Biographies & Memoirs



Pyramids of Light

Leonardo’s scientific method was based not only on the careful and systematic observation of nature—his much-exalted sperienza1—but also included a detailed and comprehensive analysis of the process of observation itself. As an artist and a scientist, his approach was predominantly visual, and he began his explorations of the “science of painting” by studying perspective: investigating how distance, light, and atmospheric conditions affect the appearance of objects. From perspective, he proceeded in two opposite directions—outward and inward, as it were. He explored the geometry of light rays, the interplay of light and shadow, and the very nature of light, and he also studied the anatomy of the eye, the physiology of vision, and the pathways of sensory impressions along the nerves to the “seat of the soul.”

To a modern intellectual, used to the exasperating fragmentation of academic disciplines, it is amazing to see how Leonardo moved swiftly from perspective and the effects of light and shade to the nature of light, the pathways of the optic nerves, and the actions of the soul. Unencumbered by the mind-body split that Descartes would introduce 150 years later, Leonardo did not separate epistemology (the theory of knowledge) from ontology (the theory of what exists in the world), nor indeed philosophy from science and art. His wide-ranging examinations of the entire process of perception led him to formulate highly original ideas about the relationship between physical reality and cognitive processes—the “actions of the soul,” in his language—which have reemerged only very recently with the development of a post-Cartesian science of cognition.2


Leonardo’s earliest studies of perception stand at the beginning of his scientific work. “All our knowledge has its origin in the senses,” he wrote in his very first Notebook, the Codex Trivulzianus,3 begun in 1484. During the subsequent years he embarked on his first studies of the anatomy of the eye and the optic nerves. At the same time, he explored the geometries of linear perspective and of light and shadow, and demonstrated his profound understanding of these concepts in his first master paintings, the Adoration of the Magiand the Virgin of the Rocks.4

Leonardo’s interest in the mathematics underlying perspective and optics intensified in the summer of 1490, when he met the mathematician Fazio Cardano at the University of Pavia.5 He had long discussions with Cardano on the subjects of linear perspective and geometrical optics, which together were known as “the science of perspective.” Soon after these discussions, Leonardo filled two Notebooks with a short treatise on perspective and with numerous diagrams of geometrical optics.6 He returned to the study of optics and vision eighteen years later, around 1508, when he explored various subtleties of visual perception. At that time, Leonardo revised his earlier notes and summarized his findings on vision in the small Manuscript D, which is similar in its brevity and elegant compact structure to the Codex on the Flight of Birds, composed around the same time.

Linear perspective was established in the early fifteenth century by the architects Brunelleschi and Alberti as a mathematical technique for representing three-dimensional images on a two-dimensional plane. In his classic work De pictura (On Painting),7 Alberti suggested that a painting should give the impression of being a window through which the artist looks at the visible world. All objects in the picture were to be systematically reduced as they receded into the distance, and all sight lines were to converge to a single “central point” (later called the “vanishing point”), which corresponded to the fixed viewpoint of the spectator.

As architectural historian James Ackerman points out, the geometry of perspective developed by the Florentine artists was the first scientific conception of three-dimensional space:

As a method of constructing an abstract space in which any body can be related mathematically to any other body, the perspective of the artists was a preamble to modern physics and astronomy. Perhaps the influence was indirect and unconsciously transmitted, but the fact remains that artists were the first to conceive a generalized mathematical model of space and that it constituted an essential step in the evolution from medieval symbolism to the modern image of the universe.8

Leonardo used Alberti’s definition of linear perspective as his starting point. “Perspective,” he states, “is nothing else than seeing a place behind a pane of glass, quite transparent, on the surface of which the objects behind that glass are to be drawn.”9 A few pages later in the same Notebook, he introduces geometric reasoning with the help of the image of a “pyramid of lines,” which was common in medieval optics.10 The first statement about perspective, too, continues with a reference to visual pyramids. “These [objects],” Leonardo explains, “can be traced through pyramids to the point of the eye, and the pyramids are intersected on the glass pane.”11


Figure 8-1: The geometry of linear perspective, Codex Atlanticus, folio 119r

To determine to what extent exactly the image of an object on the glass pane diminishes with the object’s distance from the eye, Leonardo conducted a series of experiments, in which he methodically varied the three relevant variables in all possible combinations—the height of the object, the distance from the eye, and the distance between the eye and the vertical glass pane.12 He sketched the experimental arrangements in several diagrams; for example, as shown in Figure 8-1, where the object is kept stationary and the observer’s eye, together with the glass pane in front of it, is placed in two different locations. The corresponding “pyramids” (isosceles triangles) with the two different visual angles are clearly shown.

With these experiments, Leonardo established conclusively that the height of the image on the glass pane is inversely proportional to the object’s distance from the eye, if the distance between the eye and the glass pane is kept constant. “I find by experience,” he recorded in Manuscript A, “that, if the second object is as far from the first as the first is from the eye, although they are of the same size, the second will seem half the size of the first.”13 In another entry he records a series of distances with the corresponding reductions of the object’s image, and then concludes: “As the space passed through doubles, the diminution doubles.”14

These results, obtained during the late 1480s, mark Leonardo’s first explorations of arithmetic, or “pyramidal,” progressions. To establish them, he did not really have to perform all these experiments, because the inverse linear relationship between the distance of the object from the eye and the reduction of its image on the glass pane can easily be derived with elementary Euclidean geometry. But it would be almost another ten years before Leonardo would acquire those mathematical skills.15


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.


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 Thesaurusfrom 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


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.


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 representations 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.52On 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.


The idea that light rays emanate from luminous objects in straight lines in all directions was known to Leonardo from Alhazen’s treatise on optics before he tested it experimentally. Another idea that was popular in medieval optics, which he adopted from John Pecham (who, in turn, was influenced by Alhazen), was the concept of pyramids of light filling the air with images of solid objects:

The body of the air is full of infinite pyramids composed of radiating straight lines which emanate from the edges of the surfaces of the solid bodies placed in the air; and the further they are from their cause the more acute are the pyramids, and although their converging paths intersect and interweave, nevertheless they never blend but proliferate independently, infusing all the surrounding air.53

With this poetic description, Leonardo simply rephrased Alhazen’s original insight, but he added the significant observation that the pyramids of light “intersect and interweave” without interfering with each other. In a remarkable display of systemic thinking, Leonardo used this observation as a key argument to speculate about the wave nature of light. Here is how he proceeded.

First, he combines the fact that light is radiated equally in all directions, which he has tested repeatedly, with the image of visual pyramids. He draws a diagram that shows a spherical body radiating equal pyramids (represented by triangles) in different directions, and he notes in the accompanying text that their tips are enclosed by a circle: “The equidistant perimeter of converging rays of the pyramid will give to their objects angles of equal size.”54 In other words, if observers were placed at the tips of these pyramids around the circle, their visual angles would be the same (see Figure 8-6). In the same diagram, Leonardo extends one pyramid to show that the visual angle at its apex decreases as the pyramid becomes longer.

From this exercise, he concludes that light spreads in circles, and he immediately associates this circular pattern with the circular spread of ripples of water and the spread of sound in air: “Just as the stone thrown into the water becomes the center and cause of various circles, and the sound made in the air spreads out in circles, so every object placed within the luminous air diffuses itself in circles and fills the surroundings with an infinite number of images of itself.”55

Having linked the circular pattern of the spread of light to the similar spread of ripples in water, Leonardo then sets out to study the details of the phenomenon in a pond in order to learn something about the radiation of light. In doing so, he uses, at the very beginning of his scientific explorations, a technique that would become an integral part of the scientific method in subsequent centuries. Since he cannot actually see the circular (or, more correctly, spherical) propagation of light, he takes the similar pattern in water as a model, hoping that it will reveal to him something about the nature of light under close study. And he does indeed study it very closely.

In Manuscript A, the very same Notebook that contains his analysis of perspective and many of his optical diagrams, Leonardo records his detailed investigations of the circular spread of water waves:

If you throw two small stones at the same time onto a sheet of motionless water at some distance from one another, you will see that around those two percussions two separate sets of circles are caused, which will meet as they increase in size and then interpenetrate and intersect one another, while always maintaining as their centers the places struck by the stones.56


Figure 8-6: Visual pyramids radiated from a spherical body, Ms. Ashburnham II, folio 6v

Leonardo illustrates this phenomenon with a diagram (Fig. 8-7), and to understand its exact nature, he focuses on the precise movement of the water particles, making it easier for the eye to follow them by throwing small pieces of straw into the pond and watching their movements. Here is what he observes.

Although there seems to be some demonstration of movement, the water does not depart from its place, because the openings made by the stones are closed again immediately. And that motion, caused by the sudden opening and closing of the water, makes in it a certain shaking, which one could call a tremor rather than a movement.

And so that what I say may be more evident to you, pay attention to those blades of straw which, because of their lightness, float on the water and are not moved from their original position by the wave that rolls underneath them as the circles arrive.


Figure 8-7: Intersection of circular water waves, Ms. A, folio 61r

Throughout history, countless people have thrown pebbles into ponds and watched the circular ripples they caused, but very few would have been able to match the accuracy and fine details of Leonardo’s observations. He recognized the essence of wave motion—that the water particles do not move along with the wave but merely move up and down as the wave passes by.57 What is transported along the wave is the disturbance causing the wave phenomenon—the “tremor,” as Leonardo calls it—but not any material particles: “The water, though remaining in its position, can easily take this tremor from neighboring parts and pass it on to other adjacent parts, always diminishing its power until the end.” And this is the reason, he concludes correctly, why the circular waves intersect smoothly without disturbing each other:

Therefore, the disturbance of the water being a tremor rather than a movement, the circles cannot break one another as they meet, because, water being of the same quality in all its parts, it follows that these parts transmit the tremor from one to another without moving from their place.

This smooth intersection of water waves is the key property that suggests to Leonardo that light and sound, too, propagate in waves. He has noted that the pyramids of light “intersect and interweave” without interfering with each other,58 and he applies the same reasoning to sound: “Although the voices that penetrate the air spread in circular motion from their causes, nevertheless the circles moved from different origins meet without any impediment, penetrate and pass into one another, always keeping their causes at their centers, because in all cases of motion, there is great conformity between water and air.”59 In other words, just as the intersecting circular ripples in the pond retain their distinct identities, we can see the images of different objects, or hear the sounds of different voices, and still distinguish them clearly.

From these observations, Leonardo draws the momentous conclusion that both light and sound are waves. A few years later he extends his insight to elastic waves in the earth and concludes that wave motion, caused by initial vibrations (or “tremors”), is a universal form of propagation of physical effects. “The movement of earth against earth, crushing it,” he writes, “moves the affected parts only slightly. Water struck by water creates circles round the place where it is struck; the voice in the air goes further, [and the tremor] in fire further still.”60

The realization that wave motion is a universal phenomenon in all four elements—earth, water, air, and fire (or light)—was a revolutionary insight in Leonardo’s time. It took another two hundred years before the wave-nature of light was rediscovered by Christian Huygens; the wave-nature of sound was first clearly articulated by Marin Marsenne during the first half of the seventeenth century, and earthquakes were associated with elastic waves only in the eighteenth century.61

In spite of Leonardo’s impressive insights into the nature of wave motion and its widespread occurrence in nature, it would be an overstatement to say that he developed a wave theory of light similar to that presented by Huygens two hundred years later. To do so would have meant to understand the mathematical representation of a wave and relate its amplitude, frequency, and other characteristics to observed optical phenomena. These concepts were not used in science until the seventeenth century, when the mathematical theory of functions was developed.

Leonardo gave a correct description of transverse waves, in which the direction of energy transfer (the spreading of the circles) is at right angles to the direction of the vibration (the “tremor”), but he never considered longitudinal waves, in which the vibrations and energy transfer go in the same direction. In particular, he did not realize that sound waves are longitudinal. He appreciated that waves in different media (or “elements”) travel at different velocities, but believed erroneously that the wave velocity is proportional to the power of the percussion that sets it off.62

He marveled at the swift velocity of light: “Look at the light of the candle and consider its beauty,” he wrote. “Blink your eye and look at it again. What you see of it was not there before, and what was there before is not anymore.”63 But he also realized that, however fast light moves, its velocity is not infinite. He asserted that the speed of sound is greater than that of elastic waves in earth, and that light moves faster than sound, but that the mind moves even faster than light. “The mind jumps in an instant from the East to the West,” he noted, “and all the other immaterial things have velocities that are by a long way inferior.”64

Even though Leonardo did not state explicitly that the velocity of light is finite, it is clear from his Notebooks that he held that view. This is quite extraordinary, since the traditional view, handed down from antiquity, was that the propagation of light is instantaneous. Even Huygens and Descartes subscribed to that traditional view, and it was not until the end of the seventeenth century that the finite velocity of light was established.65

Leonardo was well aware of the phenomenon of refraction (the deflection of a light ray upon passing obliquely from air into glass, for instance). He performed several ingenious experiments to explore it, without, however, relating the effect to the wave-nature of light as Descartes and others would do some 150 years later. Leonardo even used refraction in a primitive prism to split white light into components of different colors, as Isaac Newton would do again in a celebrated experiment during the 1660s. But unlike Newton, Leonardo did not go much further than accurately recording the effect.66

On the other hand, Leonardo found the correct explanation for a phenomenon that had intrigued people throughout history—the blue color of the sky. In the years of his optical experiments, he climbed one of the giant peaks of Monte Rosa and noticed the deep blue of the sky at high altitude.67During the long climb, he apparently pondered the age-old question, “Why is the sky blue?”—and with amazing intuition came up with the correct answer:

The blue displayed by the atmosphere is not its own color, but is caused by moisture that has evaporated into minute and imperceptible atoms on which the solar rays fall, rendering them luminous against the immense darkness of the region of fire that forms a covering above them. And this may be seen, as I myself saw it, by anyone who climbs Monte Rosa.68

The modern explanation of this phenomenon was given about four hundred years later by Lord Rayleigh, and the effect is now known as Rayleigh scattering. Sunlight is scattered by the molecules of the atmosphere (Leonardo’s “minute and imperceptible atoms”) in such a way that blue light is absorbed much more than other frequencies and is then radiated in different directions all around the sky. Hence, whichever way we look, we will see more of the scattered blue light than light of any other color. It is evident that Leonardo’s explanation of solar rays falling on the molecules and “rendering them luminous” is a perfectly accurate qualitative description of the effect. This must certainly rank among his finest achievements in optics.


Leonardo also explored the nature of sound, and from experiments with bells, drums, and other musical instruments, he observed that sound is always produced by “a blow on a resonant object.” He correctly deduced that this causes an oscillating movement in the surrounding air, which he called “fanning movement” (moto ventilante) in association with the oscillating movement of a handheld fan.69 “There cannot be any sound,” he concluded, “where there is not movement and percussion of air; there cannot be percussion of that air where there is no instrument.”70

Leonardo then proposed that, as in water, the initial percussion propagates in the form of circular waves, “since in all cases of movement water has great conformity with air.”71 As noted earlier, he was unaware that sound travels via longitudinal waves, but he noticed the phenomenon of resonance, demonstrating it with small pieces of straw, as he had demonstrated the transverse movement of water waves:

The blow given to the bell will make another bell similar to it respond and move somewhat. And the string of a lute, as it sounds, produces response and movement in another similar string of similar tone in another lute. And this you will perceive by placing a straw on the string which is similar to that sounded.72

The observations of resonating bells and lute strings suggested to Leonardo the general mechanism for the propagation and perception of sound—from the initial percussion and the resulting waves in the air to the resonance of the eardrum.

Lacking the appropriate mathematical language, Leonardo was not able to develop a proper wave theory of light, nor a corresponding wave theory of sound.73 He observed that the loudness of the sound generated depended on the power of percussion, but he failed to associate it with the amplitude of the sound wave; nor did he relate the pitch of sound to the wave’s frequency. However, many years later, during the time he was reviewing the contents of all his Notebooks,74 he came close to understanding the relation between pitch and frequency by studying the sound made by flies and other insects.

Whereas the common belief in his time was that flies produce sound with their mouths, Leonardo correctly observed that the sound is generated by their wings and proceeded with a clever experiment: “That flies have their voice in the wings,” he recorded, “you will see by…daubing them with a little honey in such a way that they are not entirely prevented from flying. And you will observe that the sound made by the movement of their wings…will change from high to low pitch in direct proportion to the degree that their wings are more impeded.”75

One of Leonardo’s most impressive discoveries in the field of acoustics was his observation that, “If you tap a board covered with dust, that dust will collect in diverse little hills.”76 Having enhanced the vibrations of lute strings by putting small pieces of straw on them, he now concluded correctly that the dust was flying off the vibrating parts of the board and settling at the nodes, that is, in the areas that were not vibrating. He did not stop at that observation, but carefully continued tapping the vibrating surface while observing the fine movements of the little hills of dust. Next to a sketch representing one such hillock as a pyramid, he recorded his observations. “The hills will always pour down that dust from the tips of their pyramids to their base,” he wrote. “From there, it will re-enter underneath, ascend through the center, and fall back again from the top of that little hill. And so the dust will circulate again and again…as long as the percussion continues.”77

The attention to detail in these observations is truly remarkable. The phenomenon of nodal lines of dust or sand on vibrating plates was rediscovered in 1787 by the German physicist Ernst Chladni. They are now commonly called “Chladni patterns” in physics textbooks, where it is generally not mentioned that Leonardo da Vinci discovered them almost three hundred years earlier.


To complete his science of perspective, Leonardo studied not only the external pathways of light rays, together with various optical phenomena, but also followed them right into the eye. Indeed, during the 1480s, he pursued his anatomical studies of the eye and the physiology of vision simultaneously with his investigations of perspective and the interplay of light and shadow.

At that time there was a debate among Renaissance artists and philosophers about the exact location of the tip of the visual pyramid in the eye. Most artists followed Alberti, who paid little attention to the actual physiology of vision and located the apex of the visual pyramid in a geometric point at the center of the pupil. Most philosophers, by contrast, took the position of Alhazen, who asserted that the eye’s visual faculty must reside in a finite area rather than in an infinitely small point.78

In the beginning of his investigations of perspective and the anatomy of the eye, Leonardo adopted Alberti’s view, but during the 1490s, as his research became more sophisticated, he came to embrace Alhazen’s position, arguing that “if all the images that come to the eye converged in a mathematical point, which is proved to be indivisible, then all the things seen in the universe would appear as one, and that one would be indivisible.”79

In his late optical writings in Manuscript D, finally, he asserted repeatedly and confidently that “every part of the pupil possesses the faculty of vision (virtù visiva), and…this faculty is not reduced to a point, as the perspectivists wish.”80 In this Notebook, Leonardo offers three simple but very elegant experiments, involving the shadowy perception of small objects held near the eye, as persuasive proofs of Alhazen’s position.81 From then on he distinguished between two kinds of perspective. The first, “perspective made by art,” is a geometric technique for representing objects located in three-dimensional space on a flat surface, while the second, “perspective made by nature,” needs a proper science of vision to be understood.82

Having convinced himself that in such a science of vision, the geometric apex of the visual pyramid in the eye needs to be replaced by much more complex pathways of the sensory impressions, Leonardo then traced these pathways through the lens and the eyeball to the optic nerve, and from there all the way to the center of the brain where he believed he had found the seat of the soul.

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