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

by Fritjof Capra

  THE WAVE NATURE OF LIGHT

  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.67 During 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.

  SOUND WAVES

  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.

  VISION AND THE EYE

  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.

  NINE

  The Eye, the Senses, and the Soul

  The structure of the eye and the process of vision were natural wonders for Leonardo that never ceased to amaze him. “What language can express this marvel?” he writes about the eyeball, before continuing with a rare expression of religious awe: “Certainly none. This is where human discourse turns directly to the contemplation of the divine.”1 In the Treatise on Painting, Leonardo waxes enthusiastic about the human eye: