The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance

by Fritjof Capra

  Figure 8-2: Section of the human skull, Anatomical Studies, folio 43r

  Leonardo demonstrated his thorough understanding of linear perspective not only in his art, but also in his scientific drawings. While he was conducting his experiments on the geometry of perspective, he also investigated the anatomical connections between the eye and the brain.

  He documented his findings in a series of magnificent pictures of the human skull, in which the foreshortening of visual perspective is employed to great effect (see Fig. 8-2). Leonardo combined this technique with delicate renderings of light and shade to create a vivid sense of space within the skull, in which he exhibited anatomical structures that had never been seen before and located them with complete accuracy in three dimensions.16 He used the same mastery of visual perspective and subtle renderings of light and shade in his technical drawings (see, for example, Fig. 8-3), depicting complex machines and mechanisms with an elegance and effectiveness never seen before.17

  While he skillfully used Alberti’s rules of perspective to produce radical innovations in the art of scientific illustration, Leonardo soon realized that for his paintings, these rules were too restrictive and fraught with contradictions.18

  Alberti had suggested that the geometric horizon of a painting should be at the eye level of the painted figures so as to create the illusion of a continuity between the imaginary space and that of the spectators. However, frescoes and altarpieces were often placed quite high up, which made it impossible for the spectators to look at them from a viewpoint that would make the illusion work. Moreover, Alberti’s system assumed a fixed viewpoint in front of the vanishing point, but most spectators were likely to move around and look at the painting from different angles, which would also destroy the illusion. In The Last Supper, Leonardo, well aware of the internal contradictions of linear perspective, played around with Alberti’s rules to enhance the presence of the human figures and create elaborate illusions,19 but after that he no longer painted any architectural motifs and went far beyond the linear perspective of the quattrocento.

  To refine the theory of perspective, Leonardo questioned Alberti’s simplistic assumption that the lines of all visual pyramids meet in a single mathematical point within the eye. Instead, he studied the actual physiology of visual perception. “Perspective,” he noted, “is nothing else than a thorough knowledge of the function of the eye.”20 He took into account that natural vision is binocular—produced by two moving eyes rather than the single fixed eye of Alberti’s geometry. He carefully investigated the actual pathways of the sensory impressions, and he also considered the effects of atmospheric conditions on visual perception.

  From his studies of the anatomy of the eye and the physiology of vision,21 Leonardo derived a theory of perspective that went well beyond Alberti, Piero della Francesca, and other leading artists of the early Renaissance. “There are three kinds of perspective,” he declared. “The first is concerned with the reason for the diminution [of] things as they recede from the eye. The second contains the way in which colors vary as they recede from the eye. The third and last encompasses the declaration of how objects should appear less distinct the more distant they are.” He specified that the first, traditional kind was called “linear perspective” (lineare), the second “perspective of color” (di colore), and the third the “perspective of disappearance” (di spedizione).22

  Figure 8-3: Water-powered rolling mill, Codex Atlanticus, folio 10r

  As an object recedes into the distance, its image will diminish simultaneously in those three ways. Its size will decrease, its color will become fainter, and the definition of its detail will deteriorate until all three “disappear” at the vanishing point. According to Leonardo, a painter had to master all three kinds of perspective, and in addition he had to take into account a fourth kind, the “aerial perspective” (aerea) caused by the effects of the atmosphere on colors and other aspects of visual perception.23 Leonardo demonstrated his mastery at rendering these subtle aspects of perspective in many of his paintings. Indeed, it is often the misty atmosphere and dreamy nature of their distant mountain landscapes that give his masterworks their special magic and poetic quality.

  LIGHT AND SHADOW

  Together with the effects of perspective in painting, Leonardo also explored the geometry of light, now known as geometrical optics, as well as the interplay of light and shadow under natural and artificial illumination. The study of optics had already been well developed in the Middle Ages. It had tremendous prestige among medieval philosophers, who associated light with divine power and glory.24 They knew that light traveled in straight lines, and that its paths obeyed geometrical laws as the light rays passed through lenses and were reflected in mirrors. To the medieval mind, this association of optics with the eternal mathematical laws of geometry was further proof of the divine origin of light.

  The dominant figure in medieval optics was the Arab mathematician Alhazen,25 who wrote a seven-volume work, Kitab al-Manazir (Book of Optics), published in Arabic in the eleventh century and widely available in Latin translation as Opticae Thesaurus from the thirteenth century on. Alhazen’s treatise included detailed discussions of vision and the anatomy of the eye. He introduced the idea that light rays emanate from luminous objects in straight lines in all directions and discovered the laws of reflection and refraction. He paid special attention to the problem of finding the point on a curved mirror where a ray of light will be reflected to pass from a given source to an observer, which subsequently came to be known as “Alhazen’s problem.” Alhazen’s Optics inspired several European thinkers, who added original observations of their own, including the Polish philosopher Witelo of Silesia as well as John Pecham and Roger Bacon in England. It was from these authors that Leonardo first learned about Alhazen’s pioneering work.26

  From his earliest years in Verrocchio’s workshop, Leonardo was familiar with the grinding of lenses and the use of concave mirrors to focus sunlight for welding.27 Throughout his life he tried to improve the design of these burning mirrors, and when he became seriously interested in the theory of optics, he undertook careful studies of their geometries. He was fascinated by the intricate intersections of the reflected rays, which he explored in a series of precise and beautiful diagrams, tracing their pathways from parallel beams of light through their reflections to the focal point (or points). He showed that in spherical mirrors, the rays are focused in an area along the central axis (see Fig. 8-4), whereas parabolic mirrors are true “mirrors of fire,” focusing all the rays in a single point. He also made several attempts to solve Alhazen’s problem, and late in his life, while experimenting with parabolic mirrors in Rome, found an ingenious solution by employing an instrument with hinged rods.28

  In Figure 8-4, Leonardo has constructed the reflected light rays by drawing in each point the radius of the mirror (which is perpendicular to the reflecting surface) and then using the so-called law of reflection, that the angle of incidence is equal to the angle of reflection. This law was already known to Alhazen, but Leonardo realized that it applies not only to the reflection of light, but also to the mechanical rebound of a ball thrown against a wall, and to the echo of sound.29 “The line of percussion and that of its rebound,” he writes in Manuscript A, “will make an angle on the wall…between two equal angles.” And then he adds: “The voice is similar to an object seen in a mirror.”30 Several years later he applied the same reasoning to the rebound of a jet of water from a wall, noting, however, that some of the water peels off as an eddy after the reflection.31

  Figure 8-4: Study of concave spherical mirror, Codex Arundel, folio 87v

  By far the largest part of Leonardo’s optical studies concerned the effects of light falling on objects and the nature of different kinds of shadows. As a painter, he was famous for his subtle use of light and shade,32 so it is not surprising that the longest section, part 5, of his Treatise on Painting is titled “On Shadow and Light.” Based on his earlier notes in Manuscript
C, these chapters contain practical advice to the painter on how to render gradations of light and shadow in landscapes, and on trees, drapery, and human faces, as well as abstract discussions on the nature of shadow, the difference between luster and light, the nature of contrasts, the juxtaposition of colors, and many related subjects.

  According to Leonardo, shadow is the central element in the science of painting. It allows the painter to effectively represent solid bodies in relief, emerging from the backgrounds of the painted surface. His poetic definition of shadow in the Codex Atlanticus is clearly written from the artist’s point of view:

  Every opaque body is surrounded, and its whole surface is enveloped, in shadow and light…. Besides this, shadows have in themselves various degrees of darkness, because they are caused by the absence of a variable amount of the luminous rays…. They clothe the bodies to which they are applied.33

  In order to fully understand the intricacies of the interplay between light and shadow, Leonardo designed a series of elaborate experiments with lamps shining on spheres and cylinders, their rays intersecting and being reflected to create an endless variety of shadows. As in his experiments on linear perspective, he systematically varied the relevant variables—in this case the size and shape of the lamp, the size of the illuminated object, and the distance between the two. He distinguished between “original shadows” (formed on the object itself) and “derived shadows” (cast by the object through the air and onto other surfaces).34

  Figure 8-5, for example, shows a diagram of a sphere illuminated by light falling through a window. Leonardo has traced light rays emanating from four points (labeled a, b, c, and d). He shows four gradations of primary shadows on the sphere (labeled n, o, p, and q), and the corresponding gradations of derived shadows, cast between the boundary lines of the eight light rays behind the sphere (labeled by the letters along the base of the diagram).

  In these experiments Leonardo uses extended light sources (such as windows) as well as point sources (for example, the flame of a candle), and he considers the combined effects of direct sunlight and ambient light—“the universal light of the sky,” as he calls it.35 He also introduces several lamps, studies how the gradations of the shadows change with each new lamp, and examines how the shadows move when the lamps and the object are moved. As Kenneth Clark has remarked, “The calculations are so complex and abstruse that we feel in them, almost for the first time, Leonardo’s tendency to pursue research for its own sake, rather than as an aid to his art.”36

  Figure 8-5: Gradations of primary and derived shadows, Ms. Ashburnham II, folio 13v

  OPTICS AND ASTRONOMY

  Leonardo’s optical observations also included observations of the heavenly bodies, especially the Sun and the Moon. He was well aware of the Ptolemaic system of planetary motion, but his own astronomical studies were concerned almost exclusively with the appearance of the heavenly bodies to the human eye and the diffusion of light from one body to the other. As far as we know, Leonardo saw astronomy simply as an extension of optics and the science of perspective. Indeed, he declared: “There is no part of astronomy that is not a function of visual lines and perspective.”37

  Leonardo tried to calculate the height of the Sun from two different angles of elevation, and its size by comparing it with the image in a camera obscura.38 What interested him much more, however, was the transmission of light between celestial bodies. He was familiar with the ancient division of the universe into a “celestial realm,” in which perfect bodies move according to precise, unchanging mathematical laws, and an “earthly realm,” in which natural phenomena are complex, ever changing, and imperfect.39 He also knew that Aristotle believed that the Moon and the planets were flawless spheres, each with its own luminosity. Leonardo disagreed with Aristotle on this point. Based on his observations with the naked eye, he stated correctly: “The Moon has no light of itself, but so much of it as the Sun sees, it illuminates. Of that luminosity, we see as much as faces us.”40

  Having convinced himself that the Moon is not itself luminous but reflects the light of the Sun, Leonardo went on to argue that it could not be an unblemished sphere, since it does not show a brilliant circular highlight like “the gold balls placed on the tops of the high buildings.” He hypothesized that the Moon’s patchy radiance is the result of multiple reflections of sunlight from the waves on its waters. “The skin, or surface, of the water that makes up the sea of the Moon,” he wrote, “is always ruffled, little or much, more or less; and this roughness is the cause of the proliferation of the innumerable images of the Sun, which are reflected in the ridges and concavities, and sides and fronts, of the innumerable wrinkles.”41

  He then reasoned that there could be no waves in the lunar sea unless the surface of its waters was ruffled by air, and hence he concluded that the Moon, like the Earth, has its own set of four elements.42 And in the final flourish of these interdependent observations and arguments, Leonardo pointed out that reflected sunlight from the waters of the sea must be transmitted also in the opposite direction, from the Earth to the Moon. This reasoning led him to the astonishing and prophetic statement that “to anyone standing on the Moon…this our Earth with its element of water would appear and function just as the Moon does to us.”43

  Leonardo’s ideas about astronomy, even though only partly correct, were certainly remarkable, and it is hard to believe that he was not interested in celestial mechanics at all. We know that he possessed a copy of Ptolemy’s Cosmography and that he held it in high regard. He also owned a volume by the Arab astronomer Albumazar, and several other sources on astronomy are mentioned in the Notebooks.44 But no notes on the movements of the planets have come down to us.

  It is also interesting that Leonardo did not subscribe to the ancient belief that the stars influence life on Earth. In the Renaissance, astrology enjoyed a high reputation. The professions of astronomer and astrologer were inseparably connected, and even Leonardo used the word astrologia when he referred to astronomy. Renaissance princes, including Ludovico Sforza in Milan, often consulted court astrologers about matters of health, and even about political decisions. Thus Leonardo probably kept his views about astrologers to himself at court, but in his Notebooks he showed great contempt for them, describing their practices as “that deceptive opinion by means of which (begging your pardon) a living is made from fools.”45 The main focus of Leonardo’s studies was the terrestrial realm of living, and its ever-changing forms, and he believed that its processes were not influenced by the stars but followed their own “necessities,” which he intended to understand and explain by means of reasoning, based on direct experience.

  THE NATURE OF LIGHT RAYS

  Leonardo’s studies of perspective and of light and shadow not only found artistic expression in his mastery of rendering subtle visual complexities, but also stimulated his scientific mind to investigate the very nature of the rays that carried light in pyramids from the objects to the eye. With his empirical method of systematic observation and with highly ingenious experiments that used only the most rudimentary instruments, he observed optical phenomena and formulated concepts about the nature of light that would take hundreds of years to be rediscovered.

  His starting point was the accepted contemporary knowledge that light is emitted by luminous objects in straight lines. To test this assertion, Leonardo used the principle of the camera obscura, which had been known since antiquity. Here is how he describes his experiment:

  If the front of a building, or any piazza or field, which is illuminated by the sun, has a dwelling opposite to it, and if in the front that does not face the sun you make a small round hole, all the illuminated objects will send their images through that little hole and will appear inside the dwelling on the opposite wall, which should be white. And there they will be, exactly and upside down…. If the bodies are of various colors and shapes, the rays forming the images will be of various colors and shapes, and of various colors and shapes will be the representation
s on the wall.46

  Leonardo repeats this experiment many times with various combinations of objects and with several holes in the camera obscura, as clearly illustrated on a folio in the Windsor Collection.47 Having performed a series of tests, he then confirms the traditional knowledge: “The lines from…the sun, and other luminous rays passing through the air, are obliged to keep in a straight direction.”48 He also specifies that these lines are infinitely thin, like geometrical lines. He calls them “spiritual,” by which he means simply without material substance.49 And finally, Leonardo asserts that light rays are rays of power—or, as we would say today, of energy50—which radiate from the center of a luminous body, such as the sun. “It will appear clear to the experimenters,” he writes, “that every luminous body has in itself a hidden center, from which and to which…arrive all the lines generated by the luminous surface.”51

  Thus, in essence, Leonardo identifies three basic properties of light rays: They are rays of energy generated at the center of luminous bodies; they are infinitely thin and without material substance; and they always travel in straight lines. Before the discovery of the electromagnetic nature of light in the nineteenth century, nobody could have improved on Leonardo’s description, and even then contradictions concerning the nature of light waves persisted until they were resolved by Albert Einstein in the twentieth century.52 On the other hand, the view of light rays as straight geometrical lines is still considered an excellent approximation for understanding a broad range of optical phenomena and is taught to physics students in our colleges and universities as geometrical optics.