Fritjof Capra
Page 19
In Leonardo’s science of living forms, life’s patterns of organization and its fundamental processes of metabolism and growth were the unifying conceptual threads that interlinked his knowledge of macro-and microcosm. In the macrocosm, the main themes of his science were the movements of water and air, the geological forms and transformations of the Earth, and the botanical diversity and growth patterns of plants. In the microcosm, his main focus was on the human body—its beauty and proportions, the mechanics of its movements, and how it compared to other animal bodies in motion, in particular birds in flight.
THE MOVEMENTS OF WATER
Leonardo was fascinated by water in all its manifestations. He recognized its fundamental role as life’s medium and vital fluid, as the matrix of all organic forms. “It is the expansion and humor of all living bodies,” he wrote. “Without it nothing retains its original form.”37 Throughout his life, he strove to understand the mysterious processes underlying the creation of nature’s forms by studying the movements of water through earth and air.
As an engineer, Leonardo worked extensively on schemes of canalization, irrigation, the drainage of marshes, and the uses of waterpower for pumping, milling, and sawing. Like other noted engineers in the Renaissance, he was very familiar with the beneficial as well as the destructive effects of the power of water. But he was the only one to go beyond empirical rules of hydraulic engineering and embark on sustained theoretical studies of the flow of water. His examinations and exquisite drawings of the flows of rivers, eddies, spiraling vortices, and other patterns of turbulence establish Leonardo as a pioneer in a field that did not even exist in his time—the discipline known today as fluid dynamics.
Throughout his life, Leonardo observed the flows of rivers and tides, drew beautiful and accurate maps of entire watersheds, and investigated currents in lakes and seas, flows over weirs and waterfalls, and the movement of waves as well as flows through pipes, nozzles, and orifices. His observations, drawings, and theoretical ideas would fill hundreds of pages in his Notebooks.
Through this lifelong study, Leonardo gained a full understanding of the main characteristics of fluid flow. He recognized the two principal forces operating in flowing water—the force of gravity and the fluid’s internal friction, or viscosity—and he correctly described many phenomena generated by their interplay. He also realized that water is incompressible and that, even though it assumes an infinite number of shapes, its mass is always conserved.
In a branch of science that did not even exist before him, Leonardo’s deep insights into the nature of fluid flow must be ranked as a momentous achievement. That he also drew many turbulent structures erroneously and imagined some flow phenomena that do not occur in reality does not diminish his great accomplishments, especially in view of the fact that even today scientists and mathematicians encounter considerable difficulties in their attempts to predict and model the complex details of turbulent flows.
At the center of Leonardo’s investigations of turbulence lies the water vortex, or whirlpool. Throughout the Notebooks, there are countless drawings of eddies and whirlpools of all sizes and types—in the currents of rivers and lakes, behind piers and jetties, in the basins of waterfalls, and behind objects of various shapes immersed in flowing water. These often very beautiful drawings are testimony to Leonardo’s endless fascination with the ever-changing and yet stable nature of this fundamental type of turbulence. I believe that this fascination came from a deep intuition that the dynamics of vortices, combining stability and change, embody an essential characteristic of living forms.38
Leonardo was the first to understand the detailed motions of water vortices, often drawing them accurately even in complex situations. He correctly distinguished between flat circular eddies in which the water essentially rotates as a solid body, and spiral vortices (such as the whirlpool in a bathtub) that form a hollow space, or funnel, at their center. “The spiral or rotary movement of every liquid,” he noted, “is so much swifter as it is nearer to the center of its revolution. What we are here proposing is a fact worthy of admiration, since the circular movement of the wheel is so much slower as it is nearer to the center of the rotating object.”39 Such detailed studies of vortices in turbulent water were not taken up again for another 350 years, until the physicist Hermann von Helmholtz developed a mathematical analysis of vortex motion in the mid-nineteenth century.
Figure 6-4: Turbulent wakes behind a rectangular plank, c. 1509–11, Windsor Collection, Landscapes, Plants, and Water Studies, folio 42r
Leonardo produced several elaborate drawings of highly complex patterns of turbulence, generated by placing various obstacles into flowing water. Figure 6-4, from the Windsor Collection, shows the turbulent flows around a rectangular plank inserted at two different angles. (Additional variations are suggested in the small sketches to the right of the main drawing.) The upper drawing clearly shows a pair of counter-rotating vortices at the head of a stream of random wake. The essential details of this complex pattern of turbulence are completely accurate—an amazing testimony to Leonardo’s powers of observation and conceptual clarity.
THE FORMS AND TRANSFORMATIONS OF THE LIVING EARTH
Leonardo saw water as the chief agent in the formation of the Earth’s surface. “Water wears away the mountains and fills up the valleys,” he wrote, “and if it could, it would like to reduce the Earth to a perfect sphere.”40 This awareness of the continual interaction of water and rocks impelled him to undertake extensive studies in geology, which informed the fantastic rock formations that appear so often in the shadowy backgrounds of his paintings.
His geological observations are stunning not only by their great accuracy, but also because they led him to formulate general principles that were rediscovered only centuries later and are still used by geologists today.41 Leonardo recognized temporal sequences in the strata of soil and rock, and corresponding sequences in the fossils deposited in those strata, and he recorded many fine details concerning erosion and deposits by rivers.
He was the first to postulate that the forms of the Earth are the result of processes taking place over long epochs of what we now call geological time. With this view he came close to an evolutionary perspective more than three hundred years before Charles Darwin, who also found inspiration for evolutionary thought in geology. For Leonardo, geological time began with the formation of the living Earth, a process to which he alluded in his paintings with a sense of awe and mystery.
“Describe a landscape with wind and water,” Leonardo exhorted his fellow painters, “and at the setting and rising of the sun.”42 He was a true master in rendering these atmospheric effects. Like his predecessors and contemporaries, he frequently introduced flowers and herbs into his paintings for their symbolic meanings, but unlike most of his fellow painters, he was always careful to present plants in their proper ecological habitats with seasonal appropriateness and great botanical accuracy.43
The Notebooks contain numerous drawings of trees and flowering plants indigenous to Italy, many of them masterpieces of detailed botanical imagery. Most of these drawings were made as studies for paintings, but some also include detailed notes explaining the plants’ characteristics. Unlike the formal decorative plant motifs that were common in Renaissance paintings, Leonardo’s flowers, herbs, and trees display a vitality and grace that could only be achieved by a painter who had profound botanical and ecological knowledge.
Indeed, Leonardo’s mind was not satisfied with merely depicting plants in paintings, but turned to a genuine inquiry into their intrinsic nature—the patterns of metabolism and growth that underlie their organic forms. He made detailed observations of the effects of sunlight, water, and gravity on plant growth; he examined the sap of trees and discovered that a tree’s age could be determined from the number of rings in the cross-section of its trunk; he investigated patterns of leaves and branches around their stems, known to botanists today as the study of phyllotaxis; and he related patterns of branchin
g to the activity of a tree’s “humor”—an extraordinary insight into effects of hormonal activity that became known only in the twentieth century. As in so many other fields, Leonardo carried his scientific thinking far beyond that of his peers, establishing himself as the first great theorist in botany.44
MACRO-AND MICROCOSM
Whenever Leonardo explored the forms of nature in the macrocosm, he also looked for similarities of patterns and processes in the human body. In so doing, he went beyond the general analogies between macro-and microcosm that were common knowledge in his time, drawing parallels between very sophisticated observations in both realms. He applied his knowledge of turbulent flows of water to the movement of blood in the heart and aorta.45 He saw the “vital sap” of plants as their essential life fluid and observed that it nourishes the plant tissues, as the blood nourishes the tissues of the human body. He noticed the structural similarity between the stalk (known to botanists as the funiculus) that attaches the seed of a plant to the tissues of the fruit, and the umbilical cord that attaches the human fetus to the placenta.46 He took these observations as compelling testimonies to the unity of life at all scales of nature.
Leonardo’s wide-ranging and meticulous observations of the human body must be ranked among his greatest scientific achievements. In order to study the organic forms of the human body, he dissected numerous corpses of humans and animals, and examined their bones, joints, muscles, and nerves, drawing them with an accuracy and clarity never seen before. At the same time, his anatomical drawings are superb works of art, due to his unique ability to represent forms and movements in stunning visual perspective with subtle gradations of light and shade, which gives his drawings a vivid quality rarely achieved in modern anatomical illustrations.
Looking through Leonardo’s drawings and notes in over a thousand pages of anatomical manuscripts, we can discern several broad themes. The first is that of beauty and proportion, which held great fascination for Renaissance artists. They saw proportion in painting, sculpture, and architecture as the essence of harmony and beauty, and there were many attempts to establish a canon of proportions for the human figure. Leonardo threw himself into this project with his usual vigor and attention to detail, taking a wealth of measurements to establish a comprehensive system of correspondences between all parts of the body. At the same time, he explored the relationship between proportion and beauty in his paintings. “The beautiful proportions of an angelic face in painting,” he wrote, “produce a harmonious concord, which reaches the eye simultaneously, just as [a chord in] music affects the ear.”47
The second grand theme of Leonardo’s anatomical research was the human body in motion. As noted earlier, Leonardo’s science of living forms is a science of movement and transformation, whether he studied mountains, rivers, plants, or the human body. Hence, to understand the human form meant for him to understand the body in motion. He demonstrated in countless elaborate and stunning drawings how nerves, muscles, tendons, and bones work together to move the body.
NATURE’S MECHANICAL INSTRUMENTS
Leonardo never thought of the human body as a machine.48 However, he clearly recognized that the anatomies of animals and humans involve mechanical functions. In his anatomical drawings, he sometimes replaced muscles by threads or wires to better demonstrate the directions of their forces (see Fig. I-1 on p. 10, and Fig. 9-4 on Chapter 9). He showed how joints operate like hinges and applied the principle of levers to explain the movements of the limbs. “Nature cannot give movement to animals without mechanical instruments,” he declared.49 Hence, he felt that, in order to understand the movements of the animal body, he needed to explore the laws of mechanics. Indeed, for Leonardo, this was the principal role of this branch of science: “The instrumental or mechanical science is very noble and most useful above all others, because by means of it all animated bodies that have movement perform all their operations.”50
To investigate the mechanics of muscles, tendons, and bones, Leonardo immersed himself in a long study of the “science of weights,” known today as statics, which is concerned with the analysis of loads and forces on physical systems in static equilibrium, such as balances, levers, and pulleys. In the Renaissance this knowledge was very important for architects and engineers, as it is today, and the medieval science of weights comprised a large collection of works compiled in the late thirteenth and the fourteenth centuries.
In his usual fashion, Leonardo absorbed the key ideas from the best and most original texts, commented on many of their postulates in his Notebooks, verified them experimentally, and refuted some incorrect proofs.51 The classical law of the lever, in particular, appears repeatedly in the Notebooks. In the Codex Atlanticus, for example, Leonardo states, “The ratio of the weights that hold the arms of the balance parallel to the horizon is the same as that of the arms, but it is inverse.”52
Leonardo applied this law to calculate the forces and weights necessary to establish equilibria in numerous simple and compound systems involving balances, levers, pulleys, and beams hanging from cords.53 In addition, he carefully analyzed the tensions in various segments of the cords, probably for the purpose of estimating similar tensions in the muscles and tendons of human limbs.
Leonardo applied the lever law not only to situations where the forces act in a direction perpendicular to the lever arms, but also to forces acting at various angles. The Codex Arundel and Manuscript E in particular contain numerous diagrams of varying complexities, with weights exerting forces at different angles via cords and pulleys. He recognized that in such cases, the relevant length in the lever law is not the actual length of the lever arm, but the perpendicular distance from the line of the force to the axis of rotation. He called that distance the “potential lever arm” (braccio potenziale) and marked it clearly in many diagrams. In modern statics, the potential lever arm is known as the “moment arm,” and the product of moment arm and force is called the “moment,” or “torque.” Leonardo’s discovery of the principle that the sum of the moments about any point must be zero for a system to be in static equilibrium was his most original contribution to statics. It went well beyond the medieval science of weights of his time.
LEONARDO’S MACHINES
Leonardo applied his knowledge of mechanics not only to his investigations of the movements of the human body, but also to his studies of machines. Indeed, the uniqueness of his genius lay in his synthesis of art, science, and design.54 In his lifetime, he was famous as an artist, and also as a brilliant mechanical engineer who invented and designed countless machines and mechanical devices, often involving innovations that were centuries ahead of his time.55 Today, Leonardo’s technical drawings are frequently exhibited around the world, often supplemented by wooden models that show in impressive detail how the machines work as Leonardo had intended.56
As noted earlier, Leonardo was the first to separate individual mechanisms from the machines in which they were embedded.57 In these studies, he always insisted that any improvement of existing devices must be based on sound knowledge of the principles of mechanics. He paid special attention to the transmission of power and motion from one plane into another, which was a major challenge of Renaissance engineering. In his design of a water-powered milling machine (Fig. 8-3 on Chapter 8), for example, the motion is transmitted three times between horizontal and vertical axes with the help of a combination of toothed wheels and worm gears. The corresponding transfer of power is clearly indicated by Leonardo in a small diagram below the main drawing.58
Figure 6-5: Rotary ball bearing, Codex Madrid I, folio 20v; model by Muséo Techni, Montreal, 1987
Among Leonardo’s many mechanical innovations, there are several involving the conversion of the rotary motion of a crank into a straight back-and-forth movement, which could be used, for example, in automatic manufacturing processes.59 And then there is Leonardo’s well-known, highly ingenious design of a two-wheeled hoist (Fig. 2-3 on Chapter 2), which performs the opposite conversion: The
motion of a vertical operating lever rocking back and forth is converted into the smooth hoisting of a heavy load by means of two toothed wheels and a caged lantern gear. This is one of Leonardo’s most famous technical drawings. It displays the mechanism both in its assembled form and in an exploded view that exposes the complex combination of gears and plates.60
In the Renaissance, hoists, cranes, and other large machines were made of wood, and friction between their movable parts was a major problem. Leonardo invented numerous sophisticated devices for reducing friction and wear, including automatic lubrication systems, adjustable bearings, and mobile rollers of various shapes—spheres, cylinders, truncated cones, and the like. Figure 6-5 shows an elegant example of a rotary bearing composed of eight concave-sided spindles rotating on their own axes, interspersed by balls that can rotate freely but are prevented from lateral movements by the spindles. When a platform is put on this ball bearing, friction is reduced to such an extent that the platform can be turned easily even when carrying a heavy load.
All the great Renaissance engineers were aware of the effects of friction, but Leonardo was the only one who undertook systematic empirical studies of frictional forces. He found by experiment that, when an object slides against a surface, the amount of friction is determined by three factors: the roughness of the surfaces, the weight of the object, and the slope of an inclined plane:
In order to know accurately the quantity of the weight required to move a hundred pounds over a sloping road, one must know the nature of the contact which this weight has with the surface on which it rubs in its movement, because different bodies have different frictions….
Different slopes make different degrees of resistance at their contact; because, if the weight that must be moved is upon level ground and has to be dragged, it undoubtedly will be in the first strength of resistance, because everything rests on the earth and nothing on the cord that must move it…. But you know that, if one were to draw it straight up, slightly grazing and touching a perpendicular wall, the weight is almost entirely on the cord that draws it, and only very little rests upon the wall where it rubs.61