v; fig. 186, CU628.

To accompany this passage he draws a preparatory sketch (fig. 182) which he develops (fig. 183) and labels "example." He illustrates the case of a small light source in front of a large opaque body (fig. 180) but crosses this out. He also illustrates how a large light source in front of a small object would theoretically produce a shadow converging towards f (fig. 179 cf. 176). Beneath this

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diagram he writes "proposition." Next he draws another diagram beneath which he writes "conclusion" (fig. 181, cf. 177). This demonstrates how such a light actually produce an expanding shadow. Immediately following he describes an experiment to verify this (fig. 184):

And the experiment is made as follows: let ab be the wall, cd the ball or a line and let e be the light. Measure how much [the distance] is from the light to the wall and from the line to the light. Then measure the shadow and make two lines which are equal to the distance from the wall to the light, and as large as the shadow, and in these lines observe whether the length of the line cd exceeds or is smaller than these lines.

On A109v (BN 2038 29v, TPL 615, 1492), he returns to this problem, now taking for granted his demonstrations on A97r:

How separate shadow is never similar in size to its source. If, as experience confirms, luminous rays are caused by a single point and they go increasing and spreading through the air in a circular course around this point. The further they go, the more they expand; and the thing positioned between a light and a wall is always carried by greater shadow because the rays which strike it, joined to the wall in their concourse, make it larger.

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Figs. 187-189: Concerning the link between punctiform propagation and shade. Fig. 187, CA204ra; fig. 188, CA349vd; fig. 189, CA345rb.

He pursues this theme on CA204ra (1490-1495) in a passage entitled:

The operation of light with its centre.If it were the entire light that caused shadows behind the object placed opposite this, it would hold that a body which is much smaller than a light would produce a pyramidal shadow behind it. And since experience does not show this, it must be that it is the centre of this light which performs this function.

He now draws a diagram (fig. 187) followed by a:

Proof.ab is the size of the light of a window which gives light to a stick positioned at its foot. From ac and to ad is where the window gives its light entirely. In ce, one cannot see that part of the window which is between lb and similarly df does not see am and for this reason in these 2 places the light begins to become exhausted.

On CA349vd (1490-1495), he restates what he had claimed to be the basic idea of his first book, and beneath it adds a series of basic claims and definitions (fig. 188).

No opaque body is seen which is not covered by an umbrous and illuminated surface.

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Figs. 190-200: Preliminary demonstrations of punctiform propagation. Figs. 190-193, CA144va; figs. 194-200, CA179rc.

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Figs. 201-203: Demonstrations of punctiform propagation confirming that objects have light and shade everywhere on their surfaces. Fig. 201, CA353vb; fig. 202, CA353rb; fig. 203, W19147v (K/P 22v).

The air and every transparent body makes a transit of its species (which) from the objects to the eye [in the case] of those objects that are found either in front of it or above it.Derived light is surrounded by primitive shade.Derived shade will be surrounded by derived light.Derived light is surrounded entirely or in part by primitive or derived shadows.Through its similitudes every opaque body is all in all and all in every part of the

transparent [air] that surrounds it.

On CA345rb (fig. 189, c.1508) he again alludes to the principle of punctiform propagation: "All the objects seen by a single point are seen again by the same point." Elsewhere he makes a number of sketches to illustrate how objects have light and shade everywhere. Some of these are rough (figs. 190-200). Others are more carefully drawn (figs. 201-203).

Book One. 2. Central Line

Corollary to the principle of punctiform propagation is the idea that the central line plays a determinant role in shadow projection (see below pp. ). On C3v (1490-1491), for instance, Leonardo notes that "in all the propositions that I shall make, it is understood that the middle which finds itself between bodies will be equal." On CA187va (c.1490-1491) he restates this idea more forcefully: "No shadow can imprint the true form of the umbrous body on a wall if the centre of the light is not equidistant from the extremities of this body." The nature of the central line is again considered on BM Arundel 170v (c.1492):

The centre of the length of any shadow always directs itself to the centre of the luminous body....

It is necessary that every shadow regards the centre of its light source with its own centre.

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Figs. 204-205: Elementary drawings showing the central line. Fig. 204, C17r; fig. 205, BM171r.

Illustrations of this central line occur on C17r (fig. 204, 1490-1491) and BM171r (fig. 205, c.1492) with further notes on TPl528a (1508-1510), TPL478ab (1510-1515) and CA241vd (1513-1514).

Book One. 3. Opposing Theories

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Implicit in the claim that objects have light and shade everywhere is the idea that there can be no object which does not project shade. Hence, on A102r (BN 2038 22r, figs. , 1492) Leonardo takes to task the opinions of some that a triangle [i.e. a pyramid] does not produce any shadow on a wall:

There have been some mathematicians who have firmly held that a triangle which has its base positioned towards the light does not make any shadow on a wall, which thing they prove saying as follows. No spherical object less than a light source can reach as far as the middle with its shadow. Radiant lines are rectilinear. Hence let us suppose that the light is gh, and the triangle is lmn and the wall is ik. They say that the light g sees the face of the triangle ln and the part iq of the wall. And likewise h sees the face lm as g does and in addition it seems mn and the wall pk and if all this wall is seen by the light gh, it follows that the triangle is without shadow and it cannot happen that it does not have shadow. Which thing appears credible in this case if the triangle mpg were...seen by 2 lights, g [and] h. But ip is equal to qk and each is seen by a single light. That is, ip cannot be seen by hg; k will never be seen by g. Hence pq will be twice as bright as the two visible spaces which bear shadow.

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Figs. 206-210: Demonstrations to refute the claim that some objects are without shadow. Figs. 206-207, A102r; fig. 208, A103v; figs. 209-210, CA204ra.

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Figs. 211-213: Primitive and derived light and shade. Fig. 211, BM171r; fig. 212, CA116rb; fig. 213, CU585 (TPL570).

This passage helps to explain an otherwise enigmatic note on A103v (BN 2038 23v, fig. 208, 1492): "How 2 lights, which have placed in the middle between them a body of two pyramidal sides with pyramidal bases, leave it without shadow." In the context of the earlier passage this is clearly a statement of the adversary's position. Who this adversary is, becomes evident from a further note on CA204ra (1492) in which Leonardo again launches into a demonstration without explaining the context (figs. 209-210):

Let ab be the luminous window. De produces the shadow gh. Ef produces the shadow ik. The triangle mkg is entirely luminous. Hence, the science of Marliani is false.Such a demonstration proving that an opaque object necessarily produces shade would presumably have come at the end of Leonardo's first book on light and shade.

Book Two

Shadows have in themselves various degrees of darkness, because they are caused by the absence of a variable amount of the luminous rays; and these I call primary shadows because they are first and inseparable from the body to which they belong. And on this I

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shall build the second book (CA250ra).

Primary Shade

Book two of Leonardo's treatise, devoted to primary shade and its various degrees, would probably have opened with a chapter on basic definitions such as those on BM171r (fig. 211, 1492), CA116rb (fig. 212, c.1500) and CU585 (TPL570, fig. 213, c.1505-1508) analysed earlier (pp. ).

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Figs. 214-215: Three degrees of primary shadow on C17r and CU754 (diagram also used in CU796).

The main part of book two would have been devoted to various degrees of primary shade, a topic on which there are at least six extant passages. The earliest of these, on C17r, (1490-1491), opens with a general statement:

That part of primitive and derivative shade will be less dark which is more distant from its centre.This occurs because the more the shadow removes itself from its centre, the more it is seen by a greater quantity of luminous rays and everyone knows that where there is more light there is less shade.

This general statement is followed by a specific example showing three degrees of primary shade (fig. 214):

The triangle dgr does not see anything of the light as and likewise the part of the umbrous body which is enclosed in this triangle. The triangle frk [i.e. frt] and also cri are seen by the light; am and ns and will be shadows that are brighter and more like the part of the ball which enclosed it in their angles. The triangles bhi and etk are brighter and their external boundaries are the limit of the shadow and likewise of that part of a ball which encloses it at the points of the angles because each is seen by half the light oa and sa.

Nearly two decades later he again considers three degrees of primary shade on CU754 (TPL631, fig. 215, 1508-1510):

On shadows and which are those primitive ones which will be darker on an object.Primitive shadows will be darker which are generated on the surface of a denser body and conversely, [they will be] brighter on the surface of less dense bodies. This is manifest because the species of those objects which tinge objects opposite them with their colours will impress themselves with greater vigour which find denser or more polished surfaces on these bodies.

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Figs. 216-217: Four degrees of primitive shadow on A92v and CU756.

These leads to a concrete example (fig. 215):

This is proved. And let the dense object be rs interposed between the luminous object nm and the umbrous object op. HEnce, by the seventh of the ninth, which states: the surface of every body participates in the colour of its object, we shall claim that the part bvar of this body will be illuminated because its object nm is luminous and similarly we shall state that the part opposite dcs is umbrous, because its object is dark. And thus our proposition is concluded.

Meanwhile he had been studying cases with more degrees of primary shade. On A93v (BN 2038 13v, fig. 216; CU756, fig. 217, 1492) he considers four degrees of shade:

That part of the umbrous body is less luminous which is seen by a lesser quantity of light.The part m of the body is the first degree of light because the window ad sees them all along the line af; n is the second degree because the light bd sees it along the line be; o is the third degree because the light cd sees it along the line ch; p is the penultimate [degree] because cd sees it along the line dv [and] o is the final degree because no part of the window sees it.

(figure)Figs. 218-219: Five degrees of primary shadow on A94r and CU624.

Directly opposite on A94r (BN 2038 14r, 1492) he considers a case with five degrees of shade (figs. 218=219) at greater length:

Every light that falls on umbrous bodies among equal angles holds the first degree of brightness and that [body] is darker which receives less equal angles, and light and shade function through pyramids. The angle c holds the first degree of brightness because it sees all the window ab and the entire horizon of the sky mx. The angle d is little different from c because the angles which place it in the middle are not so different in proportion as are the others below and there is lacking to it only that part of the horizon that is between y and x. Although it receives as much from the opposite side, nonetheless, its line is of little power because its angle is less than its companion. Angles e [and] c are of less light because the light ms and the light vx see them less, and the angles e [and] i are fairly difform.

The angle k and the angle f are each positioned in the middle by angles very different from one another and hence have little light because at k only the light pt is seen and at f only [the light] tq is seen. Og is the ultimate degree of light because it sees no part of the light of the horizon and these are the lines which once again recompose a pyramid similar to the pyramid c, which pyramid l finds itself in the first degree of shade because it again falls between equal angles and these angles direct themselves and regard themselves along a straight line which passes through the centre of the umbrous body and has its apex at the centre of the light. The luminous species multiplied at the boundaries of the window at the points a [and] b produce a brightness which surrounds the derived shade created by the umbrous body at the locations 4 and 6 and the dark species are multiplies at o [and] g and end at 7 and 8.

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Figs. 220-221: Seven degree of primary shadow on C14r and C21v.

Elsewhere on C14r (1490-1491) he makes a detailed drawing (fig. 220) showing seven degrees of primary shade with the brief caption: "The boundaries of umbrous bodies, because they are struck by different qualities of luminous pyramids, are surrounded by different qualities of light and shade." On C21v (1490-1491) he pursues this theme now carefully numbering the various degrees of light and shade (fig. 221), again adding the caption: "That part of the luminous body which is struck by a greater luminous angle will be more illuminated than any other." He returns to this problem on Forst II, 5r (c.1495-1497): " The shaded and illuminated parts of opaque bodies will be in the same proportion of brightness and darkness as are those of their objects."A few years later on CA199va (c.1500) he claims that the number of degrees of light and shade is infinite:

Even though practitioners put four kinds of brightness in imitating a same colour in all darkened things, [such as] trees, meadows, hair, beards, and skin, that is, first a dark fundament and 2nd, a spot that participates of the form of the parts; 3rd, a part that is clearer and brighter; 4th, lights which have their shapes clearer than other parts. But to me it appears that this variety is infinite on a continuous quantity which is divisible to infinity.

(Figure)

Fig. 222: Demonstration on ca199va that the degrees of shade are infinite.

He supports this claim with a demonstration (fig. 222):

And I prove it as follows: let ag be a continuous quantity; let d be a light which illuminates it. I say by the 4th which states that; that part of an illuminated body will be more luminous which comes closer to the cause which illuminates it. Hence g is darker than c to the extent that dg is longer than the line dc and by the conclusion that such degrees of clarity or if you wish to say obscurity are not only 4, but can imagined as being infinite, because cd is a continuous quantity and every continuous quantity is divisible to infinity. Hence the varieties of lengths are infinite, which the lines have that extend from the luminous body to the illuminated body and such is the proportion of the lights as is that of the lengths of the lines among them, which extend from the centre of the luminous body to the parts of this illuminated object.

On TPL810 (1505-1510) he takes for granted this infinite variation and in the following years makes further passing references to degrees of light and shade (TPL672, 634, 683, 1508-1510). Later, on E15r (1513-1514), he restates his earlier rule:

Painting; among bodies of varying obscurity deprived of a single light such will be the proportion between their shadows as is the proportion between their natural obscurity and the same is to be understood of their lights.

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In these later notes, he does not, however, improve on the diagrams (figs. 220-221) made in 1490-1491.

Book Three

From these primary shadows there result certain shaded rays which are diffused through the atmosphere, and these vary in character according to that of the primary shadows whence they are derived. I shall therefore call these shadows derived shadows because they are produced by other shadows; and the third book will treat of these (CA250ra).

In the case of derived shade, which was to have formed the third book on light and shade, the contrast between Leonardo's early ideas and his later studies in more marked. This third book would probably have opened with a chapter on the three traditional kinds of light, and led to a discussion of various conditions in each of the three variables: light source, object and eye (Chart 9).

Book Three. 1. Kinds of Light

Already in the third century B.C. Aristarchus of Samos3 had made a distinction between three basic kinds of shade: (1) parallel, when light source and opaque object are equal; (2) diverging, when the light source is smaller than the opaque object and (3) converging, when the light source is larger than the opaque object. Aristarchus appears to have been well known to Renaissance humanist circles.4 This distinction had, moreover, been transmitted indirectly through mediaeval commentators such as Witelo.5 By 1490 Leonardo is familiar with this distinction and he illustrates it at least ten times in the next three decades (figs. 223-232). Perhaps the earliest example is on C7v (1490-1491) where roughly drawn sketches (fig. 223) are accompanied by a clear text:

Shade and Light

The shapes of shade are three because if the material, which produces the shade is equal to the light, the shade is similar to a column and it has no end. If the material is greater than the light, its shade is equal to an inverted or contrary pyramid and its length is without end. But if the material is less than the light, the shade is equal to a pyramid and is finite as is shown in the eclipses of the moon.

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Figs. 223-228: Leonardo's illustrations of Aristarchus' three types of shade. Fig. 223, C7v ; fig. 224, C15v; fig. 225, C347ra; fig. 226, CU615; fig. 227, CU619; fig. 228, CU624.

(figure)Figs. 229-232: Further examples of these basic types of shade (constant, expanding, diminishing). Fig. 229, CA236ra; fig. 230, W12669v; fig. 231, E32v; fig. 232, CU617.

He illustrates these three kinds more carefully on C15v (fig. 224, 1490-1491), this time without text. On CA347ra (1490-1495) he draws a related series (fig. 225) with a text which he crosses out. By 1508 yhe is no longer certain about the kind of rays propagated by the sun, and hence on F77v (1508) he notes (fig. 233):

If every part of the sun sends its rays to all the surrounding objects what is that part which sends its simulacrum to the waters, that is, is it a columnar ray, or a straight pyramidal or an inverted pyramidal [ray], that is, the columnar is abcd, the truncated pyramid is acfg, the straight pyramid is ace, the inverted pyramid is fgh. Decide which carries the simulacrum of the sun to the water.

No clear decision ensues, however. When he returns to this theme on CU615 (TPL574, fig. 226, 1508-1510) he merely changes the order of presentation of the three traditional kinds of shadow asking:

Of how many shapes is derived shade?

The shapes of derived shade are three and the first is pyramidal born of an umbrous body less than the luminous body; the second is parallel born of an umbrous body equal to a luminous one; the third is infinitely expanding, and the columnar [kind] is infinite and the pyramidal [kind] is infinite also because beyond the first pyramid it makes an intersection and generates opposite the finite pyramid, an infinite pyramid, finding infinite space.

(figure)Fig. 233: Possible paths of rays propagated by the sun on F77v.

On CU619 (TPL588, fig. 227, 1508-1510) he pursues this theme:

Of the three various shapes of derivative shade.The varieties of derivative shade are three, of which one is large in its origin, and the more it is removed from such a beginning, the more it contracts. The second observes an infinite length with the same width as at its origin. The third is that which, in every degree of distance behind the width of its origin, acquires a degree of width.

This passage is directly followed by another (CU620, TP589, 1508-1510), headed:

The variety of each of the said three [kinds of] derivative shade.The derivative shade originating from an umbrous body less than the body which illuminates it, will be pyramidal and will be shorter to the extent that it is closer to the

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luminous body. But the parallel [kind] does not vary in such a case. But the expanding will be larger the closer it is to its luminous source.

Related to this is a further passage on CU625 (TPL591, 1508-1510):

That derived shadows are of three kinds.Derived shadows are of three kinds, that is, either its intersection on the wall where it percusses is greater than its base, or it will be less than this base or it will be equal. And if it is greater, it is a sign that the light which illumines the umbrous body is less than this body and if it is less, the light is larger than the body and if it is equal, the light is equal to this body.

On CU624 (TPL601, fig. 228, 1508-1510) the theme is pursued

In how many ways does the quantity of the percussion of shade vary with primitive shade.Shade, or rather the percussion of shade varies in three ways by the three kinds mentioned above, that is, converging, diverging and parallel. The diverging has a greater percussion than its primitive shade. The parallel always has its percussion equal to its primitive shade. The converging makes two sorts of percussion, namely, one [which is] converging, the other diverging. But the converging always has the percussion of its shade less than the primitive shade and its diverging part does the contrary.

On CA236ra (fdig. 229, 1508-1510) he considers a dynamic version of these three variables:

To the extent that an umbrous body smaller than a luminous body is closer to this luminous body, it will be illuminated by a smaller quantity of such a luminous body. The converse follows.

To the extent that an umbrous body larger than a luminous body approaches the luminous body it will be illuminated by a greater quantity of this luminous body.But if the umbrous body is equal to the luminous body, then the quantity that one sees behind it, will never vary in any variety of distance.

On CA195va (c.1510) he drafts another version which he later crosses out: " Why a light makes pyramidal shade after the umbrous body. Shadows are of 3 kinds of shapes of which the first is pyramidal, the 2nd parallel and the 3rd...a semi-pyramidal intersection." That same year he makes a quick sketch (fig. 230) of these three kinds of shadow on W12669v (c.1510). On E31r (TPL595, 1513-1515) he takes up the theme anew:

On simple derived shade.Simple derived shade is of three kinds, that is, the one finite in length and two infinite. The finite is pyramidal and of the infinite [kinds] one is columnar and the other is expanding and all three are rectilinear. But the converging shade, that is, the pyramidal [one] originates from an umbrous body less than the luminous body and the columnar originates from an umbrous body equal to the luminous body and the expanding from an umbrous body greater than the luminous source.

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On E32r (fig. 232, CU617, fig. 233, TPL590, 1513-1514), he pursues this theme:

On shade.Derived shadows are of three kinds of which the one is expanding, the other columnar and the third converging towards the site of the intersection of its sides which, after this intersection are infinitely long, or rectilinear. And if you said that such a shadow was terminated at the angle of conjunction of its sides and does not pass beyond, this is denied, because in the first of the above [mentioned] shadows, I have proven that that thing is entirely bounded of which no part exceeds its boundaries. The opposite of this is seen in such a shadow [however], because along with such a derivative shadow, the shape of two umbrous pyramids manifestly arise, which are conjoined in their angles.

An adversary's arguments are now considered:

Now, according to the adversary, if the first umbrous pyramid is the limit of the derived shade with its angle whence it originates, then the second umbrous pyramid, claims the adversary, is caused by the angle and not by the umbrous body.

And these arguments are promptly dismissed:

And this is denied with the help of the 2nd of this which states that shade is an accident created by umbrous bodies positioned between the site of this shade and the luminous body, and by this it is declared that the shade is not generated by the angle of derivative shade, but solely by the umbrous body, etc...If the spherical lumbrous body is illuminated by a luminous body [that is] long in shape, the shadow produced by the longest part of this luminous body will be of boundaries less well known than that which is generated by the width of the same light. And this is proved by the previous [proposition] which states that the shade created by a creator luminous body, has boundaries which are less clear and conversely, that which is illuminated by a smaller light source has boundaries which are more distinct.

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Figs. 234-240: Cases where the light source is equal to the opaque body. Figs. 234-237, CA144r; fig. 238, H76[28]v; fig. 239, CU610; fig. 240, CU605.

Read in the context of his previous notes on the subject, this passage on E32r is of particular interest, because it reveals the extent to which he transforms traditional ideas. What had begun as a passing comment of Aristarchus has now become a much more complex argument. Meanwhile he had been doing further study concerning the particular role played by each of the three variables in this process: namely, light source, object and eye, each of which is effectively a chapter in itself.

Book Three. 2. Light Source

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With respect to light sources he considers instances where they are (a) equal to, (b) smaller and (c) larger than an opaque body as well as comparative cases. We shall consider each of these in turn.

Book Three. 2a. Light Source Equal to Opaque Body

Leonardo's earliest examples of this situation are found on CA144ra (figs. 234-237, c.1492) in the form of rough sketches without text. On H76[28]v (1493-1494) he draws a clearer diagram (fig. 238) beneath which he writes two drafts:

Derived shade is never similar to the body from which it originates unless the light is the [same] shape and size as the umbrous body.Derived shade cannot be similar in shape to primitive [shade] unless it percusses between equal angles.

He returns to this theme on CU610 (TPL724, 1508-1510):

What is that umbrous body, which does not increase or decrease its umbrous or luminous parts at any distance or proximity of the body which illuminates it?When the umbrous and luminous body are both of equal size, then no distance or vicinity, which interposes itself between them will have the power of diminishing or increasing their umbrous or luminous sides.

He illustrates this with a concrete example (fig. 239):

Let nm be the umbrous body which, when taken from the site cd closer to the luminous body ab, the quantity of its shadow is neither increased nor decreased. And this happens because the luminous rays that surround it are parallel in themselves.

He pursues this theme on CU605 (TPL696b, 1508-1510):

Which luminous body is that which will never see more than half of the umbrous sphere?

When the umbrous sphere is illuminated by a luminous sphere equal in size to this umbrous one, then the umbrous and luminous part of this umbrous body will be equal.

Again he provides a concrete example (fig. 240):

Let abcd be the spherical umbrous body equal to the luminous sphere ef. I say that the umbrous part abc of the umbrous sphere is equal to the luminous part abd and this is proved as follows: the parallels es and ft are contingent on the front of the diameter ab, that is, the diameter of the umbrous sphere, which diameter passes through the centre of this sphere.

Being divided in the said diameter, it will be divided equally and the one part will be entirely umbrous and the other part entirely luminous.

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Figs. 241-244: Expanding shade. Fig. 241, Triv. 11v; fig. 242,BM170v; fig. 243, C21r; fig. 244, CU601 (TPL638).

He returns to this situation once more on CU614 (TPL567c, 1508-1510):

Which shade makes its light equal to the umbrous body in the shape of its shadows?If the umbrous body is equal to the luminous body, then the simple shade will be parallel and infinite in length. But the compound shade and light will be of a pyramidal angle with respect to the luminous body.

Book Three. 2b. Light Source Smaller than Opaque Body

Leonardo is equally interested in cases where the light source is smaller than the umbrous body. Perhaps the earliest example is that found on Triv. 11v (1487-1490) in the context of diminishing intensity of shade (fig. 241). "To the extent that ab enters cb,: he claims "to that extent will ab be darker than cd." He returns to this situation on BM Arundel 170v (fig. 242 cf. fig. 243, c. 1492) now claiming: "The light smaller than the umbrous body makes shadows bounded in this body and produces little mixed shade and sees less than half of it." This idea he develops in a later note on CU601 (fig. 244, TPL638, 1508-1510):

Which body produces a greater quantity of shade:That body will be vested with a greater quantity of shade, which is illuminated by a smaller luminous body. Let abcd be the umbrous body, g the small luminous source, which illuminates only the part abc of this umbrous body, whence the umbrous part adc remains much greater than the luminous part abc.

(figure)Figs. 245-247: Contracting shade. Fig. 245, C2v; fig. 246, CA160ra; fig. 247, Mad I 6v.

He returns to this situation once more on CU638 (TPL568, 1508-1510):

Which shade does the umbrous body larger than the luminous source make?If the umbrous body is larger than its luminous source, its simple derived shade will have its sides converging to the potential angle beyond the luminous body and the angles of the compound light and shade will regard the entire luminous body.

Hence on at least four occasions he is content merely to repeat Aristarchus' assumption that a light source larger than an object produces converging shade. This is the more striking because, as we have seen, he had designed his own experiments to demonstrate the contrary (e.g. figs. 176-181).

Book Three. 2c. Light Source Larger than Opaque Body

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Ever since Aristarchus it had been assumed that a light source larger than an umbrous body produces a converging pyramidal shadow. Leonardo illustrates this situation on C2v (fig. 245, 1490-1491) adding the caption:

If the umbrous and luminous body are of spherical rotundity, the base of the luminous pyramid will have such a proportion with its body, as the base of the umbrous pyramid will have with its umbrous body.

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Figs. 248-258: Contracting shade. Figs. 248-249, BM170v; figs. 250-252, CA199ra; figs. 253-256, CA112va; fig. 257, W19152v (K/P 118v); fig. 258, CA243ra.

On Ca160ra (fig. 246, 1490-1491) he draws a similar diagram, this time merely noting that this applies: "where the shade is less than the light." On BM170v (c.1492) he provides two diagrams (figs. 248-249) without text and on BM103v (1490-1495) he drafts a text without a diagram: "Simple derive shade born of an umbrous body less than the luminous source is of a pyramidal congregation." When he returns to this situation on Mad I 6v (c.1499-1500) he alludes both to its astronomical context and his own demonstrations to the contrary (fig. 247):

If the sun is greater than the earth, this earth makes a pyramidal shade through the air behind it. It being thus, a small ball should make a much shorter shadow beyond it when it is illuminated by the sun, and we see the opposite. But in the place of a pyramid one sees a columnar shade.

Further illustrations of this astronomical context are found on CA199ra (figs. 249-252, c.1500) and on CA112va (figs. 253-256, 1505-1508). He returns to this theme on CU603 (fig. 259, TPL639, 1508-1510) asking:

Which body takes a greater quantity of light?That body takes a greater quantity of light which is illuminated by a greater quantity of light. Let abcd be the illuminated body. Ef is that body which illuminates it. I say that since the luminous body is so much larger than the illumined body, that the illumined part bcd will be so much greater than its umbrous part bad and this is proven by the rectilinearity of the luminous rays eg [and] fg.

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Figs. 259-260: Two further demonstrations of contracting shade on CU603 and 606.

(figure)Figs. 261-264: Effects of pinhole apertures and windows on light and shade. Figs. 261-262, BM171v; fig. 263, C12r; fig. 264, CA230vc.

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Figs. 265-267: Gradations of light and shade in rooms with windows of different sizes. Fig. 265, B20v; fig. 266, A23r; fig. 267, CU133.

He pursues this theme in the second part of CU606 (fig. 260, TPL660, 1508-1510):

The greater the amount of light by which a body is illumined, the less will be the quantity of shadow which remains on this body.A is the luminous body, bc is the umbrous body, b is the part of the body which is illumined, c is that part which remains deprived of light and in this the umbrous part is greater than the luminous [part]. E is the luminous body greater than the umbrous body opposite it, fg is the umbrous body and f is the illumined part and g is the part in shade.

The accompanying diagram (fig. 260) does not show all the details described. Related diagrams occur on W19152v (K/P 118v, fig. 257, 1508-1510) and CA243ra (fig. 258, 1510-1515).

Book Three. 2d. Comparative Sizes of Light Source

Besides considering particular situations in which a light source is either equal to, smaller, or larger than an opaque body, Leonardo also makes comparative studies of light sources. His work on the camera obscura (figs. 261-262 cf. figs. ) may have prompted him to compare the nature of light and shade produced by different sized windows on B20v (fig. 265, 1490-1491). This approach is implicit in examples on A23r (CU133, TPL103, fig. 266, 1492), C12r (fig. , 1490-1491) and CA230vc (fig. 267, 1497-1500).

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Figs. 268-273: Sketches on Triv. 29r illustrating what happens when candlelight (figs. 268-269) and skylight (figs. 270-273) pass through a window.

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Figs. 274-277: Further demonstrations of what happens when small and large light sources pass through windows. Figs. 274-275, Triv. 28v; figs. 276-277, CU616.

On Triv. 29r (1497-1500) after making four preparatory sketches (figs. 268, 270-272) he compares what happens when candlelight (fig. 269) and skylight (fig. 273) pass through a window, adding the caption: "Primitive and derived shade caused by the light of a candle are larger than when caused by that of the air." The two situations which he here presents separately he combines in a single diagram (figs. 274-275) on Triv. 28v (1487-1490), now adding:

The edges of a window illuminated by two different lights of equal brightness will not send light of equal brightness within a habitation.If b is a candle and ac is our hemisphere, both illuminate the edges of the window mn but the light b only illuminates fg and the hemisphere ac illuminates as far as de.

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Nearly two decades later he returns to this comparative approach on CU616 (TPL584, 1508-1510) in a passage headed (figs. 276-277):

Of derived shade and where it is greater.That derived shade will be of greater quantity which is born from a greater quantity of light and also conversely. This is proved: ab, a small light produces derived lights cge and dfh which are small. [Now] take the following figure: nm, the light of the sky, which isuniversal, produces a large derived shadow at rtx and also the space osu, because the part pn of the sky produces this shadow rtx and likewise the space lm , a part of the sky, produces the opposite shadow [at] osu.

Meanwhile, he had also been exploring the links between the intensity of a light source and the resulting shade, as on C10r (1490-1491): "To the extent that the luminous body is of greater obscurity, to that extent will the shadows produced by the bodies illuminated by it be darker."

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Figs. 278-282: Effects of size of light source and distance on derived shade. Figs. 278-279, CA144vb; figs. 280-281, CA144ra; fig. 282, C2v.

This idea he develops on A67 (1492), CU702 (TPL620, 1508-1510) and CU860 (TPL694, 1508-1510). Rough sketches of varying light sources are found on CA144vb (figs. 278-279, 1492). On CA144ra (figs. 280-281, c.1492) he drafts two further figures accompanying which he writes:

To the extent that the diameter of the derived shade is greater than that of the primitive shade, to that extent will the primitive shade be darker...than the derived.To the extent that a more powerful light strikes dense bodies to that extent will the shadows of these bodies appear darker...and more divided by the light.

Book Three. 3. Comparative Distances and Sizes of the Object

Just as Leonardo is intent on studying the role of the light source, so too is he concerned with analysing how changes in an opaque object affect light and shade. In this respect he considers comparative distances, comparative sizes and the combined effect of the two.

Book Three. 3.1. Comparative Distances

On C2v (1490-1491) he considers the effect of distance on the intensity of derived shade (fig. 282):

To the extent that the percussion made by the umbrous concourse on the wall positioned opposite it is more distant from the luminous body and closer to its derivation, it will appear darker and with a more distinct boundary.

He returns to this idea on TPL599 (1508-1510) in a passage entitled:

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Which derived shade will show its boundaries as better known?That derived shade will show the boundaries of its percussion as betterknown, of which the umbrous body is more distant from the luminous body.

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Figs. 278-281: Diagrams analysing the effects of distance on derived shade. Figs. 283-284, A95v; fig. 285, CU641; fig. 286, CU622.

He is more interested in the effects of distance on the shape of derived shade. On Triv. 29r (1487-1490), for instance, he makes a preliminary sketch (fig. 290) with the caption: "To the extent that the larger derived shade enters into the smaller, to that extent is the cause of the lesser more luminous than the larger." On A95v (BN 2038 15v, fig. 283, CU641, TPL732, fig. 285, 1492) he analyses this problem in detail:

Every shadow with all its varieties which grows in size with distance more than its cause, has its exterior lines join together between the light and the umbrous body. This proposition appears clear and is confirmed by experience. For if ab is a windowwithout any obstruction, the luminous air that stands to the right at a, is seen to the left at d, and the air that stands to the left, illuminates to the right at the point c, and the said lines intersect at the point m.Every umbrous body finds itself between 2 pyramids, one dark, and the other luminous. The one is seen and the other not. And this only happens when the light enters through a window.

He now draws a second diagram (fig. 284 cf. CU622, fig. 286) beneath which he writes:

Recall that ab is the window and that r is the umbrous body. The light on the right at z passes the body on the left side of the umbrous body at g and goes to p. The left light K passes this said body on the right side at i and goes to m and these two lines intersect at c and there produce a pyramid. Then ab touches the umbrous body at ig and produces its pyramid at fig; f is dark because it can never see the light ab and igc is always luminous because it sees the light.

Having analysed how the pyramid of derived shade, is produced on A95v, he examines what happens to this pyramid at different distances on A90v (1492), beginning with a general claim: "Those bodies which are closer or further from their original light will produce shorter or longer derived shade." This idea he restates in terms of the size of the light source: "Among bodies equal in size that which is illuminated by a larger light source will have a shorter shadow." These claims are followed by a demonstration (fig. 287):

The above mentioned proposition is confirmed by experiment because the body mn is

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surrounded by a larger part of the light than the body pq, as is shown above. Let us say that vc ab dx is the sky that produces the original light and that st is a window where the luminous species enter and likewise that mn [and] pq and the umbrous bodies positioned opposite this light; mn will be of lesser derived shade because its original shade is little and its derived light is large because the original light cd is also large. Pq will have more derived shade because its original shade is greater [and] its derived light os less than that of the body mn, because that part of the hemisphere ab which illuminates it is less than the hemisphere cd illuminating the body mn.

This proposition recurs on CU639 (TPL725, 1492) with a slightly modified diagram (fig. 288). On Mad I 31v (1499-1500) he returns to this theme, again beginning with two general claims:

On shade.The illuminated parts of bodies of equal size are more luminous when the derived shade is shorter.On shade.The illuminated parts of bodies of equal size will have such a proportion in their brightnesses as they have in the lengths of their umbrous pyramids.

To demonstrate this a concrete example is again provided (fig. ):

The body f will be the half less illuminated than the body e, because the part of the sky which illuminates it is twice as small as that of e, as is demonstrated in [comparing] cd and ab.

On CU453 (TPL440, 1508-1510) he relates these principles to problems of painting practice:

Painting in a universal light.In the multitudes of men and animals always accustom yourself to making the parts of their shapes or bodies darker to the extent that they are lower and to the extent that they are closer to the centre of their multitude even though they are in themselves of a uniform colour and this is necessary because a smaller quantity of sky illuminating the bodies is seen in the low[er] spaces interposed between the aforesaid animals than in the upper parts of the same spaces.

A demonstration follows (fig. 291 cf. fig. 290):

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Figs. 292-293: Cases of lateral derived shade in rooms, on A95r and CU142.

This is shown by the figure placed in the margin where abcd is placed for the arc of the sky, the universal illuminator of bodies beneath it. N [and] M are the bodies which limit the space strh positioned between them, in which space one clearly sees that the site f, being only illumined by the part of the sky, cd, is illumined by a smaller part of the sky than the site e which is seen by the part of the sky ab which is three times greater than the sky dc.

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Hence it will be three times more illuminated in e than in f.

He is also interested in comparing the derived shade of objects off to the side. This situation is implicit on W12604r (fig. 294, c.1488) where he offers a:

Proof how every part of light makes one point.Although the balls a, b [and] c have light from one window, nonetheless, if you follow the lines of its shadows you will see that these make an intersection and point at the angle n.

This idea he pursues on A95r (BN 2038 15r, fig. 292, cf. CU642, TPL 293, 1492):

Every shade made by bodies is directed along the central line to a point made by the intersection of the luminous rays in the middle of the space and...the window. The reasoning presupposed above appears clearly through experience, because if you draw a site with a window to the North which is sf you will see the horizon of the East producing a line which, touching the 2 angles of the window of, will end in d and the horizon of the West will produce its line touching the other 2 angles of the window rs, and it will end in c, and this intersection comes precisely in the middle of the space and the size of the window.This reasoning will be confirmed even better if you take two sticks as in the place gh you will see the line made by the centre of the real shadow directed towards the centre m and with the horizon nf.

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Figs. 294-296: Sketches concerning lateral derived shade. Figs. 294, W12604r; fig. 295, C8r; fig. 296, BM170v.

On C8r (1490-1491) he examines in detail the case of shadows off to the side, beginning with a general claim:

Umbrous and luminous rays are of a greater power in their points than in their bases.Even though the points of luminous pyramids extend to dark sites and those of the umbrous pyramids extend to luminous places, and that among them are luminous ones. One is born of a greater base than the other. Nonetheless, if as a result of their various lengths, these luminous pyramids come to angles of equal size they will have equal light amongst them and umbrous pyramids will do the same.

A concrete example follows (fig. 295):

As is demonstrated in the intersected pyramids abc and def which, even though they originate from different sizes of base, they are, nonetheless, similar in size and in light.

He pursues this theme on BM170v (1492) beginning with the phrase: "real shade is longer the more it finds itself," which he then crosses out and writes (fig. 296):

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When the light of the air is constrained to illuminae umbrous bodies, if these umbrous bodies are equidistant from the centre of this window, that one which is positioned further off to the side will produce a greater shadow behind it.

He develops this idea on A91r (BN 2038 11r, CU643, TPL726, 1492):

Those scattered bodies situated in a habitation illuminated by a single window will produce derived shade that is more or less short, depending on whether it is more or less opposite this window.The reason why umbrous bodies which find themselves situated more directly opposite the middle of the window make shadows which are shorter than those situated in a position off to the side is that they see the window in its proper form and the bodies off to the side see it foreshortened. To the one in the middle the window appears large; and those off to the side see it [as] small.

As usual a concrete example follows (fig. 297 cf. CU643, fig. 298):

The one in the middle sees the hemisphere as large, that is, [as] ef and those to the side see it [as] small, that is, gr sees ab and likewise mn sees cd.The body in the middle because it has a greater quantity of light than those to the side is illuminated considerably lower than its centre and therefore its shade is shorter. And to the extent that ab enters ef, to that extent does the pyramid g4 enter into ly precisely.

This discussion leads directly to a consideration of the centres of derived shade (cf. above ):

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Figs. 297-298: Systematic studies of lateral derived shade on A91r and CU643.

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Figs. 299-301: Effects of distance on the shadows of objects in the open air. Fig. 299, Triv. 22v; fig. 300, W12352v; fig. 301, W12635v.

Every centre of derived shade passes through 6 centres and directs itself with the centre of the original shade and with the centre of the umbrous body and of the derived light and with the middle of the window, and ultimately with the centre of that part of the luminous body made by the celestial hemisphere.

Yh is the centre of the derived shade, lh of the original shade, l is the centre of the umbrous body, lk of the derived shade, v is the centre of the window, and e is the ultimate centre of the original light made by that part of the hemisphere of the sky which illuminates the umbrous body.

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In the left-hand margin he returns to the question of relative lengths of shade produced (fig. 297):

Among the shadows produced by equal bodies and at different distances from the aperture illuminating them, that which is longest, its body will be less luminous, and the one body will be that much more luminous than the other, to the extent that its shade is shorter than the other.The proportion that nm and vK have with st and vx, such will the shadow 4 have with x

[and] y.

His comparative studies of shadows at different distances extend to objects in the open air. On W12635v (fig. 301, c.1500), for instance, he draws two light sources illuminating an opaque body, and notes: " Whatever proportion that the line bc has with the line fc , such will the obscurity m have with the obscurity n."

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Figs. 302-307: Comparative effects of distance on derived shade. Fig. 302, CA236ra; figs. 303-305, BM100r; figs. 306-307, W19102v, (K/P 198v).

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Figs. 308-309: Demonstrations on CU728 concerning comparative sizes of objects.

Sketches on Triv. 22v (fig. 299, 1487-1490) and W12352v (fig. 300, c.1494) may well represent preparatory drafts for this diagram. He pursues this theme on CA236ra (fig. 302, 1508-1510) where he claims:

That umbrous body will have its simple derived shade with a larger base and a longer pyramid which is more remote from its luminous body. The first conclusion is tested, and let us say (that the) that the first umbrous body, a is closer to the luminous body cf than the second umbrous body br.Among bodies equal in size, the more remote will make an umbrous pyramid of a longer shape; the reverse follows, etc.

Related diagrams occur on BM100r (figs. 303-305, 1490-1495) and W19102v (K/P 198v, figs. 306-307, 1510-1515).

Book Three. 3. Comparative Sizes of Object

He is also concerned how different sizes of an object affect derived shade, as, for instance, on CU728 (figs. 308-309. TPL666, 1508-1510):

On shadow and light.That object will have its shade and light of more imperceptible boundaries which is interposed between larger dark and bright objects of continuous quantity.

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This is proved and let the object be o which is interposed between the umbrous body nm and the luminous body rs. I say that the umbrous body tinges nearly all the object with its pyramid nam and the pyramid of the luminous body rcs does the same at the opposite end.

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Figs. 310-314: Demonstrations with comparative sizes and distances. Fig. 310, C3v; figs. 311-312, CU607; fig. 313, CU602; fig. 314, CU609.

And that which is proposed is concluded by the 8th of the 5th which states that that part of the sphere will be darker which sees more of the darkness placed opposite. It follows that c is darker than any other part of this sphere.

Book Three. 3c. Comparative Sizes and Distances of Object

A next logical step in complexity would be to make comparative studies involving both different sizes and different distances. This Leonardo explores also. On C3v (fig. 310, 1490-1491), for instance, he considers a case:

When two umbrous pyramids, opposite one another, born of a same body...are such that one is doubly dark than the other and the same shape, then the two lights which are the causes thereof are such that one is double the other in diameter and at double the distance from this umbrous body.

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He returns to this theme of different sizes and distances on CU607 (TPL695, 1508-1510) in a passage headed (figs. 311-312):

Equality of shade in unequal umbrous and luminous bodies of different distances.It is possible that a same umbrous body takes equal shade from luminous bodies of different sizes.Fogre is an umbrous body of which the shadow is fgo, generated by the privation of anaspect of the luminous body de at the true distance and of the illuminating body bc at a remote distance.And this arises because both luminous bodies are equally deprived of an umbrous aspect fog through the rectilinearity of ab [and] pc.

On W12635v (c. 1500) he considers the effects of two light sources of different sizes and at different distances (figs. 315-316) accompanying which is a draft:

[If] the distance of the umbrous body has this proportion to the lights, the lights of this size will have double their shade.The proportion that the size of the light f has with the light b, such [a proportion] will the darkness of the shade d have with the shadow f.

He pursues this problem of comparative sizes and distances on CU602 (TPL722, 1508-1510) asking:

Which body is that which, when it approaches the light, its umbrous part increases?When a luminous body is less than the body illuminated by it, the shade of the illuminated body will increase to the extent that it is closer to the luminous body.

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Figs. 315-316: Derived shade of light sources of different sizes at different distances on W12635.

By way of illustration he gives a concrete example (fig. 313):Let a be the luminous body less than the umbrous body rsgl, which illuminates the entire part rsg included between its luminous rays an and am. When...by necessity of these rays, the whole of rlg remains umbrous.

Then I bring this umbrous body near the same luminous body and there will be dpeo, which is enclosed by the rectilinearity of the lines ab and ac, and is touched by these rays at the point d and the point e and the line de divides the umbrous [part] from its luminous part, [i.e.] dpe from doe, which umbrous part is necessarily greater than the umbrous [part], rlg, of the more distant body. And all arises from the luminous rays which, being rectilinear, separate themselves more distantly from the centre of such an umbrous body, to the extent that this body is closer to the luminous body.

Having considered what happens with objects larger than the light source, he examines CU609 (TPL723, 1508-1510) what occurs with objects smaller than the light source:

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What is that body which, the more it approaches the light, the more its umbrous part diminishes?When the luminous body is larger than the body illuminated by it, the shadow of the illuminated body will diminish more the closer it is to this luminous body.

This claim is again demonstrated (fig. 314):

Let ab be the luminous body larger than the umbrous body xgnh which, as it approaches the light fecd, diminishes its shadow because when it stand close to the body which illumines it, it is embraced further beyond its centre with luminous rays than when it is more remote.

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Figs. 317: Light source, eye and object on C27v. 318: Light source, object and eye on C27v.

In these examples Leonardo's systematic play with variables is again apparent: how he alters first distance, then size, then size and distance. As one might almost expect, he proceeds to study the effects of adding a further variable: the eye.

Book Three. 4. Comparative Positions of the Eye

Leonardo recognizes that the amount of shadow seen depends on the eye's position relative to the light source and the opaque body. On C27v (1490-1491), for instance, he considers the configuration: light source, eye, object (fig. 317):

PerspectiveThe eye which finds itself sending from itself visual pyramids from the same side as the luminous rays, if it is situated in the middle of these rays, it cannot see any shade on the opaque bodies positioned opposite.

Immediately following he considers the configuration: eye, object, light source (fig. 318):

PerspectiveThat spherical body which finds itself between the centre of the natural light and the centre of the visual pyramids is seen by the eye as being completely in shade with an equal luminous circle.

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Figs. 319-321: Various positions of light source, eye and object. Figs. 319, 321, C10r; fig. 320, C12v.

He develops these two basic demonstrations on C10r (1490-1491). Here the diagrams are much more elaborate (figs. 319, 321) and the accompanying texts more precise:

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All umbrous bodies, larger than the pupil, interposed between the eye and the luminous body, will show themselves as being in shade.The eye positioned between the luminous body and the bodies illuminated by this light will see the said bodies without any shade.

On C12v (1490-1491) he describes a variant of this situation (fig. 320).

The percussion of derivative shade born and caused by a spherical umbrous and luminous body and interrupted by its percussion on different bodies situated at various distances, appears round to the eye which is situated in front of it near the centre of the original light.

Some two years later he considers in somewhat more detail the configuration: light source, eye and umbrous object on A2r (fig. 322, 1492; cf. CA112va, fig. 324, c.1505-1508 and CU860, TPL694f, 1508-1510):

The umbrous body which is seen along the line of incidence of light, will not show any protruding part of itself to the eye. For example. Let the umbrous body be a. Let the light be c. Cm as well as cn are incident luminous lines, that is, lines which transfer light to the body a. The eye is at the point b. I say that [since] the light c sees the entire part mn, that those things which are in relief will be entirely illuminated. Hence the eye positioned at c cannot see shade and light. Not seeing this, each part appears to it of one colour. Whence the differences of the protruding and globulous parts do not appear.

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Figs. 322-325: Further variations of eye, object and light source. Fig. 322, A2r; fig. 323, M79v; fig. 324, CA112va;fig. 325, M80r; fig. 326, BM171r; fig. 327, M79v.

At about the same time he considers the configuration: eye, opaque body, light source on BM171r (fig. 326, c.1492): "The umbrous body situated between a light and the eye will never show a luminous part of itself unless the eye sees all the original light." When he returns to this theme some eight years later on 80r (1499-1500) he is explicit about his methodical approach (fig. 325):

On PaintingOf all the things seen, one has to consider 3 things, that is, the position of the eye that sees, the position of the thing seen and the position of the light that illuminates such a body.

Having illustrated each of these (figs. 323, 325, 327), he concludes on the folio opposite (M79v): "These show once the eye between the light and the body; 2nd, the light between the eye and the body; 3rd the body between the eye and the light." These passages may well have been drafts for his later statement on K105[25](v) (1506-1507):

On PaintingThe aspects of shadows and lights with the eye are 3, of which one is when the eye and the light are seen on the same side of a body; 2nd is when the eye is in front of the object and

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the light is behind this object; 3rd is that in which the eye is in front of the object and the light, and on the side in such a way that the line which extends from the object to the eye and from this object to the light, when joined together, will be rectangular.

The third alternative here mentioned is one he had considered as early as 1487-1490 on Triv. 10v (figs. 328-329):

The eye which finds itself between the shadow and the surrounding lights of shaded objects will see in these bodies the deepest shadows that are to be encountered with it, that is, under equal visual angles of incidence.

He alludes to it again on C27r (fig. 330, 1490-1491) under the heading of:

PerspectiveThat eye which finds itself between the light and shade surrounding the opaque bodies will see the shadows divided from the luminous side passing transversally through the centre of this body.

When he returns to this situation nearly two decades later on CU147 (fig. 331, TPL251, 1508-1510) he relates it directly to effects of relief in painting (cf. vol. 1:Pt.3 below and pp. ):

Of things positioned on a bright background and why such a use is useful in painting.When an umbrous body borders on a background [that is] of a bright colour and illuminated, then by necessity it will appear to stand out in relief and separate from this background.

That which is said happens because bodies with curved surfaces by necessity make themselves umbrous on the side opposite to which they are percussed by luminous rays, since that place is deprived of such rays, for which reason it varies a great deal from its background, and the part of that illuminated body never terminates in that illuminated background with its first [degree of] brightness. Hence between the background and the first [degree of] light of the body there is interposed a background of the body which is darker than [either] the background or than the light of the respective body.

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Figs. 328-331: Cases in which an object is half in light and half in shade. Figs. 328-329, Triv. 10v; fig. 330, C27r; fig. 331, CU147.

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Figs. 332-337: Variants where the eye is positioned obliquely relative to the light source and opaque body. Figs. 332-334, CA144vb; fig. 335, M80r; fig. 336, CA120vd; fig. 337, BM113v.

He also considers a further variant in which the eye is obliquely positioned relative to the light source and the opaque body. Rough sketches of this alternative appear without text on

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CA144vb (figs. 332-334, c.1492). On M80r (fig. 335, c.1499-1500) he returns to this variant adding a brief caption: "b is the eye, a is the thing seen, c is the light." He draws further examples of this on Ca120vd (fig. 336, c.1500) and BM113v (fig. 337, c.1510), which as will be shown (see below pp. ) had a certain importance in his astronomical studies. He pursues this theme of various positions of the eye in a series of notes in the Treatise of Painting as on CU645 (fig. 338, TPL685, 1508-1510):

On the middle included between the light and the principle shade.

Middle shade shows itself as being of greater quantity to the extent that the eye which sees it is more opposite the centre of its size. Middle shade is said to be that which tinges the surfaces of umbrous bodies behind the principal shade and is contained inside the reflection and it is darker or brighter to the extent that it is closer or further from this principal shade.Let mn be a darker shadow. The remainder always becomes brighter towards the point m and the rest of the figure does not apply to this proposition but it will serve for the succeeding one.

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Figs. 338-340: Effects of positions of the eye on derives shade. Fig. 338, CU645; fig. 339, CU647; fig. 340, CU650. On CU647 (fig. 339, TPL687, 1508-1510) he asks:

What is that site where one never sees shade on umbrous spherical bodies?The eye that is situated between the reflected pyramid of the species illuminated by umbrous bodies will never see any umbrous part of that body.Let the reflected pyramid of the illuminated species be abc and let the illuminated part of the umbrous body be the part bcd. And let the eye which stands within this pyramid be e, to which all the illuminated species bdc could never converge unless it were seen [on the same side as] the luminous point a, from which no shade is ever seen which it does not destroyimmediately. It therefore follows that e, which only sees the illuminated part odp is more deprived of seeing the boundaries of shade bc, than is a which is further away.

Having considered a case where the eye is closer to the opaque body than the light source, he asks what happens if the eye is further from the opaque body than the light source on CU650 (TPL688, 1508-1510):

What is that site or indeed that distance around a spherical body which is never deprived of shade?But when the eye is more distant from the umbrous sphere than the body which illuminates it, then it is impossible to find a site, where the eye is entirely deprived of the umbrous species of such a body.

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Figs. 341-342: How changes in the size of opaque body and light source affect derived shade. Fig. 341, CU648; fig. 342, CU649.

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This general claim is followed, as usual, by a concrete demonstration (fig. 340):

This is proved. Let bnc be the umbrous body. Let bsc be illumined object. Let o be the eye more distant from the umbrous body than the light a, which eye sees all the shade bdce. And if this eye moves circularly around this body with the same distance, it is impossible that it entirely loses all the aforesaid shade, such that, if through its movement it loses one part of this shade on one side, this is acquired by the other side through the [same] movement.

Leonardo has explored how various positions and distances of the opaque body, eye and luminous source affect the shade seen. He now adds a further variable: changes in size of the opaque body and light source. On CU648 (fig. 341, TPL734, 1508-1510) he considers cases where the luminous source is either equal in size or larger than the umbrous sphere, under the heading:

What is that light which, even if the eye is further removed from the umbrous sphere than this light, it can never see shade while standing in front of the light?When the luminous body is equal to or larger than the umbrous spherical body, then the eye which is behind this light will never see any part of the shade on the umbrous body, as a result of the difference of said luminous body.Let cedf be a spherical umbrous body; ab is the luminous source equal to the umbrous body and let cfd be the shade of this spherical body. I say that the eye l which stands behind the light ab at whatever distance one wishes, can never see any part of the shade, through the 7th of the ninth which states: Parallels never converge to a point. Since ac and bd are positioned parallel [to one another] and embrace precisely half of the sphere and [since] the lines nm...converge at the point l, this point can never see half of the sphere at its diameter cd.

Involved here are problems relating to visual perception (see below pp. ). On CU649 (fig. 342, TPL735, 1508-1510) he considers a case where the luminous source is smaller than the opaque body:

On the eye which, over a long distance will never have the view of the shade on the umbrous body occluded when the luminous source is smaller than the umbrous body.But when the luminous body is smaller than the umbrous body, there can always be found some distance where the eye can see the shade of this umbrous body.

Let opef be the umbrous body and let the light be ab smaller than this umbrous body by whatever proportion one wishes. I say, that one can never prevent the eye, n, which is behind this light, from seeing some umbrous part of the shade of the spherical umbrous body as the rectilinearity of the lines show.

Aristarchus' simple distinction between three kinds of light had served as a starting point for Leonardo. But as we have shown he considers variations in the light source, in the object, in the eye and finally in combination, to arrive at a considerably more complex picture of the situation.

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This picture will become more complex still in his fourth book, when he studies the properties of derived shade on meeting interposed objects.

Book Four

Again these derived shadows, where they are intercepted by various objects, produce effects as various as the places where they are cast. And on this I shall make the fourth book (CA250ra).

What happens when the shadow produced by one body in turn meets another opaque body? This question leads Leonardo to make a series of further detailed studies. Had he managed to compile these systematically in his fourth book on light and shade he would probably have begun with an introductory chapter (1), followed by an excursus on degrees of light (2) and on angles of light (3) which would have led to a consideration of angles of shade (4) and the role played by the position of the light source (5) and size of the light source (6). All this would have been preliminary to his basic concern, namely, consideration of how changes in position and shape of the interposed plans affect shadows (7).

Experiments had, meanwhile, made him aware that under certain conditions one light source and one opaque body could produce two shadows on an interposed plane. The how and why of this phenomenon would probably have involved a further chapter (8).

(figure)

Figs. 343-346: Basic distinctions between separate and conjoined shade, direct and oblique shade. Figs. 343-344, A102r; figs. 345-346, CU623.

The shadows produced in cases of compound shade, i.e. when more than one light source and/or more than one opaque body are involved, would have led to at least one further chapter (9, cf. Chart 9 ). Each of these will be considered in turn.

Introduction

By way of introduction to the various possible shapes of shadow Leonardo would probably have begun with a distinction such as he makes on A102r (BN 2038 22r, 1492) between "separate, and conjoined shade" (fig. 343) and "separate, inevident shade" (fig. 344). This bears comparison with his subsequent distinction made on CU623 (figs. 345-346, TPL600, 1508-1510):

In how many principle modes is the percussion of derived shade transformed?The percussion of derived shade has two varieties, that is, direct and oblique. The direct is always less in quantity than the oblique, which can extend itself to infinity.

This idea of the infinite variations of shadow is pursued on CU859 (TPL809, fig. 347, 1508-1510):

Precept A

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Lights and shadows are as various as the variations of the sites where they are found.F. When the umbrous part is augmented by a dark object, this shade will be darker than at

first to the extent that such an augmentation is less clear than the air.D. The percussion of the derived shade will never be the shape of its original primitive

[shade], if the primitive light is not the same shape as the body which makes the shadow.

(figure)

Figs. 347-348: Varieties of shade on CU859 and 588.

Accompanying this is a diagram (fig. 347) showing shade in an enclosed space. This alternative is contrasted with shade in the open air in two diagrams (fig. 348) illustrating the varieties of primitive shade on CU588 (fig. 348, TPL572, 1508-1510):

In how many ways does primitive shade vary?Primitive shade varies in two ways of which the first is simple and the second is composed. Simple is that which regards a dark place and by this such a shade is composed darkness which sees a place illuminated with various colours with the result that such a shade mixes itself with the species of the colours of the objects positioned opposite.

In the Codex Urbinas this is followed by a passage on the varieties of derived shade (CU759, TPL573, 1508-1510):

What variety does derived shade have?The varieties of derived shade are of two sorts of which the one is mixed with the air opposite the primitive shade. The other is that which percusses in the object which cuts this derived shade.

At the end of this introductory chapter he might have considered cases where primitive and derived shade are the same as on C4r (fig. 349, c.1490):

The obscurity produced in the percussion of the umbrous concourse will have conformity with its origin, which is born and terminated between nearby plane surfaces, and of the same quality and in direct opposition.

He returns to this idea in a sketch without text on Ca144VA (FIG. 350, 1492) and then in greater detail on CU710 (fig. 351, TPL581, 1508-1510), asking:

Whether primitive shade is more powerful than derived shade?Primitive shade, being simple, will be of equal darkness to simple derived shade. This is proved. And let the simple primitive shade be de and let the simple derived [shade] be fg. I say, by the fourth of this, which states: "darkness is the privation of light," [that] simple shade is therefore that which does not receive any illuminated reflection and for this reason it remains tenebrous as is de which does not see the light a. And the simple derived shade fg also does not see it and hence these shades are of equal obscurity because both the one and other are deprived of light and luminous reflection.

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Figs. 349-351: Cases in which primitive and derived shade are the same. Fig. 349, C4r; fig. 350, CA144ra; fig. 351, CU710.

(figure)

Figs. 352-353: Demonstrations relating to degrees of light. Fig. 352, Triv. 3v; fig. 353, Forst. III 58v; fig. 354, W12351r; fig. 355, I33r.

(figure)

Figs. 356-366: Concerning the properties of translucent and opaque objects.

Book Four. 2. Degree of Light

He makes several notes concerning the expansion of light and its varying degrees with distance. These could well have been intended as an introduction to analogous problems in shade. The earliest extant notes on this theme occurs on Triv. 3v (fig. 352, 1487-1490): " If a light be placed in a lanturn that is in the middle, it will enlarge its splendour; at CD its rays will be twice as large at the greater distance FB twice as far away." When he returns to the problem of degrees of light and distance on W12351r (fig. 354, C.1493-1494) he asks: "I ask how and how much one is illuminated more than the other: ab, cd and ef?" To this question he provides a reply nearly a decade later on CA150ra (1500-1503) where he discusses the properties of both translucent and opaque objects, claiming (figs. 356-366):

that part will remain more luminous, which is percussed by a greater sum of luminous rays and...conversely, it will be less luminous which is seen by a lesser quantity of these rays....All the parts of the illuminated body which see the entire circle of the luminous body will be as different in clarity among one another as they are closer to the luminous body.

On CA132vb (c.1508) he provides a more succinct answer: "That part of an illuminated object will be the more luminous which is the closer to the cause of its light," a claim that he repeats nearly verbatim on CU447 (TPL526a, 1508-1510): "That part of an object will be more illuminated which is closer to the luminous object which illuminates it." Related to this question of degrees of light is the problem how these degrees can be multiplied. On A3v (1492), for instance, he notes:

On LightsMany small lights joined together will be of greater power each in itself than when they were separate. The proof you will see if you take many lights in a straight line and you stand at a certain distance opposite the middle of this line and you note the quality of the light made by these lights and then join them together. You will see [that] the place where you stood will be more luminous that at first....

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Again it is known that the stars are of equal light to that of the moon and if it were possible to join them together that they would compose a body much larger than that of the moon, and nonetheless, even if it be a clear night and they are all shining, if the moon is not in our hemisphere, our part of the world remains dark.

(figure)

Figs. 367-373: On the multiplication of candlelight. Figs. 367-369,Ca270va; figs. 370-371, CA270ra; fig. 373, CA260ra.

He mentions the problem again on Forster III 58v (1490-1493) under the heading (figs. 353):

On the duplication of lights.If one light makes 4 ounces (and) [then] it appears that 2 of these lights together make 8 ounces.

He provides a visual demonstration of this principle on CA270va, 270ra and 270va (figs. 367-373, 1508-1510) where he compares the light of smaller candles with larger flames. On W12351r (c.1493-1494) the matter is raised as a question: If one candle consumes itself in one hour, in how much time will 3 candles together consume themselves? This theme he pursues on I33r (fig. 355, 1497), here making explicit the link between his concepts of light and the pyramidal law (cf. vol. 1, pt.2).

Of the luminous rays and the powers of their extremities.

Since the luminous ray is of pyramidal proportion and maximally when the centre is equal it will therefore happen that when 2 rays meet along a straight line parting from equal lights this ray will be doubled throughout and of equal power because where the one has the apex of its pyramid the other has its base as mn shows.

In addition to such general statements concerning the relation of degrees of light to distance and the pyramidal law, he emphasizes the connection between light intensity and luminous angles.

Book Four. 3. Angle and Intensity of Light

One of his earliest extant notes on this subject occurs on C12r (1490-1491):

That part of an illuminated wall will be the more luminous which is illuminated by a greater luminous angle. And that place [struck] by said rays will observe the accompanying quality of light less which is shadowed by a greater umbrous angle.

This idea he restates briefly on C21v (1490-1491):

That part of an umbrous body which is percussed...by a larger luminous ray will be more illuminated than any other.

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On BM103r (1490-1495) is found the draft of another version in a hand probably not Leonardo's: “That pyramid which parts from its base with more unequal and differs angles will be thinner and a more distorted demonstrator of the true size of its base.” On the verse of this folio there is another draft in this hand: “If the shade of the umbrous bodies...born of a spherical luminous source falls between equal angles and an unequal centre it will be of various shapes and various [degrees of] obscurity.” Leonardo pursues this theme on A85r (BN 2038 5r, 1492):

PaintingThat part of an object that receives over it a luminous ray between equal angles will be more luminous than [any] other part of this luminous object.And that part which is struck by a luminous ray under less equal angles will appear less luminous.

This idea he repeats more succinctly on A112v (BN 2038 33v, 1492): "That light which strikes under more equal angles is more powerful. Example of the blow." On Mad I 32r (1499-1500) he pursues this theme (fig. 376):

(figure)

Figs. 374-376: Demonstrations concerning intensity angles of light and its intensity. Fig. 374, Mad I 31v; fig. 375, CU671; fig. 376, Mad I 32r.

Lights which close themselves around the axis of the luminous ray and the base of the illuminated object will have that proportion amongst them that the bases of the compound pyramids have.

On the folio opposite (Mad I 31v, fig. 374, 1499-1500) he quantifies this problem:

DefinitionThe body f will be the half less illuminated than the body e because the part of the sky which illuminates it is twice as small as is that of c, as is shown in cd and ab.The proportion that the angle surrounding the illuminated body has to the axis of the illuminated ray...will be such as the quality of the light has.If the acute angle rcd enters r times into the angle cmn, then cm is 4 times less luminous than cn. Again if the angle dc enters 12 times into the obtuse angle ard and the angle cmn enters 3 times into the obtuse angle mno, the proportion will follow.

He returns to this theme on CU671 (TPL680, 1508-1510) under the heading:

Of the particular light of the sun or some other luminous body.That part of the illuminated body will be of greater clarity which is percussed by a luminous ray among more similar angles and least illuminated is that which finds itself among angles that are more difform than these luminous rays.

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(figure)

Figs. 377-379: Angles and light intensity on CU668.

A specific example demonstrates this claim (fig. 375):

This angle n on the side, which looks at the sun, being percussed by this sun under equal angles will be illuminated with greater power of rays than any other part of this illuminated body and the point c will be less than any other illuminated part since this point is struck by the solar body with angles that are more difform than any other part of the surface where such solar rays extend. And of the two angles, let the greater be dce and the lesser ecf and the equal angles which I have to draw first are ano and bnr which are precisely equal. And for this reason n will be illuminated more than any other part.

This connection between luminous angles and light intensity is broached afresh on CU668 (TPL718, figs. 377-379, 1508-1510):

In what surfaces is true and equal light found?That surface will be equally illuminated which is equally remote from the body which illuminates it as [for instance], if from the point a which illuminates the surface bcd, there would be drawn lines equal to this surface. Then by the definition of the circle this surface will be equally illuminated in each of its parts and if this surface were plane, as is demonstrated in the second demonstration efgd, then if the extremities of the surface are equally distant from such lines, the centre h will be the part closest to such a light and will be more illuminated than these extremities, by the extent to which it is closer to its said light e. But if the extremities of such a plane surface are removed from such a light with an unequal distance, as is shown in the third figure iklm, then the closest and the most remote part will have such a proportion in their lights as is that of their distances from the body which illuminates them.

(figure)

Figs. 380-381: Angles and light intensity on CU858.

On CU858 (TPL820, figs. 38-381, 1508-1510) he pursues the question:

On reflected lightTo the extent that the illuminated object is less luminous than its illuminating source, to that extent will its reflected part be less luminous than the illuminated part.That thing will be more illuminated which is closer to the illuminating source.To the extent that bc enters into ba to that extent will ad be more illuminated than dc.That wall which is more illuminated appears to have its shadows of greater obscurity.

On CU675 (TPL694b, 1508-1510) he asks:

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What part of a body will be more illuminated by a light of the same quality?That part of a body which is illuminated by a luminous quality will be of a more intense brightness which is percussed by a greater luminous angle.

By way of demonstration he offers a specific example (fig. 388):

This is proved. And let the hemisphere be rmc which illuminates the house klof. I say that that part of the house will be more illuminated which is percussed by a greater angle originating from a luminous source of the same quality.

Therefore at f where nfc percusses, there will be a more intense brightness of light than where the angle edc percusses and the proportion of the lights is the same as that of the angles and the proportion of the angles will be the same as is that of their bases nc and ec, of which the larger exceeds the minor in whole by part ne. And likewise at a under the eave of the roof of such a house there will be that much less light than in d to the extent that the base bc of such an angle bac is less than the base ec and thus it always follows proportionately, the light being of a same quality.

And the same which is stated above is confirmed in some object illuminated by our hemisphere and this is manifested in the part of a spherical object under the hemisphere k and f which, at the point b is illuminated by the entire part aec and at the part d by the hemisphere ef and at o by gf and in n by mf and at h by sf and thus you have understood where the first [degree of] light and the first [degree of] shade is in any body.

(figure)

Fig. 382: Angles and intensity of light on CU667.

How these statements concerning light apply also to shade is explored on CU667 (TPL755, 1508-1510) under the title:

Rule for placing the necessary shadows and lights in a figure or some body with sides.Such is the greater or lesser obscurity of shade, or indeed the greater or lesser brightness of light striking the faces of a body with sides, as is the greater or lesser size of the angle, which is enclosed between the central line of the luminous body, which percusses the centre of the illuminated side, and the surfaces of this illuminated side.

As usual this is followed by a concrete demonstration (fig. 382):

As [would be the case] if the illuminated body were an octangular column, the front of which is placed here in the margin. And let the central line be ra which extends from the centre of the luminous object r to the centre of the side sc. And again let it be that the central line rd extends itself from the centre of this luminous body to the centre of the side cf. I say that there will be such a proportion between the quality of the light which the side sc receives from this luminous body and that which the second side receives from he second side cf, as there is between the size of the angle bac and the size of the angle edf.

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These principles he summarizes in a late note on CA385vc (1510-1515):

That light is brighter which is of a greater angle.That shadow is darker which is born of a more acute angle.

Book Four. 4. Angle and Intensity of Shade

Leonardo's interest in the links between angles and intensities of shade is implicit in an early note on Triv. 28v (c.1487-1490) where he notes that (fig. 383): "to the extent that ab enters cd to that extent an will be darker than cr." On A85r (BN 2038 5r, 1492) he develops this demonstration (fig. 384):

To the extent that the shade made by the object on the wall is less than its cause, to that extent will this object be illuminated by weaker luminous rays. De is the object [and] dc is the wall. To the extent that de enters fg to that extent will there be more light in fh than in dc. To the extent that the luminous ray is weaker to that extent will it be further from its aperture.

(figure)Figs. 383-384: Links between angles and intensity of shade. Fig. 383, Triv. 28v; fig. 384, A85r.

(figure)Figs. 385-386: Demonstrations concerning angles and light intensity on CU663-664.

(figure)Figs. 387-390: Abstract and concrete demonstrations of problems of light and shade. Fig. 387, A89v; fig. 388, CU678;

fig. 389, CU675; fig. 390, G12r.

This link between angles and intensity of shade remains implicit in another note in the same manuscript, on A89v (fig. 387, BN 2038 9v, cf. CU657, TPL555a, 1492):

PaintingAmong shadows of equal quality that which is closer to the eye appears of lesser obscurity.Why is the shade eab in the first degree of obscurity and be[c] in the second [degree] and cd in the third [degree of obscurity]. The reason is that eab does not see any part of the sky. Therefore, no part of the sky sees it, and for this reason it is deprived of original light. Bc sees the part fg of the sky and is illuminated by this. Cd sees the sky hk. Since cdis seen by a greater amount of the sky than is bc it is reasonable that it be more luminous and so on up to a certain distance the wall ad will constantly become brighter until the darkness of the habitation will be overcome by the light of the window.

This principle he illustrates again on CU675 (TPL694, 1508-1510) analysed above (fig. 389, p. ) and once more on CU678 (TPL694c, 1508-1510) where he claims (fig. 388):

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The shade produced by the sun that remains under the rooves of buildings acquires darkness with every degree of height. He pursues this theme on G12r (c.1515) in the context (fig. 390):

Of universal light illuminating plants.That part of a plant will show itself as covered with shadow of less obscurity which is more distant from the earth.

This is proved. Let ap be the tree. Let nbc be the illuminated hemisphere. The part below the tree sees the earth pc, that is the part o and it sees a little of the hemisphere in cd. But the part [that is] higher in the concavity a is seen by a greater amount of the hemisphere, that is, bc, and for this reason (and because it does not see the darkness of the earth) it remains more illuminated. But if the tree is covered with leaves as is the laurel, arbutus, box or holm-oak, then it is variegated because even if it does not see the earth and it sees the darkness of the leaves, divided by many shadows which darkness reverberates to the reverse of these leaves and such trees [therefore] have shade that is darker to the extent that they are closer to the middle of the tree.

(figure)Figs. 391-392: Simple studies of shade on C3v on CU742.

In each of the six examples above Leonardo has considered various angles of shade produced by eaves of rooves or other overhanging objects. These range from concrete cases to abstract geometrical demonstrations. He is equally systematic in his approach to shade on the ground. At the simplest level he simply depicts a static situation as on C3v (fig. 391, 1490-1491). A next step is to consider the psychological aspects (cf. part 3: 4 below) of this situation as on CU742 (TPL605, 1508-1510) where he asks:

What background will render shadows darker.Among shadows of equal darkness, that will show itself as darker which is generated in a background of greater brightness. It follows that that part appears less dark which is in a darker background.

This general claim is, as usual, supported by a specific demonstration (fig. 392):

This is proved in a same shadow because its extremity, which on the one side borders on a white background, appears very dark and on the other side where it borders on itself, appears of little darkness. And let the shade of the object be bd made on dc, which appears blacker at nc because it borders on a white background ce, than at nd which borders on a dark background nc.

Having considered the static case he examines a dynamic situation on CU663 (TPL720, 1508-1510) in which the distance, angle and accordingly the shadow changes (fig. 385):

On the remoteness and distance that a man makes in going away from and approaching a same light and on the variety of its shadows.

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The shades and lights will vary in shape and quantity in a same body with the variety of the coming closer or going further which a man makes in front of this light.And this is proved. And let the man be bc who, having lights from a, will produce his shade bcf. Then the man moves from c to e and the light which stays fixed varies the shade in shape and in size, which is the 2nd shade deg.

In the following passage on CU664 (TPL721, 1508-1510) he examines a related situation. “On the variety of the shade produced by an immobile light generated in bodies that are bent...lower or higher without moving [the position of] their feet.” A specific demonstration again follows (fig. 386):

This is proven and let the immobile light be f and let the man not moving from his position by ab who bends to cb. I say that the shadow varies itself infinitely from a to c because the movement is in space. And space is a continuous quantity and consequently divisible to infinity. Therefore shade varies infinitely, that is, from the first shade aob to the second shade bcr and thus our proposition is concluded.

On CU731 (TPL5778, 1508-1510) his study of dynamic situations continues with a comparison of three cases in a single demonstration (fig. 393):

On derived shade distant from primitive shade.The boundaries of derived shade will be more confused which are more distant from primitive shade. This is proved. And let ab be the luminous body and cd the primitive shade and ed is the simple derived shade and cg is the confused boundary of the derived shade.

He returns to this problem of degrees of shade varying with angles and distance on E31v (figs. 394-395, c.1513-1514):

A long and narrow luminous source produces boundaries of derived shade which are more confused than spherical light. And it is this that contradicts the following proposition: that shade will have its boundaries more distant which is closer to the primitive shade, or if you wish to say, to the umbrous body. But the cause of this [contradiction] is the long and narrow shape, ac, of the luminous source.

Book Four. 5. Position and Intensity of Light Source

Meanwhile, Leonardo had been developing model demonstrations in which he explored the role of position and intensity of light source in relation to shade. On C8v (1490-1491), for instance, he begins with a case in which two light sources are of equal intensity (fig. 396):

That umbrous body which is positioned between two equal lights will make as many shadows as there are lights. Which lights are darker than one another to the extent that the light on the opposite side is closer to this body than the others.

(figure)Figs. 393-395: Angles, distance and shade. Fig. 393, CU731; figs. 394-395, E31v.

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(figure)Figs. 396-397: Experiments concerning intensity of light and shade on C8v.

Next he considers a case where the lights are not of equal intensity (fig. 397):

That umbrous body which is equidistantly positioned between 2 lights will make two shadows that are darker than one another to the extent that the lights producing the one is larger than the other.On C22r (1490-1491) he devises an experiment in four steps to test these factors of position

and intensity of light. As a first step he considers a case of two candles of equal intensity with an umbrous body in the centre (fig. 398): “that umbrous body will make 2 derivative shadows of equal darkness which has (in itself) 2 light sources equal in size equidistant from it.” As a second step he again takes a case of two candles of equal intensity, but with an umbrous body now off centre (fig. 399): “The proportion of the darkness of the shade ab with the shade bc will be that of the distance of the lights among themselves, that is, of nm to mf.”

(figure)Figs. 398-401: Systematic experiments concerning intensity of light and shade on C22r.

His third step is a case where the two candles are of different intensity and the umbrous body is again in the centre (fig. 400):

If one has positioned an umbrous body equidistantly between 2 lights, it will make two opposite shadows which will differ in their obscurity to the extent that the powers of the 2 light sources...which produce them, differ.

This general claim is then followed by a specific demonstration (fig. 400): “If the light xv is equal to the light vy the difference of the lights will be such as is that of the sizes.” His fourth step is a case where the two candles are of different intensity and the umbrous body is off centre (fig. 401):

But if the large light source is distant from the umbrous body and the small light source is nearby, it is certain that shades can be produced [which are] of the same darkness or brightness.

Striking here is Leonardo's scientific approach: how he systematically alters one variable while keeping the other constant, thus providing controlled situations. Lacking is a quantitative method of recording his results. Nonetheless, the way is thereby set for the Rumford's photometry experiments. When Leonardo returns to this problem on CA199ra (c.1500) quantitative considerations are alluded to. He begins by drawing a rough diagram in the right-hand margin (fig. 402) beneath which he asks: "Give me the site of the object that produces shades of equal darkness." In the text opposite he drafts a proposition:

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Of derived shades opposite, created by a same object (opaque caused by two lights opposite) interposed bet (ween 2 lights) opposite of (various sizes and distances from this object) with various distances between light of various sizes.(Given) Given Two or

This he crosses out and begins afresh with a general heading: “Of derived shades, opposite [one another], created by a same object,...which is interposed at various distances between lights of different**.” He then outlines three specific experiments:

Let there be given two opposed shades around a single object...interposed between 2 lights of double power, and that these shades are...among themselves of equal obscurity. It is asked what proportion have the spaces interposed between these lights and the said object.

You will give 2 shades of a same body interposed between 2 lights of double power and such that with those shades one is treble the darkness of the other.

You will give 2 shades born around a same body, which are of a double darkness in relation to one another and the greater darkness is towards a greater light.

Whether he actually carried out these specific experiments is uncertain. But these problems are not forgotten. On CU684 (TPL682b, 1508-1510), for instance, he considers how position and intensity of a light source affect the illumination of an umbrous body:

On universal illumination mixed with the particular [illumination] of the sun or other lights.

Without a doubt, that part of the umbrous body which is seen by a lesser quantity of a universal and particular body, will be less illuminated.

(figure)Figs. 402-404: Further experiments involving effects of distance on derived shade. Fig. 402, CA199ra; fig. 403, CU684; fig. 404, CU695.

To support this general claim he provides a specific example (fig. 403):

This is proved. And let a be the body of the sun positioned in the sky nac. I say that the point o of the umbrous body will be more illuminated by...universal light than the point r, because o sees and is seen by every part of the universal light nam and the point r is only seen by the part of the sky mc.

Moreover, o is seen by the entire quantity of the sun that is facing it, and r does not see any part of this sun.

A few paragraphs later on CU695 (TPL689, 1508-1510) he returns to this situation from the point of view of intensity of shade (fig. 404), asking:

What light makes the shades of bodies more different from their lights?

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That body will make the shades of greater darkness which is illuminated by a light of greater brightness.The point a is illuminated by the sun and the point b is illuminated by the air illuminated by the sun. And such is the proportion between the illuminated [part] a and the illuminated [part] b, is the proportion which the light of the sun has with that of the air.

On CU159 (TPL249a, 1508-1510) he provides a catalogue of various possibilities under the heading:

Of the colours of incident and reflected lights.When two lights [have] an umbrous body put in the middle [between] them, they can only vary in two ways, that is, either they will be equal in power or they will be unequal, that is, speaking of the lights in relation to one another. And if they are equal, they can vary their brightness on the object in two other ways, that is, with equal brightness or with unequal brightness. It [the brightness] will be equal when they [the light sources] are equidistant; unequal [when the light sources are] at unequal distances. [When the light sources are] equidistant they will vary in two other ways, namely, the object will be illuminated by lights equal in brightness and in distance (lights equal in power and in distance from the object opposite).*

The object situated equidistantly between two lights equal in colour and brightness can be illuminated by these two lights in two ways, namely, either equally from every side or unequally. And it will be illuminated equally by these lights when the space which remains between the two lights is of equal colour and darkness or brightness. It will be unequal when these spaces between the two lights are of different darkness.

These categories become easier to visualize when rendered in tabular form (see chart 11).

(chart)Chart 11. Categories of Light and Shade described on CU159 (TPL248a, 1508-1510).

(figure)Figs. 405-406: Elementary cases of derived shade and interposed plans on C9r and C18v.

Book Four. 6. Size/Shape of Light Source/Object

He is also concerned how the size or shape of a light source in relation to an opaque object affects derived shade. On C18v (1490-1491), for instance, he makes a general comment on this theme: "The shapes of shadows often resemble their origin, the umbrous body and often their cause, the luminous source." Directly beneath he draws an introductory example (fig. 406) with the caption:

If the shape and size of the luminous body are equal to that...of the umbrous body, the primitive and derived shade will be of the shape and size of this body, falling between equal angles.

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In the next paragraph he states why this is not always the case:

At a certain distance derived shade will never be the same shape as the umbrous body from which it originates, if the shape of the light of this illuminating body is not the same as the shape of the body of the said illuminated light.

Again he draws an example (fig. 407) followed by a caption and a restatement of his claim:

A light of a long shape will have the effect that the derived shade, originating from a round body is...wide and low even if it be percussed between equal angles.

It is impossible that the shape of derived shade is the same as that of the luminous body whence it originates unless the light, its cause, is the same shape and size as this umbrous body.

(figure)Figs. 407-408: Further cases of derived shade and interposed plane on C18v and C18r.

(figure)Figs. 409-410: Demonstrations of derived shade and an interposed plane on C12r and C8r.

Having considered the case of a long light source and a round umbrous body on C18v, he examines, on the recto of the same folio, the case of a round light source and a long umbrous body (fig. 408). He is ever playing with variables. Beneath this drawing he again adds a caption:

The umbrous percussion originating from a long umbrous body and caused by a round luminous source, at a certain distance is the shape of the umbrous body and at a certain other distance [that] of the luminous source.

On C12r (fig. 409, 1490-1491) he notes:

Not only the quantity but also the quality of the shade and their boundaries will not show themselves with their true shape in their foreshortening.

Bc being an invisible boundary, by the loss of shade, nonetheless appears visible as a result of foreshortening, not otherwise than ab and dc appear.

On C8r (1490-1491) he draws an example of a long light source and a round umbrous body (fig. 410), beneath which he asks: "Why in this case does the derived shade show itself as dark in the middle of its height ab and is not discerned at its extremities cd?". In a passage on BM103v (1490-1492) this question of the shape of light source is pursued in a series of drafts:

If a light is of equal proportion to the umbrous body positioned opposite, the derived shade will be the same as the umbrous body if it falls on a plane surface at equal angles.If a light is of a long shape and the umbrous body is round, the derived shade will be wider than it is high.

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Light which falls on a flat place under equal angles...Of the round aperture and the long light....the percussion is long.

The umbrous and luminous body of spherical shape will produce derived shade of a long shape if this falls on a flat plane at unequal angles.

He pursues these problems on CA187va (1492) in the context of his camera obscura studies, which constantly run parallel to his light and shade demonstrations (see below pp. ):

No long light will show the true form of the shade separated...from the walls by spherical bodies.This shadow is long and thin.No separate shade can stamp on walls the true form of the umbrous body if the centre of the light is not equidistant from the extremities of the body.

When he returns to these problems on CU632 (TPL607, 1508-1510) he begins with a general claim: “Shade will never have the true similitude of the contour of a body whence it originates even if it be spherical unless the light is of the shape of the umbrous body.” Directly following this he lists four specific cases:

If a light is of a long shape which extends upwards, the shades of this illuminated body extend themselves laterally.

If the length of a light is lateral the shade of the luminous body will make itself along its height. And likewise in whatever way the length of the light finds itself, the shade will always have its length intersected cross wise with the length of the light.

If the light is thicker and shorter than the luminous body the percussion of the derived shade is longer and thinner than the primitive shade.

If the length and width of the luminous body is equal to the length and width of the umbrous body, then the percussion of the derived shade will be of the same shape at its boundaries as the primitive shade.

(figure)Figs. 411-412: Simple cases of derived shadow on an olique plane on C18r and C11v.

This problem is again broached in the context of his camera obscura studies on CA195va (see below pp. , c.1510): “Why shadow is never similar to the umbrous body if the light is not equal and similar to the umbrous body and it is not stamped over a flat wall between equal angles.” On E31v (1513-1514) he returns to this theme for the last time in the extant notes:

That luminous body of a long and narrow shape makes the boundaries of derived shade more confused than he spherical light and this is what contradicts the following proposition:

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That shade will have its boundaries more noted which is closer to the primitive shade or, if you wish to say, the umbrous body, but the cause of this is the long shape of the luminous body, etc.

Book Four. 7. Position/Shape of Light Source/Object

Leonardo's expressed aim in book four is to study how shadows vary with the position and shape of the plane on which they are projected. His simplest example of this is a drawing on C18r (1490-1491) beneath which (fig. 411) he writes:

Even though the umbrous and the luminous body are of spherical rotundity and of equal size, nonetheless, its derived shade will not resemble the rotundity of the body whence it originates and will be of a long shape if it falls under unequal angles.

(figure)Figs. 413-415: Further cases of derived shadows on oblique and irregular planes on CA241vd.

One step more complex is his drawing on C11v (fig. 412, 1490-1491) where part of the plane is inclined and part is positioned upright, followed by a caption: “Of the derived shade impressed among different qualities of angles, that part which is found between straight angles will hold the first degree of darkness.” On CA241vd (1508-1510) he begins with a general claim in draft form, headed:

On the percussion of derived shade.The percussion of derived shade will never be of the same shape as the umbrous body (if the light does not have the shape) if the luminous body, cause of this shade, (is not similar to ) has not a shape equal to that of this umbrous body and if the umbrous rays...which border with the luminous rays are not equal in length amongst themselves.

In the right-hand margin beneath this he again draws the shadow of a sphere on a simple inclined plane (fig. 413, cf. fig. 411). Next he draws the shadow of a sphere on a staircase (fig. 414) and finally the shadow of a curved cylinder on another cylindrical surface (fig. 415), adding the caption:

The shadow cd from the umbrous body ab which is itself equidistant [from it], will not show itself as being equal in darkness, through being in a background of various brightnesses.

This idea he develops in the main text alongside:

Shade will never demonstrate itself...of uniform darkness in the place where it is intersected if such a place is not equidistant from the luminous body. This is proved by the 7th which states: that shade will demonstrate itself as brighter or darker which is surrounded by a background that is darker or brighter...by the 8th of this: that background...will have its part that much darker or brighter to the extent that it is more remote [from] or closer...to the luminous body and among sites equidistant from the luminous body that will show itself as

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more illuminated which receives the luminous rays among more equal angles. [It is] always [the case that] shade, impinged on some inequality of sight, will show itself with its true boundaries equal to the umbrous body, if the eye does not position itself where the centre of the luminous body was.

That shade will show itself as darker which is more remote...from its umbrous body.On C13r (1490-1491) he had, meanwhile, drawn a more complex case where a spherical light source strikes a cylindrical object, the shade of which then encounters a spherical body (fig. 416), with the caption: “That part of the umbrous body which is between illuminated bodies is more luminous. The light having been removed it will remain darker.” More complex still is the case on C11r (1490-1491) where two spherical light sources have two spherical umbrous objects positioned between them (fig. 418, cf. fig. 417). Here his caption notes: "Often it is possible that there is a derived shadow without original shade."

(figure)Figs. 419-423: Early illustrations of the phenomenon that a large one light source and a small one object can produce diverging and more than one shadow. Fig. 419, W19147v, K/P 22v; figs. 420-422, K/P 22r; fig. 423 C21v.

(figure)Figs. 424-426: Variations in light source and shade on C1r.

Book Four. 8. How/Why One Light Source and One ObjectProduce Two Shadows

Leonardo's systematic studies of how the shape/position of the light source, umbrous body and projection plane all affect the shape and quality of shadow make him increasingly aware of a curious phenomenon, namely, how one light source and one opaque body can produce two shadows. In an early diagram on W19147v (K/P 22v, 1489-1490) he shows how a light source larger than an opaque body, nonetheless produces expanding shade (fig. 419). On the recto of this folio he draws three further sketches relating to this theme (figs. 420-422).

In the Ms. C (1490-1491) he analyses this phenomenon more closely. On C1r, for instance, he draws both a small light source and a longer one to compare their effects (figs. 424-425) noting: “That inferior and superior extremity of the derived shade is less distinct than the lateral one which is caused by the light higher than it is wide.” Above this he draws a further example (fig. 426) in which a whole spectrum of shades is produced.” A variation on this theme is shown on C21v

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(1490-1491). Here a central dark shade is surrounded by a larger fainter shade (fig. 423) and beneath it, the caption: “The percussion of derived shade is always surrounded by shade mixed with the illuminated background.” Two further examples on C1v (1490-1491) illustrate how different relative sizes of light source and umbrous body can produce two circles of shade which either stand completely separate or overlap (figs. 427-428). In his own words: “The straight boundaries of bodies will appear broken which have their umbrous place rayed by the percussion of luminous rays.” On, C2r (1490-1491), the opposite folio, he begins anew with an apparently unrelated statement (cf. below p. ):

The body illuminated by solar rays which has passed through the large ramifications of trees will produce as many shades as there are branches interposed between the sun and it.

Directly beneath this he draws a further example (fig. ) in which a spherical light source and a spherical umbrous body produce two overlapping circular shadows. He then makes a more complex drawing (fig. ) in which a spherical light source standing in front of a conical pyramid produces three sets of double circular shadows, the lowest of which overlap almost completely, the middle ones less so, while the highest are entirely separate.

(figure)Figs. 427-429: How one light source and one opaque body produce two shadows. Figs. 427-428, C1v; fig. 429, C2r.

(figure)Figs. 430-431: Three different shadows produced by one light source and a cone on Ca347ra and C2r.

These drawings are followed by a series of notes which he had drafted on CA347ra (fig. , 1490-1495):

CA347ra C2rThe percussion of derived shadeoriginating in a pyramidal umbrousbody is of various darknessThe percussion of shades parting The percussion of umbrous bodiesfrom a pyramidal body is not originating from a pyramidalsimilar in shape to its origin. umbrous body will be of bifurcated

shape and of various darknesses atits points.

The light which is larger than The light which is greater than thethe point and smaller than the point and less than the base of thebase...of the pyramidal umbrous pyramidal umbrous body placedbody positioned with it, will opposite it will...produce in itshave the effect that the umbrous percussion a shadow of a bifurcatingconcourse in its percussion will shape and of various degrees ofcause shade of a bifurcating darkness.shape and of various degrees of

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darkness.If the luminous body greaterthan the umbrous [body].If the umbrous body, less than If the umbrous body less than thethe luminous, makes two shades luminous source makes two shadowsboth the umbrous body [that is] and the umbrous body is similar tothe same size and larger...than the luminous source or the largerthe luminous source produce one makes a single shadow, then it is[shade]. The pyramidal body fitting that the pyramidal bodywhich has part of it larger and which has a part which is smaller,part of it smaller than the a part which is equal and a partluminous body will make which is larger than the luminousbifurcating shade. body makes a bifurcated shade.

He pursues this question of the shade produced by conical pyramids on I 38(r) (c.1497) where he draws a schematic diagram (fig. 436), besides which he writes.

(figure)Figs. 432-437: Sketches of the sun shining through trees and pyramidal forms. Figs. 432-434, I37v; fig. 435, Author's reconstruction of 434; fig. 436, I38r; fig. 437, Author's reconstruction of 436.

Proof how at a certain distance the shadow of the pyramid does to resemble the pyramid whence it originates.

Below the diagram he adds: "Let ab be the pyramidal umbrous body. Let cd be the part which receives the shade," which text continues in the right-hand margin:

You see the shade n which is joined with its shape on the wall and likewise m, but disproportionate to the continuous proportion. You see the shade e which is joined and d which is joined even worse.

With the aid of a three-dimensional diagram (fig. 437) Leonardo's intention becomes clear. This applies also to the diagram on the folio opposite (I 37(v), fig. 434 cf. 435 ) where he is considering a curved conical pyramid, beneath which he writes: "Different shadows from pyramids equidistant from their luminous body." This is effectively a heading for the text alongside:

Even if an umbrous body is pyramidal and each of its walls is equidistant from its luminous object, nonetheless, that part of the pyramid that is most smaller than the light which illuminates it, will throw less distant shade on its cause.

In the upper part of I 37v (1497) he draws a sketch of the sun shining down on a pyramid (fig. 432) and beneath this, a sketch of the sun shining down on a tree, (fig. 433) to which he adds the caption: “The imprint of the shadow of some body of uniform thickness will never be equal to the body whence it originates.” His treatment of pyramids and trees on the same page is no coincidence. Indeed his discussions of straight and curved pyramids on I37v -I38r are probably intended as geometrical abstractions to simulate the shape of branches. This would account for his

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unexpected reference to trees on C2r (see above p. , 1490-1491) while discussing the shade of pyramids.

(figure)Figs. 438-449: How a large light and small object produce diverging or double shade. Fig. 438, BM171r; fig. 439, BM170v; figs. 440-441, CA353rb; figs. 442-443, CA155re; figs. 444-445, CA144va; figs. 446-448, CA144rb; fig. 449, CA277va.

In the years that follow the problem of how two shades are produced is often treated as a special case of the phenomenon that a light source larger than an umbrous body nonetheless produces diverging shade. On BM170v, for instance, (fig. 438, 1492 cf. BM171r, fig. 439, 1492) he draws a sketch of two diverging shades accompanying which he notes that "an object larger than the umbrous body sees more than half of it and makes much mixed shade."

On CA353rb (c.1495) he sketches both the general principle of expansion (fig. 440) and the two divergent shades in particular (fig. 441). On CA155re (1495-1497) he is content merely to sketch the general principle of expansion (figs. 442-443). Meanwhile, his search for an explanation leads him to analyse the phenomenon in terms of geometry. This begins with rough drafts as on CA144va (figs. 44-445, c.1492) where he notes:

The umbrous intersection will occur after which the divided shadows will concur in 2 different concourses...[as] if they derived from 2...divided lights.

Luminous bodies being larger than the umbrous bodies positioned opposite,-...it will happen (that the said body will be more than...than this body being) and the light will operate as if it were...divided and its shadows will divide after their intersection and their concourse will be in different places.

These drafts continue on CA144rb (figs. 447, 1492; cf. CA222ra, figs. 450-455, 1492; and CA277va, fig. 449, 1508-1510): “The boundaries (of shade made) by the size of the shadows made by a greater...less than it, umbrous bodies will spread out from their centres as if these were born of various qualities of light.”

(figure)Figs. 450-460: Further demonstrations of the principle of divergent shadows. Figs. 450-455, CA222ra; figs. 456-457, CA93vb; figs. 458-459, CA258va; fig. 460, CA258ra.

(figure)

Figs. 461-471. Eleven further demonstrations of the divergent shade problem. Figs. 461-465, CA195va; figs. 466-467, CA258va; fig. 468, CA208vb; fig. 469, CA177ve; figs. 470-471, CA195ra.

More than fifteen years pass before he returns to this problem. On CA258ra (fig. 460, 1508-1510, cf. CA93vb, figs. 456-457, C.1510) he reduces the phenomenon to its geometrical essentials. On the verso of the same folio (fig. 458-459, 466-467) he asks:

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How a derived light, even if it be generated by a single light, this light will adopt shades in it as if it were divided into two lights.The shade generated by a single light is always divided at its bifurcated point, as it if were generated by two lights.

On CA195ra (figs. 470-471, c.1510, cf. CA177va, fig. 469, 1505-1508), he again asks: “Why a single luminous source makes two shades after a single luminous body. Why a single body illuminated by a single light produces two [shades].” On CA208vb (1508-1510) he draws a nearly identical diagram (fig. 468) to make an astronomical point: "Beneath there is no part which sees the sun entirely." On CA195va (1510) he takes up anew the question of two shadows produced by a single light beginning with a draft:

Why a light makes pyramidal shade after the umbrous body.Two shades are made by a single light because all...gh sees the entire space abc and from the other side it sees the same space edf. But...

(figure)Figs. 472-473: One large light source and two shades; two small light sources and two shades on C21r.

He then begins afresh and gives a thorough explanation (fig. 463, cf. figs. 461-462, 464):

Two shades are made by a single light and a single object when the light is greater than the object and this is caused because...the entire light de illuminates...the space ders. But the space cab...in every degree that it approaches the line cb, always loses sight of this light as the motion of the line fn relative to b shows which, to the extent that n comes closer to b, the more f approaches d above with its other extremity and restricts the space of the light ed...and in a similar way the 2nd shade pno is generated and the triangle Shb is entirely seen by the light de, because half the light,...which is fe, sees with its parts...Sm and the other half of the light df, sees with its parts in nb, but the triangle hSb is seen...by two halves of this light de and hence two halves make one whole.

In the above examples we have deliberately included some cases involving a camera obscura (see pp. below) in order to give some impression of the connections between various problems in Leonardo's mind.

Book Four. 9. Compound Shade

He is also concerned with cases of compound derived shade, namely, when more than one light source and/or more than one umbrous body are involved.

Book Four. 9a. Preliminary Studies

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This interest grows partly from his attempts to show how one light source and one object can produce two shadows, as is clear from two diagrams on C21r (figs. 472-473, 1490-1491), alongside which he adds:

If the size of the luminous body surpasses that of the illuminated body, an intersection of shade will occur, beyond which divided shadows will go in two different directions as if they were derived from two different lights.

(figure)Figs. 474-475: Examples of compound light and shade on C4v and C9v.

(figure)Figs. 476-479: One opaque body and one, two and three light sources. Figs. 476-478, C22r; fig. 479, Pecham, Perspectiva communis.

In the same manuscript he compares different shadows produced by two light sources at different distances. On C4v (fig. 474, 1490-1491), for instance, he notes:

You will find that proportion of darkness between the derived shadows a [and] n as there is between the vicinity of the luminous bodies m [and] b which cause them. And if these luminous bodies are of equal size you will also find such a proportion in the sizes of the percussion of the luminous circles of the shade as is that of the distance of these luminous bodies.

On C9v (fig. 475, 1490-1491) he draws a related situation in greater detail, this time adding only a brief caption: “The percussion made by umbrous and luminous rays on a same place is mixed and of confused appearance.” These preliminary notes lead to more thorough studies.

Book Four. 9b. Multiple Lights and Objects

Here again his approach involves a systematic play with variables. At the simplest level, on C22r (1490-1491), he draws first one light source and one opaque object (fig. 476); then two light sources and one opaque object (fig. 477) and then three light sources and one opaque object (fig. 478, cf. fig. 479).

(figure)Figs. 480-481: One light source and two opaque bodies on cA144vb and C17r.

(figure)Figs. 482-483: One light source and two opaque bodies in the open air and with an interposed plane on c13r and C14r.

He also considers the case of one light source and two opaque bodies. In drawings on CA144vb (fig. 480, c.1492, possibly 1490) and on C17r (fig. 481, 1490-1491) he assumes that this will produce two converging pyramidal shadows. He changes his mind, however, and on C13r

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(1490-1491) demonstrates how intersecting shadows of differing intensities are thereby produced (fig. 482):

It is possible that mixed derived shade, caused by a single light by diverse bodies can intersect and superimpose itself on one another.

Abc is the mixed and intersected derived shade and superimposed on one another since mc is the shade of d and bn is the shade from f and to the extent that abc contains [shade], to that extent one shade superimposes itself on the other.After showing what occurs in the open air, on C14r (1490-1491), he examines what happens

when this shadow produced by one light source and two opaque objects is intersected by a wall (fig. 483):

Umbrous rays of imperfect and equal obscurity which mix themselves together double the quantity of the darkness.Reason demands that double equantity produces a double power (And for this reason two imperfect things make a perfect one).Msn and ktn are the incorporated and imperfect mixed shades and kmstn is the effectively perfect duplicated shade.

Having studied one light source and two opaque bodies on C13r, 14r, he studies the case of two light sources and one opaque body first in passing on C22r (fig. 477, 1490-1491) and then in more detail in a now partly ruined text on CA230rh (fig. , 1505-1508) entitled:

Why the shadows intersected behind the maximal shade, to the extent that they approach such maximal shade more, the more they lose in darkness.

Let aco be the maximal darkness, codnaobm are the shadows intersected on the maximal shade cao of such a shade in separating...maximal to the extent...they because white...the 2 simple lights and the...which proceed, see the...dark mixed with...surrounded by such a background.

In the Windsor Corpus this theme is pursued. On W19151v (K/P 118v/b/, 1508-1510) he makes a marginal drawing (fig. 484) alongside which he notes: “Derived shadows will be of equal darkness if they arise from lights of equal power and distance: this is proved.” On W19149v (K/P 118v/A/, 1508-1510) he redraws the diagram (fig. 485) this time providing a long explanation of its five degrees of shadow:

The greatest darkness of shadows is the simple derived shadow because it is not seen by either of the lights ab /or/ cd.

The second of lesser darkness is the derived shade efn and this is less dark by half because it is illuminated by a single light, that is, cd. And this is of uniform natural darkness because throughout only one of the two luminous bodies sees it. But it varies with accidental darkness because the more that it is distant from such a light the less it participates in its brightness.

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(figure)Figs. 484-485: Two light sources and one opaque body on K/P 118v.

(figure)Figs. 486-487: Two light sources and two opaque bodies on C19r and one light source and three opaque bodies on C13v.

The third /degree of/ darkness is the middle shade. But this is not of uniform natural darkness because the nearer it is to the simple derived shadow, the darker it is and the accidental uniformly difform uniformity is that which corrupts it, that is, the more distant it is from the two luminous sources, the darker it is.

The fourth is the shade krse, and this is so much the darker in natural darkness as it is nearer to ks because it sees less of the light ab but by accidental /shade/ it loses more darkness because that which is closer to the light cd always sees the two lights.

The fifth is of less darkness than each of the others because it always sees the whole of one of the two and the whole or part of the other and this loses more darkness to the extent that it is closer to the two lights and the more so to the extent that it is nearer to the outer side st, because it sees more of the second light ab.

Meanwhile, he had been exploring other combinations of light sources and opaque bodies. On C19r (1490-1491), he combines two light sources and two opaque bodies (fig. 486) alongside, which he explains: “To the extent that the darkness of two rays of imperfect darkness is different, to that extent the shade resulting from their mixture will differ from its original being.” Lower down on the folio he adds two further notes:

(figure)Figs. 488-490: Three light sources and one opaque body. Figs. 488-489, C6r; fig. 490, CA229rb.

It is impossible that there results a shade of darker quality...from a mixture of 2 perfect shades.It is possible that there results a perfect shade from a mixture of 2 imperfect shades....

On C13v (fig. 487, 1490-1491) he considers a case with one light source and three opaque bodies, adding the caption:

It is impossible that simple derived shades originating from different bodies and caused by a single light can ever join or touch one another.

The converse case of three light sources and one opaque body he illustrates first in two rough sketches on C6r (figs. 488-489, 1490-1491) and then more carefully on CA229rb (fig. 490, 1508-1510, cf. figs. ). Just how systematic is this play of variables becomes clear from Chart 12.

Number of Number of

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Light Sources Opaque Bodies Codex 1 1 C22r 2 1 C22r 3 1 C22r

1 1 C21r 1 2 C13r 1 3 C13rChart 12. Systematic play of variables using light sources and opaque bodies.

Having considered combinations of one, two and three lights and objects, on C13v (1490-1491), he describes an experiment with four light sources:

(figure)Fig. 491. Four light sources and four objects on C13v; fig. 492. A light source, columns and shadows on F6r.

(figure)Figs. 493-498. Columns casting shadows. Fig. 493, CA347ra; fig. 495, F1v; figs. 496-498, CA236vc; figs. 499-500, CA199va.

If in a room one has positioned 4 light sources and the sky above is completely sifted by the grid which covers it, it will hold much back and sift the grain which, in descending through the air makes the shade evident in the air, standing out and clear as is here shown.

Beneath this he draws a diagram (fig. 491) which he describes in detail:

That shade is darker which is derived from more diverse umbrous and luminous bodies.At kfs one sees the lights b, c, d which lacks only the light a, which is a quarter of the number /of the whole/. At fsm one can see only the two lights cd which is the half of all the lights. At mspn one does not see any light whence, not being able to be illuminated it is found to be the first degree of darkness.

Book Four. 9c. Columns

Related to such experiments are Leonardo's demonstrations involving columns as on F6r (1508) where a light source (fig. 492) in front of a column creates a succession of shadows. His earliest extant illustration on this theme is a rough sketch on CA347ra (fig. 493, 1490-1491) showing a light source in front of a column. Somewhat more developed, but again without text, are three sketches on CA236vc (figs. 496-498, 1508-1510) showing light sources and columns casting shadows.

On CA199va (fig. 500, cf. fig. 499, c.1500) he draws a column in isolation to illustrate that the degrees of shade on an object can be infinite (see p. above). On F1v (1508) he draws another column (fig. 495) to clarify his claim that the colour of an object is affected by the colours surrounding it. He returns to this theme on E31r (CU621, TPL594, 1513-1514) drawing first a column in isolation (figs. 501, 504) accompanying which he writes:

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(figure)Figs. 501-504: Elementary demonstrations with columns. Figs. 501-503, E31r, fig. 504, CU621.

(figure)Figs. 505-507: Preliminary demonstrations with crosses and shadows on C11v, CA229vb and CA37ra.

On pyramidal shadePyramidal shade produced by a parallel body will be thinner than the umbrous body to the extent that the simple derived shade is cut at a greater distance from its umbrous body.

Below this he draws a second column with converging shade (fig. 502) and a third with diverging shade (fig. 503) to illustrate basic types of shade (see pp. above). In themselves these demonstrations with isolated columns are of tangential interest. Their importance lays therein that they form a starting point for a series of experiments involving multiple columns which play an important role in his discussions of simple and compound shade.

Book Four. 10d. Experiments with Crosses and Columns

The earliest extant note on this theme occurs on C11v (1490-1491) where he draws (fig. 505) a long light source in front of a cruciform opaque object beneath which he adds:

If a long luminous body is the length of a cruciform umbrous body, the simple derived shade of the transverse part of the cross will not be conducive to percussion.

**CA37va

Nearly two decades pass before he returns to this problem on CA229vb (1508-1510). He now draws two separate columns/sticks which overlap one another in cross-form to produce shadows (fig. 506). What interests him in this case, is to show that there is only simple shade and not compound shade where the two shadows intersect. Or as he puts it:

The percussions of simple shadows, even if they are intersected, are not doubled because the shade of the first umbrous body remains impressed in the luminous part of the second umbrous body which intersects with the first one and since with a single light one cannot generate two simple shadows.

Two compound shadows joined together generate simple shadows. On CA37va (1508-1510) he begins to experiment systematically. He now takes a single obliquely positioned column or stick and demonstrates the shadow that it casts in the presence of one light source (fig. 508). Next he shows that with two light sources this same column produces two shadows (fig. 509) and with three light sources it produces three shadows (fig. 510). In addition he demonstrates how two light sources and two oblique columns can produce a cruciform shadow (fig. 507). This is closely related, in turn, to another demonstration that acquires great significance for him, namely, where

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two light sources in front of two columns, positioned in the form of a St. Andrew's cross, produce four shadows. Preliminary drafts for this occur on CA37ra (figs. 511-513, 1508-1510).

On BM248v (1508-1510) these drafts continue. He begins by simply drawing (fig. 514) two points from which emanate four shadows, two of which are marked a and b. In his next version (fig. 515), these two points become the base of a roughly sketched St. Andrew's cross. The four shadows are now marked A, b, a, b respectively. In a third version (fig. 516), he draws the St. Andrew's cross more carefully. The letters a and b are now linked with points representing the two light sources. The sequence of the lettering on the shadows is different however: it is now a a, b b.

Why should Leonardo be so interested in such problems? Some of the intersections of shadows produced by a St. Andrews cross result in a shadow of double intensity, while others do not. The phenomenon and the reasons for it had been a matter of debate which he hoped to set straight. Hence, having made several drafts without any accompanying text, we find him on CA177rb (1508-1510) redrawing this diagram with two columns in the form of a St. Andrew's cross (fig. 517), beneath which he outlines the problem:

Of simple shade.Why in the intersections...a /and/ b of the two compound shadows ef /and/ mc, there is generated simple shade, and likewise in ah /sic: eh/ and mg, and such simple shade is not generated in the other two intersections c /and/ d made in the same compound shadows mentioned above.

In short he is claiming that there is simple shade at the intersections a and b, and compound shade at the intersections c and d. Why this should be so he explains directly following in his:

ReplyCompound shadows are mixtures of bright and dark, and simple /shadows/ are

composed of simple darkness. Hence of the two lights n /and/ o, the one sees the compound shadows from the one side and the other sees the compound shadows from the other side. But no light sees the intersections a /and/ b and hence it is simple shade. But in compound shade both see /the/ light source.

Conscious that this claim is controversial, he introduces the opinion of an adversary:

And here a doubt arises through the adversary, because he states: in the intersection of compound shadows, the two lights causing (...) these shadows are necessarily seen and for this reason such shadows must cancel one another...such that where the two lights do not see, we say that the shade is simple and where a single one of the two lights see, we shall say that such a light is compound, and where the two lights...see, is cancelled shade because...where the two lights see no shade of any kind is generated, but is solely composed of the brightness of the background surrounding the shadows.

These opinions of the adversary he refutes:

Here it is replied that the above...mentioned /opinion/ of the adversary is true, which only makes mention of that truth which is in his favour, but if he adds the remainder, he will

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conclude that my proposal is true. And this is that, seeing the two lights, at that intersection such a shadow would be cancelled. This I confess to be true...if the two shadows were not seen in the same place because, where one shadow and one light are seen, compound shadow is generated and where two shadows and two similar lights are seen, this shadow cannot vary in any part of it, the shadows being equal and the lights being equal. And this is proved in the eighth on proportion where it is stated: such a proportion does simple power have with simple resistance, as a double power has with a double resistance.

(figure)Figs. 511-519: The development of a demonstration involving two light sources, a St. Andrew's cross and four shadows. Figs. 511-513, CA37ra; figs. 514-516, BM248v; fig. 517, CA177rb; figs. 518-519, CA241rc.

The problem continues to trouble him and on BM243r (1508-1510) he again draws (fig. 526) two light sources a and b. In place of two columns positioned in the form of a St. Andrew's cross, he merely marks the points t and s from which emanate four shadows a, b, s, b intersecting one another at m, n, r and c. This he describes in the text alongside headed:

DefinitionsThe intersection n is made by the shadow created by the light b because this light b generates the shadow tb and the shadow sb. But the intersection m is made by the light a which generates the shadow sa and the shadow ta. But if you cover the two lights a /and/ b then there are generated the two shadows n /and/ m at the same time and other than this there are generated two other simple shadows, that is r /and/ c, in which...one does not see either of the two luminous sources.The degrees of darkness that the compound shadows acquire are as many as the number of luminous bodies that see it are less.

On BM248v (1508-1510) he drafts a three step demonstration of these principles. In the right-hand margin he makes a sketch (fig. 520, cf. fig. 521) which he marks first (prima), showing how light a produces two shadows intersecting at n when light b is extinguished. He then draws another figure (fig. 522 cf. fig. 523) which he marks s (2nd) where light b in turn casts two shadows, while light a is extinguished. In the next figure, (figs. 524-525, cf. figs. 526-527) marked (3rd), he demonstrates how the shadow at g and h is doubled while the shadow at i and k is not.

(figure)Figs. 520-523: Systematic experiments in which the light source on the left, then the light source on the right is extinguished. Figs. 520, 522, BM248v; figs. 521, 523, CA241rc.

Alongside he writes a draft text:

Why the intersection n, being composed of two compound derived shadows...generates...compound shade and not simple shade as do the other intersections of the compound shade. This occurs by the 2nd of this which states

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A preliminary version of this second proposition follows: “The intersection of the two composed derived shadows originating in the intersections of the umbrous columns will not generate simple shadow which does not acquire any darkness.” This is crosses out and restates: “The intersection of derived shadows originating from the intersections of columnar umbrous bodies illuminated by a single luminous source will not generate simple shade.” Consideration of a third proposition follows:

And this arises through the 3rd /of the/ 1st which states: the intersections of simple derived shadows never acquires darknesses because...all...of the sums of darkness joined together do not acquire more darkness than a single one because if the many dark sums increase the darkness in their duplication they could not be named sums of darkness but partial darkness /instead/. But if such umbrous intersections are illuminated by a second light source positioned between the eye and the intersected bodies then...such shadows will be compound shadows and will have a uniform darkness int heir intersection as in the remainder.

The text that follows serves as caption for his third diagram (fig. 525):

By the 1st and the 2nd above, the intersections i /and/ k will not be of double darkness as they are double in quantity but by this 3rd /proposition/ the intersections g /and/ h will be of double darkness and quantity.

The folio ends with definitions of simple and compound shade (cf. p. above). These drafts on BM248v (1508-1510) serve as a starting point for a more detailed analysis on Ca241rc (1508-1510) which begins with a general description of:

ShadeWhen the intersection of two umbrous columnar bodies generates its derived

shadows, with two luminous sources then it is necessary that four derived shadows are generated,...which shadows are compound and these shadows intersect in four places and of these there are two which compose simple shadow and two remain compound shadow and these 2 simple /shadows/ are generated where the two lights cannot see, and the compound shades are generated where one of the two lights cannot be illuminated. But the intersections...of the compound shade are always generated by a single light source and the simple /shadows/ by two luminous bodies and the right intersection of the compound shade is generated by the left light and the left intersection is generated by the right light. But the two intersections of the simple shadows, both the upper as well as the lower one are generated by the two luminous sources: that is /both/ the right light and the left light.

(figure)Figs. 524-527: Demonstration concerning compound shade. Figs. 524-525, BM248v; fig. 526, BM243r; fig. 527, CA241rc.

This claim is, as usual, followed by a demonstration (figs. 519, 521, 523):

This is proved.

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Let...S be the intersection of the two columnar umbrous bodies af and bL and let the derived shadows, generated by such (shadows) umbrous bodies, be aa and ab generated by the two superior luminous sources a /and/ b,...and the same...2 luminous sources generate the other two derived shadows bb and ba. But each of the shadows which intersect one another do not originate from these two lights.

And this is demonstrated because, removing the light b, the shadow aa remains and the shadow ba intersected at the point x and hence these two shadows, through the removal of such a light, remain simple shadows, being generated by a single luminous source a...and, by the ninth of this, no...quantity of simple shadows generated by more...umbrous bodies,...by means of a single luminous source can generate...intersections among them, if the umbrous bodies, the causes of the shadows do not intersect among themselves.

The results and implications of these experiments concerning shadows at the outer intersections are now summarized (i.e. v and x in fig. 527):

Therefore we have demonstrated that the 2 lateral intersections do not generate any other darkness of shadow than that which is beyond such an intersection and this originates because the shadow is as diminished at such an intersection...in umbrous bodies as it is beyond this intersection because the shadow of the first umbrous body stamps itself on the back of the second umbrous body, which is in contact at the intersection, and for this reason it does not descend to the pavement...where it intersects the other derived shade. And if you make the cruciform figure without superimposing such an intersection then this cross will be diminished throughout and the derived shade will be diminished in its intersection and for this reason it cannot acquire darkness in its intersection, etc.

(figure)Figs. 528-532: Three light sources in front of a St. Andrew's cross producing six shadows. Figs. 528, 532, CA37va; figs. 529-530, CA177ve; fig. 531, BM243r.

In the final section of his demonstration he considers the shadows at the two inner intersections (r and t in fig. 527):

But the shadows bb and aa which are double in their intersections r /and/ t, being created by the two luminous sources a /and/ b, in their intersections lose the luminous sources and the remainder only loses one of the two luminous sources.... But the intersections v /and/ x do not lose any luminous source, because each of them always sees one of the luminous sources.

Having examined cases with one and two light sources the systematic Leonardo cannot resist studying the effects of three light sources positioned in front of a St. Andrew's cross. Preliminary sketches for this occur on CA177ve (figs. 529-530, 1505-1510), BM243r (fig. 531, 1508-1510) and CA37va (figs. 528, 532, 1508-1510). On CA229rb (1508-1510) he goes further. First he shows (fig. 533) how one light source in front of a St. Andrew's cross formation produces

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two intersecting shadows. Then he shows (fig. 534) how two light sources in front of the same cross produce four intersecting shadows.

(figure)Figs. 533-535: One, two, and three light sources in front of a St. Andrew's cross on CA229rb.

Finally he shows (fig. 535) how three light sources in front of such a cross produce six intersecting shadows. Alongside this he asks:

If three lights with two columnar umbrous bodies in their intersections produce...six shadows with 9 intersections, why is it that the first two exterior intersections do not double the darkness at q, in their intersection as do the other 7 intersections?

This he crosses out. In the next column he begins anew, ignoring this complex question and discussing instead a simpler case (fig. 534 cf. figs. 511-519) which he had dealt with previously:

On simple shadowThe light nm...is the cause that the columnar umbrous body os...makes the two

shadows of and og and the lights...m /and/ n do the same to the body hp in generating the shadows...ph and PK and for this reason, the simple derived shadows b /and/ c are generated in such intersections b/and/ c and not at the intersections d /and/ e because b /and/ c and not see either one of the two lights but d /and/ e see...these two lights..., that is, one sees the shadow og, and the other sees of which are two compound shadows. Therefore the intersection of such compound shadows sees...or rather is seen by two lights;...but...the two lights do not see this intersection, because such a shadow would be destroyed. But each shadow sees a single light in its entire length and each shadow, joined together with the other will not make simple shadow, because simple is that which does not see...any luminous source and which, in such...an intersection, each in itself only sees its own luminous body and the one can never see the luminous source of the other. It follows that such shadow remains without doubling its darkness and without being destroyed by being seen by the two lights.

This is clearly a restatement of his answer to an adversary on CA177rb (1508-1510) cited above. An understanding of these passages in turn renders intelligible other diagrams without text. On CA37ra and CA37va (1508-1510), for instance, he makes a series of four sketches (figs. 536-539) which show how two light sources in front of a St. Andrew's cross produce four shadows. In a preliminary sketch on CA37ra (fig. 528, 1508-1510) he shows how three light sources in front of such a cross produce six shadows. This idea he develops in a series of four sketches (figs. 532, 540, 542-543) without text on CA37va (1508-1510).

He also considers different numbers of light sources in front of a more complex object which he represents simply as four points (cf. fig. 514 where he represents a St. Andrew's cross as two points). On BM243r (1508-1510), for instance, he shows how two light sources (fig. 544) in front of such points produce eight shadows. On CA229rb (1508-1510) he illustrates how three light sources (fig. 545) in front of four points produce twelve shadows. In a further sketch on

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BM243r (1508-1510) he demonstrates (fig. 546) how four light sources in front of four points produce sixteen shadows.

On this same folio, BM243r he also considers the configurations of shade produced by three light sources in front of a s St. Andrew's cross coupled with a vertical pole to produce a figure . In one sketch (fig. 547) he marks the three light sources as the points a, b and c, draws in the figure , adds the resulting nine shadows and identifies those which have been produced by light sources a, b or c respectively. In a second sketch he draws (fig. 548) effectively the same situation, with the exception that the three light sources are no longer as far apart. Also on this folio are four preparatory sketches (figs. 549-552) in various degrees of completion.

(figure)Figs. 536-543: Demonstrations of shadows produced by a St. Andrew's cross and either two or three light sources. Fig.536, CA37ra; figs. 537-540, 542-543; fig. 541, CA177va. (figure)Figs. 544-546: Demonstrations involving three or four light sources. Fig. 544, BM243r; fig. 545, CA229rb; fig. 546, BM243r.

(figure)Figs. 547-552: Shadows produced by three light sources in front of a shape on BM243r.

The most basic of these (fig. 551) is, in turn, related to a sketch (fig. 553) on CA37ra (1508-1510). Here he shows how three light sources, in front of a figure, produce three shadows at the right-hand column. In a second sketch (fig. 554) he draws this situation again but for purposes of comparison he now shows how the other two columns would each produce two shadows if they had two light sources. In the next sketch (fig. 555) two columns have three light sources and only the column on the far left has two light sources. In a final sketch (fig. 556) all three columns have three light sources and therefore produce nine shadows. In this case, however, the light sources are positioned further off to the right.

Book Four. 10. Conclusion

Although the extant notes contain no concluding remarks concerning this fourth book, the chief ideas of this section can, nonetheless, be readily summarized. Leonardo has shown that derived shade varies (a) with the light source: its degree, angle, position, size and shape, (b) with the shape of the umbrous body and (c) with the shape of the projection plane on which the derived shade falls.

His attempts to understand how/why a single light source produces two shadows lead him to systematic studies concerning the properties of compound shade: i.e. situations in which one light source is in combination with one, two or three umbrous bodies and conversely where one umbrous body is in combination with one, two or three light sources.

(figure)Figs. 553-556: Further demonstrations relating to crosses and shadows.

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Equally systematic are his experiments with one, two, three or four light sources in front of columns in the form of a St. Andrew's cross, in order to determine at what points shadow becomes doubled. As will be seen (below pp. ) these experiments are, in turn, paralleled by others involving one, two, three or four pinhole apertures in a camera obscura. The more we penetrate his thought, the more systematic we find his approach to be.

BOOK FIVE

And since all round the derived shadows, where the derived shadows are intercepted, there is always a space where the light falls and by reflected dispersion is thrown back towards its cause, it meets the original shadow and mingles with it and modifies it somewhat in its nature. And on this I shall build the fifth book.(CA250 ra).

On the basis of this outline on CA250ra a tentative reconstruction of Leonardo's fifth book on light and shade can be suggested. It would probably have opened with basic propositions concerning reflect light and shade (Bk.V: Chapter 1). This might have been followed by notes on lustre (V:2) and elementary demonstrations of reflection (V:3). Separate chapters on reflection involving interposed rods (V:4) and interposed walls (V:5) could have followed. A series of theoretical demonstrations (V:6) might have ended this section. We shall examine each of these in turn.

Book Five. 1. Basic Propositions

Ludwig's edition of the Treatise of Painting contains two sections, one with seventeen propositions (TPL156-172), the other with eleven propositions (TPL780-790) devoted to problems of reflection in light and shade. Some of these pertain to books six and seven and will be discussed later. Others are of an introductory nature and concern us here (see Chart 13). On CU162 (TPL171, 1508-1510), for instance, Leonardo compares the nature of reflected and direct light in a passage entitled:

On the colours of reflections.All reflected colours are of less luminosity than direct light and incident light has as much proportion with reflected light, as is that /proportion/ which there is between the luminosity of their causes.

He is conscious that reflection is not always possible and on A94v (BN 2038 14v, CU158, TPL157, C.1492) describes these situations under the heading:

Where there can be no luminous reflectionAll dense objects invest their surfaces with various qualities of light and shade. Lights are of two natures....

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1. Straight vs. 2. Which Objects 3. Surface 4. Universal/ Reflected Light Participates Particular Light

171 156 162 157 164 158 166 168

169 170 781 782/784b 785 783/790

Book V V VI V

5. Role of 6. Role of 7. Reflected Distance Angle Light/Background

159 161 160 163

168 167 172

789 780 786 787

Book VII V V

Chart 13: The above provides a summary of the chief themes in the two sections on reflection in the Treatise of Painting (156-172, 780-790). The numbers refer to Ludwig's edition. TPL788 has not been included because it has nothing to do with reflection.

But to return to the promised definition. I say that luminous reflection is not from that part of the body which faces umbrous bodies, as are dark places, meadows of various herbs, green or dry trees which, even if part of each branch facing the original light is vested with the quality of this light, nonetheless, there are so many shadows made by each branch in itself, and so many shadows made by one branch on the other that there results in sum such a darkness that the light is as nothing. Hence objects such as this cannot give any reflected light to bodies /placed/ opposite /them/.

Those bodies in which reflection is possible he describes briefly on A94v (BN 2038 14v, CU157, TPL156, c.1492):

On ReflectionReflections are caused by bodies of bright and flat quality and semidense surfaces which,

percussed by light, like the bouncing of a ball, repercuss them at the first object.

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On CU167 (TPL158, 1505-1510) he mentions the nature of such reflecting bodies:

On reflectionsReflections participate that much more or less with the object where they are generated, than with the object which generates them, to the extent that the body where they are generated is of a more polished surface than that which generates them.

A series of five general rules, four of them numbered, on CU172, 170 (TPL168-169, 1505-1510) might also have formed part of this introductory chapter:

On reflections1. The surface of bodies participate more in the colours of those objects which refelct their similitudes in it at more angles.2. Of the colours of objects, which reflect their similitudes in the surfaces of the bodies positioned opposite at equal angles, that will be more powerful which will have its ray reflected a shorter length.3. Among colours of objects that are reflected at equal angles and at equal distance in the surfaces of the bodies positioned opposite, that one will be darker which is a brighter colour.4. That body reflects its colour more intensely onto the body positioned opposite, which does not have around it other colours than of its species.ReflectionBut that reflection will be of a more confused colour which is generated by various colours of objects.

In the late period there is a further elementary note on G11v (1510-1515) entitled:

On shade in bodies.When you draw dark shadows in umbrous bodies always draw the cause of such a darkness and you will do the same with the reflections because the dark shadows originate from dark objections and the reflections from objects of small brightness, that is, of diminished lights, and such is the proportion between the illuminated part and the part brightened by the reflection as there is between the cause of the light of this body and the cause of this reflection.

(figure)Figs. 557-559: Highlights and lustre on A113r, CU799 and H90/42/v.

In addition, a series of passages on reflections in relation to backgrounds (TPL160, 163, 167, 172, 780, 786, and 787), to be discussed later (see below pp. ), might have formed part of this opening chapter of book five.

Book Five. 2. Lustre

Lustre is one of the basic phenomena of reflected light. The Treatise of Painting contains a section of nine passages (TPL771-779) devoted to this theme. There are also other notes scattered

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throughout the manuscripts. It is likely that these would have served as basis of a second chapter on reflection. On A113r (BN 2038 32r, CU799, TPL746, 1492), for instance, Leonardo writes:

On the highlights which turn and move as the eye seeing this body is moved.Let us suppose that the said body is this round one drawn here on the side (fig. 557, cf. fig. 558) and that the light is the point a and that part of the illuminated body is bc and that the eye is at the point d. I say that the lustre, because it is all in all and all in a part, that standing at the point d, the lustre will appear at the point c and to the extent that the eye f moves from d to a to that extent will this lustre move from c to n.

On H90/42/v (1492) he broaches the topic again: “The lights of lights, that is the lustre of some object will not be situated in the middle of the illuminated part. But will make as many mutations as the eye regarding this.” Definitions of lustre follow on CU774 (TPL775, 1508-1510) and E31v (CU780, TPL776, 151301514, see above pp. ). On E31v (CU772, TPL777, 1513-1514) he also asks:

Which bodies are those which have light without lustre?

Opaque bodies which have dense and round surfaces will never generate lustre in any illuminated part of them.

Immediately following on E31v (CU781, TPL778, 1513-1514) he asks the converse:

Which bodies are those which have lustre and not a luminous part?

Dense opaque bodies with dense surfaces are those which have all the lustre in as many places of the illuminated part as are the sites which can receive the angle of incidence of the light and of the eye, but because such a surface mirrors all the things surrounding it, the illuminated object will not be recognized in that part of the illuminated body.

On CU776, (TPL774, 1508-1510) he makes notes

On the size of lustres on their terse bodies.Of lustres generated on spheres equidistant from the eye, that will be of a smaller shape which is generated on a sphere of less size.This is seen in the little grains of mercury which are a nearly imperceptible size and their lustres are equal to the size of these grains and this arises because the visual power of the pupil is greater than this little grain and for this reason it surrounds it as was said.

A diverse series of notes follow on TPL779 (Mad.II 26r, 1503-1504) under the heading:

Of lustre.

Lustre participates more in the colour of the light that illuminates the body which is lustrous than in the colour of this body and this arise in terse surfaces.

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The lustre of many umbrous bodies is integrally of the colour of the illuminated body as is that of bronzed gold and silver and other metals and similar bodies.

The lustre of foliage, glass and jewels participate little in the colour of the object where they originate and considerably in the colour of the body that illuminates them.

The lustre made in the depth of dense transparent objects are in the first degree of beauty of such a colour as is seen in a pale red ruby, glass and similar things. This occurs because between the eye and this lustre there is interposed all the natural colour of the transparent body.

The reflected lights of dense and lustrous bodies are of much more beauty than the natural colour of these bodies as is seen in the pleats that occur in gold that is filigreed and in other similar bodies where the one surface reflects on the other positioned opposite it and the other reflects onto it and thus they do successively ad infinitum.

No lustrous and transparent body can show on it the shadows received by any object, as is seen in the shadows of bridges of revers which are never seen except in turbid waters, and in clear /waters/ they do not appear.

Lustre is found in as many sites as the places where it is seen are various.

The eye and the object standing /still/ without motion, the lustre will move on the object together with the light which causes it. The light and the object standing still, the lustre will move on the object together with the motion of the eye that sees it.

Lustre originates on the polished surfaces of a body that takes more light the more it is dense and polished.

This section of the Treatise of Painting also contains three further passages (L771-773) concerning lustre in relation to background (see below pp. ). He returns to this theme of lustre briefly in the Manuscript G (1510-1515) where he gives a further definition on G24r (see above pp. ) and on G3v notes:

The shadows that are in the transparent leaves seen from the reverse are directed by this foliage which is transparent from the reverse side along with the luminous part, but the lustre cannot be transparent.

Book Five. 3. Elementary Demonstrations

A series of basic demonstrations concerning the nature and effects of reflection, scattered through the notebooks, might well have been intended as the basis of a third chapter. In an early passage on C25r (1490-1491), for example, he compares given perceptual effects with those of reflections mirrored in water (fig. 560):

(figure)

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Figs. 560-562: Elementary definitions of reflection on C25r, CU212, and CU545.

Shade and LightIf the visual line which sees the shade by the light of the candle is at an equal angle to that...of the shade, it will appear to make shade under the body which causes it like that made by the images of bodies mirrored in the water, which appear to be that much below to the extent that they are above, and thus this shadow does the same such that it appears with its limit to be that much below the plane where it is generated, as the summit of the body which generates it is above this plane, as appears to be on the wall that cb, shade, is as much as ab and that cb stands under ab.Reflection in water interests him and becomes the subject of two further propositions. One,

on CU212 (fig. 561, TPL227, 1505-1510) is entitled:

Of things mirrored in the water of countrysides and first of the air.

That air alone will be that which gives of itself an image in the surface of the water, which reflects from the surface of the water to the eye at equal angles, that is, that the angle of incidence is equal to the angle of reflection.

A second passage on reflections in water follows on CU545 (fig. 562, TPL505, 1510-1515) under the heading:On the shadows made by bridges on their water.

The shadows of bridges will never be seen on their waters if the water does not lose its function of mirroring as a result of turbulence. And this is proved because clear water,...with a lustrous and polished surface, mirrors the bridge in all the places interposed at equal angles between the eye and the bridge and mirrors the air below the bridge where the shade of this bridge is. Which turbid water cannot do because it does not mirror, but it /nonetheless/ receives shade well, as would a dusty street.

As early as 1490 he had become interested in the properties of light entering a narrow shaft. He makes preliminary sketches in this regard on c12r (1490-1491, fig. 563) and A91v (BN 2038 11v, fig. 564, 1492) without text. This leads to a further diagram on A92r (BN 2038 12r, fig. 565, 1492) again without text which shows light bouncing back and forth down into what is probably a well. Some fifteen years later he illustrates a related situation to make a somewhat different point on CU706, (fig. 566, TPL619, 1508-1510) in a passage entitled:

On derived shade generated in other derived shade.

The derived shade originating from the sun can be made on derived shade generated by the air.This is proved. And let the shade of the object m which is generated by the air ef be int he space dcb. And let it be that the object n, by means of the sun g, makes the shadow abc and from the remainder of the shdow dme, which in such a site neither sees the air ef nor the sun. Therefore it is double shade because it is generated by the two objects, that is n and m.

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On CU865 (TPL789, c.1510) again uses the example of a well to illustrate a problem of battle painting (fig. 567):

(figure)]Figs. 566-568: Reflected derived shade in an enclosed space or a well on CU706, 865 and 694.

Of the illumination of the lower parts of bodies which are close together as are men in a battle.

With men and horses engaged in battle, their parts will be that much darker to the extent that they are closer to the earth which sustains them. And this is proved by the walls of wells which are that much darker the deeper they are and this arises because the deepest part of the wells sees and is seen by a smaller part of the luminous air than any other part and the pavements, which are of the same colour as the legs of the aforesaid men and horses, are always more illuminated under equal angles than the other aforesaid legs.

The reflective nature of light in such a narrow shaft is again discussed on CU694 (TPL587, 1508-1510) in a passage headed (fig. 568):

On the simple shade of prime darkness.

Simple shade is that which cannot be seen by any reflected light but will only be augmented by a shadow opposite.

Let the spherical body be g put in the concavity bcef and let the particular light be a which percusses at b and reflects at d and rebounds with the second reflection onto the spherical body g which, on the one side has simple shade at the angle e, which does not see either the incident light or the light reflected at any degree of reflection. Therefore the shadow of the sphere receives the reflection of the simple shade ue and for this reason it is called simple shade.

A related situation of light entering a cave is described on Forst I 10r (fig. 569, c.1505):

(figure)Figs. 569-570)Further cases of reflection on Forst I 10r and CU160.

Abcd is a cave which is opened on two fronts and has particular light more powerful than the reflected light rs by 2/3. Therefore the particular light will penetrate 2/3 of the cave and the reflected light rs will take the third nd.

The connection between brightness of reflection and angle of reflection, raised above in connection with a well on CU865 (TPL789) is further explored in a passage on CU160 (TPL161, 1505-1510) entitled:

What part of the reflection will be brighter

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That part of a reflection will be more illuminated which receives its light under more equal angles /both/ from the luminous body as well as from its percussion.

A demonstration follows by way of support (fig. 570):

This is proved. And let the luminous body be n and let ab be the part of the illuminated body which rebounds through all the concavity opposite which is shady. And let it be that the light reflected at e is percussed at equal angles. And below this it will not be reflected at equal angles as the angle cab shows which is more obtuse than the angle eba. But the angles afb, even if it is among angles of less equality than the angle e, this has the base ab which has angles more equal than this angle e, this has the base ab which has angles more equal than this angle c. And hence it is brighter at f than at e. And that which is closer to the thing which illuminates it, will also be brighter, by the sixth which states: that part of the umbrous body is more illuminated which is closer to its luminous body.

Book Five. 4. Interposed Rods

One step more complex are his demonstrations of reflected light and shade involving an interposed rod as on C5r (fig. 571, 1490-1491, cf. fig. ) beneath which he writes: "The more that derived shade approaches its penultimate extremities to that extent will the darkness appear greater." Towards the end of the section that follows he reminds himself: "And this wishes to be at the beginning of the demonstration:"

Let Ab be part of the primitive shade. Let bc be the...primitive light. Let d be the place of intersection. Let the derived shade be fg. Let fe be the derived light.

This introduces the demonstration proper:

Behind the intersection gz is only seen by the part of the shade;...through the intersection yz takes the shade mn and the shade am directly, whence it has two times as much shade as gz. Through the intersection yx sees no and nma directly whence xy demonstrates itself as having three times as much shade as zg. Through the intersection xf sees ob and it sees onma directly whence we shall say that the shade in between f/and/ x is four times darker than the shade zg because it is seen by four times as much shade.

On CA357rb (c.1490) he drafts a series of similar sketches (figs. 577-584) accompanying which he writes: "That part of an illuminated place will be more luminous where the rays concur with a greater angle." This idea he develops on CA31vb (c.1495) where he redraws the diagram (fig. 572, cf. fig. 571) claiming:

That place is darker which is seen by a greater sum of umbrous rays.That place which is percussed by a greater angle of umbrous rays will be darker.a is twice as dark as b because it arises from a double base at an equal distance.

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(figure)Figs. 571-576: Interposed rods and reflected light. Fig. 571, C5r; fig. 572, C31vb; figs. 573-574, CA144va; figs. 575-576, W12352v.

(figure)Figs. 577-584: Preliminary sketches involving interposed rods and reflected light on CA357rb.

(figure)Figs. 585-590: Further cases of interposed rods and reflected light.Figs. 585-586, CA357rb; fig. 587, CA31vb; fig. 588, CA18rab; figs. 589-590, CA224rb.

Directly following he demonstrates the converse:

That place will be more luminous which is repercussed by a greater sum of luminous rays.D is the beginning of the shade df and it tinges a little at c; de is the middle shade df and it tinges more at the percussion b. Df is an entirely umbrous interval and it tinges the place a entirely with itself.

At the top of this folio is a related phrase: "...is tinted...by the brightness or darkness of the umbrous and luminous bodies placed opposite." Within the next two decades he returns to this diagram, now without text on several occasions: CA144va (figs. 573-574, c.1492); W12352v (figs. 575-576, C.1494); CA18rab (fig. 588, 1500-1504) and CA224rb (figs. 589-590)

Related to these demonstrations is a more complex series involving the modification of shadow through light that has been reflected backwards. On C5r (1490-1491), for instance, he draws (fig. 593, cf. figs. 591-592) a light source K casting three rays which are then reflected backwards from the ground bf. Above this diagram he notes: "That luminous body will appear...brighter...which is surrounded by darker shadows." Beneath the diagram is a longer passage based on drafts on CA144va (1490, cf. p. 1492):

CA144va C5rThe umbrous and luminous quantity,even though it is reduced to/being/ small as a result offoreshortening, does not diminishin brightness or darkness.

The width and length of shade orlight, even though their fore-shortenings appear narrower orshorter will neither diminish norgrow in its brightness or darkness.

The length and width of shade andlight even though it appears of less

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quantity through foreshortening willnonetheless not appear diminished inthe quality...of brightness ordarkness.

The width and length of shade or The width and length of shade andlight, even though they are light, even though it makes itselfnarrower or shorter through... narrower and shorter through fore-foreshortenings, nevertheless, shortening does not diminish orwill neither diminish nor augment the quality and quantityincrease the quality and of its brightness.quantity of its brightnessor darkness.

But the function of such light The function of shade and lightdiminished by foreshortening diminished by foreshortening willwill be to illuminated or to be to shade and...to illuminate theobscure the counterposed object body positioned opposite dependingin that quantity and quality on the quality and quantity thatwhich appears from that body. appears in this body.

(figure)Figs. 591-594: Interposed objects and reflected light and shade. Figs. 591-592, CA144va; fig. 593, C5r; fig. 594, C21r.

On C21r (1490-1491) he draws a related figure (fig. 594). A spherical light source now casts twelve rays onto the ground which reflect backwards and modify the nature of the shade produced by an opaque body. Accompanying this is the claim:

That part of the derived shade will be darker which is closer to its source.The /phenomenon in the/ above proposition occurs where a greater luminous angle is joined with a thinner...umbrous body. This luminous body /then/ overcomes it and effectively converts it into its luminous nature. And likewise it is proposed that where the greatest umbrous angle joins itself with the thinnest luminous /body/, the umbrous body will effectively convert the adjacent luminous body into its nature.

This is followed by a demonstration:

At h the angles of the umbrous and luminous pyramids are joined. The umbrous is mnh. The luminous is oph. The umbrous [pyramid] is the smallest that there is among bcdefgh born of mn and again at h is the largest among the luminous pyramids bcdefgh born at op.

Which leads him to conclude:

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The greatest luminous body always has as its companion the least umbrous angle and similarly you will find that the largest umbrous angle always borders with the smallest umbrous body.

He considers a more complex situation on C8v (1490-1491) under the heading:

That part of a wall will be darker or more luminous which is obscured or illuminated by a greater dark or luminous angle...

(figure)Figs. 595-596: Reflected light and shade on C8v and C4r.

Beneath this he draws (fig. 595) a luminous sphere from which emanate eight rays that are reflected on the wall ae and modify the shadows produced by the interposed plane fg. As he explains:

The above mentioned proposition is clearly proved in this way. Let us say that mq is the luminous body and so fg will be the umbrous body and let ae be the said wall on which the above mentioned angles percuss representing there the nature and quality of their bases. Now a is more luminous...than b because the base of the angle a is larger than that of b and therefore it makes a greater angle...which is amq. And the pyramid bpm is narrower and moe will be thinner /still/ and so on, step by step, the more one approaches e, the more the pyramids are thinner and darker.

By way of a corollary he claims:

That point of the wall will be of less brightness (d) in which...the size of the umbrous pyramid is greater than the size of the luminous /pyramids).

This he demonstrates:

At the point a the luminous pyramid will be of as much power as the umbrous /one/ because the base fg is similar to the base rf. And at the point d the luminous pyramid is that much thinner than the umbrous one to the extent that the base sf is less than the base fg.

As an afterthought he adds:

Divide the above mentioned proposition into two figures, that is, one with umbrous and luminous pyramids and the other with the luminous /ones/.

Closely related is another diagram on C4r (1490-1491) beneath which (fig. 596) he again begins with a general claim:

The larger the luminous body, the more the course of the umbrous and luminous rays will be mixed together. The effect of the above mentioned proposition occurs because where

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one finds that there is a greater sum of luminous rays there is greater light and where there is less light, results where the umbrous bodies come to mix together.

Which leads, as usual, to a demonstration:

M sees and is seen by the entire luminous body ag whence we shall say that among the percussions of luminous rays from m /to/ s, m holds the principle degree of light, n sees af which are 5/6 of the light; o sees ae which are 2/3 of the light; p sees ad which is half the light; q sees ac which is a third, re is seen by ab that is a 1/6 of the light; s sees a the limit of the light and here begins the real and simple shadow.

And this results because the points of the luminous pyramids mnopqrs which are born on the luminous body ag, to the extent that they are narrower, are derived from a lesser base and they make less light in equal distance.

Book Five. 5. Interposed Walls

The reflections of light and shade produced by objects of various shapes positioned in front of a wall also interest him considerably. On C4v (fig. 599, 1490-1491), for instance, he notes that:

The concourse of shade born and terminated between near and plane surfaces of the same quality and directly opposite will have a darker end than beginning which will terminate in the percussion of luminous rays.

On CA144va (1490? Pedretti claims C.1492) he sketches what is probably a draft (fig. 597) for the above diagram. A slight variant of this diagram (fig. 598) recurs over fifteen years later on CU712 (TPL602, 1508-1510) where he describes:

How the derived shade being surrounded either entirely or in part by an illuminated background, is darker than the primitive /shade/.Derived shade which is all or partly surrounded by a luminous body will always be darker than the primitive shade which is on a plane surface.

This he demonstrates:

Let a be the light and let bc be the object which retains the primitive shade and let the wall de be that which receives the derived shade in the part nm and a remainder of it, dn and me will remain illumined by a. And the light dn is reflected in the primitive light bc and the light me does the same. Therefore the derived shade nm, not seeing the light a, remains dark and the primitive /shade/ is illuminated by the illuminated background which surrounds the derived shade. And hence the derived shade is darker than the primitive shade.

(figure)Figs. 597-601: Interposed walls and reflected light and shade. Fig. 597, CA144va; fig. 598, CU712; fig. 599, C4v; fig. 600, CU713; fig. 601, CU715.

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(figure)Figs. 602-605: Spheres, interposed walls and reflected light and shade. Fig. 602, C17r; figs. 603-604, C16v; fig. 605, C20v.

On CU715 (fig. 601, TPL580, 1508-1510) he considers what happens if the interposed plane is tilted, asking:

What is augmented shade?Augmented shade is that in which only its derived shade is reflected.Let a be the luminous body. et bc be the primitive or original shade and dg will be the originated shade.

What happens when this flat interposed plane is substituted by a spherical object he considers on CU713 (fig. 600, TPL603, 1508-1510):

How the primitive light which is not joined with a flat surface will not be of equal darkness.This is proved: and let bcd be the primitive shade joined to the object in which is seen the derived shade fg and again one sees its illuminated background ef /and/ gh. I say that such a body will be more illuminated at the extremity b than at the middle d, because at b the primitive light a is seen and the derived light fg is not joined because fbd is the angle of contingence made by the straight line fb and by the curve bd. And all the remainder of such a body is seen by the derived shade fg, more or less according to whether the line fg can be made lower with a triangle with a greater or lesser angle.

The characteristics of reflected shade produced by a spherical body in front of a wall had concerned him at some length in the Manuscript C. On C17r, for instance, he makes a preliminary drawing (fig. 602), which he then develops on C16v (figs. 603=604, 1490-1491). Between these diagrams on C16v he adds a marginal note:

The rays doubled through intersection in lights and shadows are also of double brightness and obscurity.The umbrous part of this superior body is brighter in mhn than in tqp because in this part of the two reflected derived lights intersect, that is, ab and dc, as appears in the triangle mno and at tm one will only see ab and not dc.

In the main body of the text he pursues his ideas on reflected shade in a "proposition" which he had drafted on CA144va (c.1490? Pedretti claims c.1492):

CA144va C16vThe primitive and derived reflection The primitive and derivedsurrounding dense and spherical reflected light surroundingbodies will make that the boundaries dense and spherical bodiesof primitive shade...of this body will be the cause that theare that much better understood boundaries of the primitivefrom the one extremity than from shade are that much more

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the other to the extent that the distinct and bounded with itsprimitive light is brighter than nearby illuminated part, tothe reflected. the extent that the derived

light is brighter than thederived shade.

This proposition leads to a "comment":

That is said to be primitive light which illuminates bodies primarily and derived shade is said to be that which rebounds from these bodies in those parts which are distant from this primitive light.

Which then introduces a (fig. 603):

DemonstrationLet k be the primitive light which illuminates the umbrous body at tp and the places ab/and/ cd. From ab /and/ cd there departs derived shade and it rebounds onto the body opposite at mn and the entire part of the body at h will be more luminous than at q because it is seen by a double light, that is ab and dc. Whence q is not seen except by simple light and therefore it is dark.

Immediately following he adds another proposition and commentary:

PropositionThat part of primitive shade is more luminous which can see the middles of the derived lights equally.CommentaryIt can clearly be recognized that that part of umbrous bodies seen by a greater quantity of light is more luminous and maximally if that part is illuminated by two lights as is seen in reflected lights which, put in the middle of themselves the derived shadow made through them by the dense bodies opposite.

This is again followed by a (fig. 604):

DemonstrationLet n be the part of the body which is more luminous than any other in this (body) because it is seen equally by the first 2 powers of the lights positioned opposite it, that is b, greater than the power of the light ac and similarly e is greater than df and all the 2 see (in) the said ndc and similarly af because they are extremities, only less powerful and these see the body in ro and ut, this place being seen by less light. Darker is the part illuminated by these, the entire triangle is seen by double lights of various qualities of brightness.

This leads to a third: "proposition": again based on a draft on CA144va:

(figure)Figs. 606-607: Reflected light and shade on C16v and CA144vb.

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CA144va C16vEvery luminous body with all of it Every luminous body with alland with its part illuminates the and with part of itself part and all of the object. illuminates the part and all

of the object positionedopposite it.

This is followed by a

DefinitionThis proposition is fairly obvious and that which cannot be denied (denied) is that where the entire pupil of the eye looks at this place it does not look at this place (its). Every part of this is a place seen by this pupil /and it/ does the same towards the pupil /i.e. every part of this place also sees the pupil/.

Which claim is again supported (fig. 606 cf. 607) by a:

DemonstrationLet ac be the luminous body. Let df be the illuminated object which, even though it is composed of infinite points, we shall only make a test of three, that is d, e /and f. Now you see that e is seen by the part b of the luminous body and by all of ac as is demonstrated by the lines ae and ce and by the centric line be and also at the point d, all of ac is seen, and the centre ube and you will find the same at f and the same occurs throughout all the parts of the object df.

On C20v he develops these diagrams (fig. 605, 1490-1491), this time under the heading:

To the extent that the derived light has less light than the original /light/ to that extent will its pyramids illuminate the place percussed by them.

(figure)Figs. 606-614: The development of an idea. Figs. 608-609, CA144rb; fig. 610, CA144va; fig. 611, CA144vb; fig. 612, CA144ra; fig. 613, CA144va; fig. 614-616, CA144rb.

This idea he reformulates beneath the diagram: "Pyramids illuminate the place percussed by them less to the extent that the angles of these are thinner." This is one of the rare cases where a proposition is not followed by a demonstration.

On CA144vb (c.1490? cf. P.1492) he drafts a related diagram (fig. 606) which he develops on C16v (fig. 607, 1490-1491). Alongside the diagram on CA144vb, he writes "The boundaries of." This he crosses out and begins again. "The boundaries of shade of spherical bodies are." This he crosses out also and begins afresh. "The boundaries of shade on dense and spherical bodies are that much more kn/own.../ and the surroundings than of the reflection." Still unsatisfied he crosses this out too and finally drafts a coherent passage:

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The primitive and derived light, surrounding the dense and spherical bodies will have the effect...that the boundaries of the primitive shade...of this body is that much more known from one extremity than from the other to the extent that the primitive light is brighter than the reflected.

The problem of varying degrees of light continues to trouble him. On CA144ra, rb, va, vb (figs. 604-614, 1490? Pedretti claims 1492) he drafts a further series of diagrams one of which is accompanied by a rough text:

All the lines of shadows are straight, because...among the luminous lines even from r...ts is more than another luminous part...of the half triangle lt because it is seen...no straight and from og and cf one...on which are 2/8 of all the li/ght/...ls is...this same luminous body because it is se/en/...by nf, cg straight and from goec...above, which are again 2/8 of the whole....Whence we shall say that there is less light...the triangle __x and beyond the shadows 3/8 exactly.

(figure)

Figs. 617-618: Demonstrations of reflect mixed and reflected light and shade on C3r and C4r.

Beneath this he drafts another section which he later crosses out:

Ts is seen by the whole of no directly and by...ogef inverted, but by the same...is from cf and for this /reason/ it is illuminated more than the other part of the half triangle because it is seen by 5/8 of the light.S is seen by gn and df.

The diagram on CA144rb he develops on C3r (fig. 617, 1490-1491) which, as he explains, involves an astronomical phenomenon that had been under discussion in the optical tradition:

That body will appear less bright which is surrounded by a more luminous background.I have found that those stars which are closer to the horizon appear of a greater size than the others because these see and are seen by a greater sum of the solar body than when they are seen above us. And by seeing more sun they have more light and the body which will be more luminous will show itself as larger as the sun above us demonstrates in fog, which seems greater without fog and in fog diminishes.

This diagram on C3r is a record of what happens when a luminous body casts light on an opaque body in the open air. On C4v (fig. 618) he adds an interposed wall which causes a complex array of reflected light and shade. Here he develops the ideas he had drafted on cA144rb.

That part of the surface that is percussed by a greater angle by the species of the objects positioned opposite will tinge itself more in the colour of these.8 below is a greater angle than 4 since its base an is greater than its base en by 4.This figure below wishes to be terminated by an and 4 and 8.

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Beneath the diagram is a further proposition:

That part of the illuminated body which surrounds the percussion of the shadow is more luminous which is closer to the percussion.

Let 4 be the part of the illuminated object 4, 8 which surrounds...the percussion of shade 9/?/ and 4 and this place 4 is more luminous because it sees a lesser sum of shade than does the place 8 because 4 only sees the shade in and 8 sees and is percussed by the shade ae and by the shade in which is 2 times as dark and this same occurs when you put the air with sun in a place with light and shade.

Book Five. 6. Theoretical Demonstrations

Among the most fascinating aspects of Leonardo's approach is the way in which he returns to problems at various levels of abstraction. Hence, having provided concrete demonstrations he may well go on to give more abstract geometrical illustrations of the same principle. We have seen, for example, how he discussed the quality of darkness in experimental terms (see above pp. ). On CU689 (fig. 619, TPL641, 150-8-1510) he gives a geometrical demonstration:

Of the quality of darkness of shadows.The darkness of derived shadows is infinitely variable with so much greater or less power as the distances in which the percussions of the derived shadows are caused are greater or less. Percussions of the derived shadows are caused are greater or less.

(figure)Figs. 619-621: Theoretical demonstrations of light and shade on CU689, 681 and 682.

This is proved. And let the sun be a which generates the shadow nphi in which enters the light of the air which surrounds the solar rays, that is, eb/to/ rs above and /from/ fc to rs below, and it will brighten this shade which is most dark in the space npo where it sees neither sun nor air nor its extremities b /or/ c.

On CU723 (fig. 630, TPL783, 1508-1510) he considers the problem in semi-abstract terms:

How reflection is generated in universal lights.Reflection is generated in bodies illuminated by universal lights when a part of the illuminated body reflects its larger light in that place where a smaller part of the same light sees: As /is the case when/ the sky ef, seeing the place d, and a greater part of the same sky sees h, then the derived light h will reflect in d.But of this a separate treatise will be made at the appropriate place.

Although there is no evidence of the intended treatise itself, he clearly pursued the problem. On CA207ra (1508-1510) he notes in passing "the consummation of shadows by degrees in universal light," and on CU681 (TPL643, 1508-1510) he provides a fully abstract geometrical demonstration of reflection in universal lights (fig. 620) under the heading:

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Precept of painting.In universal lights, shadows occupy little space in the surfaces of their bodies. And this arises because a great sum of the light of our hemisphere tinges as far as the lowest parts of umbrous bodies if it is not impeded with its horizon and maximally if it is suspended from the earth.Let f be the umbrous body, e the earth, abcd is our hemisphere from which the darkness of the earth vx obscures that much of the umbrous body, as it sees; as the horizon which sees the same parts illuminates the same places and confounds the umbrous species of the aforesaid earth, which will be in a position to make such dark shadows under the object were it not impeded.

This problem preoccupies him. On CA207ra (1508-1510) he drafts another version:

...part of the bod/y/...It follows that...of the umbrous body will be more illuminated which sees a greater sum of the body that illuminates it.

This he develops on CU682 (fig. 621, TPL681, 1508-1510) in a passage entitled:

Of the universal light of the air where the sun does not percuss.

That thing will show itself as more illuminated which is seen by a greater quantity of the luminous body. By that which has been said e will be more illuminated than a because e sees a greater amount of sky, seeing rs which a does not see, it seeing only the sky bcd.

On CU722 (TPL782, 1508-1510) he pursues this geometrical analysis of conditions in universal light, beginning with a general statement under the heading:

Why reflections are little seen or not at all in universal lights.

The reflections of umbrous bodies are little seen or not at all in universal lights. And this occurs because such a universal light sufficiently surrounds and covers each of these bodies, the surface of which, as has been proved, participates in the colour of its objects.

(figure)Figs. 622-626: Abstract geometrical demonstrations of reflected light and shade. Fig. 622, CA207ra; fig. 723, CU675; fig. 624, CA207ra; fig. 625, CU690; fig. 626, G3v.

By way of illustration he carefully describes a geometrical diagram (fig. 632):

As would be /the case/ if the body a were illuminated by its hemisphere gcd and shaded by the earth gfd. Here the surface of such a body is illuminated and shaded by the air and by the earth which stands as its object and it is more or less illuminated or shaded to the extent that it is seen by a lesser or greater amount of a luminous or dark body. As is seen at the point k /which/ is seen by the entire part hci of the hemisphere and is not seen by any part of the darkness of the earth. Therefore it follows that k is more illuminated than a where only

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the part cd of the hemisphere is seen and such an illumination is corrupted by the darkness of the earth rd, all of which sees and is seen by the point a, as is proved in perspective. And if we wish to speak of the point b, we shall find that this is less illuminated than the point a, such that this b sees half of the hemisphere that a sees; that is, /a/ sees all of cd and b only sees ed, which is half of cd. And it sees all the darkness of the earth which a sees, that is, the earth rd and to this is added the part rf which is darker, because in this, the light of the hemisphere ec is lacking, which is not lacking to the earth rd.

Which leads him to conclude:

Therefore, for such a reason, this body cannot have reflection, because the reflection of light is after the principal shade of the bodies and here the principal shadow is in the point where such a body is in contact with the plane of the earth, which is why it is entirely deprived of light.

(figure)Figs. 627-632: Further abstract demonstrations of reflections. Fig. 627, CU676; fig. 628, CU207ra; fig. 629, CU677; fig. 630, CU723; fig. 631, CA207ra; fig. 632, CU722.

The mode of analysis here used - a geometrical sphere surrounded by a hemisphere -appeals to him and becomes the starting point for no less than ten further demonstrations. On CU687 (TPL736, 1508-1510), for instance, he considers which part of an opaque spherical body will be darker (see above pp. ) under the heading:

Of the shadow of the opaque spherical body positioned in the air.That part of the opaque spherical body will be darker which is seen by a greater sum of darkness.

He now describes the geometrical diagram (fig. 641, cf. fig. 640):

Let the dark object be the plane dc and let the luminous hemisphere be dnc, and let the spherical body interposed between the light of the hemisphere and the darkness of the earth be bqop. I say that the part oqp is more obscure than any part of such a spherical body because it alone sees all the sides of the darkness of the earth dc opposite. And every other side of it sees less.

Directly following is a proof:

This is proved by one of the elements which states: the line produced from the centre of the circle to the angle of contingence will be perpendicular and will fall between two right-angles. It follows that the line which comes from the centre x of the sphere, terminates at sc under right-angles at the point o /and/ sees all the darkness of the earth dc and likewise such an o is seen by this earth. And p opposite does the same for the same reasons.

And likewise q and every part which puts itself between in the space op. But q is of more excellent darkness being in the middle over the earth which o or p is not, being closer to the

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extremities of such darkness of the earth and begin to see the horizon of this hemisphere and mix themselves with its light.

As a corollary to this proposition, he considers the case of an opaque body on the ground, first in draft form on CA207ra (1508-1510) and then on CU688 (TPL737, 1508-1510):

CA207ra TPL737On the shadow of the opaquespherical body positioned over theearth

But the shadow of the opaque But the shadow of the opaquebody...darker than that of the spherical body positioned in contactopaque...in the air, because, with the earth will be of greaterother than receiving the... darkness than the foregoing, whichearth positioned opposite it, only sees it as its object.it also receives...that whichmakes it above this earth.

This claim on CU688 (TPL737) is again supported by a geometrical demonstration (fig. 633):

This is proved and let the opaque spherical body be nms positioned over the earth ac of the point s and the arc abc.

And the same as was said above is confirmed in a body illuminated in our hemisphere and here it manifests itself in the part of the spherical body under the phemisphere k and f which, in the point b, is illuminated by all the part aec and in the part rd by the hemisphere ef and in o by gf and in n by mf and in h by sf. And thus you have understood where the first light and the first shadow in a body is.

This idea he reformulates in the next passage CU676 (fig. 627, TPL694a, 1508-1510):

That part of an umbrous body will be more luminous which is illuminated by a greater sum of light.

Hence, placing the body abc as umbrous body and dfn as luminous body, that is the illuminated hemisphere, in the part c there is twice as much light as in the part b, and 3/4 more than in a, because c is illuminated by the sky dgfe and b by the sky df which is the half less than de and the part a is only illuminated by the quarter part of de, that is, by gd.

This passage is followed, in turn, by a geometrical demonstration (CU677, TPL694b, 1508-1510) of another principle that interests him (see below pp. ):

The surface of every body participates in the colour of its object.Let d be the opaque body. Let an be the luminous body, let ac be a body of a dark colour, let cd be the illuminated plane of the hemisphere afmn. By the aforesaid, r will be more

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illuminated than o; o than s; s than t and the parts which are facing the dark body, ac, will do the same as will those which are facing the illuminated place cd and from this originate light and shade and reflected light.

The accompanying diagram (fig. 629) may be based on the draft on CA207ra (fig. 628, 1508-1510) beneath which he notes: "n does not make shadow on the earth." In this series one demonstration builds on the other in the manner of a proposition in Euclidean geometry. Leonardo is set on translating his experimental results into a systematic geometrical language. To this end he makes further drafts on CA207ra (1508-1510) which lead to another series of demonstrations on CU690, 679, 686 (TPL748-750, 1508-1510). On CU690 (TPL748b, 1508-1510) he begins by raising a question which he had already answered elsewhere in concrete terms (see above pp. ): "Which part of the spherical body is less illuminated?". His preliminary answer on CU690 (TPL748b) is again based on drafts on CA207va (1508-1510):

CA207va TPL748bThat umbrous body will have alesser...quantity of itselfilluminated which part...is seenby a smaller...quantity of theluminous body.

That part of the umbrous body will That part of the umbrous bodybe that much less illuminated will be less illuminated whichwhich sees a smaller part of the is seen by a smaller part ofbody which illuminates it. the luminous body.

This is followed by a geometrical demonstration (fig. 625 cf. fig. 624):

This is proved. And let the umbrous body be asqr and let the hemisphere of the luminous source be ncedf. I say that the part a and the part o, being seen by equal arcs bced and cdef, are seen by an equal quality of light and for this reason are equally illuminated by these. But r, seen by a smaller arc edf, receives less light and p which only sees df/which/ is less than edf and for this /reason/ it remains less luminous. And q also remains less luminous which sees only the extremity of the horizon f.

On CU679 (TPL749, 1508-1510) he poses the converse question: "Which part of the spherical body is more illuminated?". The general claim that follows is again based on an earlier draft:

CA207ra TPL749And that part of spherical bodies And that part which iswhich is illuminated will be of a illuminated by sphericalgreater brightness, than that bodies will be of a morewhich is accompanied by a lesser intense brightness which hassum of umbrous species. a smaller sum of umbrous

species accompanying it.

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His claim is again supported by a geometrical demonstration:

This is proved. And let fno be the spherical umbrous body and abc is the luminous hemisphere and the plane ac is the darkness of the earth. Therefore I say that the part of the sphere, fn will be of a more intense brightness because it does not see any part of the earth ac and it is, in itself, of equal brightness, being illuminated by the equal arcs of the hemisphere abc, that is the arc are is equal to the arc rbc and the arc gsc and by a proposition (concettione) that states when two things are equal to a 3rd they are equal among themselves. Therefore p, f and n are equal in brightness.

He returns to the question he had asked on CU690 (TPL748a) in the next proposition in the treatise of painting, i.e. CU686 (TPL656):

Which part of the opaque body is less illuminated?That part of the opaque spherical body is of darker shadow which is seen by a lesser sum of luminous rays.

He is conscious that he has already dealt with the problem. What challenges him is the idea of an alternative proof:

Even though this has great resemblance with the 1st above, I will not be content unless I prove it, because this proof is somewhat different.

A geometrical proof follows (fig. 634, cf.633):

And let the umbrous body be fno and the hemisphere is abc and the darkness of the earth is the line ac. I say firstly that the superior part of the spherical body fpm will be equally illuminated by all the hemisphere abc and likewise I demonstrate it for the three given equal portions, that is are which illuminates the point f and rbs which illuminates p and gsc which illuminates n. Therefore by the 7th of the 9th it is concluded that fpn, the superior part of the spherical body is of equal brightness. Which 7th of the 9th states that all the parts of bodies which are illuminated equidistantly be equal and similar lights will, by necessity always be of equal brightness, which condition occurs at fpn.

One alternative demonstration is not enough (fig. 636, cf.fig. 635):

There follows a second demonstration.Let abc be the umbrous spherical body. Let dfe be the illuminating hemisphere. D is the earth which causes shadow. I say by the foregoing /proposition/, that the entire part anb of the sphere is deprived of shadow because it is not seen by the darkness of the earth and all the remainder of the surface of such a sphere is umbrous with more or less darkness, depending on whether a greater or lesser sum of darkness of the earth accompanies a greater or lesser sum of darkness of the hemisphere. Therefore, the point c, which sees a lesser sum of such a hemisphere and a greater sum of earth will be darker than any other part of the

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shadow, that is, it only sees rd and se of the hemisphere and it sees all the earth de. And the brightest is ab because it only sees the extremities of the earth d, e.

On CA207ra (1508-1510) he makes further drafts:

That part of a body illuminates.The illuminated part of a spherical body will be of that much less...shape,...to the extent that it will be seen by a smaller amount of the...luminous body.That part which...is illuminated by some spherical body,...will be less..., to the extent that it is seen by a lesser sum of the luminous body....That part which is illuminated by some spherical...will be that much less to the extent that it is seen by a smaller light.That p/art/ which is illuminated in some spherical body, will be that much less to the extent

that the part of the luminous body which sees it will be....

These drafts, which he crosses out, serve as starting point of his next proposition on CU686 (TPL750, 1508-1510):

That part which is illuminated by some spherical body will be that much smaller to the extent that the part of the luminous body which sees it is smaller.

On CA207ra (1508-1510) he also drafts a corresponding demonstration (fig. 635):

n has so much...darkness through the object to the extent that it has light and it is shown that ab and ed are equal to bd and do not make shadow on the earth.

(figure)Figs. 633-651: Abstract geometrical demonstrations of light and shade. Fig. 633, CU688; fig. 634, CU679; fig. 635, CA207ra; fig. 636, CU686; fig. 637, CU686; fig. 638, CA207ra; fig. 639, CU686; fig. 640, CA207ra; fig. 641, CU687.

This demonstration is developed on CU686 (fig. 736, cf. fig. 635), although the lettering is different:

This is proved. Let ah be the umbrous body. Let cie be our hemisphere. It follows that part a of the umbrous body will be less illuminated, being seen by a smaller part of the umbrous body, that is, by a lesser part of the day of this our hemisphere, as the two parts bc and de show.

He also claims the converse (fig. 639, cf. fig. 638).

Therefore that part of a spherical body which is illuminated will be of a larger shape which is illuminated by a greater sum of the luminous body.This is proved by the converse of the foregoing.If the minimal light bc and de of our hemisphere illuminate a minimal part of the spherical body ah, the same light will illuminate the maximal part of this spherical body, that is, if

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bc /and/ df of the following body, that is, if bc /and/ df of the following figure illuminate only the part nmr, the rest of the hemisphere, joined with its part bc /and/ df will illuminate the remainder of the aforementioned spherical body.Which is why even though bc /and df illuminate nmr it also illuminates the part kn on the side of the spherical body and the other /part/ lr on the side opposite.

This series ends with a plea in defence of such geometrical demonstrations directed against an adversary:

Here the adversary who does not want such science, says that the practice of drawing natural things suffices. To which it is replied that there is nothing which deceives more than trusting one's judgment without any other reasoning (raggione) as is always proved by experience, the enemy of alchemists, necromancers and other simple spirits (ingiegni)>

Read in context this oft cited passage is all the more fascinating because it reveals an important link between experience/experiment and geometry in Leonardo's approach. In his conception of science neither practice nor theory is sufficient in itself. Science involves a process of translating particular experience into a universal language of geometry. This is why, when he asks a question, one demonstration is never enough. He needs to provide various demonstrations in order to create bridges between concrete experience and abstract geometry. This is a theme to which we shall return in the eiplogue - see pp.** below.

BOOK SIX

Besides this in the sixth book I shall investigate the many and various diversities of reflections of these rays which will modify the original /shadow/ by /imparting/ some of the various colours from the different objects whence these reflected rays are derived. (CA250ra)

That reflected light and shade should influence the colours of surrounding objects was by no means a new idea. (Pseudo-) Aristotle in De Coloribus had, for instance, pointed out:

Lastly we never see a colour in absolute purity: it is always blent, if not with another colour, then with rays of light or with shadows and so it assumes a new tint...This is...why reflections in mirrors resemble the colour of mirrors.6

This aspect of colours had also been considered by later authors such as Ptolemy7 and Alhazen.8 How Leonardo intended to organize his own scattered notes on this theme is not clear. It is likely, however, that his projected sixth book would have included sections on reflections from 1) mirrors, 2) water, 3) white objects, 4) faces, 5) landscape and verdure, 6) a series of demonstrations show how yellow and azure combined produce green, 7) another set of demonstrations involves walls and lights of different colours, which become a starting point for his parallels, 8) between mixing lights and mixing pigments. There 9) further demonstrations also. Together these form the basis for his 10) precepts and 11) general statements concerning reflected light, shade and colour. Each of these aspects will be considered in turn.

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Fig. 642: Mirror reflection on Forst III 54r.

Book Six. 1. Mirrors

In De Coloribus Aristotle had noted that reflections in mirrors resemble the colour of mirrors.9 Ptolemy, in his Optics h ad pointed out that the colour of a mirror affects the colour of things seen.10 This phenomenon had also been mentioned by authors such as Heliodorus of Larissa11, Alhazen12 and Witelo.13 Leonardo's first extant reference to this question is on Forst III 54r (fig. 642, c.1493) under the heading:

MirrorIf the illuminated object is of the size of the illuminating object and of that where this light is reflected, the quality will have such a proportion with the medium light as the second light will have with the first, these bodies being level and white.

On BM57r (1497-1500) he restates this idea more succinctly:

Whence they say that it transmutes itself in as many natures as the places where it passes are various. And as the mirror transmutes itself into the colour of its objects, so too does this transmute itself into the nature of the place where it passes.

A slightly different version occurs on BM58v (1505-1508):

And as the mirror transmutes itself into the colour of the objects which pass in front of it, it has nothing in itself, but moves or takes everything and transmutes itself into as many various natures as the places where it passes are various.

On CU167 (TPL158, 1505-1510) he notes that the phenomenon depends on the degree to which the reflecting surface is polished:

On reflectionsReflections participate that much more or less of the thing where they are generated than of the thing which generates them to the extent that the object where they are generated is of a more polished surface than that which generates them.

On CU211 (TPL256, 1508-1510) he pursues the question:

Of the colours reflected on the lustres of various colours.

The reflected object always participates of the colour of the body which reflects it. The mirror is tinged in part by the colour reflected by it and participates that much more of the one than the other, to the extent that the object which is mirrored is more or less powerful than the colour of the mirror. And that object appears of a more powerful colour in a mirror which participates more of the colour of this mirror.

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He mentions the phenomenon once more on BM211v (1508-1512): “The image impressed in the mirror participates in the colour of the aforesaid mirror.”

Book Six. 2. Water

He studies the physics of reflections in water in connection with his mirror studies (see below pp. ). In addition to this there are at least four passages where he explores the properties of reflected colour in water. The simplest of these, on CU542 (TPL521, 1505-1510) is headed:

On objects reflected in water.Of objects reflected in water that will be more similar in colour to the reflected object which is reflected in clearer water.

On CU5453 (TPL522, 1505-1510) he considers reflection in water:

On objects reflected in turbulent waters.Objects reflected in turbulent waters always participate in the colour of that object which renders such water turbulent.

He considers a more complex situation on CU213 (TPL237, 1505-1510) under the heading:

Of the reflection and colour of the water of the sea seen from various aspects.The sea with waves does not have universal colour, but he who sees it from firm ground sees it of a dark colour and that much darker to the extent that it is closer to the horizon and he sees some brightness or lustres which move slowly in the manner of a herd of white pigs and he who sees the sea /while/ standing in the high sea/s/, sees it as azure.

And this occurs because from the land the sea appears dark because you see in it the waves which reflect the darkness of the earth and from the high sea/s/ they appear azure because you see in the waves the azure air which these waves reflect.

(figure)Fig. 643: Reflection from water on CU1007.

In the late period he considers reflected colours in water once more on CU1007 (TPL943, 1510-1515), this time in connection with painting:

Where the horizon is reflected in the waves.By the sixth of this the horizon is reflected on the size seen by the horizon and the eye as the horizon f demonstrates, seen by the side of the wave bc and this side is also seen by the eye.Again you, /o/ painter, who have to draw the inundation of the water, recall that the colour of the water will not be seen by you as being other than bright and dark, whatever the brightness or darkness of the site may be where you are, mixed together with the colour of the other things which are behind you.

Book Six. 3. White Objects

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Leon Battista Alberti, in his On Painting, had described how:

reflected rays carry with themselves the colour they find on the plane. You may have noticed that anyone who walks through a meadow in the sun appears greenish in the face.12

Leonardo adapts this example and develops it on A100r (BN 2038 20r, 1492) under the heading:

How white bodies must be represented.If you represent a white body surrounded by much air, because white does not have colour in itself, it is tinged and transmuted in part by the colour which is its object. If you see a woman dressed in white in a countryside, which is seen by the sun her colour will be bright in such a way that she will in part, like the sun, hurt the sight. And that part which is seen by the air or luminous body by the rays of the sun, interwoven and penetrated by it, because the air in itself is azure, that part of the woman seen by this air appears to tend towards azure. If the nearby surface of the earth is covered in meadows and the woman finds herself in that meadow illuminated by the sun and this sun sees all the parts of this which can be seen of the meadow, it will be tinged by the reflected rays in the colour of this meadow and thus it goes transmuting into the colour of nearby luminous and non-luminous objects.

White objects interest him particularly because he considers them to have no colour of their own (see above p. ) and therefore most apt to adopt the colours of surrounding objects. Hence, on A19v (1492), for instance, he begins with a:

PropositionEvery body without colour is coloured entirely or in part by the colour positioned opposite.This is seen by experience, because every body that reflects is tinged in the colour which is its object. And that body which is tinged in part, if it is white, then that part which is illuminated by red appears red and by every other luminous or umbrous colour.

He then gives a second in which he mentions white walls:

PropositionEvery opaque body without colour participates in that colour which it has for object. This happens on a white wall.

On A20r, the folio opposite, he restates this idea: "Every white and opaque body is tinged in part by the image of the colours that are its object." He mentions this quality of white objects again in the third of a series of drafts on W19141r (K/P99r, 1506-1508):

The surface of every opaque body will participate in the colour of its object.The surface of the opaque body is tinged by the colour of its object with that much more power to the extent that the rays of the species of these objects strike these bodies between more equal angles.And the surface of an opaque body is tinged more by the colour of its object to the extent that such a surface is whiter and the colour of its object is more luminous or illuminated.

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On F75r (CU204, TPL247, 1508) he restates the idea, now referring to it as a fourth proposition:

PaintingSince white is not a colour but in power receptive of every colour, when this is in a high landscape all its shadows are azure and this originates by the fourth which states: the surface of every opaque body participates in the colour of its object.Therefore such a white, being deprived of the light of the sun through the interposition of some object placed between the sun and it, therefore all the white that sees the sun and the participating air remains the colour of the sun and that part which does not see the sun remains umbrous and participating in the colour of the air and if such a white does not see the verdure of the countryside stretching to the horizon, nor the whiteness of such an horizon, without doubt this white would appear to be the simple colour which the air demonstrates itself to be.

He reformulates this principle on CU206 (TPL196, 1505-1510) again referring to it as a fourth proposition (see Chart 16 ):

Colour of the shadow of white.The shadow of white seen by the sun and by the air has its shadows tending towards azure. And this occurs because white has no colour in itself but receives some colour and by the 4th of this which states: the surface of every body participates in the colour of its object, it is necessary that that part of the white surface participates in the colour of the air /which is/ its object.

On CU465 (TPL471, 1508-1510) he develops the principle into two propositions under the heading:

Paintinga. The surface of every opaque body participates in the colour of its object and all the more to the extent that this surface approaches a greater whiteness.b. The surface of every opaque body participates in the colour of the transparent medium interposed between the eye and this surface and the more so, to the extent that it is denser, and a greater space is interposed between the eye and the said surface.

He pursues this question of white bodies on CU753 (TPL628, 1508-1510):

That the shadows must always participate in the colour of the umbrous body.Nothing appears its natural whiteness because the sites, in which these things are seen, render it that much more or less white to the eye, to the extent that such a site is more or less dark. And this is taught by the moon, which by day shows itself of little brightness and at night with such splendour that it renders from itself the image of the sun and by day with its dispelling of shadows.

To explain this he offers two reasons:

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And this arises from two things. And the first is the comparison which has in it the nature of showing things that much more perfect in the species of their colours to the extent that they are more disform. And the second is that the pupil is larger at night than by day as is proved and the larger pupil sees a luminous body of greater quantity of more excellent splendour than the smaller pupil, as is proved by him who looks at the stars through a small aperture made in a piece of cardboard (see below pp. ).

He returns to the characteristics of white subjects once more on CU785 (TPL704, 1508-1510):

Which object will tinge the white surfaces of opaque bodies more with its similitudes?That object will tinge the surfaces of white opaque bodies more with its similitudes which is by nature more remote from white. That which demonstrates itself as being the most remote from white is black and this is that by which the surface of a white opaque body is more tinged than by any other colour of other objects.

Book Six. 4. Faces

What applies to white colours, applies equally to flesh colours, as is clear from a passage on CU174 (TPL162, 1505-1510) headed:

On the colours reflected from the fleshThe reflections from flesh which have the light of other flesh are redder and of a more excellent flesh colour than any other part of the flesh that there is on a man. And this occurs by third of the second book which states: the surface of every opaque body which participates in the colour of its object, is that much greater to the extent that such an object is closer and that much less to the extent that it is more remote and to the extent that it is larger because, being larger, it impedes the species of the surrounding objects which are often of various colours, which corrupt the first species /which are/ closer, in the case of small bodies.But it is, nonetheless, possible that the reflection of a nearby small colour tinges more than a large remote colour by the sixth of perspective which states: large things can be at so great a distance that they appear considerably less than the little ones from nearby (see below pp. )/.

He outlines the consequences this has for his painting practice on CU175 (TPL170b, 1508-1510) under the heading:

On ReflectionsThat colour which is closer to the reflection will be more tinged by this reflection and conversely.Thus you, /O/ painter, need to do in the reflection of the faces of figures /with/ the colour of the parts of vestments which are close to the parts of the flesh /and the more so with/ those which are closer, but not to separate them with too much pronunciation if you need not.

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He discusses the problem of faces and reflected light again on CU798 (TPL644a, 1508-1510) in a passage entitled:

On the shadows which are not accompanied by the illuminated part.Very rare are those shadows of opaque bodies which are the true shadows of their illuminated part.This is proved by the 7th of the 4th which states that the surface of every umbrous body participates in the colour of its object. Therefore the illuminated colour of faces, having as its object a black colour will participate in the black shadows and yellow, green and azure will do the same as will every other colour positioned opposite it. And this occurs for the reason that every body sends its similitude through all the surrounding air as is proved in perspective and as is seen by the experience of the sun of which all the objects positioned opposite it participate in its light and reflect this to the other objects as is seen by the moon and the other stars, which reflect to us the light given them by the sun.And shadows will do the same, because these invest all the things which they strike with their darkness.

This leads to a more general formulation on CU797 (TPL645, 1508-1510):

Of the light of umbrous bodies which are practically never of the true colour of the illuminated body.We can say that it is practically never that the surface of illuminated bodies is the true colour of this body.

In the demonstration that follows he again refers to the principle of colour participating as the seventh of the fourth (see Chart 16 ):

That seventh of the fourth states the cause of this and also demonstrates that when a face positioned in a dark place is illuminated on one side by a ray of the air and is, on the other, struck by the ray of a candle, it undoubtedly appears to be of two colours. And before the air sees such a face, the light of the candle will appear its given colour and likewise the intervening air.

Another demonstration follows which, in turn, relates to his camera obscura experiments (see below pp. ):

If you take a white band and put it in a dark place and you take a light through an aperture, namely, from the sun, from fire and from the air, such a band will be of three colours.

On CU801 (TPL708, 1508-1510) he discusses the effects of reflected colour on both the clothes and faces of persons:

What the shadows do with the lights in comparison.Black clothes make persons stand out in greater relief than white clothes and this arises through the 3rd of the 9th which states: the surface of every opaque body participates in the colour of its object. It therefore follows that the parts of the face which see and are seen by

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black objects show themselves as participating in this black and for this /reason/ the shadows will be dark and there is a great difference between /these and/ the parts of this face which are illuminated.But white clothes will make the shadows of the face participate in such a whiteness and for this reason the parts of the face will show themselves to you as being of little relief, the bright and the dark having between them little difference from the bright and the dark, it follows that in this case the shadow of the face will not be the true shadow of such skin.

(figure)Figs. 644-646: Reflected light and colour on A113v, CU199 and CA305va.

In this context Mona Lisa's dark clothes make more sense.

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Book Six. 5. Landscape and Verdure

Practical experiences in Nature also serve to demonstrate effects of light and shade on colour. On A113v (BN 2038 32v, CU199, TPL209, 1492), for instance, he cites an example of reflected sunlight in the mountains (fig. 644, cf. fig. 645):

Which part of the colour should reasonably be more beautiful.

If a is the light /and/ b is illuminated along a line from this light, c which cannot see this light sees only the illuminated part, which part let us say that it is red. It being thus, the light which it will throw to the side will be similar to its cause and it will tinge red the /rock/ face uc. And if c is also red you will see that it is much more beautiful than in b and if c were yellow you will see it create a colour between yellow and red.

(figure)Figs. 647-650: Reflected light and colour. Figs. 647-648, CA144vb; fig. 649, CU203; fig. 650, CU711.

He considers another case of reflected light and colour in the mountains on CU203 (TPL250, fig. 649, 1509-1510, cf. fig. 647):

Of the colours of shadowsIt often happens that the shadows in umbrous bodies are not the same as the colours of the lights: either the shadows are greenish or the lights reddish even though the body is of the same colour.This happens because the light looks above the object from the east and illuminates the object with the colour of its splendour and from the west there is some other object illuminated by the same light which is of another colour. Which first object where it bounces with its reflected rays towards the east and percusses on the side of the first object facing it and it takes its rays from it and they remain firmly together with their colour and splendour.I have many times looked at a white object and red lights and bluish shadows. And this occurs in the mountains of snow where the sun is in the west and the horizon shows itself aflame.

A more complex example of reflected light, shade and colour in the mountains occurs on CU793 (TPL654i, 1508-1510):

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On lights and the shades and colours of these.No body will ever show itself entirely in its natural colour.That which is proposed can happen through two different causes of which the first occurs through the interposition of the medium which is included between the object and the eye. The second is when the things which illuminate the said body retain in themselves the quality of some colour.

That part of of a body would show itself of its natural colour which was illuminated by a luminous body without colour and if one did not see any other object than the aforesaid light in such an illuminated object. That this can never happen can be seen unless it were a deep blue colour positioned towards the sky on a very high mountain such that in this place one cannot see another object and the sun, in setting, is covered by low clouds and such that the cloth is of the colour of the air.

But in this case I contradict myself because the red also grows in beauty when the sun, which illuminates it, reddens in the west along with the clouds which are interposed between it. Although in this case one could again accept it for true because if this redness illuminated by the reddening light shows more beauty than elsewhere, it is a sign that the lights of colours other than red take on its natural beauty.

On CU166 (TPL762, 1508-1510) he cites another case involving objects in the countryside:

On the sites of lights and shadows of things seen in the countryside.

When the eye sees all the parts of the bodies seen by the sun it will see all the bodies without shadow. This is proved by the 9th which states: the surface of every opaque body participates in the colour of its object. Therefore, the sun being the object of all those parts of the surfaces of bodies which see it, these parts of the surfaces will participate in the brightness of the sun which illuminates it. Look at these bodies and /you will see/ that it is impossible that one can see another part of such bodies other than that which is seen by the sun. Therefore you will see neither primitive nor derived shade on any of the aforesaid bodies.

The reflected light of green meadows which Alberti had mentioned12, interest Leonardo also. On CA305va (c.1508), for instance, he notes (fig. 646): "If ab is green then by reflection nb is also green," and on Mad II 127v (1503-1504; CU225b-225b-226a, TPL767, 1508-1510), he explores the consequences of this phenomenon for his painting practice:

Of the consistency of shadows accompanied by their lights.

In this part you should have great respect for the things surrounding these bodies which you wish to draw by the first of the 4th which proves that the surface of every umbrous body participates in the colour of its object. But you should arrange artfully to make opposite the shadows of green bodies, green things such as green meadows and similar appropriate things such that the shadows participating in the colour of such an object do not come to

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generate and appear the shadow of a body other than green, because if you put illuminated red opposite a shadow which is green in itself, this shadow will redden and will make a colour of shade which will be most unbecoming and very different from the true green, and that which is said of such a colour is intended for all the others.

As his studies of Nature continue he becomes more aware of the natural variety of colours, as for instance, on BM114v (c.1510):

The trees of the countryside are of various kinds of green, because some blacken such as firs, pines, cypresses, laurels, box and the like; some tend to yellow as walnuts, pears, vines and young verdure. Some become yellowish with darkness such as chestnuts /and/ holm-oak. Some turn red in autumn as the service-tree, the pomegranate, vine and cherry and some are whitish such as the willow, olive, reeds and the like.

This awareness leads to further advice concerning painting practice on CU979 (TPL920, 1508-1510):

How to compose the fundament of colours of plants in a painting.

The way of composing the fundaments of colours of plants which border on the air is to make them as you see them at night in little brightness, because you see them equally of a dark colour mixed with the brightness of the air and thus you will see their simple shape clearly, without the impediment of various colours of green, bright or dark.

From 1508 onwards his interest in Nature focusses on the characteristics of individual plants and leaves. On G28v (CU935, TPL872c, c.1510-1515), for example, he examines reflected light on dark leaves:

The lights of those leaves will be more the colour of the air which is mirrored in them which are of a darker colour. And this is caused because the brightness of the illuminated part /together/ with the dark...composes an azure colour and such brightness arises from the azure of the air which, is reflected in the polished surface of such leaves and augments the azure which the said brightness usually generates with dark things.

(figure)Fig. 651: Reflected light and colour in leaves on G8v.

A more detailed analysis of reflected light and colour in leaves follows on G3r-2v (1510-1515):

Even if the leaves of a polished surface are in large part of a same colour on top and their reverse side, it occurs that the part which is seen by the air participates in the colour of this air and it appears to participate in the colour of of this air the more, to the extent that the eye is closer and sees it in more foreshortened form and universally its shadows show themselves as darker on top than on the reverse through the comparison which is made by the lustres which border on such shade.

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The reverse of the leaf, even if its colour be the same as that on the top will show itself as being of a more beautiful colour, which colour has a green participating in yellow and this happens when such a folio is interposed between/the eye and the light which illuminates it from the opposite side.He pursues this theme on G8v (TPL896, fig. 651, c. 1510-1515):

If the light comes from m and the eye is at n, this eye will see the colour of the leaves ab all participating in the colour of m, that is of the air. And lbc will be seen from behind as transparent with a beautiful green colour participating in yellow.

If m is the luminous body illuminating the leaf s, all the eyes that see the reverse of this leaf will see it as a beautiful green since it is transparent.

Other examples of reflected light, shade and colour with respect to verdure and landscape have been cited elsewhere (eg. CU782, TPL779 see above p. ; CU936, TPL875; CU980, TPL905; CU961, TPL911 see below pp. ).

Book Six. 6. Yellow, Azure and Green

His studies of Nature also lead him to study mixtures of colours produced by smoke from chimneys, as he records on CU179 (TPL205, 1505-1510):

Of the changes of transparent colours thrown or mixed on various colours with their different veilings.When a transparent body is over another colour and varied from it, it composes a mixed colour different than any of the simple ones composing it. This is seen in the smoke issuing from chimneys which, when it meets the black of this chimney, becomes azure and when it rises and meets the azure of the air, it appears grey (berrettini) or reddish and likewise purple (paonazzo) placed on azure makes itself a violet colour and when the azure is set on yellow it will make green and crocus on white makes yellow and white on darkness makes a darkness that is that much more beautiful to the extent that bright and dark are more excellent.

In this list of examples the combination of azure and yellow to produce green is mentioned in passing. For our purposes this example is of particular interest because it later becomes one of Leonardo's basic demonstrations to show that the surfaces of opaque bodies are tinged by the colours of surrounding objects. On CU790 (fig. 652, TPL701, 1508-1510), for example, he cites it in a passage headed:

(figure)Figs. 651-654: Experiments with coloured shadows on CU790, CU168, and CU169.

On the colours of the species of objects which tinge the surfaces of opaque bodies.Many are the times that the surfaces of opaque bodies in being tinged by the colours of their objects receive colours which are not in these objects.

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This is proved. Let cd be the opaque body and let ab be its object which we shall take as being of a yellow colour and the opaque body azure. I say that the entire part of the surface dnc of such an opaque body which is in itself azure will demonstrate itself to be green. And it would do the same if the opaque body were yellow and the object azure. And this occurs because various colours when they are mixed transmute themselves into a third participating of both one and the other and for this /reason/ yellow mixed with azure produces green, which green is a compound of its components which is comprehended clearly by the speculative painter.

In the above passage he refers to this phenomenon occurring many times. On CU168 (TPL166, 1508-1510), he claims that the phenomenon occurs in almost all cases (fig. 653):

Why the times are very rare when he reflections of colours are the colour of the body to which they attach themselves.

Very rare are the times that reflections are the proper colour of the body to which they are attached. And let the spherical body be dfge and let it be yellow and let the object the back of which reflects its colour, which is azure, be ubc. I say that the part of the spherical body that is percussed by such a reflection will tinge itself in a green colour bc being illuminated by the air and the sun.

On CU201 (TPL214, 1505-1510) he again cites the example of blue and yellow combining to produce green. This time he refers to the phenomenon as happening with certainty:

On the surface of every umbrous body.The surface of every opaque body will participate in the colour of its object. Umbrous bodies demonstrate this with certainty, as none of the aforesaid bodies shows its figure or colour if the medium interposed between the luminous body and the illuminated body is not illuminated. Let us therefore say that the opaque body is yellow and the luminous body is azure, I say that the illuminated part will be green, which green is composed of yellow and blue.

When he next cites this case on CU169 (fig. 654, TPL615, 1508-1510), he refers to it occurring in all cases:

How no reflected colour is simple but is mixed with the species of other colours.No colour which is reflected on the surface of another body tinges this surface with its proper colour but will be mixed with the concourse of the other reflected colours which rebound in the same place.As is the case with the yellow colour a which reflects in part of the spherical body coe and in the same place the azure colour b is reflected. I say that through this mixed reflection of yellow and blue, that the percussion of its concourse will tinge the spherical body. If it is white in itself it will be of a green colour, because it has been proven that yellow and azure mixed together compose a most beautiful green.

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On CU196 (TPL248e, 1508-1510) he mentions the need to compare ordinary light with reflected colour:

On ColoursThe light of fire tinges everything in yellow. But this does not appear to be correct unless it is compared with things illuminated by the air and this comparison can be seen close to the end of the day or indeed after sunrise and again where, in a dark room, an aperture shines on the object with daylight and again an aperture with candlelight and in such a place the differences will certainly be seen clearly and distinctly.Here he is alluding to camera obscura experiments he himself had made (see below p. ).

In the paragraph that follows, he again cites the mixing of azure and yellow to produce green:

But without such a comparison their differences will never be known except in the colours which have more similarities but are recognized as white by bright yellow, blue by azure and how to mix together azure and yellow and these compose a beautiful green and if yellow is then mixed with this green it is made more beautiful.

On W19151v (K/P118v(B), 1508-1510) he notes that this phenomenon of azure and yellow mixing to produce green can also be demonstrated using panes of coloured glass:

If the rays of the sun pass through two panes of glass which are in contact with one another, of which panes the one is azure and the other is yellow, the ray passing through this will not tinge either in azure or in yellow, but in a most beautiful green.

By now the phenomenon intrigues him the more because he realizes that within the eye blues and yellow together do not produce an impression of green, which he interprets as evidence that images do not interfere at the aperture of the eye (see below p. ). The mixture of azure and yellow to produce green also becomes relevant for his plant studies, as on G28v (CU938, TPL873d, c. 1510):

Of lights of green foliage tending towards yellow.But the leaves of verdure tending towards yellow in their reflection of the air do not have to make a lustre participating of azure, because everything which appears in a mirror participates in the colour of such a mirror. Therefore the azure of the air, reflecting in the yellow of the foliage appears green, because azure and yellow mixed together compose a most beautiful green. Hence the lustres of bright foliage tending towards the colour yellow will be a greenish-yellow.

He pursues this theme on CU936 (TPL875, 1508-1510) in a passage headed:

The leaves of plants are commonly of a polished surface as a result of which they partly mirror the colour of the air, which air participates in white when mixed with thin and transparent clouds. The surfaces of these leaves when they are of a dark nature such as those of elms, when not dusty, will render their lustres in a colour participating in azure. And this occurs through the 7th of the 4th which shows: white mixed with dark composes azure.

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And such leaves have their lustres that much more azure to the extent that the air which is mirrored in them is more pure and azure. But if such leaves are young, as in the tips of branches in the month of May then they will be green with a participation of yellow. And if their lustres are generated by the azure air which is mirrored in them, then their lustres will be green by the 3rd of this 4th which states: a yellow colour mixed with azure always generates a green colour.

Which leads him to conclude:The lustres of all leaves of dense surfaces participate in the colour of the air and to the extent that they are dark leaves, the more they will serve as mirrors and consequently such lustres will participate more in azure.

Some five years later on CA45va (figs. 655-756, c.1515) he reconsiders the mixture of yellow and blue asking:

Why shadows are made by a luminous body, tinger or surrounder of shadows.The shadow made by a luminous body and which is yellow sees the shadow as being azure, because there is the shadow of the body a, made on the pavement at b in which it is seen by the azure luminous body. And likewise the shadow made by the luminous body d which which is azure is yellow at the site uc being seen by the yellow luminous body and the background bc surrounding these shadows other than its natural colour, will be tinged by a colour mixed with yellow and blue because it is seen and illuminated by a yellow luminous body and an azure luminous body at the same time.

(figure)Figs. 655-658: Demonstrations involving the mixture of yellow and blue to produce green. Figs. 655-657, CA45va; fig. 658, CA181ra.

Shadows of various colours depending on the lights seen by them.

Immediately preceding this passage he notes: "That which makes shade does not see it because shadows are made by the luminous body tinging or surrounding these shadows." This idea he restates directly following the passage: "Shadows of various colours /vary/ depending on the lights seen by them. That light which makes shadow does not see it." On CA181ra (fig. 658, c.1516-1517) he returns once more to his demonstration how azure and yellow mix to produce green, now referring to it as the second proposition:

PaintingThe surface of every body participates in the colour of the object.The colours of illuminated objects impress themselves on the surface of one another in as many sites as are the varieties of the situations of such objects.O is the illuminated azure object and it alone sees without other company the space bc of the white sphere abcdef and it tinges it with azure colour:...m is the azure object which illuminates the space ab in the company of the azure o and it tinges it in a green colour (by the 2nd of this which proves that azure and yellow make a most beautiful green, etc.).

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He goes on to relate this to his camera obscura experiments (see below p. ) which he intends to include in his book of painting:

And the remainder will be said in the book of painting and in this it will be proved that, making the species and colours of bodies illuminated by the sun enter through a small round hole in a dark place on a flat white wall white in itself etc. But everything will be upside down.

These eleven demonstrations involving a mixture of azure and yellow to produce to produce green might seem more than sufficient to establish that "the surface of every opaque or umbrous body participates in the colour of its objects." But Leonardo, fascinated and almost obsessed with the phenomenon also uses a series of other demonstrations.

Book Six. 7. Walls

Among these are a number of experiments involving walls and planes of different colours. On A112v (BN 2038 33v, TPL668e, 1492), for instance, he describes a spherical object positioned on a red plane opposite a green wall:

On shade and light.Every part of the surface which surrounds bodies is transmuted in part into the colour of that which is positioned as its object.Example.If you place a spherical body in the middle of various objects, that is, which from one side is the light of the sun and from the opposite side there is a wall illumined by the sun which is green or another colour /and/ the plane where it is positioned is red. From the two transverse sides it is dark. You will see the natural colour of the said body participate in the colours which are its object. The luminous will be the most powerful. The second will be that of the illuminated wall; the third that of the shadow. There then remains a quantity which participates in the colour of its extremities.

(figure)Fig. 659: Demonstration of mixture of lights on CU478.

Some twelve years later, on Mad II 125r (cf. Mad II 26r below pp. ), he records another experience involving green shadows on a white wall on All Saints Day (2 November), 1504 at Piombio:

I have just seen the green shadows made by the cords, mast and lanteen yards on the side of a white wall, the sun going down in the west. And this occurred because the surface of this wall did not tinge itself with the colour of the sun, /but/ tinged itself with the colour of the sea that was its object.

On CU478 (fig. 659, TPL467, 1508-1510) he derives a more complex play of colours on white walls:

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Why towards evening the shadows of bodies generated on a white wall are azure.The shadows of bodies generated by the redness of the sun near the horizon are always azure. And this arises through the 11th where it is stated: The surface of every body participates with the colour of its object. Therefore the whiteness of the wall being deprived of every colour, is tinged by the colour of its objects, which objects are in this case the sun and the sky, since the sun reddens towards evening and the sky demonstrates azure. And where there is shadow the sun does not see, by the 8th of shadows, which states: No luminous body ever sees the shadow figured by it. And where the sun does not see on such a wall, there it is seen by the sky.

(figure)Fig. 660: Experiment with mixture of blue and red light, on W19151v (K/P 118v(B)).

Therefore by the said 11th of shade, derived shade will have percussion of an azure colour on the white wall and the background of this shadow seen by the redness of the sun will participate in a red colour.

He considers the play of blue and red light on a wall at greater length on W19151v (fig. 660, K/P 118v(B), 1508-1510) in a passage headed:

On the colours of simple derived shadows.The colours of derived shadows always participate in the colours of the bodies which light them up. To prove this, let an opaque body be interposed between the wall sctd and the lights, de, blue and ab, red. I say that de, the blue light, sees all the wall sctd except op which the shadow of the opaque body qr occupies, as the straight lines dqo and erp show. And the same happens to the light ab which sees the whole wall sctd except the place occupied by the shadow /of/ qr as the lines aqn and brm show. Therefore one concludes that the shadow nb sees the blue light de and not being able to see the red light ab, nb remains a blue shadow in a field of red, mixed with blue, because the field sctd sees both lights, but in the shadows it sees only one light. For this reason such a shadow is a half-shadow because if such a shadow was seen by no light at all it would be a maximal shadow, etc. /i.e. darkness/. But the shadow op does not see the blue light because the body qr interposes and prevents it there. Only the red light ab is seen there and this tinges it with the colour red and so this rosy shadow remains in a field of mixed blue and red.

The shadow of qr on op being caused by the blue light de is red and the shadow of this cast by the red light ab is blue at nm. Therefore we shall say that the blue light in this case makes a red derived shadow from the opaque body qr and the red light makes the same opaque body cast a blue derived shadow. But the primary shadow is not of this colour but it is a mixture of red and blue.

On CU202 (TPL239, 1505-1510), he pursues this theme, beginning with a general claim:

Of the colour of the shadows of some body.The colour of some body will never be the true or proper shadow if the object which it shades is not of the colour of the body shaded by it.

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This he then demonstrates with a case where azure light is reflected from a green wall:

Let us say, for example, that I have a house the walls of which are green. I say that if azure be seen in such a place which is illuminated by the brightness of the azure of the air, that such an illuminated wall will be a most beautiful azure and the shadow will be ugly and not the true shadow of such beauty of azure, because it is corrupted by the green which is reflected in it and it would be worse if such a wall were tan /coloured/.

Book Six. 8. Light and Pyramids

These demonstrations of reflected light and colour involving walls of different colours become a starting point for his analogies between mixing coloured lights and mixing pigment colours (see above p. on CU469 [TPL433, 1508-1510]):

(figure)Fig. 661: Concerning the mixing of colours on CU469.

Whether the surface of every opaque object participates in the colour of its object.

You need to understand that if a white object is put between two walls of which one is white and the other is black, that you will find such a proportion between the umbrous and the luminous part of this body as is that of the said walls and if the wall is of an azure colour it would do the same. Whence, having to paint it, you will do as follows:

Take the black to shade the azure object which is similar to the black or indeed the shadow of the wall which you assume that it has to reflect in your object and you will do thus, wishing to do it with a certain and true science, and you will accustom yourself to doing it in this way.

When you make your walls of whatever colour you want, take a little spoon, a little larger than that of a tea spoon and larger or smaller depending on large or small works in which you have to exercise such operations.

And such a spoon will have its extremities of equal height and with this you will measure the degrees of quantities of colours which you adopt in your mixtures. As would be if in the said walls that you have made the first shade were of 3 degrees of darkness and one degree of brightness, that is 3 level spoons as one does in the measures of grain and if these three spoons were of simple black and one spoon of white, you would have made a composition of a certain quantity without a doubt.

Immediately following he considers a more complex situation (fig. 661):

Now you have made one wall white and one dark and you have to put an azure object amongst them which object you wish to have the true shades and lights as are fitting for such an azure /object/.

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Therefore you put this azure /object/ to one side which you wish to remain without shade and put black on the side. Then take 3 spoons of black and mix them with one spoon of luminous azure and put this with the darkest shade.

Having done this see whether the object is spherical or columnar or square or how it is. And if it is spherical, draw a line from the extremity of the dark wall to the centre of this spherical object and where these lines intersect on the surface of such an object, there the greatest shades terminate at equal angles.

Then it begins to become brighter again as would be at mo which leaves as much of the dark to the extent that this participates in the superior wall ad, which colour will mix with the first shadow of ab with the same distinctness.

He pursues this analogy between mixing coloured lights and mixing pigment colours on CU869 (TPL756, 1508-1510):

Rule for taking the true brightnesses of lights on the sides of the aforesaid body.

Let there be taken a colour similar to the colour of the body which you wish to imitate and let the colour of the principal light be taken, with which you wish to illuminate this body.

Then if you find that the above mentioned angle is twice the lesser angle then you take a part of the natural colour of the body which you wish to imitate and give it two parts of light which you wish it to receive and you will have placed the light double the lesser light.

Then, in order to make half the light, take only one part of this natural colour of the aforesaid body and add to it only one part of the said light and thus you will have made on a same colour a light which is double the other, because on a quantity of this colour is given a similar quantity of light and to the other quantity two quantities of such a light are given.

And if you wish to measure these quantities of colour exactly, you will have a small spoon with which you can take your equal quantities and when you have taken your colour with it, you level it with a little ruler, as one does with the measures of grain when this grain is sold.

Book Six. 9. Further Demonstrations

In the meantime, Leonardo has been recording further demonstrations to show the nature of reflected light, shade and colour. On A93v (BN 2038 13v, fig. 216, CU756, TPL728, 1492), for instance, he describes how:

Every shadow made by an umbrous body less than the original light will send its derived shade tinged with the colour of its origin:

Let the origin of the shade ef be n and let the origin be tinged in its colour by h and let it be o and this is similarly tinged in its colour and likewise the colour of vh is tinged in the

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colour of p because it originates from it and the shadow of the triangle zky is tinged in the colour of Q because it is derived from it. To the extent that cd enters into ad to that extent is nrs darker than m and all the rest of the background /is/ without shade. Fg is the first degree of light because there all the window ad illuminates and likewise in the umbrous body me is of similar brightness. Aky is a triangle which contains in itself the first degree of shade because in this triangle the light ad does not reach; xh is the 2nd degree of shade because it is only illuminated by a 1/3 of the window, that is, cd; he is the third degree of shade because it sees 2/3 of the window bd. Ef is the ultimate degree of shade because the ultimate degree of light from the window illuminates the place from f.

A few folios later, on A98v (BN 2038 18v, CU284, TPL146, c.1492), he cites another demonstration involving firelight:

How one should represent a night /scene/.That thing which is entirely deprived of light is all darkness. The night being in a similar condition and you wish to represent a story, you will make that there is a large fire and that that thing which is closer to that fire is tinged more in its colour because that thing which is closer to its object, participates more in its nature. And making it a red colour, you will make all the things illuminated by it also tend to redden and that those which are further from the said fire can be tinged by the black colour of the night. The figures that are made in front of the fire appear dark in the brightness of this fire because that part of this thing which you see is tinged by the darkness of the night and not by the brightness of the fire and those that are to the side are half dark and half reddening and those which can be seen beyond the boundaries of the flames will be entirely illuminated by the reddening light a against a black background.

(figure)Fig. 662: Mixing of light and colour on CU766.

On CU766 (TPL702, 1508-1510) he opens with a general claim under the heading:

On false colour of shadows in opaque bodies.When an opaque body makes its shadows on the surface of another opaque body, which is illuminated by two various luminous sources, then such a shadow will not show itself to be of the same opaque body, but something else.

This he illustrates with a demonstration using a candle flame (fig. 662):

This is proved. Let nde be an opaque body and let it be white...and let it be illuminated by the air ab and by the fire cq. Then let there be interposed between the fire and the opaque body, op the shadow of which will cut the surface at dn. Now at this dn the redness of the fire is no longer illuminated but /rather/ the azure of the air. Whence dn participates in azure and nf sees the fire. Therefore the azure shade terminates below the redness of the fire above such an opaque body and above it terminates with the colour of violet, that is, which at de is illuminated by a mixture composed of the azure of the air ab and of the redness of the fire qc which is almost the colour of fire.

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And thus we have proved that such a shadow is false, that it is neither the shadow of white nor that of red which surrounds it.

On CU741 (TPL438b, 1508) he offers a further demonstration involving sunlight:

But a colour will never be seen simply. This is proved by the ninth which states: the surface of every body participates in the colour of its object, even if it is the surface of a transparent body as is air, water and the like. Since the air takes the light of the sun and the darkness from the privation of this sun, therefore it is tinged in as many various colours as are those which are interposed between the eye and them. Since the air does not have in itself a colour any more than the water does, but the humidity which mixes itself with this upper middle region is that which makes it expand and expanding, the solar rays which percuss it, illuminate, it and the air below this said middle region remains tenebrous and since light and dark compose an azure colour it is this azure in which the air tinges itself with this much more or less darkness to the extent that the air is mixed with less or greater humidity.

Another demonstration on CU796 (TPL633, 1508-1510) is headed simply (fig. 215):

What part of the surface of an umbrous body is it where the colours of objects mix.

Throughout the entire part of the surface of an umbrous body which is seen by the colours of more objects the species of the aforesaid colours will be mixed.

Hence the part abcd of the umbrous body is mixed with light and shade because in such a place it is seen by the light nm and by the darkness op.

On W19152r (K/P 188r, 1508-1510) he discusses the principle again, this time in connection with his explanation of how rays enter the eye (see below p. ):

Nature of the rays which are composed of the species of bodies and their intersections.The rectilinearity of the rays which carry through the air the shape and colour of bodies whence they part do not tinge the air of themselves, nor can they tinge one another in the contact of their intersection. But they only tinge the place where they lose their being, because such a place sees and is seen by the origin of these rays and no other thing which surrounds this origin can be seen by that place where that ray being intersected remains destroyed, there leaving the prey carried by it.And this is proved by the 4th of the colours of bodies where it is stated: the surface of every opaque body participates in the colour of its object. Hence it is concluded that the place which sees and is seen by the origin of such species through the ray carrying the species is tinged by the colour of that object.

In a demonstration on CU711 (TPL554, fig. 650, cf. fig. 648, 1508-1510) he compares the effects of primitive and derived shade:

Which is darker primitive shade or derived shade?

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Primitive shade is always darker than derived shade if it is not corrupted by a reflected light which makes itself a background of the percussion of this derived shade. Let bcde be the luminous body. Let a be the light which causes the primitive shade bec and produces the derived /shade/ bechi. I say that if fh and ig is not reflected, which reflects and corrupts the primitive shade at be with fh and at ce with ig, that this primitive shade will remain darker than the percussion of the derived shade, the one being shade and the other made on a surface of equal darkness of colour or of equal brightness.

He also examines how the rarity or density of an object affects the reflected colours of objects on CU754 (TPL631, 1508-1510):

Of shadows and which are those primitive ones which will be darker on a body.

Primitive shadows will be darker which are generated on the surfaces of bodies which are denser and conversely, will be brighter on the surfaces of bodies which are rarer. This is clear because the species of those objects which tinge the bodies positioned opposite with their colours will impress themselves with greater vigour which are found on a denser or more polished surface on these bodies.

This he again demonstrates with a concrete example (fig. 215):

This is proved. And let the dense body be rs interposed between the luminous object nm and the umbrous body op. By the seventh of the ninth which states: the surface of every body participates in the colour of its object, we shall therefore state that the part bar of this body is illuminated, because its object nm is luminous. And in like fashion we shall state that the part opposite dcs is umbrous because its object is dark. And thus our proposition is concluded.

As will be shown, this interest in reflective properties of rare and dense objects relates to his studies of the moon (see below p. ). On CU755 (TPL632, 1508-1510) he uses the same diagram as that in CU754 (fig. 215) to demonstrate another aspect of the phenomenon.

Which part of the surface of a body is better impressed with the colour of its object.That part of the surface of a dense body participates more intensely in the colour of its /object/ which is less seen by other objects and other colours. Let us therefore use the same figure for our proposition and let it be that the surface of the above mentioned body arb is not seen by the darkness op, /then/ it will be entirely deprived of shade and similarly if the surface csd is not seen by the luminous body nm, it will be entirely deprived of light.

Four further demonstrations on CU677 (TPL694b); CU722 (TPL782, 1508-1510) (see above pp. ) and CU720 (TPL781); CU724 (TPL698, 1508-1510) (see below pp. ), have been cited elsewhere.

Book Six. 10 Precepts

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One result of this volley of demonstrations is a series of pithy precepts and rules. Among the earliest of these is a note on Forst III 74v (1493): "The surface of any umbrous body will participate in the colour of bodies opposite it," which he restates on the adjacent folio, Forst III 75r (1493): "The surface of any opaque body participates in and is tinged by the colour of the bodies positioned opposite it."

On CU794 (TPL655, 1508-1510), he pursues this theme:

On the shadow and lights in objects.The surface of every umbrous body participates in the colour of its object.The painter must take great care in situating his things between objects of various powers of light and various illuminated colours, since every body surrounded by these never shows itself fully in its true colour.

On CU860 (TPL694f, 1508-1510) he lists nine propositions, three of which deal with reflected light, shade and colour:

1. Part of the surface of every body will participate in as many various colours as are those which stand as its object.7. All illuminated things participate in the colour of the illuminating object.8. Objects in shade retain the colour of the thing that obscures them.

These develop into four propositions on CU172 (TPL168, 1505-1510) under the heading:

Of reflections1st The surfaces of bodies participate more int he colours of those objects which reflect

their image in them amidst more equal angles.2nd On the colours of objects which reflect their images on the surfaces of bodies positioned opposite under equal angles, that will be more powerful which will have its reflected ray of a shorter length.3rd Among the colours of objects reflected at equal angles and at an equal distance on the

surface of bodies positioned opposite, that will be more powerful which will be a brighter colour.

4th That object reflects its colour in the body positioned opposite which does not have around it other colours than of its own species.

A few of his pithy statements concerning reflected light, shade and colour relate directly to his camera obscura studies. On CA230rb (1505-1508), for instance, he notes: "The surface of every body participates in the colour of its object" (see below p. ). On CA37ra (1508-1510) he refers to a second proposition: "The surface of every opaque body participates in the colour of its object" (see below p. ) and on CA195va (c.1510) he claims: "This is proved by the fourth of this which states: the surface of every opaque body participates in the colour of its object" (see below p. ).He mentions the phenomenon of reflected light, shade and colour once more on W19076r (K/P 167r, c. 1513): "the boundaries of derived shade are surrounded by the colours of the illuminated objects surrounding the luminous body, the cause of this shade," and again in summary form on the same folio: "shade always participates in the colour of its object."

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This he restates on E32v (1513-1514) under the heading: "On shade: The surface of every opaque body participates in the colour of its object." Another restatement occurs on G37r (1510-1515), this time under the heading: "On painting: The colour of the illuminated object participates in the colour of the object illuminating it," which idea he develops in a final passage on G53v (1510-1515):

The surface of every body participates in the colour which illuminates it and in the colour of the air that is interposed between the eye and this body, that is, of the colour of the transparent medium which is interposed between the object and the eye and among colours of a same quality the second will be the same colour as the first and this arises through the multiplication of the colour of the medium interposed between the object and the eye.

Book Six. 1. General Statements

In addition to the above demonstrations and precepts he makes a series of general statements on the nature of reflected light, shade and colour, as for example, the passage on CU768 (TPL815, 1508-1510) entitled:

PreceptBodies illumined by various qualities of colours of lights do not have the illuminated parts of their surfaces corresponding to the colours of their shaded parts.Very rare are the times when the colours of the surfaces of opaque bodies have the necessary colours of the shadows corresponding to the colours of their illuminated parts.That which is proposed arises because objects which make the shadows on such bodies are not of the natural colour of these bodies, nor of the same natural colour of the illuminator of this body.

He mentions reflected colours again in passing on CU265 (TPL119a, 1505-1510):

ReflectionBut that reflection will be of a more confused colour which is generated by various colours of objects.

On CA207ra (1508-1510) he makes another comment in passing concerning reflected light: "Every reflective body which moves in front of another reflective body which is immobile will infuse itself on this," a theme which he pursues on CU162 (TPL171, 1508-1510):

On the colours of reflections.All reflected colours are of less luminosity than direct light and the incident light has such a proportion with the reflected light as the luminosities have amongst themselves with respect to their causes.

On CU758 (TPL608, 1508-1510) he makes another brief note under the heading:

On the boundaries that surround derived shadows in their percussions.

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The boundaries of simple derived shadows in their percussions are always surrounded by the colours of illuminated things which send their rays from the same side of the luminous body, which illuminates the umbrous body generator of this shade.

He pursues this theme of coloured on CU788 (TPL609, 1508-1510):How every umbrous body generates as many shadows as are the luminous parts that surround it.Umbrous bodies generate as many sorts of shade around their bases and are of as many colours as are the illuminated colours opposite which surround it, but are that much more powerful the one than the other to the extent that the luminous source opposite it is of greater splendour. And this is taught by various lights positioned around a given umbrous body.

Which leads to a more radical claim on CU714 (TPL579a, 1508-1510):

Nature or condition of shade.No shade is without reflection, which reflection increases or weakens it. And that reflection increases it which is born of an object darker than this shadow. And this other reflection weakens it, which is born of an object brighter than this shadow.

On E17r (1513-1514) there is a further summary note on reflected shade with respect to painting practice:

PaintingIf your drawing note how among the shadows there are shadows of an imperceptible darkness and shape and this is proved by the third which states: the globulous surfaces are of as many various darknesses and brightnesses as are the varieties of darknesses and brightnesses which stand as its object.

(chart)

13. Conclusions

When Leonardo becomes enthusiastic about an idea he repeats it in almost all possible combinations. His "all in all..." passages were a first example of this. His notion that "colour participates," just examined, is another case. He devotes no less than fifty passages to this phrase (see Chart 14) and, in addition twenty-eight others where the concept is expressed more generally (see Chart 15).

As of 1503-1504 he refers to this idea as a proposition, suggesting that it is to form part of a more coherent treatise. In the course of the next six years he refers to such a proposition no less than thirteen times. What is noteworthy, however, is how the number keeps changing: what begins as a first proposition, becomes the fourth, seventh and ultimately the eleventh proposition (see Chart 16). This re-shuffling gives some impression of the energy with which Leonardo reformulates and reorganizes his ideas. Closely linked to the above passages on reflected light,

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shade and colour is a further series which involve the distance factor. These he intended to present in his seventh book on light and shade.

BOOK SEVEN

will treat of the various distances that may exist between the spot where the reflected rays fall and that where they originate, and the various shades of colour which they will acquire in falling on opaque bodies. (CA250ra)

The above outline of book seven continues the themes of book six: reflected light, shade and colour, with the addition of a distance factor. Concerning this he again has demonstrations and general statements. Further examples overlap with his studies of perspective of colour and diminution of form. Each of these will be considered in turn.

Book Seven. 1. Demonstrations

Amongst the earliest of these is a passage on CU858 (TPL820, 1505-1510) under the heading (figs. 381-38):

Of reflected lightTo the extent that the illuminated object is less luminous than that which illuminates it, its reflected part will be that much less luminous than the illuminated part.That object will be more illuminated which is closer to the illuminating source.To the extent that bc enters ba to that extent will it be more illuminated in ad than in dc.

He considers the distance factor again on F1v (fig. 495, 1508):

(figure)Figs. 663-666: Effects of distance on reflected light and colour. Fig. 663, H66/187/r; fig. 664, CU720; ;fig. 665, CU724; fig. 666, CU161.

The surface of every opaque body participates in the colour of its object.That part of the surface of opaque bodies participates more in the colour of its object which is closer.LEt ab /aob/ be an opaque body and cd a luminous object and ef an umbrous object. I say that the middle of this opaque body o participates equally in the one and the other object and the part ao is more luminous than the part ob and the closer it is to the luminous body the more it will be illuminated and hence the darker part of this body which which is closer to the umbrous body will make itself darker.

He expresses a similar idea on CU720 (fig. 664, TPL781, 1508-1510) under the heading:

Where the reflection must be darkerIf the light s illuminates the body rhp, primitive light brighter above towards the light than below where this body is positioned on the flat /ground/ by the 4th of this which states: the surface of every body participates in the colour of its object. Therefore the derived shade

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which stamps itself on the pavement at the site mp rebounds on the side of the umbrous body op and the derived light which tinges such a shade, that is mn, rebounds to or. And this is the cause by such umbrous bodies never have a luminous reflection at boundaries which the umbrous body has with its pavement.The accompanying diagram (fig. 664) can be seen as a development of a sketch on H66/18/r

(fig. 663, 1494) where he had noted: "The derived shade that borders with the primitive will be darker than this primitive." He develops this theme on CU724 (TPL698, 1508-1510), beginning a general claim:

Of the various darknesses of shadows of bodies imitated in pictures.The surface of every opaque body participates in the colour of its object and more or less to the extent that the body is closer or further.

Part one of his demonstration follows (fig. 665):

The first part of this is proved. And let abc be the surface of the opaque body which we shall assume is of a white surface and that the object rs is black and the object nm is also black. And by the 9th of this which proves that every body fills the surrounding air with the species of its colour, and with the similitude of its coloured body.Therefore, rs, a black body will fill the air which stands in front of it with dark colour which will terminate in gab, part of the opaque object abc, which part will tinge itself in the colour of its object rs and the white colour of the other object nm will whiten the entire part abc of the opaque body. Therefore, in the opaque body one will find all of ag in a simple participation of /the/ black /object/ urs and in bc in simple white and in ab which is seen by both the white and the black object, there will be a colour composed of white and black, that is, a surface of a mixed colour.

Part two of the demonstration considers the distance factor:

For the second part of the said proposition it will be much darker at a than at b because a is closer to the black object rs than is b and this is shown by the definition of the circle in geometry as is drawn /alongside, our fig. 665/. And besides this at the angle b, being the smallest angle that there is, as is proved in geometry, at the angle of contingence b, one cannot see other than the extremity of the body rs at the point r and other than this there is joined to b the brightness of the white object nm which, even if it were black, being further away from ub than a from rs, as was proved, b still would never be as dark as is that of a.

A slightly more complex demonstration, involving two spheres (fig. 661) follows on CU161 (TPL164, 1505-1510) in a passage entitled:

On doubled and trebled reflections.Doubled reflections have more power than simple reflections and the shadows that interpose themselves between the incident light and these reflections are of little darkness.Let a be the light source, BC a wall which receives light from this luminous body. Dre and nso are the parts of the two spherical bodies illuminated by direct light. Npm and dhe are the parts of these bodies illuminated by reflections. The reflection dhe is the simple

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reflection; npm is the doubled reflection. And the simple reflection is said to be that which is only seen by one illuminating body and the doubled /reflection/ is seen by two illuminated bodies. And the simple /reflection/, dhe, is made by the illuminated body Bg; the doubled /reflection/, npm, is composed of the illuminated body BK and the illuminated body dre. And its shadow which is interposed between the incident light n and the reflected light np is of little darkness.

(figure)Figs. 667-668: Concerning the distance factor in reflected light and colour on CU717 and CU670.

He again considers the distance factor in reflected light on CU717 (TPL751, 1508-1510):

On the proportion that the luminous parts of bodies have with their reflections.The illuminated part of incident light will have that proportion to that which is illuminated by reflected light, as is the /proportion of the/ incident light to this reflected light.

A demonstration follows (fig. 667):

This is proved. Let ab be the incident light which illuminates the sphere cd at cnd and passes with its rays to the object ef and from there it is reflected at cmd. I say that if the light ab has two degrees of power and ef has one, which is half of two, that the reflected light cmd will be one half the light cnd.

On CU670 (fig. 668, TPL635, 1508-1510) he presents a further demonstration in purely geometrical terms, asking:

Which part of an illuminated surface will be of greater brightness?That part of an illuminated body will be more luminous which is closer to the object which illuminates it.This is proved. And let the illuminated part of the object be ucx and let the object which illuminates it be ab. I say that the point c is more illuminated than any other part of such a body because the luminous angle acb which percusses it is greater than any other angle that can be generated on such a surface.

He pursues the theme on CU818 (TPL786, 1508-1510):

Of the eye which stands in the bright /air/ and looks at a dark place.In the darkness no second colour is of the same brightness as the first even if they are similar. This is proved by the 4th of this where it is stated: the surface of this body will be more tinged by the transparent medium interposed between the eye and this body, of which the interposed medium is of greater size.Therefore it remains concluded that a second colour put in a dark transparent medium will have more darkness interposed between it and the eye than the first colour, which which is found closer to the same eye. And such will be the proportion of darkness to darkness of these colours as there is from quantity to quantity of the dark mediums by which they are tinged.

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Book Seven. 2. General Statements

In addition to such demonstrations he also makes a number of general statements concerning reflected colours and distance as, for instance, on CU200 (TPL192, 1505-1510) under the heading:

Of the colour of the shade of a colour.The colour of the shade of some colour always participates in the colour of its object and that much more or less to the extent that this object is closer or further from this shade and to the extent that it is more or less luminous.

On CU171 (TPL216b, 1505-1510) he restates this idea in terms of a question and answer:

What part of a body will tinge itself more in the colour of its object?The surface of every body will participate more intensely in the colour of that object which is closer.This occurs because the nearby object occupies a greater multitude of variety of species which, coming to the surface of a body will corrupt the surface of such an object, which it would not do if such a colour were remote and occupying such species, this colour will show its nature more integrally in this opaque body.

He considers the distance factor again on CU811 (TPL629, 1508-1510), under the heading:

Of white things remote from the eye.The white thing remote from the eye, the more it is removed the more it loses its whiteness and the more so, to the extent that the sun illuminates it, because it participates in the colour of the sun, mixed with the colour of the air which is interposed between the eye and the white object. Which air, if the sun is in the east, shows itself a turbid red through the vapours, which rise around it. But if the eye turns itself to the east it will only see the shadows of white participating in an azure colour.

While Leonardo is interested in light and shade for its own sake, as a problem of physics, he is also concerned with its applications to painting. In the period after 1505 this artistic motive becomes explicit in passages such as CU175 (TPL170b, 1505-1510):

On reflectionThat colour which is closer to the reflection will tinge this reflection more with itself and conversely.Hence, /O/ painter, make sure that in the reflections of the faces of figures you use the colours of parts of clothes which are close to the parts of the skin which are nearest. But do not separate them too markedly, if it is not necessary.

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He pursues this artistic interest in reflected light and shade on CU163 (TPL159, 1505-1510):

On the reflections of lights that surround shadows.

The reflections of illuminated parts rebounding in the shadow positioned opposite alleviate their darkness more or less depending on whether they are more or less close or more or less bright. This is taken into consideration by many, and there are many others who avoid it and each /party/ laughs at the other.

But you, in order to avoid the calumnies of both (one and the other, put into operation the one and the other when they are necessary. But make /sure/ that their causes are known/ cf. G11v, that is, that these clearly see the cause of reflections and their colours and equally clearly the cause of the things that do not reflect. And doing this you will neither be entirely scorned nor praised by their various judgments, which persons if they are not entirely ignorant, it will be necessary that they praise you on the whole both one party and the other.

Leonardo's artistic interest in reflected light, shade and colour also explains why he includes some of his general statements on this topic (eg. G37r, CA181ra) under the heading of "painting" and, in addition, accounts for certain links with his studies of perspective.

Book Seven. 3. Links with Perspective

Leonardo's studies of perspective of colour and diminution of form have been analysed elsewhere (see Vol. 1, Part 3.1-2). Here, a few examples will serve to draw attention to links between these studies and his interest in reflected light, shade and colour. On CU234 (TPL241b, 1505-1510), for instance, he discusses explicitly a connection between the principle that colour participates and the perspective of colours:

Perspective of ColoursThe first colours must be simple and the degrees of their diminution must correspond with the degrees of distance, that is, that the sizes of things participate more in the nature of a point to the extent that they are closer to it i.e. the vanishing point/ and that colours have to participate in the colour of the horizon to the extent that they are closer to it.

Intimately connected with his colour perspective is an azure rule concerning distant objects which, in turn, relates to his principle that colour participates as on CU814, (TPL630, 1508-1510):

On the shadows of remote things and their colours.

The shadows of remote things participate that much more in azure colour to the extent that they are darker and more remote. And this occurs through the interposition of the brightness of the air which puts itself between the darkness of the umbrous bodies interposed between the sun and the eye which sees it. But if the eye turns opposite the sun, it will not see similar azure.

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This connection between reflected colour and his azure rule applies to verdure, as on G15r (CU980, TPL905, 1510-1515):

On the shadow of verdure.The shadow of verdure always participates in azure and likewise every shadow of every other thing and it takes more to the extent that it is more distant from the eye and less to the extent that it is closer.

The same connection also applies to rocks, as on W12414 (c.1511):

the rocks of this mountain will naturally retain a colour tending towards azure and the air which is interposed makes it even more azure and maximally in their shadows...

And it applies equally to landscapes as on CU961 (TPL911, 1510-1515):

On landscapes.The umbrous parts of distant landscapes will participate more in an azure colour than the illuminated parts. This is proved by the definition of azure in which the air is tinged, deprived of colour, which /air/ if it did not have darkness over it would remain white because of itself the azure of the air is composed of light and darkness.

As early as 1492, on A100v (GBN 2038 20v), Leonardo also makes explicit further links between reflected colour and the disappearance of shadow at a distance:

How the shades are lost over a long distance.Shades are lost over a long distance because the large quantity of the luminous air found between the eye and the thing seen tinges its shadows with this thing in its colour.

He develops this idea on CU792 (TPL705, 1508-1510) in a passage entitled:

On the accidents of surfaces of bodies.The surface of every opaque body participates in the colour of its object, which colour will be that much more evident on this surface, to the extent that the surface of such a body is whiter and to the extent that such a colour is closer.

This leads to a clear connection between his principle of reflected colour and disappearance of form perspective, as on CU870 (TPL690, 1508-1510):

When objects close to one another and small are seen at a long distance, such that the distinctness of their shapes is lost, then a mixture of their species is caused which will participate more in that colour which is invested by a greater amount of such objects.

Which connection he restates more subtly in passages such as E17r (CU448, TPL472, 1513-1514):

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The surface of every opaque body participates in the colour of its object. But with that much greater or lesser impression to the extent that this object is closer or more remote or of greater or less power.

In the end his principles concerning reflected light, shade and colour become so closely entwined with his perspectival studies that numerous passages apply equally to both domains, which again brings into focus a basic feature of his thought, where one thing literally leads to another.

BOOK EIGHT

MOVEMENT OF SHADOWS

Introduction

In his list on CA277va (c.12513-1514), Leonardo indicates a number of other books, (ie. chapters in the modern sense), which he intends to write concerning light and shade (see Chart 10). Among these is a proposed chapter on the movement of shadows, for which he writes a series of preparatory notes. On CU658 (TPL582, 1508-1510), for instance, he outlines five basic situations which concern him:

On the motions of shadows.The motions of shadows are of five natures of which let us say that the first is that which moves the derived shade together with its umbrous body and the light causing this shade remains immobile. And let us say that the second is that of which the shadow and the light moves but the umbrous body is immobile and the third will be that of which the umbrous and the luminous body moves but the luminous body with more slowness than the umbrous body. In the fourth motion of this shade, the luminous body moves with more speed than the umbrous body and in the fifth the motions of the umbrous and luminous body are equal among one another.And of this will be treated distinctly in its place.

On CU646 (TPL686, 1508-1510) he outlines a further situation in which the eye moves while the umbrous body and light remain constant. Taken together these six situations provide a probable framework for his intended chapter. Each of them will be considered in turn.

(chart)

Chart 17. Six basic situations concerning movement of shadows based on CU658 (TPL582) and CU646 (TPL686).

(figure)Figs. 669-691: Demonstrations of movement of shadows, where derived shade and umbrous body move, while light remains immobile on C3v, A110r and CU622

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Book Eight. 1. Derived shade and umbrous body move while light constant

In the early period he considers only the case in which a light source is constant while the umbrous body and derived shade moves, as, for instance, on C3v (fig. 669, 1490-1491):

If the body be moved slowly in front of the luminous body and the percussion of this object is far from this object, the motion of the derived shade will have such a proportion with the primitive shade as the space which is between the object and the light, has with that which there is between the object and the percussion of the shade, in /such a/ way that moving the object slowly the shadow is speedy.

He continues this theme on A110r (BN 2038 30r, fig. 670, CU703, TPL611, 1492) under the heading:

Of the shade made by a body situated between 2 equal lights.

That body which is found positioned between 2 equal lights will move by itself 2 shadows which direct themselves along a line from the 2 lights and if you remove said body and you bring it closer to the one of the lights, its shadow which will direct itself to the nearest light will be of lesser darkness than that which directs itself to the further light.

He provides a further illustration of this situation on CA370ra (1497):

Of the shadow which moves with a man.You will see forms and figures of men or animals which follow these animals and men wherever they flee and such is the motion of the one as of the other. But it appears a wondrous thing from the various sizes in which it transmutes itself.

(figure)Figs. 672-673: Further cases where derived shade and umbrous body move while light remains immobile on CU659 and 660.

Of the shadows of the sun and of mirroring in the water at the same time.Many times you will see one man becoming 3 (-) and all follow after him, /and/ often the most certain one abandons him.

Approximately a decade later he again considers this situation where the light source is constant while the derived shade and umbrous body move on CU659 (fig. 672, TPL575b, 1508-1510) in a passage headed:

Of the shade which moves with greater velocity than its umbrous body.It is possible that derived shade is many times more speedy than its primitive shade.This is proved and let a be the luminous body, let b be the umbrous body which moves from b to c along the line bd and at the same time the derived shade of the body f moves the entire space fe which space can receive the space bc in itself thousands of times.

He considers the converse of this case on CU660 (fig. 673, TPL576, 1508-1510):

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On the derived shade which is muchslower than primitive shade.It is also possible that the derived shade is much slower than the primitive shade.

This is proved. And let it be that the umbrous body bc moves the whole space ce over the plane ne and that its derived shade be on the opposite plane de. I say that the primitive shade bc will move the whole space bd, while the derived shade does not depart from de.

On E2v (1513-1514) he drafts a further passage on the problem under the heading:

Of shadow or its movement.Of 2 umbrous bodies which are one behind the other between the window and the wall with some space interposed,...the shade.... The umbrous body which is close to the side of the wall will be mobile, if the umbrous body near the window is in a transverse motion to this window.

To illustrate this he gives a concrete example:

This is proved. And let the two umbrous bodies be ab interposed between the window nm and the wall op with some space interposed between them and the same space ab. I say that if the umbrous body moves towards s that the shadow of the umbrous body b which is c will move to...d.

As will be shown (see below pp. ) this situation overlaps with his camera obscura studies. In Manuscript E 30v (CU662, TPL593, 1513-1514) he pursues this theme under the heading:

On the motion of shadow.The motion of shadow is always more speedy than the motion of the body that generates it, /if/ the luminous body /is/ immobile.

This claim is followed by a demonstration (fig. 671):

This is proved and let the luminous body be a and the umbrous body b and the shade d. I say that the umbrous body moves from b to c in the same time that the shadow d moves to e and there is that proportion from speed to speed made at the same time which there is from length of motion to length of motion. Therefore in the proportion made from the umbrous body b to c with the length of motion made from shade d to e, such /a proportion/ will the aforesaid speeds of motion have between them.

(figure)Figs. 674-675: Further cases where derived shade and umbrous body move while light remains immobile on E30v and G92v.

Immediately following he examines cases where the light moves (a) as fast as the umbrous body (Situation 5), (b) more quickly than the umbrous body (Situation 3), and (c) more slowly than the umbrous body (Situation 4, see below). The diagram accompanying the main demonstration on

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E30v (fig. ) is abstract. On G92v (1510-1515) he considers a concrete example based on everyday experience (fig. ):

On the speed of clouds.The passage of clouds is in itself less rapid than its shade which moves on the earth. This is proved. And let e be the solar body. Let a be the cloud and its shade c. Therefore, moving the cloud from a to b, the shadow will move from c to d. Whence it follows that the shadows which go from the earth to the cloud made by lines concurring towards the centre of the sun, we shall say by the fourth of this that this is true which is proposed. Which fourth states that the intersections equidistant from the angle of the two converging lines will be that much less to the extent that they are closer to the place of convergence. Hence the clouds being without doubt closer to the sun than their shade, the shade will make a greater voyage over the earth than the cloud through the air in the same time.

Book Eight. 2. Derived shade and light move while umbrous body is immobile

On CU665 (TPL810, 1505-1510) he explores a second situation under the heading:

(figure)Fig. 676: Movement of shadow on CU665.

Of the illuminated body which turns round without changing position and it receives a same light from various sides and varies infinitely.

The shadows of lights which invest an irregular body in the countryside will be of that many more darknesses and that many more figures to the extent of the variety that this body makes in its turning motion. And it is as much to turn the body around when the light stands firm as to turn the light around an immobile object.

This he illustrates, as usual, with a concrete demonstration (fig. 676):

This is proved and let en be the immobile body and let the mobile light be b which moves from b to a. I say that when the light is in b the shade of the protrusion d will extend from d to f which, in moving the light from b to a is changed from f to e and thus the said shade is changed in quantity and shape because the place where this is found is not of same shape as the place where it was divided. And such a mutation of shape and quantity is infinitely variable because if all the site which was at first occupied by the shade and is in itself completely various and of continuous quantity and every continuous quantity is divisible to infinity, therefore it is concluded that the quantity of the shadow and its shape is variable to infinity.

Book Eight. 3. Umbrous body moves faster and light moves slower

He refers to this third situation only in passing on E30v (CU662, TPL593, 1513-1514): “But if the luminous body is slower than the umbrous body then the shade will be more speedy than the umbrous body.”

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Book Eight. 4. Umbrous body moves slower and light moves faster

This situation too is referred to only in passing on E30v (CU662, TPL593, 1513-1514): “And if the luminous body is more speedy than the umbrous body, then the speed of the shade will be slower than the speed of the umbrous body.”

Book Eight. 5. Umbrous body moves as fast as light

This situation he describes on CU661 (TPL577, 1508-1510) under the heading:

Of the derived shade which is equal to the primitive shade.

The motion of the derived shade will be equal to the motion of the primitive shade when the luminous body, cause of the shade, will be of a motion equal to the motion of the umbrous body or, if you wish to say, of the primitive shade. Otherwise it is impossible because one who walks towards the west from morning to evening will have the shade going in front of the walker in the first part of the day. In the last half of the day the shadow will be much quicker in fleeting behind than the umbrous body in going forward.

He alludes to this situation again on E30v (CU662, TPL593, 1513-1514):

But if the luminous body is equal in speed to the motion of the umbrous body, then the shade and the umbrous body will be of equal motion relative to one another.

Book Eight. 6. Eye moves while umbrous body and light immobile

He also considers a case in which the eye moves while the umbrous body and light remain constant on CU646 (TPL686, 1508-1510) in a passage entitled:

(figure)Fig. 677: Situation where the eye moves while the umbrous body and light are immobile on CU646.

On the site of the eye which sees more or less shadow depending on the motion that it makes around the umbrous body.The proportions of the quantities that the umbrous and illuminated parts of umbrous bodies have vary as much as the variety of the sites of the eye that sees them.

A specific example (fig. 677) follows:

This is proved. Let amnu be the umbrous body. Let p be the luminous source which embraces it with its rays pr and ps /and/ illuminates the part uman and the remainder num remains dark.

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And let the eye which sees such a body be q which, with its visual rays embraces this umbrous body and sees all of dmo in which glance it sees dm, the illuminated part, considerably less than mo; the umbrous part, as is proved in the pyramid dqo, intersected at kh, equidistant from its base, divided at point i.

And thus will vary similarly in so many ways darkness at the eye which sees it as are the varieties of the sites of the aforesaid eye.

This factor of the eye's position he again considers on CU770 (TPL676b, 1508-1510) in a passage entitled:

On umbrous bodies which are polished and lustrous.In umbrous bodies which have polished and lustrous surfaces, which have particular light, they will vary in their shadows and lustres in as many sites as are the mutations of the pupil (luce) of the eye which sees them.

In this case the particular light can be immobile and the eye mobile and also conversely which is the same with respect to the changes of lustres and shadows on the surface of these bodies.

Conclusions

A comprehensive study of Leonardo's scattered notes on light and shade reveals elements of an unexpectedly systematic approach and confirms that he wrote sufficient material for his projected seven books on light and shade (CA250va, 1490) to permit a reconstruction of their contents. Book One opens with a consideration of the nature of light and shade, its punctiform propagation, the role played by central lines and sets out to dismiss opposing theories. Book Two provides definitions of primary shade and describes its degrees.

Book Three is devoted to derived shade. It begins with a classification of three basic kinds of shade based on Aristarchus: a first where light source and object are the same size; second, where the light source is larger than the object and third, where the light source is smaller than the object. Derived shade involves three variables (light source, object and eye), each of which Leonardo studies in terms of comparative sizes, distances and positions. While Book Three considers the nature of derived shade in the open, Book Four concentrates on the effects produced when this shade strikes surfaces of different shapes and sizes in various positions and at various distances. This leads to a series of studies on how one light source and one object can produce two shadows. It leads also to an examination of compound shade which, for Leonardo, involves multiple light sources and/or multiple objects.

In the period after 1501 this culminates in a series of experiments involving interposed columns both in isolation and combined in the manner of a St. Andrew's cross. These experiments again reveal a systematic approach involving one, two, three and four light sources, and are of further interest because they are parallelled by another series of experiments involving one, two, three and four pinhole apertures in a camera obscura (see below p. and pp. ).

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Book Five considers further effects when this derived shade strikes objects and is reflected in such a way that it mixes with the surrounding light. His demonstrations on this theme include interposed rods and walls as well as theoretical situations. In 1490 Leonardo plans to write a sixth book on how reflected and shade alter the colours of surrounding objects, but his extent notes on this theme do not begin until 1492. In that year he begins with demonstrations using white objects, faces and landscapes. In the period after 1505 he explores this theme in more detail with demonstrations involving mirrors, water, the combination of yellow and azure to produce green, and reflections from different coloured walls. This leads to both a series of precepts and general statements. As in the case of Book Six, the ideas for Book Seven were also conceived of in 1490, but not written until later. Here, Leonardo considers the effects of distance on reflected colour, concerning which he has several demonstrations and a number of general statements. Especially in the period after 1505 there are increasing links between these principles of reflected colour at a distance and his studies of perspective of colour and diminution of form.

In addition to these seven books outlined on CA250va (1490) Leonardo subsequently notes a series of other books that he intends to write on light and shade. The most comprehensive of these lists (CA277va, 1513-1514) mentions sixteen books. Some of these are clearly developments of themes from the earlier seven books (cf. Charts 9 and 10). For instance Book Three, "On the shape of shadows," corresponds to aspects of Book Four, Chapters Six and Seven in the reconstructed earlier version.

Book Four, "On quality," (cf. the definition thereof on CU841), is very probably a development of Book Four, Chapters Two to Four in the reconstructed version, while Book Five, on CA277va, "On quantity" (cf. again the definition thereof on CU841), is probably a development of aspects of Book Four, Chapters Six and Seven in the earlier version Books Ten and Eleven, "On darkness" and "On light" are probably a development of Book One in the earlier version and of his definitions of darkness and light (see above pp. ). Book One, "On the usefulness of shadows," (CA277va), develops a theme only mentioned in passing in his outline on CA250ra (1490), and is probably based on his eulogies of chiaroscuro (see Vol. 1, Part 3.4, pp. ). Book Two, "On the motion of shadows," involves a theme not mentioned in the earlier outline. But, with the aid of other notes (CU658 and 686) a reconstruction has again been possible (our book eight). In the case of Book Nine, "On decompounded shade," the title is too sketchy to permit a serious attempt at reconstruction.

The remaining five books mentioned on CA277va (numbers 6, 12, 13, 14 and 16) are related to his camera obscura studies. For instance, Book Six, "On boundaries" is probably based on those camera obscura experiments by means of which he demonstrates the production of a spectrum of boundaries (see below pp. ). Book Twelve, "On light penetrating through apertures of different shapes," involves experiments with apertures in the form of triangles, squares, slits and crosses (see below pp. ). Book Thirteen, "On light passing through various numbers of apertures," is corollary to this and involves a further series of experiments with one to thirty-two apertures (see below pp. ). Book Fourteen, "On the composition of multiple luminous rays" may be based on other demonstrations involving coloured lights passing through a different number of apertures (see below pp. ). Finally Book Sixteen, "Whether parallel rays can come from a single light and penetrate through some apertures," also appears to be based o n camera obscura demonstrations (see below pp. ).

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Hence, notwithstanding some areas of uncertainty, there is a wealth of evidence that Leonardo studied questions of light and shade much more systematically than the surface disorder of the notebooks suggests. While he is initially concerned with these problems as a painter, he gradually develops an interest in the physics of light and shade for its own sake. His analogy between light and sight helps furnish a further motive. since the pupil can be treated as an aperture and the interior of the eye as a camera obscura, the study of light and shade in connection with camera obscuras can offer insight into the nature of the visual process. This leads to further demonstrations and experiments, which are the theme of our next chapter.

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PART TWO: CHAPTER THREE THE CAMERA OBSCURA

1. Introduction 2. Astronomical Context 3. Inversion of Images 4. Non-interference5. Images all in all 6. Intensity of Light and Shade or Image 7. Contrary Motion8. Size of Aperture 9. Shape of Aperture 10. Number of Apertures 11. Apertures and Interposed Bodies 12. Spectrum of Boundaries 13. Camera Obscuras and the Eye14. Conclusions

(figure)Figs. 678-679: Pinhole apertures and camera obscuras in observations of the sun on Triv 6v and A20v.

1. Introduction

Leonardo has commonly been credited with the invention of the camera obscura.1. This is not true. Unequivocal descriptions of the instrument go back at least until the ninth century AD2 and by the thirteenth century it had assumed an important function in astronomy.3 On the fourth of June, 1285, for instance, William of St. Cloud used a camera obscura to observe an eclipse of the sun.4 This use of the camera obscura develops in the fourteenth century and in the latter fifteenth century provides one source of Leonardo's interest in the instrument. He uses it, for example, to estimate the size and distance of the sun and moon.

The aperture of the camera obscura is, for Leonardo, analogous to the aperture of the pupil and this leads him to study various characteristics of apertures: how images passing through them are inverted, how they do not interfere with one another, how such images are "all in all and all in every part"; how they can vary their intensity, and how they move in a contrary direction beyond the aperture. He considers the effect of changing the size of the aperture and examines in some detail both the properties of a single aperture with a changing number of sides and the characteristics of multiple pinhole images. From a note on CA277va (1513-1514) cited above (see p. ), he clearly intended to adopt these findings for two additional books on light and shade.

Indeed many of the experiments that he had made with umbrous bodies in the open he repeats in combination with a camera obscura, now focussing on a particular phenomenon: how the boundaries between light and shade are actually a series of subtle graduations. These late studies of 1508-1510, as will be shown, have important consequences for his theories of vision and perception. But before considering these, we need to examine the details of his camera obscura studies.

2. Astronomical Context.

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Leonardo's earliest extant reference to the use of an aperture in observing eclipses is on Triv.6v (fig. 678, 1487-1490):

Way of seeing the sun eclipsed without hurting the eye. Take a card and make the aperture with a needle and through these /pinhole/ apertures look at the sun.In this case the image is seen directly and the aperture serves merely to screen off excessive

light. On CA270vb (c. 1490) he describes a case where the image is seen indirectly under the heading:

How bodies...send...their form and heat and power beyond themselves.When the sun, through eclipses, remains in the form of the moon, take a thin sheet of iron and in this make a little hole and turn the face of this sheet towards the sun...holding a piece of cardboard a 1/2 braccia behind this and you will see the similitudes of the sun come in a lunar shape, similar to its shape and colour.

Immediately following he offers a (cf. fig. 679).

Second example.This said sheet /of iron/ will also do the same at night with the body of the moon and also with the stars. But from the sheet /of iron/ to the cardboard there is by no means to be any other aperture other than this little hole and this is similar to a square box, of which the faces above and below and the two on the side of the card are of solid wood; that in front has the sheet /of iron/ and that behind, a thin white cardboard or paper pasted to the edges of the wood.

Finally he provides an illustration that simulates the effects produced by these natural phenomena:

Third example.Again take a candle of wax, which makes a long light and placed in front of this aperture, the said light will appear on the paper opposite in a long form and similar to the form of its cause, but upside down.

This example, described on CA270vb, is illustrated in diagrams on CA126ra (fig. 156), CA125vb (fig. 694), and CU789 (fig. 704). The case of the moon is also considered on A64v (1492):

Because all the effects of luminous bodies are demonstrative of their causes, the moon in the form of a boat, having passed through the aperture, will produce at the object /i.e. the wall/ a boat shape.

On A61v (1492) he pursues this theme, now adding an illustration (fig. ):

That perforation of round quality which is half closed will appear in the form of ab and the part c will be the light and n will be the closed off part and this same happens to the luminous half-moon.

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The problem of the moon's shape continues to perplex him. On CA243ra (1510-1515) he notes:

Since over a long distance a long luminous source makes itself round to us and /yet/ the horns of the moon do not observe this rule and even the light from nearby observes the demonstration of its point.

(figure)Figs. 680-681: Use of camera obscura for astronomy on A21r (1492) and by Mario Bettini (1642).

(figure)Figs. 682-683: Uses of camera obscura for astronomy on BM174v and CA243rb.

The use of the camera obscura in determining the diameter of the sun had been discussed by late mediaeval authors such as Levi ben Gerson5 (c.1370). This also interests Leonardo as is shown by a diagram on A20v (fig. 679, cf. A21r, fig. 680 and Mario Bettini, fig. 681) beneath which he adds:

Way of knowing how large the sun is. Make that from a /to/ b there are hundred braccia and make that the aperture where the solar rays pass is 1/16 of a braccia and note how much the ray has expanded in percussion.

The problem is not forgotten. On CA225rb (1497-1500) he reminds himself in passing of "the measurement of the sun promised me by Master Giovanni the Frenchman." On BM174v (1500-1505) Leonardo describes a more complex procedure involving a combination of camera obscura and mirror (fig. 682):

ab is the aperture through which the sun passes. And if you could measure the size of the solar rays at nm, you could see very well the true lines of the concourse of these solar rays, the mirror standing in ab, and then make the rays reflected at equal angles towards nm. But in order that you do not then distort (torse) it at nm, take it inside the aperture at cd, which can be measured in the percussion of the solar ray, and then place your mirror at the distance ab and there make fall the rays db /and/ ca and then rebound under equal angles towards cd. And this is the true method: but you need to operate such a mirror at exactly the same month, day and hour and you will do it better than at any other time because in such a distance of the sun, such a pyramid is caused.

The precise function of this procedure is not explained.On Leicester 1r (1506-1509) he alludes to but again does not elaborate on a:

Record of how I at first demonstrated the distance of the sun from the earth and with one of its rays which have passed through an aperture in a dark place find its true quantity again and besides this, through the centre of the water to find the size of the earth.

To this problem of sizes and distances of the planets he returns on CA243rb (1510-1515) now providing a more detailed explanation:

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If you have the distance of a body you will have the size of the visual pyramid which you will cut near the eye on the window (pariete) and then you remove the eye to that extent, such that the intersection is doubled, and note the space from the first to the 2nd intersection and say: if in so much...space the diameter of the moon increases so much above the first intersection, what will it do in all the space that there is from the eye to the moon? This will make the true diameter of this moon.

The method here described is identical to the surveying procedure used to demonstrate principles of linear perspective on CA42rc, (cf. vol. 1, fig. 122). On CA151va (1500-1506) he gives an alternative method of measuring the distance of the sun, this time using a staff again familiar from the surveying tradition. Lower down on CA243rb he draws a rough sketch of a camera obscura (fig. 683) beneath which he notes "Measure of the size of the sun, knowing the distance." On CA297va (1497-1500) he drafts a passage concerning the use of apertures in a meteorological context:

The solar rays, penetrating the apertures...which are interposed between the various...globosities of clouds, illuminate with their straightness all...the passage interposed...between the earth and this aperture...and tinge from themselves all the sites where they intersect.

(figure)Figs. 684-685: Apertures in clouds and camera obscuras on CU476 and CA248va.

This he drafts again directly below:

The solar rays penetrating the apertures interposed between the various globosities of clouds...make a straight and spreading course towards the earth where they are intersected...illuminating...with their...brightness all the penetrated air.

These drafts serve in turn as the basis for a passage on CU476 (fig. 684, TPL447, 1510-1515):

On the solar rays that penetrate the apertures of clouds.

The solar rays penetrating the apertures positioned between the various densities and globosities of clouds illuminate all the sites where they intersect and even illuminate the darknesses or tinge with themselves all the dark places that are behind them, which darknesses show themselves to be between the intervals of these solar rays.

Also in this late period, on CA248va (fig. 685, 1510-1515), he again mentions the relative intensity of the sun's illuminating in a camera obscura:

ab is brighter than cd. But the point t being illuminated by the narrow aperture by the part of the sun o will be that much less illuminated than being illuminated by the diameter ab to the extent that this o is less than this diameter ab.

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These astronomical and meteorological uses of the camera obscura constitute but a small part of Leonardo's concern for this instrument. He is fascinated by the analogies that it offers with the pupil (see below pp. ) and therefore employs the camera obscura to demonstrate such optical principles as inversion of images, their non-interference, and their existence "all in all and all in every part." We shall examine each of these in turn.

(figure)Figs. 686-687: Use of a camera obscura to demonstrate the inversion of images on W19147v (K/P 22v): fig. 688, another example on C6r.

3. Inversion of Images

Leonardo believes that images are inverted in passing through the aperture of the pupil. As early as 1489-1490 he employs a camera obscura to demonstrate this principle in a passage on W19147v (K/P 22v):

But if the plane of this interruption has in it a small aperture which enters into a dark home /that is/ dark not by colour but through privation of light, you will see the lines enter through this said aperture /and/ carry on the second wall the entire form of its origin, both with respect to colour and form. But everything will be upside down....

A specific example follows (fig. 686):

Let ab be the origin of the lines. Let de be the first wall; let c be the aperture where the intersection of the lines is. Let fg be the last wall. On the last wall and percussion you will find that a remains below at the place g and g below rises above to the place f.

He pursues this eye-camera obscura analogy on C6r (fig. 688, 1490-1491):

All the things that the eye sees beyond little apertures are seen upside down by this eye and are known as right side up.Let ad be the pupil (luce) that sees through the aperture n. The line eh is seen by the lower part of the eye; dh is seen by the upper part of the pupil.

(figure)Figs. 689-696: Demonstrations of inversion in camera obscuras. Fig. 689, Forst. III 29v; figs. 690-691, CA373rb; fig. 692, CA155rde; fig. 693, BM170v; fig. 694, CA125vb; figs. 695-696, CA345vb.

He illustrates this inversion principle in sketches without text on Forst III 29v (fig. 869, 1490-1493), CA155rde (fig. 692, 1495-1497) and BM170v (fig. 692, 1492); then mentions it again on BM232v (1490-1495):

The bases of inverted pyramids, if they are in a dark place, will show upside down the shape and cause of their source.

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A further illustration occurs on CA125vb (fig. 694, 1492) accompanying which he drafts an explanation:

The sun and every luminous body, which sends its rays through an aperture smaller than it in size, will send the rays upside down behind this aperture and you will see the experience with a lighted candle, taking its rays beyond an aperture smaller than it.

On CA126ra (c. 1492) the problem is further illustrated (fig. 156). Thereafter, more than a decade passes before he broaches the problem again on CA345vb (figs. 695-696, 1505-1508):

And they /the images/ impress themselves on the wall opposite the said point, perforated in a thin wall, and for this reason the eastern part will impress itself in the western part of such a wall and the western in the eastern and likewise the northern in the southern and conversely, etc.

In the Manuscript D he discusses the problem of inversion at length. A first passage on D10r (c. 1508) is entitled:

Of the species of objects that pass through narrow apertures into a dark place.

It is impossible that the species of bodies that penetrate through apertures into a dark place do not reverse themselves. This is proved by the 3rd of this which states (the particles of each umbrous ray are always rectilinear).

(figure)Figs. 697-698: Demonstrations concerning inversion of images on D10r and D8r.

To demonstrate this he provides a concrete example (fig. 697):

Therefore the part b of the object ab, passing through the aperture n into the dark place oqpr, will impress itself on the wall pr on the site c and the opposite extremity a of the same object ab will impress itself on the wall cr /sic: pr/ at the point r /sic:c/ and thus the right extremity of such an object makes itself left and the elft makes itself right, etc.

On D8r he pursues this analogy between eye and camera obscura under the heading:

How the species of objects received by the eye intersect inside the albugineous humour.

The experience which shows that objects send their intersected species or similitudes inside the eye in the albugineous humour is shown when the species of illuminated objects pass through some small round aperture into a habitation that is very dark, then you will receive such species on a white piece of paper...placed in such a habitation somewhat near this aperture and you will see all the aforesaid objects on this paper with their proper shapes and

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colours but they will be smaller and inverted as a result of the said intersection. Which images if they originate in the eye illuminated by the sun appear properly depicted on this paper which would be very thin and seen from behind.

A concrete demonstration follows (fig. 698):

And let the said aperture be made in a very thin sheet of iron. Let a, b, c, d /and/ e be the said objects illuminated by the sun. Let or be the face of the dark habitation in which is the said aperture nm. Let st be this /piece of paper where the rays of the species of these objects are intersected upside down, /and/ because their rays are straight a /on the/ right makes itself left at k and e /on the/ left makes itself on the right at f and it does the same inside the pupil.

(figure)

He again mentions this way in which images are inverted in passing on W19150v (K/P 118v(a), fig. 699, 1508-1510): "No image of however small a body penetrates the eye without being turned upside down....." On CA241vc (also 1508-1510) he produces two further drafts:

Every umbrous and luminous species which penetrates through the apertures...behind such a penetration turns (upside down after such a penetration) in contrary aspects all the parts of its size.Every umbrous and luminous species interests after the penetration made behind the apertures, turning in contrary aspect every part of their size.

These he crosses out and reformulates:

The rays of umbrous and luminous species intersecting after the penetration made by them inside the apertures, turn in contrary aspect every part of their size.

4. Non-Interference

The non-interference of images is another phenomenon which he demonstrates using the camera obscura. On A93r (BN2038 13r, 1492), for instance, he shows how a red, white and yellow light can intersect without interference (fig. 700). Similar demonstrations occur on CA256rc (figs. 701-702, c.1492) involving a red, green and yellow light and a red, white and green light. Accompanying these are a series of draft notes concerning the intensity of colour, light and shade passing through apertures:

(figure)Figs. 700-702: Demonstrations concerning non-interference of images. Fig. 700, A93r; figs. 701-702, CA256rc.

(figure)Figs. 703-705: Further demonstrations concerning non-interference of colours in a camera obscura on K/P 118r, CU789, K/P 118v.

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That colour which is more illuminated will show itself better in the percussion made by its rays within the aperture.

The luminous rays will make the shadows of bodies greater which oppose themselves between the aperture and the percussion, which bodies are touched by a less luminous ray.To the extent that the umbrous body is closer...to the percussion of the rays, the more its shadow will observe the form of its derivation.The qualities of rays are 2, that is: luminous and umbrous.The percussion of the luminous ray is surrounded by those images of things which surround the luminous body.

On 19112r (K/P 118r, 1508-1510) he returns to the theme of non-interference of colours in a camera obscura, this time using yellow and blue lights (fig. 703). Slightly more complex is a demonstration on CU789 (fig. 704, TPL707, 1508-1510) where he shows how light from a candle and from the air produces different colours on an interposed object:

Of the colours of lights illuminating umbrous bodies.

The umbrous body positioned between nearby walls in a dark place which is illuminated on one side by a minimal light of a candle and is illuminated on its opposite side by a minimal aperture of air, if it is white, then such a body will show itself yellow on one side and azure on the other, the eye standing in a place illuminated by the air.

On CU797 (TPL645, 1508-1510) he had used a similar demonstration to establish that the colour of shadows participates with the colour of surrounding objects (see above pp. ). On W19150v (fig. 705, K/P 118v, 1508-1510) he pursues both the themes of colour participating and non-interference of images under the heading:

Of the rays which carry the images of bodies through the air.

All the smallest parts of the images penetrate each other without occupying the space of one another.

And let r be one of the sides of the opening. Opposite this, let s be an eye which sees the lower extremity u of the line no, which extremity cannot send its similitude from itself to such an eye s such that it does not touch the extremity r and m, the middle of this line does the same and the same happens to the upper extremity n /going/ to the eye /at/ v. And if such an extremity is r, the eye v does not see the colour green of o at the edge of the aperture, but only the red of n by the 7th of this where it is stated that every similitude sends its species beyond itself by the shortest line which, by necessity, is straight etc.

On the recto of this same folio (K/P 118r) this phenomenon of the non-interference of images is discussed further in a passage entitled:

On the nature of the rays of which the images of bodies are composed and their intersections.

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The straightness of the rays which carry through the air the shape and colour of the bodies whence they depart do not tinge the air with themselves nor, furthermore, can they tinge one another in the contact of their intersection.

(figures)Figs. 706-707: Three light sources through one aperture and through two apertures on W19150v (K/P 118v).

This claim he qualifies:

But they only tinge the place where they lose their being because such a place sees and is seen by the origin of these rays and no other object which surrounds this object can be seen from the place where such a ray, being cut off is destroyed, leaving there the spoil /i.e. the image/ carried by it.

And this is proved by the 4th on the colours of bodies where it is stated that the surface of every opaque body participates in the colour of its object. Therefore it is concluded that the place by means of which the ray which carries the images sees and is seen by the origin of such a species is tinged by the colour of that object.

5. Images All in All

The way in which Leonardo uses a camera obscura to establish that images are all in all and all in every part has been analysed previously (see above pp. ). This was chiefly in terms of passages on A9v (fig. 133, 1492) and W19150v (figs. 706-706, K/P 118v, 1508-1510). This principle is also implicit in a sketch on BM171r (fig. 708, 1492) beneath which he writes:

Every surface is the boundary of a dark body /and/ multiplies in that boundary the species of the things positioned opposite it and if they are bright it sends them inside as bright.

(figure)Figs. 708-711: Camera obscura demonstrations on BM171r, W12353v, CA91vb and CA112va.

(figure)Figs. 712-721: Camera obscura demonstrations. Figs. 712-719, CA238rb; fig. 720, CA133va; fig. 721, CA238vb.

(figure)Figs. 722-728: Further camera obscura demonstrations. Figs. 722-726, CA238rb; fig. 727, CA382vb; fig. 728, K/P 118v.

(figure)Figs. 729-732: Camera obscuras on CA155rd.

(figure)

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Figs. 733-736: Camera obscura demonstrations. Fig. 733, CA256rc; fig. 734, W12353; figs. 735-736, C14v.

It is equally implicit in other sketches on W12352v (fig. 709, 1494), CA91vb (fig. 710, C.1500), CA112va (fig. 711, 1505-1508), as well as in a series of sketches (figs. 712-726) on CA238rb, vb (1505-1508) accompanying which is only an interrupted text: “Every part of ab...is in every part...d, but more...illuminate...short ray...where the spe/cies.../ powerful occupy...the less powerful.” On CA155rd (figs. 729-732, 1497-1500) he implicitly demonstrates this principles again in a series of four diagrams accompanying which he merely notes:

For the adversary...the long light ab would only illuminate the point c and experience illuminates it in des.abfik is according the adversary; abcgh is mine.

This principle is illustrated roughly once more in a diagram without text on W12353 (fig. 734, 1508-1511).

6. Intensity of Light and Shade or Image

Images in the camera obscura may be "all in all and all in every part." Nonetheless, Leonardo is convinced that they can vary in their intensity, and he uses the camera obscura to demonstrate this. On C14v (1490-1491), for instance, he draws a rough diagram (fig. 735) which he then develops (fig. 736). Accompanying this he opens with a general comment:

That part of the air will participate more in its natural darkness which is percussed by a more acute luminous angle.

It is clearly comprehended that where /there is/ a lesser luminous...angle there will be less light because the pyramid of this angle has a smaller base and because of this smaller base, a smaller number of luminous rays concur to its point.

This is followed by a specific demonstration which refers back to the diagram (fig. 736):

The angle a has a larger base than the angle b, the base of a is...sf and that of b is gh. Therefore a has a base that is a quarter larger than b /and/ has a quarter more light. Again c /and/ d hold a similar difference amongst one another because c sees ik which is half of the light ef and from d it sees the quarter lm.

This theme of differing intensities of light within the camera obscura is mentioned in passing on CA256rc (fig. 733, 1492): “The luminous rays make the shadows of bodies greater which are opposite the aperture and the percussion, which bodies are touched by a less luminous ray.” On CA238rb (1505-1508) following the all in all passage cited previously (see above p. ), he again takes up this theme of differing intensities of light and shade (fig. 723):

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By the simple proof of the lines intersected at the boundary of the umbrous body all the percussion of the luminous species would be of equal brightness...such that ab, a quarter of the luminous body responds to gh;...a quarter of the percussion at bc with ih which is similarly a quarter of the luminous body and of the percussion and the other 2 quarters do likewise: whence kf would be of equal light. But experience does not confirm it, whence other....

He now draws a second diagram (fig. 724) which he explains:

Therefore experience showing how the percussion of luminous rays acquires degrees of darkness in every part of height and this not being concluded by the first figure the second concludes it, because all the light ae sees i and 3/4 of this light be sees h and half the light ce sees g and a quarter of the light...de sees f. Hence f is less luminous and 3/4 so than i.

Another diagram follows (fig. 726 cf. 727-728) accompanying which he notes: "When the motion is from n to m, the shadow will descend from a to b." On this single folio CA238rb (1505-1508) he has thus used a camera obscura to demonstrate that (1) images are all in all and all in every part (see above pp. ); (2) that images vary in their intensity and (3) that images inside a camera obscura are inverted mind these three demonstrations are closely related. It is no wonder, then, that the remaining preparatory sketches on this folio (figs. 715-719, 722, 725) which are without text have a certain ambiguity about them: they could serve to support any of or all three of these demonstrations.

On C12v (fig. 737, 1490-1491), in the course of his studies of light and shade, he had illustrated how a light source in front of two opaque bodies produces concentric rings of light and shade of different intensity. The text on C12v is closely related to a draft, possibly in another hand on BM101r (1490-1495):

BM101r C12v

That umbrous body of spherical That umbrous body of sphericalrotundity will make circular rotundity will make circular mixedmixed shade which will be be- shade which has placed between ittween it /and/ the sun a body and the sun an umbrous body of itsplaced opposite it similar to quality.its quality /and/ quantity.

This demonstration of concentric rings caused by opaque bodies in the open air is the more interesting because it is paralleled by further demonstrations involving camera obscuras. On CA242v (1497-1500), for instance, he draws (fig. 738) sunlight entering through an aperture which is intersected at various distances. Directly beneath he describes the first percussion:

The first percussion of the solar ray is illuminated toward its centre b by the simple solar body b and at ab and bc it is illuminated by the air opnm...which mixes such a percussion, which is not simple light of the sun and towards the extremities a /and c, it is not illuminated except from the sides which are toward the centre of the sun such that between

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the lack /of light/ in such a site and the light of the sun, through the light of the air these spaces ab and bc become considerably darker towards their extremities a /and c.

He then describes the second percussion

At the 2nd percussion of the solar ray ac sees the entire solar body and would be very luminous if the darkness of the lateral air...did not corrupt such brightness with its darkness; that is, bt etc is seen other than by the sight of sun by ohegm.

ab and cd lack the light of the sun in each degree of distance from its centre r and towards a /and/ d, the extremities are only illuminated by the centre of the sun as the triangle adr shows.

And the said space ab is seen by op and the space cd by nm which obscure it.

A description of the third percussion is given short schrift: "At the 3rd figure bc is seen by all the body of the sun."

(figure)Figs. 738-740: Camera obscuras and concentric rings of light and shade on CA272v, CA262ra and CA238rb.

He pursues this theme on CA262ra (1497-1500). Here he carefully redraws his diagram showing various percussions of sunlight within a camera obscura (fig. 739). Directly above this he drafts a general claim: “The solar ray which penetrates inside the apertures (of the eye) of houses, in each degree of its length changes quality as quantity.” It is noteworthy how he here writes "apertures of the eye" which he then crosses out to write "apertures of houses." The correction is significant because on the same folio he also discusses different kinds of pupils (see pp. ). The camera obscura-eye analogy is very important for him. Not content with his first draft he crosses it out and begins afresh: “The solar ray which, through a narrow aperture made in a thin wall, penetrates a dark place, in each degree of its length m...” Here the text breaks off. Beneath the diagram he starts anew, now with a description of the 6th percussion.

In the 6th demonstration the sun is more powerful in ab than in cd because ab sees and is seen by the entire solar body and cd is seen by half.

The triangle cem carries the entire figure of the sun to m: whence there is the first degree of luminosity in this m. And the triangle del carries the half less light to the site 1 than in m, because only half of the sun shows itself there. In the triangle egk there is carried to this k all the luminosity of the sky eg and to i is carried the quality...of the base of the triangle fgi which is half of eg.

(figure)Figs. 741-748: Demonstrations of contrary motion using camera obscuras. Fig. 741, C3r; figs. 741-742, CA133va; figs. 744-748, CA357rb.

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(figure)Figs. 749-751: Demonstrations of contrary movement of images in camera obscuras on W19149r (K/P 118r).

He returns to this theme in a sketch without text on CA238rb (fig. 740, 1505-15087). In the Manuscript F similar sketches becoming a starting point for theories of the pupil (figs. ** , see below pp. ** ).

7. Contrary Motion

Leonardo's use of the camera obscura to demonstrate the phenomenon of contrary motion can be traced back to a note on C3r (fig. 741, 1490-1491):

The movement of the percussion of the sun which passes through the aperture of a wall and repercusses on the other side will make its growth towards the bottom. And this occurs because with the sun rising....If the sun bc sees all of ef when it has risen to ab it will see as far as fn and from this it arises that apertures of the sun grow towards the bottom.It is impossible that in bifureated and mixed derived shade there is a part where the entire umbrous body can be seen.

He pursues this theme in a series of sketches without text on CA357rb (figs. 744-748, c. 1490). On CM171v (1492) the principle is again mentioned:

All the similitudes of things which pass through a window out of the open air into the constrained air of the wall are seen in a contrary site: that thing which in the open air moved from east to west will appear as a shadow in the illuminated wall of constrained air / to be/ of contrary movement.

Further sketches (figs. 742-743) without text follow on CA133va (1497-1500). Approximately a decade later he demonstrates this principle of contrary movement of images in a camera obscura by moving the edges of the aperture on W19149r (K/P 118r, figs. 749-751, 1508-1510, see above pp. ). In this same passage he also broaches the themes of aperture size, rectilinear propagation and the principle that images are all in all in every part.

(figure)Gis. 752-756: Experiments concerning contrary motion of images in a camera obscura on CA277va.

The principle of contrary movement is further examined on CA277va (1508-1510). Herehis approach is experimental and systematic. He begins with a situation (fig. 752-753) where a stick, situated in a high position in front of a near wall, casts a shadow low down on the far wall. The accompanying text is headed:

Operation of compound shade.The operations of compound shade are always of contrary motion.

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That is, if it is touched by an opaque body, the concourse of the luminous rays,...before they come to their intersection, all the shadows of this interrupting body of the superior ray are demonstrated beyond such an intersection in the percussion of the inferior ray and just as the...superior ray...makes itself inferior after the intersection, so too the motions that the umbrous body makes inside such a superior ray will show themselves of contrary motion behind such an intersection. And this is shown in the intersection of compound shade on the pavement or on the wall percussed by the sun or other luminous body.

Having considered the contrary motion of this stick's shadow when the stick is positioned in front of the first wall (fig. 754), he explores what happens if the stick is positioned behind this first wall:

But if the luminous ray is...impeded by an opaque body...somewhat behind its intersection, then...the percussion of the derived shade of the opaque body will make a motion similar to the motion of its opaque body.

(figure)Fig. 757: Demonstration of contrary motion of images in a camera obscura on E2v.

Next he turns to a case where the stick is positioned in the same plane as the first wall (fig. 755, cf. fig. 756):

And if such rays are impeded at the actual site of their intersection, then the shadows of the opaque body will be two...and they will move with contrary motions with respect to one another, before they come to unite.

This passage ends with a general comment:

The compound derived shade is the cause that the percussion of the solar ray which passes through some angle does not impress...these angles but portions of...that much larger or sm/aller/ to the extent that such impressions are more remote or close to these angles.

He returns to this theme of contrary movement on E2v (1513-1514) in a passage headed:

On shadow and its movement.Of two umbrous bodies which are one behind the other behind a window and a wall is interposed at a given space.... The shadow of the umbrous body that is close to the wall will move if the umbrous body closer to the window is in transverse motion /relative/ to this window.

This he demonstrates with a concrete example (fig. 757):

This is proved. And let the two umbrous bodies be a /and/ b interposed behind the window nm and let the wall be op interposed with some space interposed between them which is the space ab /he means bc/. I say that if the umbrous body a moves towards s that the shadow of the umbrous body b which is c will move towards...d.

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(figure)Figs. 758-765: Demonstrations with camera obscuras having different sizes of apertures. Fig. 758, CA373rb; fig. 759, CA256rc; fig. 760, H227 inf. 47v-48v; fig. 761, a2r; fig. 762, CA373rb; fig. 763, CA256rc; fig. 764, H227 inf. 47v; fig. 765, A85r.

8. Size of Aperture

Concerned as he is with studying the variables of a given problem it is not surprising to find him exploring the role played by different sizes of aperture. On CA373rb (1490-1495) he merely makes two preliminary diagrams without text (figs. 758, 762). These he develops in two diagrams (figs. 759, 763) on CA256rc (1492) where he draws a thin and a thick wall alongside which he writes:

Among apertures of equal...size that which is in a larger wall will render a darker...and smaller percussion.

On H227 in f. 47v=48r he takes these ideas further. Here he draws a thick wall (fig. 760) accompanying which he notes: "The aperture that is positioned in a thick wall will give little light to the site where it reaches." He also draws a thin wall (fig. 764) with the comment: "That aperture that is positioned in a thinner wall will give more light to the place where it reaches." He pursues this theme on A2r (fig. 761, 1492) where he draws a relatively thin wall and considers changing intensities of light inside a camera obscura under the heading:

Quality of lightsTo the extent that ab enters into cd so many times is it more luminous than cd and similarly as many times as the point e enters into cd so many times is it more luminous than cd. And this light is good for those /things/ which entail subtle work.

On A85r (BN2038 5r, 1492) he draws a related diagram (fig. 765) alongside which he drafts an explanation:

That air which is luminous penetrates through perforated walls and passes inside dark habitations will make the place that much less dark to the extent that this perforation enters into the walls that surround and cover the pavement, to the extent that this perforation is less than the walls that surround and cover the pavement.

Not content, he crosses this out and reformulates it under the heading of:

PaintingThat luminous air which penetrates and passes through perforated walls into dark habitations will make the place that much less...dark, to the extent that this perforation enters into the walls that surround and cover their pavement.

He redraws the diagrams of A2r and A85r on CA262v (figs. 766-768, 1497-1500) this time with no accompanying text. About a decade later he takes up this theme of apertures of different

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sizes once more in W19152v (K/P 118v, 1508-1510), beginning with a general claim: “Images which pass through apertures into a dark place intersect their sides that much nearer to the aperture to the extent that this aperture is of lesser width.” By way of illustration he draws three diagrams and first discusses the case (fig. 769) on the far left in which the opaque objects ab and ik produce shadows which pass through the aperture de:

This is proved: And let ab be the umbrous body which sends not its shade but the image of its darkness through the aperture de which is the width of this umbrous body. And its sides ab being rectilinear (as was proved) it is necessary that they intersect between the umbrous body and the aperture but that much closer to the aperture to the extent that this aperture is less wide than the umbrous body.

(figure)Gis. 769-771: Demonstrations of concerning different sizes of aperture in camera obscuras on W19152v (K/P 118v).

He proceeds to discuss the two figures to the right of this, namely, abc, which is on the far right, (fig. 771) and nmo which is on the near right (fig. 770, or as he puts it, to the left relative to abc):

as is shown on your...right side and /the figure/ to the left /of it/ in the two figures abc /and/ nmo where the aperture /of the figure on the/ right being equal in width to the umbrous body ab, which intersection of such an umbrous body makes itself in the middle between the aperture and the umbrous body in the point c, which the figure to the left /fig. 770/ cannot do, the aperture o being considerably smaller than the umbrous body nm.

He then considers further properties of the diagram at the far left (fig. 769):

It is impossible that the images of bodies can be seen between the bodies and the openings through which the images of these bodies penetrate. And this is apparent because where the air is illuminated such visible images are not generated.

This discussion of where image formation occurs leads him to mention where images are doubled:

Images doubled ;by the reciprocal penetration of each other always double their darkness. To prove this let such a doubling be deh which, although it sees only between the bodies bi, this does not stop its being seen from fg or from fm. It is composed of the images a, ik which are infused with one another in deh.

Here the physics of light and shade dovetails with the problem of image formation in a camera obscura which interests him because of its relevance to vision (see below pp. ). In Leonardo's mind one topic constantly leads to another.

(figure)Figs. 772-776: Demonstrations concerning sizes of apertures in camera obscuras on CA385va.

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These interweaving analogies he develops on CA385vc (1510-1515) where he sketches two examples with two opaque bodies (figs. 772-773) and three cases with three such bodies at various distances (figs. 774-776). Other sketches on the same folio (figs.** ) make explicit the camera obscura-eye analogy and leave no doubt concerning the parallels intended between physics of light and shade and physiology of vision.

9. Shape of the Aperture

When light passes through an aperture does the resulting image on the wall resemble the shape of the aperture or the light source? Already in Antiquity this had been a problem as is evident from two questions posed in the Problemata attributed to Aristotle:

Why does the sun penetrating through quadrilaterals form not rectilinear shapes but circles, as for instance when it passes through wicker work?6Why is it that during eclipses of the sun if one views them through a sieve or a leaf - for example, that of a plane tree or any other broad-leaved tree - or through the two hands with the fingers interlaced, the rays are crescent-shaped in the direction of the earth?7

Alhazen, in the eleventh century, had mentioned the problem of light passing through apertures.8 Witelo, in the thirteenth century had considered briefly light passing through square, round and angular apertures.9 With Pecham this phenomenon emerged as a serious problem. He devoted two of his longest propositions to the properties of images passing through triangular apertures.10

(figure)Figs. 777-778: What happens to round images passing through triangular apertures in Pecham's Perspectiva communis and Leonardo's A82v.

(figure)Figs. 779-782: Round images passing through triangular apertures. Fig. 779, CA144vb; figs. 780-782, CA236ra.

Leonardo stands clearly within this tradition. A diagram on A82v (fig. 778, 1492) bears comparison with standard diagram from Pecham's work (fig. 777). But whereas his predecessors had been content to consider isolated cases, Leonardo is more systematic. In his notebooks he illustrates a series of situations showing how, with greater distance, the image gradually loses the shape of the aperture and takes on the shape of the light source. When the light source, aperture and projection plane are very close, the light passing through a triangular aperture produces a triangular image. This limiting case (fig. 783) he considers in some detail a situation in which a triangular image begins to become curved (figs. 780-782, cf. fig. 779):

The exterior sides of compound derived shade are always seen by all the luminous body.But none of the interior sides of compound derived shade sees any part of the luminous body.

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The more that the converse pyramid, created by decomposed (derived) shade is removed from its angle the more it will see of the luminous body but it will never see half.Or so much more than half to the extent that the luminous body is greater than the umbrous body.The percussion of the luminous ray, which...penetrates inside the concavity of the angle will never impress this angle, but in place...of this it impresses a portion of a circle.But if the angle is convex then the impression of the obtuse angle will make an acute angle.

(figure)(Figs. 783-788: Stages in the transformation of a triangular image to a round image. Fig. 783, Author's reconstruction: fig. 784, CA277va; fig. 785, C10v; fig. 786, Forst III 29v; fig. 787, H227 inf. 50v-51r; fig. 788, H227 inf. 49r.

How the shadow of an obtuse angle makes itself acute with curved sides.Of the curved and acute sides the derived shadow of the...obtuse angle is made convex from rectilinear sides.And this is proved. And let the luminous object be ah of which all its rays see the obtuse angle c of the straight sided triangle kgc and thus the left ray a passes the angle c and bends to the side.As the distance increases each point of the triangle generates a circle in the form of the light source which results in a triangular configuration of three circles.

He illustrates this situation on CA277va(fig. 784, 1508-1510). On C10v (1490-1491) he shows (fig. 785) a next step in this process where the circles have begun to overlap. The accompanying text summarizes the phenomenon:

It is impossible that the ray born of a spherical...luminous source, can at a distance go /on/ conducting in its percussion the similitude of the quality of any angle in the angular aperture through which is passes.

On Forster III 29v (fig. 786, c. 1493) he demonstrates a next stage in this process of transformation. The distance is now greater and the three circles have begun to overlap more. The accompanying text is brief: "The angle is terminated in a point. In the point are intersected the images of bodies." A next stage, where the distance is greater and the circles overlap even more is recorded on Manuscript H 227 inf. 50v-51r (fig. 787) accompanying which is a thorough explanation:

If the entire body of the sun sees all the square aperture it is necessary that every minimal part of this aperture sees all the sun and transfers it all behind the first wall where it terminates the course of the solar rays. Therefore no angle can appear over a long distance...in the solar sphere.

The point of the triangle B is centre of the circle D. A is the centre of the circle E and similarly C comes to be centre of F and if make a triangular aperture in a plate of thin iron of similar size and you make the rays of the sun pass inside and receive them in an object

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that much distant from a similar triangle that when the rays dilate in the size of the circle CDF , you will see the little triangle make itself in a spherical form.

When the distance is greater still the image takes on entirely the shape of the original light source and is fully round, in spite of having passed through a triangular aperture.11 This situation he describes in draft on CA135va(1490-1492) and develops on H227 inf. 49r (fig. 788):

CA135va H227 Inf. 49rThe further from the intersection The further from the intersection/that/ the second pyramid /is/, ... that the base of the secondthe more it expands at the object. pyramid made the object isIts circles created at the angle generated, the more it expands.of the aperture enter more into And its circles created in thethe body (the one than the other) angles of the apertures areand to the extent that they incorporated together more, andincorporate one another, the more the more that they are incorpora-the solar ray remains round at ted together, the rounder thethe object. base of the solar pyramid remains.

When the rays have made so long apath that they enter four hundred timesinto the greater diameter of theaperture, the rays will carry to theobject a spherical body and the formof the aperture will be lost.

(figure)Figs. 789-790: Eye looking through a semi-circular and triangular aperture on A61v and H71/23/r.

(figure)Fig. 791: Shadow produced by a round light source striking a rectangular occlusion on C12r;Fig. 792:Image produced by a rectangular light source passing through a round aperture on CA256rc.

(figure)Figs. 791-800: Transformation from a square image to a round image generated by a round light source and square aperture. Fig. 793, Author's reconstruction; fig. 794, Forst II 5v; fig. 795, H227 inf. 48r-48v; fig. 796, CA135va; fig. 797, H227 inf. 48v; ig. 798, CA135va; figs. 799-800, H227 inf.

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Given his interest in the light-sight analogy (see above pp. ) it is not surprising to find him on H71 /23/(r) (fig. 790, 1494) examining what happens when an eye looks through a triangular aperture. "The eye," he concludes, "does not comprehend the nearby luminous object."He is equally interested in the properties of square apertures and occluding objects. On C12r (1490-1491), for instance, he (fig. 791) asks: "What shadow a square umbrous body will make with a spherical luminous source?" On A20r (1492) he notes how: "the solar rays repercussed on the square mirror will rebound in a circular form on the distant object." On CA256rc(c. 1492) he considers a variant situation in which (fig. 792) a rectangular light source passes through a circular aperture and projects a rectangular image.

When a circular light source, a square aperture and a projection plane are close to one another, the projected image takes on the squareness of the aperture. This limiting case (fig. 793) Leonardo does not illustrate. As the distance increases each of the four corners of the aperture generates intersecting circles. This situation he illustrates (figs. 794-797) and describes at length in a passage recorded on H227v inf., 48r-48v:

It is possible that the sun, having passed through four apertures, composes a spherical body on the object at a long distance. Let the four apertures be A, B, C, /and/ D. When the circles created by the sun by the said four apertures have expanded so much over a long distance that each one intersects itself with the nearby one in such a way that N /and/ M touch one another, then the four circles compose a single circle. The first degree of the light that is in the four circles is at E, because there the four luminous circles are superimposed on one another. F, G, H /and/ I are one quarter less bright then E, because there only three circles are superimposed on one another. P, K, L /and/ O are the half less light than E, because there two circles are superimposed. Q, R, S /and/ T are 3/4 less light /i.e. 1/4 of the light/ than E, because there is only one circle.

The passage ends with a description of what happens when the distance is increased:

Which /circles/ in going a long distance are lost because they are converted to darkness and thus P, K, L /and/ O become rounded and complete the spherical body and finally over a long distance the square E is converted to a circle and all the other parts of less duplicated light are lost.

How these circles gradually come together as the distance increases he sketches roughly on CA135va (fig. 798, 1492), and then with more care in a diagram recorded on H227 inf. 48v (figs. 799-800) accompanying which he writes:

Proof in what way the square is made in the form of a sphere by the solar rays at the object.The point A spreads to M /and/ N and at a greater distance it spreads to OR and it sees the point of the angle which is extended to AS and then at AT when it has reached AV. The line above RD is consummated and the summits of the spheres will touch one another through the intersections of the circles and then the square is reduced to a circle.

This description of how a square is transformed into a circle effectively provides a visual demonstration of the age old problem of quadrature of the circle. Which raises the question: did

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Leonardo perhaps see in these optical experiments a case study in principles of practical geometry? This would, for example, account for a striking resemblance between the diagrams (figs. 795-795) just analysed and a diagram on Forst II 5v (fig. 794, c. 1495-1497), above which he writes "True proof of the square" and below which he adds:

If 4 circles are situated on the he line of a single circle with their centres in such a way that the circumference line of each one is made on the centres of each one, it is certain that they are equal and the circle where such an intersection is made, remains divided in 4 equal parts and is one half /the size of/ each of the four circles and inside each circle a square of equal angles and sides is produced.

Whether or not he saw these connections with transformational geometry, he was clearly fascinated in studying various stages in the process of a square image becoming round. On CA135va (fig. 798, 1492), for instance, he records a further stage in which the four circles nearly overlap one another completely. Accompanying this he drafts another passage which again emerges in more polished form on H227 inf. 49r (fig. 799):

CA135va H227 inf. 49rIt is a necessary thing that the It is a necessary thing that theintersections of...round intersection of the round pyramidpyramids is made in a single is made in a single closed pointpoint which is closed...of a of a non-transparent circumference.non-transparent circumference. Therefore the half-round pyramidIt is necessary that the inter- will divide the point onlysection made by the half-round surrounded by the half part, whichpyramid is amde in a point point you will find in the rightclosed only...by the half part. angle and if you approach it to

the angle of the eye it willappear to you in the form of ahalf-circle.

On H227 inf. 50v (fig. 800) he considers a case where the distance is greater still and the image becomes completely round, losing all trace of the square aperture:

Of luminous bodies.Because the effects have similitude with their causes the sun, being a spherical body, it is necessary that the solar rays over a long distance do not retain the form of any angular aperture whence they pass, but rather, that they demonstrate after this the form of their cause in the first percussion.

An understanding of this phenomenon how the image of a round light source passing through a square aperture is square at a close distance and round at a greater distance helps us, in turn, make sense of a passage on A64v (1492) that is puzzling if read out of context:

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The intersection of luminous rays made at the faces of the square aperture is produced beyond the said faces. And the intersection made at the angles is made in the size of its angle.(Every aperture carries its form to the object over a long distance.)

No aperture can transmute the concourse of luminous rays in such a way that over a long distance they do not bring to the object the similitude of their cause....

(figure)Figs. 801-808: Effects on images of slit-shaped apertures. Fig. 801, H227 inf. 51r-51v; figs. 802-804;, CA135va; fig. 805, H227 inf. 81r-51v; fig. 806, A64v; fig. 807, CA135va; fig. 808, , H227 inf. 49v.

Having considered the characteristics of triangular and square apertures, he explores slit-shaped apertures. On CA135va (1492), for example, he draws (fig. 802) a case where each of the two end-points of the slit produces a circle. He then redraws the slit (fig. 803) showing how the two circles overlap more at a greater distance. These diagrams are without text. When he pursues the problem on H227 inf. 51r-51v (fig. 801) he adds an explanation:

The solar ray passing through an angular aperture in the percussion made by it on the wall will not carry the true similitude of the aperture.Proof how the angles are the cause of making that the apertures render on the object the solar rays in spherical form.The aperture composed of four faces makes four angles A, B, N (and/ M and these angles are the cause that the pyramids which have passed through them expand over a long distance in such a way that, occupying the faces they will make in the object the spherical light.

On CA135va (fig. 804, 1492) he also sketches a related case where two rectangular faces are positioned such that they produce an open slit. This he again draws and describes on H277 inf. 51r-51v (fig. 805):

Where the aperture has its faces without angles as PQ shows the solar rays having passed through, this will make on the object precisely the shape of the aperture with two lines. Therefore, if the sun, having passed where there are angles, it is made round, and where there are not angles, /it is/ never /made/ so round. This clearly shows that the angles are cause.

(figure)Figs. 809-814: Steps in the transformation from a cross shape to a round shape. Fig. 809, C9r; fig. 810-812, CA135va; fig. 813, H227 inf. 49v-50r; fig. 814, C10v.

On A64r 91492) he sketches (fig. 806) and comments briefly on another variant in which one slit is placed above another: "It is impossible that the luminous rays which have passed through parallels demonstrate to the object the form of their cause." On CA135va (1492) he sketches yet another variant in which the two slits are side by side (fig. 807). This he develops on H227 inf. 49v (fig. 808), explaining:

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The sun having passed through narrow apertures which are divided by a small interval, it is necessary that in their percussion it demonstrates the two heads of these apertures in the form of two half circles which intersect one another.

Having placed the two slits above one another and beside one another Leonardo explores the next logical combination in which the two slits intersect one another to form a cross-shaped aperture. On C9r (1490-1491) he illustrates (fig. 809) and describes a case where light source, aperture and projection plane are near one another, with the result that the image resembles the aperture:

The luminous ray which has passed through a small aperture and the stamp of its percussion having been interrupted at a nearby opposition will be more similar to the aperture...through which it passes than the luminous body whence it originates.

On CA135va (1492) he sketches how each of the ends of the cross shaped image acquires a rounded shape (fig. 810). Next he shows how, at a greater distance, four circles emerge (fig. 811) which, at a greater distance still overlap more (fig. 812). Accompanying this he indicates the distances involved in quantitative terms:

The transit of solar rays through an angular aperture is necessary in some space.

At the distance of 20 braccia it will lose the parts e, f, g /and/ h and at 30 /braccia there will be produced a spherical body by the parts a, b, c /and d because the part which is towards the centre is more powerful.

On H227 inf. 49v 49v-50r, he illustrates (fig. 813) a situation where the distance is more than 30 braccia and the four circles are nearly coincident with one another. This he discusses in detail beginning with a general statement:

If each aperture sees the entire body of the sun, it sees all of its parts, which parts are received by all the aperture, and through all /of it/ and all in the parts. Therefore the part of the aperture, even if it is acute and apt to give passage to the sum of the rays that have parted from all parts of the sun which compose opposite (...) in the first percussion a spherical form, similar to their cause.

A specific description of the diagram (fig. 813) follows:

C is the centre of KR; D is the centre of L; E is /the centre/ of MZ; F of HY:G of OX; H of PV; A of QF; B of IS such that the eight exterior angles of the cross /- shaped/ aperture, the solar rays passing inside them compose in the object a round brightness, which brightness is composed of eight circles which make themselves the centre of eight angles of the aperture and each circle has in it fourteen intersections made by its seven companions, which are ninety-eight intersections in all.

At a still greater distance the image loses all trace of the cross-shaped aperture and becomes

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perfectly round like the light source. This situation he demonstrates (fig. 814) on C10v (1490-1491) beneath which he adds:

(figure)Figs. 815-819: Intersecting circles produced by a square aperture, a slit, a triangular, cruciform and star shaped aperture. Figs. 815-818, CA187ra; fig. 819, C7v.

The...luminous ray which has passed through some aperture of a strange form, if the stamp of its percussion be lengthened, will be similar to the luminous body whence it originates.../rather/ than the aperture through which it passes.

Leonardo has analysed in detail the properties of images passing through apertures in the form of triangles, squares, slits, double-slits and crosses. But he is not content to stop here. On C7v (1490-1491) he alludes to a more complex situation:

PerspectiveIf you wish to make the rays of the sun pass through an aperture in the form of a star, you will see beautiful effects of perspective...in the percussion made by the sun which has passed.

Accompanying this he sketches in rough form an eight sided star (fig. 819). The problem is not forgotten. Within two years he describes this eight-sided star at length in connection with eight apertures on CA187ra (c.1492):

Remember that you note the quality and quantity of the shadows.

If you wish to see the clear and well-defined boundary of separation of simple shadows from the mixed ones you will have /the equivalent of a/ cloths for seiving made of paper soaked in turpentine and oil, in which the light of the sun shines and on the inside...you place a thin-board perforated by equal apertures made in a circle in 8 parts equidistant from one another and let the diameter of this circle be one...1/2 braccia and at a half a braccio...from the centre of this circle you will place, away from it /and/ facing you, a dense spherical body. Then you will place between your eye and the said body a thin folio of stationery which touches the spherical body, which is an inch in diameter and looking at its shadow behind it on the paper, the shadow of this body will appear to you precisely in the form of the image shown.

And if you wish to see the simple shadows with all the minuteness of degrees make a star of 8 rays which are as exactly large at the extremities as at the beginning and set this facing the sun, placing behind it the spherical umbrous body and then the paper and then your eye as was said above.

This entire description is written in the margin surrounding a carefully drawn diagram beneath which he writes:

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This shadow is made by a spherical umbrous body illuminated by a light made in /the form of a/ star, which has its rays of equal size.

In the lower right-hand part of this folio he has drawn (fig. 837) a draft of this octagonal star shaped aperture. Directly above this are four intersecting circles such as would result from a square aperture (fig. 815, cf. figs. 794-797). Beneath the principal diagram showing the shadows produced by the octagonal star (fig. 838) are three other diagrams of intersecting circles: two circles, as would result from a slit (fig. 816, cf. fig. 803), three intersecting circles, as would result from a triangular aperture (fig. 817, cf. figs. 784-785), and four intersecting circles as would result from a cross-shaped aperture (fig. 818, cf. figs. 811-812). In short, this octagonally shaped aperture marks the culmination of a series of experiments.

(figure)Figs. 820-823: What happens to a round light passing through a single aperture. Figs. 820-822, CA277va; fig. 823, CA241rd.

(figure)Figs. 824-825: What happens to a round light passing through two apertures on CA277va and CA241rd.

On CA256rc (c.1492) he drafts a summary of these results: "In the percussion of rays is demonstrated part of the nature of its cause." Which idea he then reformulates in a passage headed:

On the nature of aperturesAn aperture is composed of a number of sides and that which has fewer will demonstrate the truth of things less.That which has more is better and maximally when the parts of the sides are equidistant from the centre of the aperture.

10. Number of Apertures If we return to read more carefully the passage on CA187ra (c. 1492), we find that he not

only mentions an eight-sided aperture, but also eight apertures equidistantly arranged in a circle. This is not an oversight on his part. For just as he has been studying the properties of multi-sided apertures, so too has he been exploring the comparable properties of multiple pinhole apertures. On CA277va (1508-1510), for instance, where he outlines his new plan for arranging the work on light and shade (cf. Chart 10 above), he illustrates the image cast by one pinhole aperture (fig. 822), two (fig. 824), three (fig. 829), four pinhole apertures (fig. 834), and a rough sketch with the image cast by perhaps as many as eight pinhole apertures (fig. 838, although this could well represent an advanced stage in the rounding produced by three apertures, fig. 830, cf. 831).

These draft sketches on CA277va (figs. 822, 824, 829, 1508-1510) are developed on CA241rd (figs. 823, 825, 836, 1508-1510), this time accompanied by text. In the upper right-hand margin he notes in passing: "Many minimal lights in the long distance will continue and make themselves noticeable." In the main body of the text he drafts general rules of light and shade (see below, p. ). Alongside the drawings he discusses the:

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(figure)Figs. 826-832: Intersecting circles produced by three apertures. Figs. 826-831, CA277va, fig. 832.

(figure)Figs. 833-836: Effects produced by a light source passing through four apertures. Fig. 833, A177rb; fig. 834, CA177ve; figs. 835-836, CA241rd.

Nature of the light which penetrates apertures.

Of the light which penetrates apertures it is to be doubted whether, with the dilation of its rays, it recomposes as much size of impressions behind such an aperture, as the width of the body causing these rays. And other than this, whether such a dilation is of a power equal to the luminous body.

To the first doubt it is replied that the dilation made by the rays behind their intersection, recomposes such a size behind the aperture,...as was that which it had in front of the aperture: the space from the luminous body to the aperture being that which /there/ is from the impression of these rays to this aperture. And this is proved by the rectilinearity of luminous rays, concerning which it was proved that there is such a proportion from size to size...as there is from distance to distance of their intersections.

But the power does not go with the same proportion...as was proved, where it is stated:...such is the proportion of heat to heat...and of brightness to brightness of the luminous rays in the same centre, as there is from distance to distance from their origin. Therefore it is proved that the luminous ray loses that much in heat and brightness, to the extent that it is removed from its luminous body.

It is true that compound...derived shadows, that originate from the edges of such apertures, break such a rule through their intersections and...this is treated fully in the second book of shade.

(figure)Figs. 837-838: Effects produced by a round light source passing through an eight sided aperture.

Having answered his first two questions he considers the proportions of light involved under the heading:

Of the proportion that the impressions of light have placed partly one above the other.

Such are the proportions of lights that are generated in the impressions of luminous rays...in part superimposed on one another as is that which the number of impressions have, which are superimposed among one another.

To demonstrate this he now describes his figures (figs. 823, 825, 835, 836):

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This is proved in the 2nd: And let the luminous rays be mb and mc which penetrate through the aperture op to the impression bc, which impressions are superimposed in part at the space n. I say that...the illuminated space, n, will be doubly bright than the remainder of the two impressions b /and/ c because n is seen twice by the luminous source m, and b and c are seen a single time. And by the second of this: such is the number of its luminous sources illuminating it at equal distances. And the same recurs in the 3rd figure where qmop have one degree of brightness, dpgh have two, enci have three and a 4. Therefore we shall say that the degrees of light will be as many as the number of apertures.

Further examples of four apertures occur on CA177rb (fig. 833); CA177vc (fig. 840, 1508-1510) he makes a rough sketch involving perhaps as many as eight apertures. On CA187ra (cf. fig. 838, 1492) he explicitly describes the use of eight apertures and on CA385vc (fig. 841, 1510-1515) he carefully draws apertures and the eight circles thereby produced. On the same folio he sketches two other cases with 18 apertures (figs. 842-843). This theme of multiple apertures is developed on CA241vc (1508-1510). Here he draws three intersecting circles which frame 24 apertures (figs. 845). Beneath this he draws another diagram (fig. 846) with 24 apertures to show how these, at a greater distance, produce 24 interlacing circles.

Accompanying these diagrams on CA241vc is a text that develops the ideas of CA241rc:

(figure)Figs. 839-843: Effects of light produced by multiple apertures. Figs. 839-840, CA277va; figs. 841-843, CA385vc.

DOn lightOf the proportion that there is from illuminated object to illuminated object by a same luminous light.Such will be the proportion...of the brightnesses that the illuminated sites of a same luminous body have as is that of the number of apertures through which this luminous body illuminates the aforesaid site. This is proved by the third placed behind this folio at the foot.12

This leads to a second general rule:

Of multiplied brightness taken from a single luminous body.

The brightness of a same luminous body at an equal distance will make itself of more power, to the extent that the number of the apertures, whence it penetrates to its impression (onto a same place).And this is proved in the 3rd, behind this face, etc.But it is also proved with the 13th of the other book where it is stated: that part of a site will be more illuminated which is seen by a greater number of luminous bodies.

(figure)Figs. 844-846: Effects produced by 24 apertures on CA241vc.

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On CA229vb (1505-1508) he takes this theme further. He begins with a rough sketch showing two circles inscribed within a larger one (fig. 847): this might represent a situation involving two apertures. He then draws (fig. 848) four apertures and the four circles thereby produced. There follow other examples which are multiples of four, beginning with a sketch (fig. 849) of 16 (4 x 4) apertures with a hint of the circles they produce. Next he draws (fig. 850) 12 points on a half-circle which would amount to 24 (4 x 6) apertures in all. A case (fig. 851) involving 21 (4 x 7) apertures follows. Finally he draws (fig. 852) a series of eight circles which span but a quarter of the circumference of a circle that would contain 32 (4 x 8) apertures. Of these various examples only the case involving 28 apertures (fig. 851) is accompanied with a text:

A piece of iron is perforated with the perforations of a sieve and it takes the rays of the sun in such a way that all the perforations become enlarged, as the circle an and in the middle a makes the multiplication of rays placed one over the other which occupy the space of the lesser...circle m which will be warm and lucid.

We have already noted Leonardo's implicit comparison between multiple sided apertures (triangles, squares, crosses, octagons) and multiple-apertures (1, 2, 3, 4, 8, 16, 24, 32 pinholes). On looking more closely at CA229rb, vb as a whole another, theme of comparison becomes apparent: he is analysing multiple-apertures on the same folio that he is exploring multiple shadows produced by a St. Andrew's cross. This is not a coincidence. His analyses of the multiple shadows produced by a St. Andrew's cross occur on CA37ra, 177rb, 177ve, 241rcd, and CA229rb, vb (see above pp. ). These are the very folios on which he also explores multiple aperture problems of light (see Chart 18).

(figure)Figs. 847-852: Effects of multiple apertures on CA229vb.

CODEX MULTIPLE APERTURES MULTIPLE-SIDED MULTIPLEAPERTURES SHADOWS

CA177rb 4 4CA177ve 4 6CA241rcd 1,2,3,4 2,4CA241vc 24 1 (on multiple

surfaces)CA229rb,vb 4,16,24,32 2,4,6CA277va 1,2,3,4, 3 1CA37ra 1 (in various degrees) 2,4,6CA385vc 8, 18 1,2,3,CA187ra 8 2,3,4,8 2,3,4,8

Chart 18: Links between Leonardo's work on multiple apertures, multiple-sided apertures and multiple shadows.

(figure)Figs. 853-863: Draft sketches showing effects of light which passes through a slit and encounters a sphere. Figs. 853-862, CA187va; fig. 863, CA187rab.

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(figure)Figs. 864-866: Development of a demonstration on CA187vab, A89v and A89r.

11. Apertures and Interposed Bodies

We have examined how Leonardo explores the properties of light when it passes through apertures of various shapes such as slits and crosses. A next stage in complexity would be to study such apertures in combination with various shaped opaque bodies. This he does on CA187va (c. 1492) where he makes a series of preliminary sketches to show what happens when light passing through a slight-shaped opening encounters a spherical object (figs. 853-854, 856-862). Above these diagrams he makes two drafts of an explanation:

These umbrous bodies will make their derived shade more or less, depending on whether they are more or less far from the light.

These bodies will make their derived shade more or less short, depending on whether they are closer or further (from the light of the window) from their light.

Unsatisfied, he crosses out these drafts. He turns the sheet ninety degrees and makes two further sketches showing how light, having passed through a slit, and encountering a spherical object, produces a combination of simple and mixed shadow (figs. 855, 864). In the left-hand margin he drafts a phrase: "That light which," then stops short. Alongside the lower diagram he notes: "This light is long and then." Directly beneath this he writes: “No separate shade can stamp on the wall the true form of the umbrous body, if the centre of the light is not equidistant from the extremities of this body.” In the upper left-hand margin he now claims: “No long light will send the true form of the separate shadows (to the wall) from the spherical bodies to the wall.” On CA187ra (1492) he redraws his sketch of light passing through a slit, which encounters a sphere and casts shadows on the wall (fig. 863). Alongside this figure he writes: "Remind yourself that you note the qualities and the quantities of the shadows." On A89v (BN 2038 9v, 1492) he redraws the situation (fig. 865) and on A89r (BN 2038 9r, 1492) he develops it into a beautiful diagram without text (fig. 866). As is so often the case, he expects that his visual statement will speak for itself.

Closer attention to the other sketches on CA187ra (1492) again reveals Leonardo's delight in playing with variables. Having shown what happens when a slit-shaped light encounters a spherical opaque body (fig. 863), he considers what occurs when a spherical light encounters a slit-shaped opaque body (fig. 867). Not content to stop here he lets light pass first through a round and then through a slit-shaped aperture (figs. 868-869) and contrasts this with light which passes first through a slit-shape and then through a round aperture (fig. 870). Next he replaces this slit-shape with a cross shape (figs. 871-875).

This final example can be seen as a starting point for his illustration more than fifteen years later on CA207ra (fig. 876, c. 1508-1510) to "make a crucifix enter a room." Here he takes a blank wall on which he marks a crucifix. Opposite this wall he positions an aperture which is in a room. The sunlight reflects the light of the wall, enters through the camera obscura and casts the image of a cross into the room, (cf. Kircher's later example, fig. 877).

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(figure)Figs. 867-875: Effects of light and shade involving combinations of apertures and/or occluding objects on CA187ra.

(figure)Fig. 876: Reflection of a cross shape through a camera obscura on CA207ra. Fig. 877: Development of this principle in Athanasius Kircher's Ars magna lucis et umbra (1646).

(figure)Figs. 878-879: Apertures, occluding objects and shade on triv. 22v and C11r.

(figure)Figs. 880-882: Sunlight, bubbles in water and cross shaped images on F28v.

A more complex play with cross-shaped images is suggested on Triv. 22v (fig. 878, 1487-1490) which may be the basis of his diagram on C11r (fig. 879, 1490-1491) where light passes through a cross-shaped aperture, encounters a transparent sphere and then casts a rounded cross-shaped shadow. Accompanying this he notes:

The shape of the derived shade will always have conformity with the form of the original shade.The light in the form of a cross is the cause why the umbrous body of spherical rotundity will cause from it shadows in the shape of a cross.

In 1508, he returns to this problem of cross-shaped images, now in an unexpected context. On F28v (fig. 880) he observes that:

The ray of the sun, having passed through the bubbles of the surface of the water sends to the bottom of this water an image of this bubble that has the form of a cross. I have not yet investigated the cause, but I believe...that it is because of the other little bubbles...which are joined to this larger bubble.

By way of illustration he makes two sketches to show how smaller bubbles13 surrounding the larger bubble (figs. 881-882) might serve to generate a cross-shape. On CA236rd (1508-1510) he makes a note: “on the shadows situated at the bottom of the water and which send their species to the eye through water and through the air,” but proceeds to discuss refraction (see below p. ). He appears not to have pursued the problem of cross-shaped bubbles as he had hoped.

(figure)Figs. 883-887: Slit shaped apertures and shade on CA258va.

In the period 1508-1510 he does return, however, to problems of slit-shaped apertures and opaque bodies on CA258va. Here he begins (fig. 883) with light passing through a slit-shaped aperture which encounters a narrow opaque body and casts a shadow on the ground at ninety degrees to this. Directly beneath he explains:

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When the light of a long shape generates derived shade in rectangular conjunction with primitive shade, then the derived shade, in every degree of its length...diminishes its first darkness.

And this is proved by the 4th of this, where it is stated: so much of the darkness of derived shade is lost to the extent that it makes itself remote from its primitive shade.

Next he considers a situation where this shadow is cast at more than ninety degrees (fig. 884):

To the extent that the angle which is generated at the conjunction of the derived shade with its primitive shade is larger, to that extent the boundary of the derived shade is ?. And this arises by the said 4th because: to the extent that the angle created by the conjunction of the derived shade...with the primitive shade is of greater size then the opposite extremities of such shade will be...more distant from one another and by the 4th, the derived shade will be of less darkness.

Immediately following he turns to the converse:

There follows the converse of the said.The extremity of the derived shade will be that much darker...to the extent that the angle which is generated in the conjunction of the primitive shade with the derived shade is more acute. And this is proved by the converse of the fourth which states: ...that part of the derived shade will be darker which is closer to its primitive shade.

(figure)Fig. 888: A slit-shaped aperture, a slit-shaped object and its shade on CU630.

To illustrate this he moves an interposed stick through various degrees of obliquity (fig. 885). Finally he considers a case where a slit-shaped aperture, thin opaque body and the resulting shadow are all in the same plane (figs. 886-887)

Therefore, the more acute angle being the cause of making its sides closer, it is necessary that the primitive shade and derived shade of which the sides are composed are again much closer to one another.

This situation interests him and on CU630 (TPL627, 1508-1510) he examines it in more detail under the heading:

Of the derived shade created by light of a long shape which percusses an object similar to it.When the light which passes through an aperture of a long and narrow shape percusses the umbrous body similar to it in shape and position, then the shade will have the shape of the umbrous body.

A specific demonstration follows (fig. 888):

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This is proved. Let the aperture, through which the light penetrates into a dark place, be ab and let the columnar object equal to and of the same shape as the aperture be cd. And let ef be the percussion of the umbrous ray of the said object cd. I say that such a shade cannot be /either/ greater or less than this aperture at any distance, the light being conditioned in the said way. And this remains proved by the fourth of this which states that all umbrous and luminous rays are rectilinear.

(figure)Figs. 887-891: Combinations of apertures and occluding objects on C9v, W12352v and CA236rc.

Having studied in isolation the effects of different shapes of umbrous bodies and apertures, he examines various situations where these factors act in combination. On C9v (1490-1491), for instance, he draws a light source (fig. 889) the rays of which, on meeting an opaque body, cast a shadow which passes through an aperture. On the far side of this aperture are two further light sources g and h which cast rays intersecting this shadow. Directly beneath this diagram he adds a brief text:

The simple percussion of derived shade will not change its darkness even though its umbrous rays are changed and mix in the air with luminous rays. The figure on the right is well placed over this said proposition.

On W12352v (c. 1494) he draws another diagram (fig. 890) of a luminous body the rays of which meet an opaque body and cause shadows which then pass through an aperture. Here there is no accompanying text. But then on CA236rc (1508-1510) he redraws the diagram carefully (fig. 891) and adds a full explanation under the heading:

What difference there is between shadow and image.

The difference that there is between simple shade...of the opaque body to the image of this body is that such simple shade does not penetrate inside minimal apertures as...does the image of the same umbrous body.

This is proved. And let the umbrous body be cd and let the luminous body accompanying the umbrous body in the generation of shadows be ab and let the aperture be r through which the said...species penetrate into the dark place vmhn. I say that the simple derived shade, cdp is first intersected at this p which comes to the aperture r and spreads in such a way that it cannot penetrate through this aperture.

(figure)Figs. 891-894: Apertures and occluding objects in combination on CA216rb.

(figure)Figs. 895-901: Effects of spherical and rectangular occlusions on shade on CA238vb.

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Meanwhile, on CA216rb (c. 1495) he had been exploring more complex variants of this situation. In a first diagram (fig. 892) a light source is left undrawn and an opaque object casts its shadow through two apertures onto a wall. In a second diagram (fig. 893) there are again two apertures, but now there is an opaque body in front of these apertures and a smaller opaque body behind them. Their shadows combine to produce a series of four intersecting circles. Finally there is a third diagram (fig. 894) which has the same elements and differs only in that relative sizes of the opaque bodies are changed. There is no text accompanying these diagrams.

On CA238vb (1505-1508) he takes a flat rectangular board and a round ball. These he places in near proximity to one another in order that they effectively function as an aperture. He then examines the effects produced by moving the light source and altering relative positions of the board and ball. Accompanying the series of diagrams that result (figs. 895-963) he drafts a number of only half intelligible notes which are here translated without comment:

The more [fig. 900] it is closer to the intersection of the rays, the more m makes the function than the circles and likewise it will do the converse the more it approaches the percussion of these rays.Here [fig. 721] the solar rays make an intersection at the upper limit of the ball and at the lower limit of the axis.But those which are interrupted...are those which are intersected at the board and, after such an intersection, are interrupted by the upper limit of the ball.

All [fig. 901] the luminous rays that are cut by n are lacking at m.

(figure)Figs. 902-909: Demonstrations with occluding surfaces. Figs. 902 -903, CA238v; fig. 904-909, CA133va.

(The umbrous body outside the window)And such an interruption of rays are of those which are intersected at the upper limit of the ball.(The solar rays)(Here the shadow of the board carried by solar rays).Of which it happens (that the)(Here) many correlates, that is derivatives.(Here the...solar rays which terminate the shade)When the solar rays, after their intersection at the upper part of some sphere have to terminate the inferior shade of the figure of straight lines.Of the circle.The shadow /fig. 895/ n is always greater or less depending on...its direct (saetta) vicinity to the shadow of the ball and in its growing and diminishing there will always be beyond the...shadow of the ball in...the shape of a semi-circle as can be seen (/se/ en by him who interposes the eye to the ray of the sun when) by him who puts the shadow of the ball opposite him, near the eye, when the stick touches...the boundary, the shade...of the ball and

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from such a boundary the said stick slowly moves towards the sun, not deviating from the contact of the ball.The umbrous body inside the window, the sun always....

The derived shade of the spherical body illuminated by a light equal to it...will diminish strongly if the percussion is at the limit of the shade of another spherical body (that) that touches the first spherical body.

ae /fig. 897/ is the lower limit of the shadow of the board.

Related diagrams are to be found on CA238rb (figs. 712-719, 1505-1508) and CA133va (figs. 904-909, 1497-1500. With two exceptions these are without text. Beneath one (fig. 908), he points out that "the line ab is the boundary of the luminous body." Below the largest diagram (fig. 909) he writes: "When n touches m, f will touch g." On such folios which represent an interim stage int eh development of his ideas, rough sketches suffice. Careful explanation is not yet necessary.

12. Spectrum of Boundaries

Leonardo's studies of a camera obscura in combination with opaque bodies lead him to abandon his early assumptions concerning clearly defined boundaries and to emphasize instead a spectrum of gradations between light and shade. This he does in terms of demonstrations involving a series of basic arguments: (1) that derived shade has less power to the extent that it is more distant from its primitive shade; (2) conversely, that derived shade has more power when it is closer to its source; (3) to what extent one can speak of uniformity of derived shade; (4) that primitive and derived shade mix with distance; (5) where primitive and derived shade are joined together; (6) where shade is greater; (7) where primitive and derived shade are not joined; (8) implications for the perception of backgrounds and (9) simplified gradations of shade.

We shall consider his demonstrations for each of these arguments in turn and show how these interests lead directly from the physics of light and shade to problems of vision and perception.

12.1 Derived Shade is Less Powerful When More Distant From Its Primitive Shade

The idea that derived shade loses strength with distance is clearly expressed on CA258va (1508-1510):

Derived shade is that...much less...powerful than primitive /shade/, to the extent that it is more distant from this primitive shade. There follows the converse. And derived shade is that much more similar to the primitive shade to the extent that such derived shade is closer to this primitive shade.

He mentions this idea in passing on CU705 (TPL553d, 1508-1510): "The darkness of the derived shade diminishes to the extent that it is more remote from the primitive shade." On CU707 (TPL561, 1508-1510) Leonardo returns to this problem in greater detail in a pass age entitled:

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On compound derived shade.Compound derived shade loses that much more of its darkness to the extent that it is more remote from simple derived shade. This is proved by the ninth which states: that shade will make itself of less darkness, which will be seen by a greater quantity of the luminous body.

A concrete demonstration (fig. 924) is cited in support:

Therefore let ab be the luminous source and lo the umbrous body and abf be the luminous pyramid and lok the pyramid of simple derived shade. I say that...g will be a quarter less illuminated than at f because at f one sees all the light ab and at g a quarter of the light ab is missing such that only cb which is three-quarters of the luminous body ab, is that which illuminates g. And at h one sees the half db of the luminous body ab. Therefore, h has half of the light f and at i one sees a quarter of this light ab, that is, eb. Therefore i is three-quarters less luminous than f. And at k one does not see any part of this light. Therefore there is a privation of light and the beginning of simple derived shade. And thus we have defined compound derived shade.

12:2 Derived Shade is More Powerful When Closer to Primitive Shade

On CA144va (c. 1492) he drafts this idea: “The closer that the derived shade is to the primitive to that extent is it darker...and its boundaries are less /than the/...luminous part that surrounds it.” This he crosses out. On CU730 (TPL598, 1508-1510) he takes up this claim afresh under the heading:

Whether the derived shade is darker in one place than in another.

Derived shade will be that much darker to the extent that it is closer to its umbrous body, or closer to its primitive shade and for this reason it arises that its boundaries are better known in the /ir/ origin than in other parts distant from this origin.

He reformulates the idea on CU699 (TPL606, 1508-1510) under the heading:

Of the boundaries of derived shade.That boundary of derived shade will be darker and better known which is closer to its primitive shade. This is proved by the 5th of this, which states: in the contact that derived shade has with its primitive shade, there...the conjunction of the simple with the compound shadows are not noticeable because, beginning in an angle, they begin in a point, as is proved in the definition of an angle where it is stated: the angle is the concourse of two straight lines in a same point.

Immediately following the objects of an adversary are mentioned and answered:

Even though the adversary says that such lines composing an angle can be curved, this is partly accepted and partly denied, because such lines could be of a same curvature and equally distant from the centre...of that circle surrounded by lines, whence...the contact of those two lines would make a single line and would be like the contact of two straight lines

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in a same directness which, also would not compose an angle but a single straight line. But let us say that an angle is the contact of two straight lines situated outside a same rigour. And of the curves let us say that a curvilinear angle is composed of curved lines with various distances from their circle.

12.3 Uniformity of Darkness

The above two demonstrations serve as basis for his comments on CA258va (1508-1510):

On the Uniformity of Derived Shade.But the derived shade is of more uniform darkness, which has a...more uniform distance from its primitive shadow. And this is proved by the 4th and by the 5th of this which states in the 4th: that part of the derived shade is darker which is closer to the primitive shade and by the 5th: that part of the derived shade will be of lesser darkness which is more distant from the primitive shade. It follows, from these two contraries, that that which is of a uniform distance from such primitive shade is of uniform darkness.

12.4 Primitive and Derived Shade Mix With Distance

The concept that primitive and derived shade mix with distance is implicit in a statement on CA256rc (c. 1492): “To the extent that the umbrous body is closer to the percussion of the rays its shadow will observe the form of its derivation more.” On CA144va (c. 1492) he drafts an idea: “(That part of the derived shade will mix itself less with its boundaries in the light that surrounds it which is closer to the primitive shade.)” This he crosses out and makes two further drafts:

To the extent that the derived shade is more distant from the primitive, to that extent will it mix its extremities more with the luminous body that surrounds it.

To the extent that the light is more distant from the luminous body, the extremities of its shadow and the light will be mixed together more.

One reason for this claim stems from everyday experience as is clear from a passage on A92v (BN2038, 2v, 1492): “How the shadows are confused over a long distance is proved in the shadows of the moon which are never seen.” On CU636 (TPL438a, 1505-1510) he returns to the general problem in passing: "and the derived shade mixes itself the more with the light to the extent that it is more distant from the umbrous body." Which idea he reformulates on CU699 (TPL606, 1508-1510): "That shade is more distinct and defined which is closer to its origin, and the more distant is the least defined," and on CA371rb (1510-1515) he expresses it differently again:

The more distant that the derived shade is from the primitive shade the more it varies from this primitive shade with its boundaries.

12:5 How Primitive and Derived Shade are Joined Together

Related to the foregoing is a demonstration on CU697 (TPL562, 1508-1510) entitled:

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How primitive and derived shade are joined together.

Derived shade is always joined with primitive shade. This conclusion is proved per se, because primitive shades makes the basis of derived shade but they only vary insomuch that primitive shade of itself tinges the body to which it is joined and the derived shade is spread through all the air penetrated by it.

By way of illustration he gives a concrete example (fig. 925):

This is proved and let the luminous body be f and let the umbrous body be aobc, and the primitive shade which is joined to such an umbrous body is the part abc. And the derived shade abcd originates together with the primitive and such a shade is said /to be/ simple in which no part of the luminous body can see.

This theme he pursues on CU708 (TPL563, 1508-1510) headed:

How simple shade is conjoined with compound shade.

Simple shade is always conjoined with compound shade. This is proved by the foregoing where it is stated. Primitive shade makes its base of derived shade and since simple and composed shade are born in a same body joined to one another, it is necessary that the effect participates of the cause. And because the compound shade in itself is nothing other than diminution of light and begins at the beginning of the luminous body and finishes together with the boundary of this luminous body it follows that such shade is generated in the middle between simple shade and simple light.

A demonstration without an illustration follows:

This is proved and let the luminous body body be abc and the umbrous body de and let the simple derived shade be def and let the compound derived shade be fek.

And leads to a second claim:

But the compound derived shade always sees a part of the luminous body, greater or less, depending on the greater or lesser distances that its parts have from the simple derived shade.

Which, in turn, is demonstrated, again without an illustration:

This is proved and let such a shade be efk which, with half of its size sees fk, that is, ik sees half of the luminous body ab which is ac, /and/ this is the brighter part of this compound shade. And the other darker half of the same compound /shade/ which is fi sees cb, the second half of this luminous body. And thus we have determined the two parts of the compound derived shade, the one brighter or less dark than the other.

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12.6 Where the Shade is Greater

On Forst III 87v (c. 1493) Leonardo mentions how the extremities of shade are affected by light:

The luminous or illuminated object bordering on the shade intersects as much as it cuts.

As much of the extremities of the shade of bodies will be lacking as is touched by an illuminated or luminous object.

On H66/18/(r) (January 1494) he notes: "that part of the derived shade will be less dark which is more distant from its extremities." He returns to this idea in two drafts on CA190rb (1505-1508):

The boundaries of all colour which pass through apertures are more evident than their middles.

...The boundaries of the species of each colour that penetrates through a narrow aperture into a dark place, are always of a more powerful colour than its middle.

On this same folio he also drafts another phrase: "That object will make itself darker which is...seen by a greater amount of darkness." The way in which this and related themes are associated in Leonardo's mind is seen clearly on CA230rb (1505-1508) which opens with a series of general claims and a questions:

To the extent that the umbrous body is closer to the luminous body, to that extent is the maximal whiteness more remote from the maximal derived shade surrounded by it.

The boundaries of the maximal derived shade is darker than its middle.

That part of the derived shade will be darker which sees a greater sum of darkness and that will be of less darkness which sees a darkness of less quantity.

The surface of every opaque body participates in the colour of its object.

The medium of uniform transparency gives passage to some species of a given colour or shape without occupation of the site in this medium.

Why shades tinge dense bodies and not rare ones?

This is followed by a further question:

(figure)Figs. 910-915: Gradations of light and shade in camera obscuras on CA230rb.

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Why the shadows intersected behind the maximal shade, lose more darkness to the extent that they approach such a maximal shade?

This is answered with the help of a demonstration (fig. 911):

Let aco be the maximal darkness, codn /and/ aobm are the shadows intersected on the maximal shadow cno. I say that such a shadow, in separating itself from the maximal shade, the further it is removed, the darker it becomes with some space. And this shade increases because the whiteness of the two simple lights and the darkness which proceeds sees the dark background mixed with the light surrounding such a background.

The accompanying diagram (fig. 911, cf. figs. 910, 912) recalls his studies of divergent shade (see above pp. ). Above this diagram he adds a brief caption: "To the extent that g /and/ i are less, to that extent are the whites surrounding maximal shade narrower." To the left of this he draws a further diagram (fig. 913, cf. figs. 912, 914-915), beneath which he explains:

The pyramid Ste is tinged by the colour of its objects and thus makes its background. And for this /reason/ this pyramid is, in itself, variable in its part with various darknesses, such that it sees where it is darker and where it is less dark than the dark object cb.

Which explanation continues in the next column to the left:

Ab sees the end of the shade cd and therefore the derived shade is of little darkness; ec sees all the shade cb and for this reason there is shade of much darkness; nm sees half the shade cb and for this reason it is shade of middle darkness.

And by such a demonstration we have proved that the maximal derived shade is darker in the extremities than in the centre.

Tresf are of the observed darkness because in every part of their length they see a same darkness cbho.

He restate this conclusion in passing on CU699 (TP606, 1508-1510); "The shade will show itself as darker towards the extremities than towards its centre," and sets out to demonstrate it afresh on CA195va (fig. 930, c. 1510):

And in their boundaries colours are more intensive and brighter than their parts.

This is proved by the 4th of this which states: The surface of every opaque body participates in the colour of its object. It follows that the line ap...tinges with itself the surface np and this given line carries with it the boundary of brightness ca with the dark ab and in this line one does not see any part of the dark line, but all brightness. But if you remove yourself more from the boundary of the derived shade, then such brightness gh is tinged by the shade ak and thus the illuminated part is corrupted by the percussion or the mixture of the umbrous image ak which a could not do in gp and maximally in p. And thus is proven our intent to show that the limit(s) of the bright image with the derived shade is

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brighter in act and not in appearance as the boundary wishes than in the remainder of the other background.

Later on the same folio he pursues this theme asking (fig. 930):

Why the boundary of derived shade remains intersected after the pyramid of maximal darkness...and why such an intersected shade is dark towards the angle of the umbrous pyramid and outside it is bright /?/.

(figure)Figs. 916-917: Gradations of light and shade in a camera obscura on CA297va.

First reply.

The exterior limit of the derived shade which is intersected will be that much darker outside than towards the middle because in hx, the more...you remove yourself from x towards s, the more you will find light at al and the contrary you will find at...the opposite side, because the more you remove yourself from p towards o, the more...the darkness ab is demonstrated and this is said concerning the background of intersected derived shade.

Such investigations lead him to examine precisely where gradations of light and shade are brighter or darker. On CA297va (1497-1500), for instance, he makes a preliminary sketch (fig. 916) which he then redraws (fig. 917) and describes:

The line ed sees the luminous body in every part of its length and the line bk sees...the middle of the same luminous body.

And the 3rd line pq sees the entire umbrous body cp and all the luminous body ac.

By that which was said above...the space qg will be that much less dark to the extent that it comes closer to the line dg and the space gf will be that much less bright.

Roughly a decade later he takes up this theme afresh on CA37ra (1508-1510). He now draws two preliminary diagrams (figs. 918-919) and then a third (fig. 920, cf. figs. 921-922). As usual, the accompanying text opens with a general statement:

Speak first of the qualities of divided lights of compound shadow frbc born of particular light.

(figure)Figs. 918-922: Gradations of light and shade in camera obscuras. Figs. 918-920, CA37ra; fig. 921, CA385vc; fig. 922, CA277ra.

The compound shade frbc is conditioned in such a way that to the extent that it is more remote from its intrinsic side, to that extent does it lose its darkness.

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A demonstration follows (fig. 920):

This is proven. Therefore let the luminous source be da and the umbrous body fa and let ae be one of the side walls of the window, that is, da. I say by the 2nd that...the surface of every body participates in the colour of its object. Hence the side rc which is seen by the darkness ae participates in this darkness. And similarly the extrinsic side, which is seen by the light da, participates in this light and thus we have demonstrated such an extremity.

He now writes a new heading: "Of the middle contained by the extremities." He is, however, unsatisfied and crosses out the entire passage. In the right-hand margin he begins afresh:

This divides itself into 4: 1st: of the extremities containing the compound shade. 2nd: of the compound shade within the extremities.

Again he breaks off and in the lower centre of the folio he notes in passing: "Where the shade is greater or less or equal to the umbrous body, its origin." He now turns the folio to the side, draws a considerably more complex diagram (fig. 923) and analyses it in a passage headed:

(figure)Fig. 923: Compound shade on CA37ra.

Of the shade bch.

This is proved because the shade opch is that much darker to the extent that it comes closer to the line ph and is that much brighter to the extent that it comes closer to the line oc and let the light ab be a window and let the dark wall where this window is positioned be bs, that is one of the sides of the wall.

Therefore we shall say that the line ph is darker than another part of the space opch because this line sees and is seen by all the umbrous space of the wall bs. But...the line oc is brighter than any other part of this space...opch because this line sees all the luminous space abe.

He pursues this theme of various gradations of brightness and darkness on CA258va (1508-1510) beginning with two demonstrations (figs. 956):

Abo is illuminated by the entire light cdo but more...at a where it sees all of cd, than at b where the same dc finishes its sight in which a suddenly finishing this sight of the light cd, there begins the sight of the darkness de and the background is tinged by these bright and dark images.

The space opas begins dark at ps...because it sees the darkness de and goes on becoming brighter towards s to that point where it always acquires a greater sight of the light dc and this brightness having finished, the background oan begins to become dark again, because this background is seen by the darkness d and it makes itself that much darker to the extent that it approaches on more and it does the same from the opposite side.

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An interim paragraph follows in which he introduces the question of maximum brightness.

Having proved the cause of the shape and darkness of which the simple derived shade is composed and, other than this, having proved the shape and darkness of compound shade, surrounding this simple shade...it remains to prove the maximum brightness of the background surrounding this compound shade.../by means of/ which we shall also prove the necessity of the maximum brightness of the aforesaid background.

To this end a further demonstration follows:

Therefore let the line oa be the boundary of the compound shade oba which, as was said, is seen by all the light cd and is illuminated the more to the extent that it is closer to the line oa where it sees all the light cd and it is illuminated the less to the extent that it comes closer to the opposite side qs. Therefore the line oa is the brightest part of this derived shade because such a line is continuous with od, the boundary of the light cd, behind which line oa the remainder of the background begins to brighten again, that is, the background aon, which background acquires that much more darkness and removes itself more from that line of brightness. And this is proved because, through such distance one always sees a greater amount of the dark background to the side of the light cd, that is, the darkness de.

He returns to this theme on CU669 (TPL719, 1508-1510) under the heading:

(figure)Figs. 924-927: Demonstrations of compound shade in camera obscuras. Fig. 924, CU707; fig. 925, CU697; fig. 926, CU669 and fig. 927, K/P 178r.

Of the brightness of derived light

The most excellent brightness of derived light is where it sees all the luminous body with half of its right or left umbrous background.

The diagram for the demonstration that follows is reminiscent of earlier discussion in this context (fig. 926, figs. 924-925):

This is proved and let the luminous body body be bc and let its right and left umbrous field be dc and ab. And let the umbrous body less than the luminous body be nm and the wall ps is where the umbrous and luminous species are impressed.

I therefore say that on this wall ps at the point r there will be a more excellent brightness of light than in any other part of this pavement. This is shown because at r one sees all the luminous body bc with half of the dark background ad, that is cd, as the rectilinear concourses of the umbrous pyramid(s) cds and the luminous pyramid bcr show. Therefore, at r one sees as much quantity of the dark background cd as there is of the luminous source bc. But at the point s one sees the umbrous /part/ ab and one also sees the umbrous /part/ cd, which two dark spaces amount to double that of the luminous body bc. But the more

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you move from s to r, the more you will lose of the darkness /of/ ab. Therefore, from us to r the pavement sr will always brighten. Again the more you move from r to o the less you will see of the luminous source. And for this reason the pavement ro becomes darker the more one approaches o.

And through such a discourse we have proved that r is the brightest part of the pavement qs.

(figure)Figs. 928-929: Demonstrations where primitive and derived shade are not joined on CA258va and CA195va.

12.7 Where Primitive and Derived Shade are not Joined

On CA258va (1508-1510), having discussed conditions under which primitive and derived shadow are joined he considers (fig. 928):

Of the shadow that does not join the derived and the primitive.

This figure is that...which is described here below...and it is said of the part of the triangle hnp, that is, its sides qop and qnm seen by the light vxs and yor which is that much more or less illuminated to the extent that it is closer or further from the line go or, if you wish, qm.

The fourth which is lacking below at de is...the space psb which becomes that much brighter again to the extent that it removes itself from the angle p...and this proved by the sixth which states: that part of the umbrous body will be...of less darkness which is illuminated by a greater quantity of the luminous body. Therefore our proposition is concluded because, to the extent...that the sides of the triangle psb remove themselves from the...point p, to that extent do they see the light of cd more and to that extent are they seen by a greater sum of light etc.

A similar diagram and demonstration are found on CA195va (fig. 929, 1508-1510) where he observes:

In the triangle grt is the triangle aco which is luminous and it also sees the opposite luminous triangle enp. Therefore this triangle grt will be twice as luminous as in the two lateral triangles ogr and pgt where its light, even though it has the same derivation, is simple, and the other is composed of two lights. Therefore this illuminated triangle...is that which separates the two shadows opqr and optn from one another.

(figure)Figs. 930-931: Reconstruction of CA195va by Pedretti.

These diagrams are the more interesting because they return to a problem that had perplexed him in his earlier studies of light and shade, namely, what causes the shadow of an opaque body smaller than the light source to be divergent.

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On CA195va (1510) he draws (fig. 930, cf. 931) a camera obscura in which the entering light encounters two opaque bodies and produces complex gradations of light and shade, which he describes briefly: “op sees and is seen by ab and is tinged by its colour and on the side p is seen the beginning of the brightness of the air which brightens the place where its image percusses.” Hence this combination of camera obscura and opaque objects provides yet another demonstration for his "colour participates" argument (see above pp. ).

12.8 Implications for The Perception of Backgrounds

At the same time this demonstration serves as a starting point for a further argument.

Why black painted bordering on white does not show itself as blacker than where it borders on black, nor white shows itself more...white bordering on black as do the species which have passed through an aperture or through the limit of some opaque obstacle.

This arises because the species tinge the place they intersect with their colour and when the difform species see a same site, they make a mixture of their colours, which mixture participates more or less in one colour than another, to the extent that the one colour is a greater quantity than the other.

(figure)Figs. 932-937: Gradations of shade in camera obscuras on CA354rb.

This particular demonstration is of considerable interest because, as will be shown (see below pp.** ) he had made various experiments to establish the contrary, namely, that, white on a black background appears whiter and black on a white background appears darker. On this same folio he explicitly compares the effects of a camera obscura with those of the pupil in the eye. Problems relating to physics of light and shade, the physiology of vision and perception are all intimately connected in Leonardo's mind. As a result what had traditionally been philosophical and psychological questions of vision and perception now emerge as problems of physics. Problems of optics are no longer a matter of theoretical debate but open to practical verification by experiment. He returns to this situation of a sphere placed within a camera obscura once more on W19086r (K/P178r, fig. 927, c. 1513) where he notes that:

...Among bodies of equal size and distance that tinges the body positioned opposite more with its species which is more luminous. Of bodies of equal brightness that tinges the surface of its object more which is of larger form, all being of equal distance. Of bodies of equal brightness and size the closest tinges its object more.

12.9 Simplified Gradations of Shade

Parallel with these demonstrations is a further series which omits the interposed opaque sphere and reduces the problem of gradations of shade within the camera obscura to its essentials. Preliminary drawings (figs. 932-941) on this theme are found on CA345rb (1505-1508) amidst discussions of species being everywhere in the air (cf. pp.** ) and how things cannot be seen without apertures (cf. pp.** ). Among these ten drawings, only one is explained (fig. 941):

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(figure)Figs. 938-941: Further gradations of shade in camera obscuras on CA345rb.

dACB /is a/ triangle, /which/ through the luminous base ab illuminates the angle c in the maximum degree of illumination. Dfe /is a/ triangle /which/ has that much less light in angle f than the light c, to the extent that de, the base of f is less than ab the base of c.

On CA190rb (1505-1508) this theme of gradations of light/shade within a camera obscura is developed. In the right-hand column he begins with a preliminary sketch (fig. 942), beneath which he draws a camera obscura with various gradations (fig. 943). To this diagram he adds six letters. These, however, are not explained. Beneath the diagram he merely notes: "That object will be darker which is seen by a greater sum of darkness." He now draws two further diagrams showing gradations of shade in a camera obscura (figs. 944, 947) and in the passage that follows describes the one on the right under the heading:

How and where the dark object mixes...its derived shade with the derived light of the luminous body.

The derived light of the dark walls...lateral with the brightness of the window are those which with their various darknesses are mixed with the derived light of this window and with various darkness except for the maximal light c.

A precise description of the figure now follows:

This is proved: and let da be the primitive shade, which sees all and makes the point e dark with its derived shade, as is demonstrated by the triangle aed of which the angle e sees all the dark base da and the point v is seen by the darkness as, part of ad and since the whole is more than the part, it will be darker than v which sees only a part. Applying the above conclusion to the figure, t.../this/ will be less dark than v, because the base of the triangle f is part of the base of the triangle t and...c is the limit of the derived shade and maximal beginning of the maximally illuminated part.

Here the right-hand column ends. In the upper left-hand column he drafts two further diagrams (figs. 945-946) beneath which he drafts an explanation of the left-hand figure:

The simple light...absees all in the point m and is not...in any other part of the space hs, as the rectitude of the sides...of which the triangle aem is composed, which are in contact with the limits of the aperture fg. L lacks a quarter of the light ab. Therefore it is seen by the remainder of the light bc. K lacks half of the light ar. Therefore it is seen by the other half and 1 is only seen by a quarter of the light ab, that is by de and h is seen by the limit of the light...e is the beginning of the maximal shade...er.

h sees the weak limit of the light and sees...the maximum darkness of the maximum shade such that in this h one sees entirely shade.

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Here his manner of referring to different fractions of light and shade strikes us as familiar. We have encountered it on more than one occasion (see above pp. ). His references to maximal light and shade we have also encountered elsewhere (CA258va, CA230rb, CA345rb). But if the initial thoughts remain similar, their applications are, nonetheless, quite different.

This diagram in the upper left-hand margin is probably a draft for the left-hand diagram (fig. 947) in the right-hand column, which he describes after he has crossed out his draft:

Why the derived light that passes through an aperture into a dark place does not make percussion of uniform brightness.

Let ab be the primitive light of a window. Let rs be the aperture where the derived light penetrates the dark place xtov. Let oc be the percussion of the derived light on the dark wall ov or the pavement of this place. I say that in such a percussion...oc...made by the luminous ray...will not be illuminated by uniform brightness. And this is proved by the 4th which states: that thing will be more illuminated which over an equal distance is seen by a greater quantity...of luminous body. Therefore, being in the percussion of the luminous ray oc, the part c seen by all the luminous source ab, it is necessary that the point c will be maximally luminous and the more illuminated than the point e which is seen by db, part of this luminous body and likewise...the point g will be less luminous than t because it is illuminated by fb, part of db and similarly m will be less luminous than g because it is seen by ub part of fb, whence it follows that the point o is the limit of such...an illuminated /object/ and is...the beginning...of maximal darkness of the maximal derived shade because the point o, besides being the limit of the luminous object ab, as has been demonstrated, sees the entire umbrous body bp, etc.

It is proved how the point o receives in itself the percussion of the maximal shade, part of the darkness of the maximal shade.

(figure)Figs. 948-950: Gradations of light and shade in camera obscuras and the eye on CA190vb.

Here the text breaks off and he gives instructions to turn the "page" to CA190vb (1505-1508) which opens:

O mathematicians throw light on such error.

Spirit has no voice because where there is voice there is body and where there is body there is occupation of place which impedes the eye from seeing the things positioned beyond such a place. Therefore such a body fills of itself all the surrounding air, that is with its species.

This is reminiscent of a passage on CA345 (see above pp. ) which also occurs in connection with a camera obscura passage. The lower part of CA190vb contains various diagrams relating to the inversion of images within the eye (figs. ) to be discussed later in section three. Amidst these diagrams he draws another preliminary sketch of a camera obscura with its gradations of shade (fig. 948), beneath which he draws two more elaborate versions (figs. 949-950), the latter

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of which appears intended to serve as an imitation eye. Alongside this figure he adds a text which is interrupted:

The images of objects are of two natures of which the first...receives the true image of the real thing, the 2nd...retains the same but with confused boundaries of their shape and the first passes with parallel lines onto the surfaces of plane mirrors and the second passes through the apertures of thin walls in a dark place where it enters but...

Here the transition from physics of light and shade in a camera obscura to problems of vision and perception remains implicit.

(figure)Figs. 951-952: Gradations in a camera obscura and an eye on D10v.

13. Camera Obscuras and the Eye

On D10v (1508) this analogy is taken one step further. Here towards the centre of the right-hand column he draws a camera obscura with various gradations of shade (fig. 951). Above this he writes: "first." Above this, in turn he draws an eye in which various rays are being inverted at the pupil (fig. 952). This figure is headed: "second." Between these two figures he adds a brief marginal note:

The boundaries of bodies are little known because such boundaries are made in surfaces reduced to lines which being indivisible are imperceptible.

Lower down the same right-hand column this perceptual problem is pursued:

But the extremities of things drawn (because they are joined to the background where they are drawn, where they figure) are not subjected to this lack, and for this reason paintings that are close to the eye have to be painted with boundaries which are less known than the boundaries of these things that are distant and this you will recognize perceptibly in judging the upper boundary of an object near the eye and then removed from it.

Here the bridge between Leonardo's physics of light and shade and his physiology of vision is manifest. Indeed it is clear how his camera obscura studies which make him aware of differing gradations of light and shade influence both his theories of perception and painting. Leonardo returns to these themes briefly on CA195va (c. 1510) which, as has been noted, is another of those folios on which the camera obscura-eye analogy is explicit (see above pp. ). In the lower left-hand portion of this sheet is a rough sketch (fig. 953) of a camera obscura with five gradations. In the lower centre is a slightly more developed version (fig. 954) with seven gradations and near the bottom is an example with nine gradations (fig. 955). Each of these three possibilities is duly recorded in a brief note: "Make five or 9...or 7 spaces in ir in order that the white no stands in the middle." Beneath this is a further passage which partly explains the bottom diagram (fig. 955):

(figure)

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adsees...rm and the extremity of the light a sees r and it illuminates it little because in the extremities of the light there is little light but in n is seen all the light ad, simple light and yet it is enough light...m...sees ad, light and dc, shadow, begins...will corrupt.../the/ light....

Even if this text is interrupted, the accompanying diagrams remain of considerable interest because they reveal that Leonardo is trying to quantify gradations of shade. He wants, as far as possible, to measure what had previously been a purely subjective problem and thereby he brings the field of optics one step closer to its modern position as a branch of mathematical physics.

14. Conclusions

Although it is generally known that Leonardo worked with the camera obscura and compared the inversion of images in this instrument with those of the eye, scholars often refer to these facts as if they were only mentioned in passing in the notebooks. Our comprehensive study of the topic has shown that Leonardo devoted no less than 270 diagrams to the theme of camera obscuras and that these interests grow in part out of the astronomical tradition.

He uses the camera obscura to demonstrate not only the inversion of images but also that images passing through an aperture do not interfere with one another, that images are all in all and all in every part, that pinhole apertures produce different intensities of light and shade and that inverted images demonstrate a contrary motion.

Mediaeval optical writers had given considerable attention to the images of round light sources passing through triangular and other complex apertures. Leonardo studies the problem systematically in the case of triangular, square, octangular, slit-shaped and cross-shaped apertures. He demonstrates that whether the shape of the projection resembles the aperture or light source depends on the relative distance of these factors. He does not attempt to arrive at a formula for these relationships but he does give some quantitative references to his experiments.

In addition he studies situations with 1, 2, 3, 4, 8, 16, 24 and 32 pinhole apertures. He also studies the effects of light which passes through apertures of different sizes and encounters various interposed objects. Such experiences lead him to new studies of gradations of shade which prompt further analogies with problems of visual perception: why, for instance, the eye cannot perceive clearly the boundaries of nearby objects.

The great importance of these extensive studies of the camera obscura is that they bring various questions concerning the nature of light and shade and vision into the experimental domain of physics. Optics is no longer a problem for philosophical discussion: it is now a domain which requires scientific demonstration. In the section that follows we shall see how this mentality also leads Leonardo to make physical models of the eye. If the answers he finds are not always correct, the new kinds of answers he seeks are nonetheless important.

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