Leonardo Studies II · Chapter 17 of 18
Part Five, Chapter 1: 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. ).
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. ).
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.
As in the case of Leonardo's work on light and shade, his studies of astronomy follow a plan which is sufficiently detialed to permit a reconstruction of his proposed treatise. This treatise confirms the scope of the work that he envisaged, ranging from illusions in the eye, the nature of basic elements, to relationships of earth, moon and sun and ultimately, the equivalence of planets and stars.
The treatise opens (chapter one) with a demonstration that the eye is a source of astronomical illusions or, as he puts it on CU15 (1500-1505), that "the science of astronomy is born of the eye." A consideration of astronomical illusions due to atmosphere refraction follows (chapter two). Some of his contemporaries hold that the moon is either a convex mirror or a body with variable density and transparency. To counter these theories he examines the reflective properties of different kinds of mirrors (chapter three) as well as of dense and transparent bodies (chapter four). In this context he also considers the nature of visual pyramids (chapter five).
The prevailing cosmology of the time assumes that the earth is at the centre of the universe and that if there were solid elements on the moon or other planets, these would fall from the heavens towards the centre of the earth which is also centre of the world. To confute this theory Leonardo makes his own study of the nature of the elements (chapter six) and concludes that the earth is not at the centre of the world but only at the centre of its elements. Similarly the moon and other planets are also at the centre of their own elements (chapter seven).
An outline of the physics of light and shade (chapter eight) next leads to the claim that the sun is the only light source in the universe (chapter nine). From a distance the earth's light is due to sunlight reflected from its oceans. Because these waters have receded since the time of the flood, the earth's light is now less (chapter ten). In like manner, the moon also has oceans which reflect sunlight (chapter eleven), as well as elements (chapter twelve). Indeed, the functions of earth and moon, in terms of their having days, nights, summers, and winters are equivalent (chapter thirteen), and if seen from a still greater distance, the earth functions as a star (chapter fourteen).
Leonardo's treatise On the Earth and its Waters thus emerges as a synthesis relating microcosm and macrocosm, beginning with the sight of a single eye and ending with a vision fo the universe. Vasari1 had a good reason to emphasize Leonardo's work in astronomy.
In one late note Leonardo does on to claim that the sun does not move (K/P 127r, W12669v) but he does not pursue the idea.2 His concept of the universe remains geocentric. Even so he implicitly challenges traditional cosomology. He has abandoned a heirarchical notion of the elements with a chain of being (Lovejoy3) from the baseness of earth to the purity of the heavens. Earth and star are now equivalent. He has opened the way for an infinite universe, but it remains for Copernicus, Brahe, Kepler and Galileo to explore its dimensions.
A survey of the literature shows that previous scholars have concentrated on details of Leonardo da Vinci's optical researches, and have taken into account little more than five percent of the extant diagrams. The present study, with 2,150 diagrams, is the first comprehensive treatment of the subject. By way of introduction the chief themes of classical and mediaeval optics are outlined. A gradual shift towards optics as a problem of physics is identified. Leonardo's role in this shift is suggested. In the Epilogue to Volume One attention was drawn to Leonardo's visual literalism in general. In Volume Two, which is a case study of how he approaches a specific branch of science, it is shown that this mentality underlies his physics of light and shade.
Writers in Antiquity such as Aristotle, Seneca and Vitruvius, had used verbal metaphors and analogies in their claims concerning the natural world. For instance they compared both sight and sound to the blow of a hammer or a bell, and compared the propagation of light and sound to circular waves produced by a pebble when thrown into water. Leonardo uses the same similes but with an important twist. He explores the similes literally (a heightened nominalism?) and records them visually. Hence traditional verbal comparisons between percussion, light, sight and sound become translated into diagrams.
This visualisation of verbal concepts is part of Leonardo's quest to analyse Nature in terms of mechanical principles. As he explains on K/P153r (W19060r), he has drawn up the rules of the four powers of Nature in order to account for movement in animals in mechanical terms. This is also why he plans that his "book of elements of mechanics" should precede "the demonstration and the force of man and of other animals." Leonardo thus points to a strictly mechanical model of Nature as later codified by Huygens and Newton.
Nonetheless, a careful analysis of Leonardo's definitions of basic concepts such as point and line confirms that he has neither Huygens' purely atomistic corpuscular theory nor Newton's wave theory. Using Euclid's Elements as his starting point, Leonardo attempts to make a non-material point the basis of his physics of light and shade. From this analysis it becomes clear that Leonardo must have been familiar with a number of traditional sources, hardly surprising for a man who studied with the humanist Latin scholar, Nicolo Perotti,1 and who, by 1504, had a library of 119 books.
Nonetheless, the question of how Leonardo used his sources, permits no simple answer. On occasion, as in the case of Pecham or Alberti he copies out or translates passages of earlier authors. When he reads Francesco di Giorgio Martini he adds his comments in the margin. We have no evidence, however, that he made a systematic study or commentary of the optical writings of Euclid, Alhazen, Witelo or Pecham.
Detailed study of his work on light and shade confirms that although his notes are scattered and repetitious, they contain the outlines of coherent treatises which, in a number of instances, involves a systematic experimental approach now associated with modern science. He isolates key variables in a problem, keeps all but one of these constant and examines the consequences of changing one variable, step by step.
He applies this experimental approach to his study of one to four light sources, to interposed objects of different shapes such as a pole and cross, and to his studies of the camera obscura, which he examines the effects of light passing through triangular, square, slit-form, cruciform, octagonal and other apertures. In contrast to his predecessors who had examined isolated aspects of these problems, Leonardo considers a series of situations. He frequently offers concrete and abstract solutions to a given problem. He does not, however, arrive at general laws. Algebra as a means of expressing formulae remains foreign to him.
He devotes over 270 diagrams to the camera obscura alone. This is because the instrument serves to demonstrate a number of basic optical phenomena - such as inversion and non-interference of images or their property of being "all in all and all in every part." Because the camera obscura simulates basic aspects of the opticsl process, his well-known comparison of camera obscura and eye takes on new meaning. Far from being a passing simile, it is a bold claim based on a series of empirical studies. For instance, he studies slit-form apertures in camera obscuras becasue he wishes to understand the properties of cats' eyes which are slit-formed.
He also makes glass models of the eye in order to simulate aspects of the visual process. These models are not entirely accurate and even lead him to wrong conclusions, such as the conviction that a double inversion of images occurs in the eye. The real importance of these models lies, however, not in the conclusions which they prompt, but rather in the approach which they assume, namely, a redefinition of vision strictly as a problem of physics.
Leonardo's treatment of various parts of the eye varies considerably, ranging from passing comments in the case of the optic chiasma to very detailed discussions concerning the nature and function of the pupil. Although he plans a thorough anatomical study of the eye and other parts of the visual process, his extant notes suggest that he did not carry out his intent in full. As early as 1489-1490 he studies the sockets of the eye and optic foramen. In the period 1506-1508 he makes experiments using wax injections to determine the shape of the ventricles in the brain and also makes careful studies of the optic nerves and eyeball. There is no conclusive evidence, however, that he ever made an actual dissection of the eyeball.
With respect to the visual process he considers various arguments in favour of an extromission theory of vision, but then rejects these and later unequivocably supports an intromission theory of vision. In his early writings images are discussed in purely theoretical terms. By 1508, the question of image formation has become a physical problem.
Both the Greek and Latin terms for images (Eidolon, imago, simulacrum) had referred indiscriminately to (a) literary images (b) mental visual images such as dreams and hallucinations and (c) visual images. Leonardo's model-making as well as his visual testing of verbal images cuts through this polyvalence of meaning and prepares the way for Kepler's subsequent distinction between imagines rerum (subjective images) and picturae rerum (objective images on walls, etc.).
His theories of the visual process change with time. At first he accepts that images coverge towards a single point. By 1492 he is exploring the possibility that images intersect and subsequently diverge again. Experiments with small objects inf ront of the eye and with objects seen beyond a pinhole aperture convince him that the visual power is spread throughout the eye. These experiments, along with his models of the eye, lead him to conclude that a double inversion of images occurs in the visual process.
With respect to visual appearances and illusions there are may parallels between propositions in Euclid's Optics and claims made by Leonardo in his notes. Even so, there are a number of Euclid's propositions which Leonardo does not consider. Euclid's Optics was devoted primarily to the perception of isolated objects. Leonardo, by contrast, studies the perception of objects in context, including effects of background on apparent size and brightness. Some of his examples have their roots in mediaeval procedents. Other experiments appear to be his own and are again characterized by a step by step method.
These trends are also evident in his study of optimal and minimal conditions of vision. His treatment of problems such as the central ray, the limits of the visual field, perception of objects smaller than the eye and diplopia also buids on the mediaeval optical tradition. But here again his use of diagrams, models and experiments sets him apart from his predecessors.
In part four it is shown that manuscripts D and F are advanced drafts in which a series of ideas are restated, often more than once. It is found that the sequence of his arguments does not follow a straightforward page sequence. This is important because it establishes that the inherent order of Leonardo's ideas cannot be achieved by a simple reshuffling of pages and is only possible through a thematic treatment such as in this study.
An analysis of Manuscripts D and F confirms that Leonardo's optic studies have a practical goal of understanding illusions in astronomy. This leads to an outline of his treatise On the Earth and its Waters in which he uses everyday experiences of the sun's image reflecting in water on earth as a basis for his claims that the sun's image is equally reflected by the oceans of the moon and that the planets are therefore, ultimately equivalent in their functions. He concludes that from a great distance, the earth is effectively a star.
In the Appendices Leonardo's writings on optical instruments are analysed. In the case of spectacles, it is shown that he considered both bi-convex and bi-concave lenses. His notes concerning early forms of telescopes are assessed and his various lens grinding devices are described. Leonardo gives relatively little attention to plane mirrors. By contrast, he makes detailed studies of convex and concave mirrors in order to determine the location of the angle of incidence. For this difficult task, today remembered as Alhazen's problem, he appears to have relied very little on Alhazen and focussed instead on his own experimental evidence. He does not, however, achieve his own stated aim to discover a general rule.Leonardo's notes on refraction, the rainbow, and other meteoreological effects do not incorporate the findings of his late mediaeval predecessors.
A complex picture of Leonardo's position within the optical tradition thus emerges. In some cases he did not know of, or ignored the work of his predecessors. In many cases he used their examples as a starting point for more systematic studies. This systematic aspect of his work helps explain the great increase in visual expressions - sketches, diagrams, drawings and paintings - at the end of the 15th Century. With the development of methods for rendering individual organs and objects came a challenge of drawing from four, six or eight viewpoints and in as many as eight or ten layers. Hence there evolved also a commitment to consider a problem in as many situations as possible. Instead of mentioning camera obscuras in passing, Leonardo draws over 270 examples. Instead of copying a traditional model of the eye, he draws at least 30 variants. A systematic play with variables thus generates an unprecedented number of images both in science and in art.
The explosion in artistic activity, now associated with the Renaissance, is thus, in large part, an active process on the part of a handful of exceptionally conscious individuals. This bears emphasis in order to balance a fashion which explains this increase in the scope of art as a passive response to economic and social factors, one argument being: newly affluent bankers and capitalists wanted novel means of displaying their wealth and thus created a demand for more art.
In Leonardo's case the situation is more complex. Some of the commissions for which he is specifically paid he never completes. The practical tasks for which he is paid, as engineer, architect and military advisor, he does well, but these do not explain his universal fame. His patrons, the Medici, the Sforza and ultimately Francis I have the wisdom to leave him alone sufficiently to enable his exploring many "useless" pursuits: anatomical drawing, perspective, movement of water, flight of birds, principles of mechanical motion, geometrical transformations and a theory of four powers of Nature. Paradoxically it is precisely these activities which his contemporaries would have dismissed as useless and a waste of time that later become cornerstones of early modern science and art.
Optics was among these useless activities. There was no profit to be gained from studying the physics of light and shade, drawing models of the eye or examining illusions of sight. Yet such studies are a central dimension both of his own development and that of the Renaissance. The Sforza court where Leonardo works for more than fifteen years is an experiment in creating a context where individuals pursue study for its own sake, and results in studies which, in the long run, prove more useful than the most practical machine or financially attractive investment.
On the other hand it is mistaken to imagine that Leonardo is centuries ahead of his time. In the case of optics his insights and experiments may often presage the later work of Kepler, Huygens or Newton, but he lacks their mathematical formulation of optical principles. He lays the cornerstones for a new study of optics strictly as a problem of physics, independent of philosophy and theology. But it remains for others to complete the construction that he begins.
The full story of how that edifice is built requires much further study. This contribution has analysed only writings unquestionably linked with Leonardo. Other scholars, will need to correlate these with the Zaccolini manuscripts to which Pedretti drew attention, and which Clearfield Bell has examined in part. It will then be necessary to reassess Leonardo's influence on Jerome Cardan, Fracastoro, Maurolyco, G - B della Porta and other sources that lead to Kepler, Snell, Grimaldi, Huygens, Hooke and Newton in the seventeeenth century.
Many of us imagine that a genius is one who is ahead of his time, an individual who warrants study because he has quick ways of reaching answers, possesses an easy way to truth. Perhaps, however, a genius is precisely he who knows that restatement, repitition and even blind alleys are essential aspects of profound searching. If so a genius confronts us not with the simplicity of intellectual pregress, but with its complexity, prompts us to reflect upon the cumulative nature of knowledge and the continuity underlying innovation. He knows that there are ultimately no short cuts and, like Leonardo, rejects abbreviators.
E. H. Gombrich4