This article presents some quotes carefully extracted from the book ‘The science of Leonardo: inside the mind of the great genius of the Renaissance’, Fritjof Capra, Anchor Books, 1st Edition, December 2008.
1) …in the collection of his notes on painting, known as Trattato della pittura (Treatise on Painting), he writes: The science of painting extends to all the colors of the surface of bodies, and to shapes of the bodies enclosed by those surfaces… [Painting] with philosophic and subtle speculation considers all the qualities of forms… Truly this is science, the legitimate daughter of nature, because painting is born of nature.
2) Nature as a whole was alive for Leonardo. He saw the patterns and processes in the microcosm as being similar to those in the macrocosm. He frequently drew analogies between human anatomy and the structure of the Earth, as in the following beautiful passage from Codex Leiscester: We may say that the Earth has vital force of growth, and that its flesh is the soil; its bones are the successive strata of the rocks which form the mountains; its cartilage is the porous rock, its blood the veins of the waters. The lake of blood that lies around the heart is the ocean. Its breathing is the increase and decrease of the blood in the pulses, just as the in the Earth it is the ebb and flow of the sea.
3) As a painter, Leonardo felt that he should use gestures to portray the frames of mind and emotions that provoked them. He asserted that, in the painting of a human figure, the most important task was to “express in gesture the passion of its soul.” Indeed, to portray the body’s expression of the human spirit was the artist’s highest aspiration, in Leonardo’s view. And it was one in which he himself excelled, as the paintings of his mature period attest. As art historian Irma Richter explains in the introductory comments to her classical selections from the Notebooks, for Leonardo, “the human body was an outward and visible expression of the soul; it was shaped by its spirit.” We shall see that this vision of soul and spirit, unmarred by the mind-body split that René Descartes would introduce in the seventeenth century, is perfectly consistent with the conception of the “embodied mind” in today’s cognitive science.
4) Leonardo did not pursue science and engineering to dominate nature, as Francis Bacon would advocate a century later. He had a deep respect for life, a special compassion for animals, and great awe and reverence for nature’s complexity and abundance. While a brilliant inventor and designer himself, he always thought that nature’s ingenuity was vastly superior to human design. He felt that he would be wise to respect nature and learn from her. It is an attitude that has reemerged today in the practice of ecological design. Leonardo’s synthesis of art and science is infused with a deep awareness of ecology and systems thinking. It is not surprising that he spoke with great disdain of the so-called “abbreviators”, the reductionists of his time: The abbreviators of works do injury to knowledge and to love… Of what value is he who, in order to abbreviate the parts of those things of which he professes to give complete knowledge, leaves out the greater part of the things of which the whole is composed?… Oh human stupidity!… You don’t see that you are falling into the same error as one who strips a tree of its adornment of branches full of leaves, intermingled with fragrant flowers and fruit, in order to demonstrate that the tree is good for making planks.
5) Leonardo’s physical beauty in his youth and middle aged years must have been exceptional, as it is mentioned by all his contemporary commentators, even though this was not customary at the time. An anonymous writer called the Anonimo Gaddiano exclaimed, “He was so unusual and many-sided that nature seemed to have produced a miracle in him, not only in the beauty of his person, but in many gifts with which she endowed him and which he fully mastered.” Others marveled at the unique combination of physical strength and grace seemed to embody. Many authors, including Vasari, referred to him with the ultimate epithet – il divino.
6) Throughout his life, Leonardo displayed an air of serene self-confidence, which helped him to overcome professional setbacks and disappointments with equanimity and allowed him to calmly pursue his research even during times of great political turbulence. He was aware of his unique genius and skill, yet he never boasted about them. Nowhere in his Notebooks does he vaunt the originality of his inventions and discoveries, nor does he flaunt the superiority of his ideas, even as he explains how they differ from traditional beliefs. This lack of arrogance and ego was remarkable indeed. Another quality that distinguished him was his passion for life and for all living things.
7) The artist’s fascination with the grotesque forms also led him to devise the most extravagant, and often quite macabre, practical jokes, which delighted the courtiers in Milan and Rome. At the papal court in Rome, Vasari tells us that Leonardo obtained a large lizard to which he attached “with a mixture of quicksilver some wings, made from the scales stripped from other lizards, which quivered as it walked along. Then, after he had given it eyes, horns, and a beard he tamed the creature, and keeping it in a box he used to show it to his friends and frightened the life out of them.”
8) During Leonardo’s time, the term “genius” did not have our modern meaning of a person endowed with extraordinary intellectual and creative powers. The latin word genius originated in Roman religion, where it donate the spirit of the gens, the family. It was understood as a guardian spirit, first associated with individuals and then also with people and places. The extraordinary achievements of artists and scientists were attributed to their genius, or attendant spirit. This meaning of genius was prevalent throughout the Middle Ages and the Renaissance. In the eighteen century, the meaning of the word changed to its familiar modern meaning to denote these individuals themselves, as in the phrase “Newton is a genius”.
9) The first is an intense curiosity and great enthusiasm for discovery and understanding. This was indeed an outstanding quality of Leonardo, whom Kenneth Clark called “the most relentlessly curious man in history.” Another striking sign of genius is an extraordinary capacity of intense concentration over long periods of time. Isaac Newton apparently was able to hold a mathematical problem in his mind for weeks until it surrendered to his mental powers. When asked how he made his remarkable discoveries, Newton is reported to have replied, “I keep the subject constantly before me and wait until the first dawnings open little by little into the full light.” Leonardo seems to have worked in a very similar way, and most of the time not only on one but on several problems simultaneously.
10) Indeed the Italian humanists were so bold as to compare artistic creations to the creations of God. This comparison was first applied to the creativity of poets, and was then extended, especially by Leonardo, to the painter’s creative power: If the painter wants to see beauties that make him fall in love, he is the lord who can generate them, and if he wants to see monstrous things that frighten, or funny things that make him laugh, or things that truly arouse compassion, he is their lord and God… In fact, whatever there is in the universe, by essence, presence, or imagination, he has it first in his mind and than in his hands.
11) He wished to achieve relief through the scientific use of the light and shade. According to Leonardo, such an achievement is “the soul of painting”. Leonardo’s technique of using light and shade to give his figures “great vigor and relief,” as Vasari put it, culminated in his celebrated creation of sfumato, the subtle melting of shades that eventually became the unifying principle of the paintings. “Leonardo’s sfumato was the power behind the poetry of his paintings,” Arasse claimed, “and the mystery that seems to emanate from them.”
12) Leonardo could have not developed his mastery of chiaroscuro, nor his characteristic sfumato style, without a major advance in Renaissance paint – the use of oil-based paints. Oil painting makes it possible to put layers of paint on top of each other without blurring the colors (provided the layers are allowed to dry individually), to go back over work again, and to mix paints at ease, all of which were essential for Leonardo to achieve his special effects of relief and sfumato.
13) Over the years, Leonardo achieved a sublime mastery in applying the finest layers of paint to create the luminous color tones that give his paintings their special magic. As Serge Bramly describes it, “The light passes through his paintings as if through stained glass, straight on to the printed surface beneath, which reflects it back, thus creating the impression that it emanates from the figures themselves.
14) On the other hand, Leonardo’s completed masterpieces always involved radical innovations at several levels – artistic, philosophical, and scientific. For example, the Virgin of the Rocks was not only revolutionary in its rendering of light and dark. It also represented a complex and controversial meditation on the destiny of Christ, expressed through the gestures and relative positions of the four protagonists, as well as in the intricate symbolism of the surrounding rocks and vegetation.
15) In a similar vein, Vasari refers to Leonardo as “Florentine painter and sculptor” in the title of his biography. And yet, we have no known sculpture from Leonardo’s hand. His reputation rests on a single piece of work: a monumental bronze horse that was never cast, but which occupied Leonardo intensely for over ten years.
16) Good designers have the ability to think systematically and to synthesize. They excel at visualizing things, at organizing known elements into new configurations, at creating new relationships; and they are skillful in conveying these mental processes in the form of drawings almost as rapidly as they occur. Leonardo, off course, possessed all these abilities to a very high degree. In addition, he had an uncanny knack of perceiving and solving technical problems – another key characteristic of a good designer – so much so, in fact, that it was almost second nature to him.
17) What made Leonardo unique as a designer and engineer, however, was that many of the novel designs he presented in his Notebooks involved technological advances that would not be realized until several centuries latter. And second, he was the only man among the famous Renaissance engineers who made the transition from engineering to science. Like painting, engineering became a “mental discourse” for him. To know how something worked was not enough for Leonardo, he also needed to know why. Thus an inevitable process was set in motion, which led him from technology and engineering to pure science. As art historian Kenneth Clark notes, we can see the process at work in Leonardo’s manuscripts: First, there are questions about the construction of certain machines, then… questions about the first principles of dynamics; finally questions which had never been asked before about winds, clouds, the age of the earth, generation, the human heart. Mere curiosity has become profound scientific research, independent of the technical interests which had preceded it.
18) In other words, the problems Leonardo addresses are theoretical problems of architectural design. The questions he asks are the same questions he explores throughout his science of organic forms – questions about patterns, spatial organization, rhythm, and flow. The notes accompanying his drawings (written in his customary mirror writing, and hence intended for himself) can be seen as fragments of a treatise on architecture that Leonardo, according to Heydenreich, may have intended to compose.
19) In view of Leonardo’s central focus on understanding nature’s forms, both in the macro- and the microcosm, it is not surprising that he emphasized similarities between architectural structures and structures in nature, especially in human anatomy. In fact, this linking of architecture and anatomy goes back to antiquity and was common among Renaissance architects, who recognized the analogy between a good architect and a good doctor. As Leonardo explained, “Doctors, teachers, and those who nurse the sick should understand what man is, what is life, what is health, and in what manner a parity and concordance of the elements maintains it… The same is also needed for the ailing cathedral, that is, a doctor-architect who understands well what buildings is and from what rules the correct way of building derives.”
20) Leonardo’s science, by contrast, cannot be reduced to a single foundation, as we have seen. Its strength does not derive from a single trunk, but from the complex interconnectedness of the branches of many trees. For Leonardo, recognizing the numerous patterns of relationships in nature was the hallmark of a universal science. Today, we, too, sense a greater need for such universal, or systemic, knowledge, which is one of the reasons why Leonardo’s unified vision of the world is so relevant to our time.
21) Leonardo showed greatly artistic talent early in his youth; his synthesis of art and science was also foreshadowed early on. This is vividly illustrated in a story related by Vasari. When Piero da Vinci was asked by a peasant to have a “buckler” (a small wooden shield) decorated with a painting in Florence, he did not give the shield to a Florentine artist but instead asked his son to paint something on it. Leonardo decided to paint a terrifying monster. “To do what he wanted,” writes Vasari, “Leonardo carried into a room of his own, which no one else entered except himself, a number of lizards, crickets, serpents, butterflies, locusts, bats, and various strange creatures of this nature. From all these he took and assembled different parts to create a fearsome and horrible monster… He depicted the creature emerging from a dark cleft of a rock, belching forth venom from its open throat, fire from its yes, and smoke from its nostrils in so macabre a fashion that the effect was altogether monstrous and horrible. Leonardo took so long over the work that the stench of dead animals in his room became unbearable, although he himself failed to notice because of his great love of painting.” When Ser Piero came to see the finished painting, Leonardo went back into the room, put the buckler on an easel in the light, and shaded the window. Then he asked Piero to come and see it. When his eyes fell on it, Piero was completely taken by surprise and gave a sudden start, not realizing that he was looking at the buckler and that the form he saw was, in fact, painted on it. As he backed away, Leonardo stopped him and said: ‘This work certainly serves its purpose. It has produced the right reaction, so now you can take it away.’”
22) Other inventions he created from that time involved fire and a hot air. In addition to the self-regulating spit mentioned earlier, Leonardo invented a method of creating a vacuum to raise water by means of a fire burning in a closed bucket, based on the observation that a burning flame consumes air. During these early years he also developed his first versions of a diving apparatus. During a visit to Vinci he designed an olive press with more efficient leverage than the presses used at the time. While he was engaged in these multiple projects of invention, design, and engineering, Leonardo also painted his Annunciation, two Madonnas, and the portrait of Ginevra de’ Benci.
23) He drew [a] long series of diagrams showing the effect of light falling on spheres and cylinders, crossing, reflecting, intersecting with endlessly variety… 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.
24) He was asked by Ludovico to paint a portrait of the Moor’s mistress, the young and lovely Cecilia Gallerani. Leonardo painted her holding an ermine, a symbol of purity and moderation which, because of its Greek name, gale, was also a veiled allusion to her name, Gallerani. Lady with an Ermine, as it is called today, was a highly original portrait in which Leonardo invented a new pose, with the model looking over her shoulder with an air of surprise and subdued delight, caused, perhaps, by the unexpected arrival of her lover. Her gesture is graceful and elegant, and is echoed in the animal’s twisting movement.
25) For Leonardo himself, the 1940s were a period of intense creative activity. With two major projects – the equestrian statue and The Last Supper – his artistic career was at its peak, he was consulted repeatedly as an expert on architectural design, and he embarked on extensive and systematic research in mathematics, optics, mechanics and the theory of human fly.
26) Leonardo’s research in statics and dynamics was concerned not only with the workings of machines but also, and even more important, with understanding the human body and its movements. For example, he investigated the body’s ability to generate various amount of forces in several positions. One of the key aims was to find out how a human pilot might generate enough force to lift a flying machine off the ground by flapping its mechanical wings. In his studies of machines during that period, Leonardo began to separate individual mechanisms – levers, gears, bearings, couplings, etc. – from the machines in which they were embedded. This conceptual separation did not arise again in engineering until the eighteenth century. In fact, Leonardo planned (and may even have written) a treatise on Elements of Machines, perhaps influenced by his discussions with Fazio Cardano of Euclid’s celebrated Elements of Geometry in Pavia.
27) Leonardo’s Last Supper, generally considered the first painting of the High Renaissance (the period of art between, approximately 1495 and 1520), is dramatically different from earlier representations of the subject. Indeed, it became famous throughout Europe immediately after his completion and was copied innumerable times. The firstly highly imaginative feature one notices is the way Leonardo integrated the fresco into the architecture of the refectory. Demonstrating his mastery of geometry, Leonardo contrived a series of visual paradoxes to create an elaborate illusion – a complex perspective that made the room of the Last Supper look like a refectory itself, in which the monks ate their meals. One consequence of this complex perspective is that from every viewing position in the room, the spectator is drawn into the drama of the picture’s narrative with equal force. And dramatic it is. Whereas traditionally the Last Supper was pictured at the moment of communion, a moment of calm, individual meditation for each apostle, Leonardo chose the ominous moment when Jesus says, “One of you will betray me.” The words of Christ have stirred up the solemn company, creating powerful waves of emotion. However, the effect is far from chaotic. The apostles are clearly organized into four groups of three figures, with Judas forming one of the groups together with Peter and John. This is another striking compositional innovation. Traditionally, Judas was pictured sitting on the other side of the table, facing the apostles, with his back to the spectator. Leonardo had no need to identify the traitor by isolating him in this way. By given the apostles carefully expressive gestures, which together cover a wide range of emotions, the artist made sure that we immediately recognize Judas, as he shrinks back into the dark of John’s shadow, nervously clutching his bag of silver. The depiction of the apostles as embodiments of individual emotional states and the integration of Judas into the dramatic narrative were so revolutionary that after Leonardo, no self-respecting artist could go back to the previous static configuration.
28) Soon after they began their study sessions, Leonardo and Fra Luca decided to collaborate on a book, titled De divina proportione, to be written by Pacioli and illustrated by Leonardo. The book, presented to Ludovico as a lavish manuscript and eventually published in Venice, contains an extensive review of the role of proportion in architecture and anatomy – and in particular of the golden section, or “divine proportion” – as well as detailed discussions of the five regular polyhedra known as the Platonic solids. It features over sixty illustrations by Leonardo, including superb drawings of the Platonic solids in both solid and skeletal forms, testimony to his exceptional ability to visualize abstract geometric forms. What further distinguishes this work is that it is the only collection of drawings by Leonardo published during his lifetime.
29) In the Madonna and Child with Saint Anne, as the paint is called today, Leonardo had again broken new ground with both his composition and the theological interpretation of a traditional religious theme. Rather than presenting Mary and her mother, Saint Anne, in static configuration – seated next to each other with Jesus in Mary’s arms between them, or with Saint Anne seated higher in a majestic, hierarchical composition – Leonardo upset tradition by adding a lamb as a fourth figure. Jesus, having slipped to the ground, reaches for the lamb as Mary tries to restrain him, and Saint Anne seems to hold her back. The theological message embodied in Leonardo’s highly original composition can be seen as a continuation of his long meditation on the destiny of Christ, which he had begun with the Virgin of the Rocks. Mary, in an anxious gesture, attempts to pull her soon away from the lamb, the symbol of Passion, while Saint Anne, representing Mother Church, knows that Mary’s gesture is futile – the Passion is Christ’s destiny and cannot be avoid.
30) When he had built flight machines in Milan and tested them in his workshop in Corte Vecchia, Leonardo’s main concern had been to find out how human pilot could flap mechanical wings with enough force and velocity to compress the air underneath and be lifted up. For these tests he had designed various types of wings modeled after those of birds, bats, and flying fish. Now, ten years later, he embarked on careful and methodical observations of the flight of birds. He spent hours in the hills surrounding Florence, near Fiesole, observing the behavior of birds in flight, and filled several Notebooks with drawings and comments that analyzed the birds’ turning maneuvers, their ability to maintain their equilibrium in the wind, and the detailed mechanisms of active flight. His aim was to design a flying machine that would be able, like a bird, to maneuver with agility, keep its balance in the wind, and move its wings with enough force to allow it to fly.
31) In his Anatomical Studies, Leonardo gives a vivid description of the dreadful conditions under which he had to work. As there were no chemicals to preserve the cadavers, they would begin to decompose before he had time to examine and draw them properly. To avoid accusations of heresy, he worked at night, lighting his dissection room by candles, which must have made the experience even more macabre. “You will perhaps be impeded by the fear of living through the night hours in the company of these corpses, quartered and flayed and frightening to behold.”
32) One will see darkly gloomy air beaten by the rush of different and convoluting winds, which are mingled with the weight of continuous rain, and which are carrying helter-skelter an infinite number of branches torn from the trees, entangled with countless autumn leaves. The ancient trees will be seen uprooted and thorn to pieces by the fury of the winds… Oh how many will you see closing their ears with their hands to shut out the tremendous noises made in the darkened air by the raging of the winds… Others, with gestures of hopelessness, took their own lives, despairing of being able to endure such suffering; and of these, some flung themselves from high rocks, others strangled themselves from high rocks, others strangled themselves with their own hands.
33) The drawings that illustrate his apocalyptic narrative are dark, violent, menacing, and disturbing. Nonetheless, they are astonishingly accurate in their renderings of water and air turbulence. Throughout his life, Leonardo had carefully studied the forms of waves, eddies, waterfalls, vortices, and air currents. Here, in old age, he summed up his knowledge of turbulence. Beyond their expressive emotional power, the deluged drawings can be seen as sophisticated mathematical diagrams, presenting a visual catalog of turbulent flows that would not look out of place in a modern textbook on fluid dynamics.
34) In Leonardo’s mind, his science of living forms was certainly an integrated whole. At the end of his life, his problems were no longer conceptual; they were simply the limitations of time and energy. As he wrote several years before his death, “I have been impeded neither by avarice nor by negligence, but only by time.” And yet, Leonardo never gave up. In June 1518 he wrote what may have been the last entry in his Notebooks: “I shall go on.”
35) Nor was he perturbed by contemplating his approaching death. “Just well-spent day brings a happy sleep,” he had written thirty years earlier, “so a well-employed life brings a happy death.”
36) A few days after completing his will, on May 2, 1519, Leonardo da Vinci died in the manor of Cloux – according to legend, in the arms of the king of France.
37) To appreciate Leonardo’s science, it is important to understand the cultural and intellectual context in which he created it. Scientific ideas do not occur in a vacuum. They are always shaped by the technologies available at the time. The entire constellation of concepts, values, perceptions, and practices – the “scientific paradigm” in the terminology of science historian Thomas Kuhn – provides the context that is necessary for scientists to pose the great questions, organize their subjects, and define legitimate problems and solutions. All science is built upon such an intellectual and cultural foundation. Hence, when we recognize ancient or medieval ideas reflected in Leonardo’s scientific writings, this do not mean that he was less of a scientist, Leonardo consulted the traditional texts and used their conceptual framework as his starting point. He then tested the traditional ideas against his own scientific observations. And, in accordance with scientific method, he did not hesitate to modify the old theories when his experiments contradicted them.
38) The leading figure in the movement to weave the philosophy of Aristotle into Christian teachings was Saint Thomas Aquinas, one of the towering intellects of the Middle Ages. Aquinas taught that there could be no conflict between faith and reason, but the two books on which they were based – the Bible and the “book of nature” – were both authored by God. Aquinas produced a vast body of precise, detailed, and systematic philosophical writings in which he integrated Aristotle’s encyclopedic works and medieval Christian theology into a magnificent whole. The dark side of this seamless fusion of science and theology was that any contradiction by future scientists would necessarily have to be seen as heresy. In this way, Thomas Aquinas enshrined in his writings the potential for conflicts between science and religion – which indeed arose three centuries later in Leonardo’s anatomical research, reached a dramatic climax with the trial of Galileo, and have continued to the present day.
39) A few years later, at the height of his anatomical work in Milan, Leonardo added a technical note about the reproduction of his drawings to his famous assertion of the superiority of drawing over writing. He insisted that his anatomical drawings should be printed from copper plates, which would be more expensive than woodcuts but much more effective in rendering the fine details of his work. “I beg you who come after me”, he wrote on the sheet that contains his magnificent drawings of the vertebral column, “not let avarice constrain you to make the prints in [wood].”
40) The conception of the Renaissance worldview was the conceptions of the universe that had been developed in classical Greek science: that the world was a kosmos, an ordered and harmonious structure. From its beginnings in the sixth century B.C., Greek philosophy and the science understood the order of the cosmos to be that of a living organism rather than a mechanical system. This meant that all its parts had an innate purpose to contribute to the harmonious functioning of the whole, and that objects moved naturally toward their proper places in the universe. Such an explanation of natural phenomena in terms of their goals, or purposes, is known as teleology, from the Greek telos (purpose). It permeated virtually all of Greek philosophy and science. The view of the cosmos as an organism also implied for the Greeks that its general properties are reflected in each of its parts. This analogy between macrocosm and microcosm, in particular between the Earth and the human body, was articulated most eloquently by Plato in his Timaeus in the fourth century B.C., but it can also be found in the teachings of the Pythagoreans in other earlier schools. Over time, this idea acquired the authority of common knowledge, which continued throughout the Middle Ages and into Renaissance. In early Greek philosophy, the ultimate moving force and source of all life was indentified with the soul, and its metaphor was that of the breath of life. Indeed, the root meaning of both the Greek psyche and the Latin anima is “breath”. Closely associated with that moving force – the breath of life that leaves the body at death – was the idea of knowing. For the early Greek philosophers, the soul was both the source of movement and life, and that which perceives and knows. Because of the fundamental analogy between micro- and macrocosm, the individual soul was thought to be part of the force that moves the entire universe, and accordingly the knowing of an individual was seen as part of a universal process of knowing. Plato called it the anima mundi, the “world soul”.
41) The culmination of the early phase of Greek mathematics was reached around 300 B.C. with Euclid, who presented all of the geometry and other mathematics known in his days in a systematic, orderly sequence in his celebrated Elements. The thirteen volumes of this classical textbook were not only widely read during the Renaissance, but remained the foundation for the teaching of geometry until the end of the nineteenth century.
42) Health, according to the Hippocratic writings, requires a state of balance among environmental influences, the way in which we live, and the various components of human nature. One of the most important volumes in the Hippocratic Corpus, the book on Airs, Waters and Places, represents what we might now call a treatise on human ecology. It shows in greater detail how the well-being of individuals is influenced by environmental factors – the quality of air, water, and food, the topography of the land, and general living habits. During the last two decades of the fifteenth century, this and several other volumes from the Hippocratic Corpus were available to scholars in Latin, most of them derived from Arabic translations.
43) Leonardo da Vinci shared with his fellow humanists their great confidence in the capabilities of the human individual, their passion for voyages and exploration, and their excitement about the rediscovery of the classical texts of antiquity. But he differed dramatically from most of them by refusing to blindly accept the teachings of the classical authorities. He studied them carefully, but then he tested them by subjecting them to rigorous comparisons with his own experiments and his direct observations of nature. In doing so, I would argue, Leonardo single-handedly developed a new approach to knowledge, known today as scientific method.
44) All scientific models and theories are limited and approximate. This realization has become crucial to the contemporary understanding of science. Twentieth-century science has shown repeatedly that all natural phenomena are ultimately interconnected, and that their essential properties, in fact, derive from their relationships to other things. Hence, in order to explain any one of them completely, we have to understand all the others, which is obviously impossible. This insight has forced us to abandon the Cartesian belief in the certainty of scientific knowledge and to realize that science, to put into bluntly, we never deal with truth, in the sense of a precise correspondence between our descriptions and the described phenomena. We always deal with limited and approximate knowledge. This may sound frustrating, but for many scientists the fact that we can formulate approximate models and theories to describe an endless web of interconnected phenomena, and that we are able to systematically improve our models and approximations over time, is a source of confidence and strength. As the great biochemist Louis Pasteur put it, “Science advances through tentative answers to a series of more and more subtle questions which reach deeper and deeper into the essence of all natural phenomena.”
45) “All our knowledge has its origins in the senses,” he noted in his first Notebook, the Codex Trivulzianos. “Wisdom is the daughter of experience,” we read in the Codex Forster, and in his Treatise on Painting, Leonardo asserted: “To me it seems that those sciences are vain and full of errors that are not born of experience, mother of all certainty… that is to say, which do not at their beginning middle, or end pass through any of the five senses.” Such an approach to the study of nature was unheard-of in Leonardo’s day, and would fully emerge again only in the seventeenth century, the era of the Scientific Revolution.
46) He recognized that learning from skilled masters was important in the arts, but he also observed that such masters were rare. “The surer way,” he suggested, “is to go to the objects of nature, rather than those that are imitated with great deterioration, and so acquire sad habits; for he who can go to the well does not go to the water jar.”
47) He was deeply aware of the fundamental interconnectedness of all phenomena and of the interdependence and mutual generation of all parts of an organic whole, which Immanuel Kant in the eighteenth century would define as “self-organization.” In the Codex Atlanticus, Leonardo eloquently summarized his profound understanding of life’s basic processes by paraphrasing a statement by the Ionian philosopher Anaxagoras: “Everything comes from everything, and everything is made of everything, and everything turns into everything, because that which exists in the elements is made up of the elements.”
48) Only in the twentieth century did the limits of Newtonian science become fully apparent, and the mechanistic Cartesian worldview begin to give way to a holistic and ecological view not unlike that developed by Leonardo da Vinci. With the rise of systemic thinking and its emphasis on networks, complexity, and patterns of organization, we can know more fully appreciate the power of Leonardo’s science and its relevance for our modern era. Leonardo’s science is a science of qualities, of shapes and proportions, rather than absolute quantities. He preferred to depict the forms of nature in his drawings rather than describe their shapes, and he analyzed them in terms of their proportions rather than measured quantities. Proportion was seen by Renaissance artists as the essence of harmony and beauty. Leonardo filled many pages of his Notebooks with elaborate diagrams of proportions between the various parts of the human figure, and he drew corresponding diagrams to analyze the body of the horse.
49) Leonardo was always impressed by the great diversity and variety of living forms. “Nature is so delightful and abundant in its variations,” he wrote in a passage about how to paint trees, “that among trees of the same kind there would not be found one plant that resembles another nearby, and this is not only of the plant as a whole, but among the branches, the leaves, and the fruit, not one will be found that looks precisely like another.”
50) 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 the humor of all living bodies,” he wrote. Without it nothing retains its original form.” 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.
51) 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 basin 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 scalability and change, embody an essential characteristic of the living forms.
52) 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 fourteenth centuries.
53) 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. 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.
54) Based on these designs, British engineers recently built a glider and tested it successfully in a flight from the chalk cliffs in southeast England know as the Sussex Downs. This maiden flight of “Leonardo’s glider,” reportedly, exceeded the first attempts by the Wright brothers in 1900. Although the machines with movable mechanical wings were not destined to fly, the models built from Leonardo’s designs are extraordinary testimonies to his genius as a scientist and engineer. In the words of art historian Martin Kemp: “Using mechanical systems, the wings flap with much of the sinuous and menacing grace of a gigantic bird prey… [Leonardo’s] designs retain their conceptual power as archetypal expressions of man’s desire to emulate the birds, and remain capable of inspiring a sense of wonder even in a modern audience, for whom the sight of tons of metal flying through the air has become a matter of routine.”
55) Leonardo’s careful and patient studies of the movements of the heart and the flow of blood, undertaken in old age, are the culmination of his anatomical work. He not only understood and pictured the heart like no one before him, but also observed subtleties in its actions and in the flow of blood that would elude medical researchers for centuries.
56) Leonardo’s success in cardiac anatomy [is] so great that there are aspects of the work which are not yet equaled by modern anatomical illustration… His consistent practice of illustration of the heart and its valves, both in systole and in diastole, with a comparison of the position of the parts, has rarely if ever been performed in any anatomical textbook.
57) Leonardo’s embryological drawings are graceful and touching revelations of the mysteries surrounding the origins of human life. In the words of physician Sherwin Nuland, “[His] depiction of a five-month fetus in the womb is a thing of beauty… It stands as a masterwork of art, and, considering the very little that was at the time understood of embryology, a masterwork of science perception as well.” Leonardo knew very well that, ultimately, the nature and origin of life would remain a mystery, no matter how brilliant his scientific mind was. “Nature is full of infinite causes that have never occurred in experience,” he declared in his late forties, and as he got older his sense of mystery deepened. Nearly all the figures in his last paintings have that smile that expresses the ineffable, often combined with a pointing finger. “Mystery to Leonardo”, wrote Kenneth Clark, “was a shadow, a smile and a finger pointing into darkness.”
58) Leonardo’s approach to mathematics was that of a scientist, not a mathematician. He wanted to use mathematical language to provide consistency and rigor to the descriptions of his scientific observations. However, in his time there was no mathematical language appropriate to express the kind of science he was pursuing – explorations of the forms of nature in their movements and transformations. And so Leonardo used his powers of visualization and his great intuition to experiment with new techniques that foreshadowed branches of mathematics that would not be developed until centuries later. These include the theory of functions and fields of integral calculus and topology.
59) The really important mathematics for him was geometry, which is evident from his praise of the eye as “the prince of mathematics.”
60) Like most mathematicians of his time, Leonardo frequently used geometrical figures to represent algebraic relationships. A simple but very ingenious example is his pervasive use of triangles and pyramids to illustrate arithmetic progressions and, more generally, what we now call linear functions. He was familiar with the use of pyramids to represent linear proportions from his studies of perspective, where he observed that “All the things transmit to the eye their image by means of a pyramid of lines. By ‘pyramid of lines’ I mean those lines which, starting from the edges of the surface of each object, converge from a distance and meet in a single point… placed in the eye.”
61) Leonardo realized very early on that the mathematics of his time was inappropriate for recording the most important results of his scientific research – the description of nature’s living forms in their ceaseless movements and transmutations. Instead of mathematics, he frequently used his exceptional drawing facility to graphically document his observations in pictures that are often strikingly beautiful while, at the same time, they take the place of mathematical diagrams. His celebrated drawing of “Water falling upon water”, for example, is not a realistic snapshot of a jet of water falling into a pond, but an elaborate diagram of Leonardo’s analysis of several types of turbulence caused by the impact of the jet.
62) Arasse makes an interesting point: Whenever Leonardo rendered objects in their sharp outlines, these pictures represented conceptual models rather than realistic images. And whenever he produced realistic images of objects, he blurred the outlines with his famous sfumato technique, in order to represent them as they actually appear to the human eye.
63) What Leonardo found especially attractive in geometry was its ability to deal with continuous variables. “The mathematical sciences… are only two,” he wrote in the Codex Madrid, “of which the first is arithmetic, the second is geometry. One encompasses the discontinuous quantities [i.e., variables] the other the continuous.” It was evident to Leonardo that a mathematic of continuous quantities would be needed to describe the incessant movements and transformations in nature. In the seventeenth century, mathematicians developed the theory of functions and the differential calculus for that very purpose. Instead of these sophisticated mathematical tools, Leonardo had only geometry at his disposal, but he expanded it and experimented with new interpretations and new forms of geometry that foreshadowed subsequent developments.
64) In the course of his explorations of circles and squares, Leonardo tried his hand at the problem of squaring the circle, which had fascinated mathematicians since antiquity. In its classical form, the challenge is to construct a square with an area equal of that of a given circle, and to do so by using only ruler and compass. We know today that this is not possible, but countless professional and amateur mathematicians have tried. Leonardo worked on the problem repeatedly over a period of more than a dozen years. In one particular attempt, he worked by candlelight through the night, and by dawn he believed that he had finally found the solution. “On the night of St. Andrew,” he excitedly recorded in his Notebook, “I found the end of squaring the circle; and at the end of the light of the candle, of the night, and of the paper on which I was writing, it was completed; at the end of the hour.” However, as the day progressed, he came to the realization that this attempt, too, was futile.
65) When we look at Leonardo’s geometry from the point of view of present-day mathematics, and in particular from the perspective of complexity theory, we can see that he developed the beginnings of the branch of mathematics now known as topology. Like Leonardo’s geometry, topology is a geometry of continuous transformations, or mappings, in which certain properties of geometric figures are preserved. For example, a sphere can be transformed into a cube or a cylinder, all of which have similar continuous surfaces. A doughnut (torus), by contrast, is topologically different because of the hole in its center. The torus can be transformed, for example, into a coffee cup where the hole now appears in the handle. In the words of historian of mathematics Morris Kline: Topology is concerned with those properties of geometric figures that remain invariant when the figures are bent, stretched, shrunk, or deformed in any way that does not create new points or fuse existing points. The transformations presupposes, in other words, that there is a one-to-one correspondence between the points of the original figure and the points of the transformed figure, and that transformation carries nearby points into nearby points. This latter property is called continuity.
66) During the last twelve years of his life, Leonardo spent a great deal of time mapping and exploring the transformations of his “geometry done with motion.” Several times he wrote of his intention to present the results of these studies in one or more treatises. During the years he spent in Rome, and while he was summing up his knowledge of complex turbulent flows in his famous deluge drawings, Leonardo produced a magnificent compendium of topological transformations, titled De ludo geometrico (On the Game of Geometry), on large double folio in the Codex Atlanticus. He drew 176 diagrams displaying a bewildering variety of geometric forms, built from intersecting circles, triangles and squares – row after row of crescents, rosettes and other floral patterns, paired leaves, pinwheels, and curvilinear stars. Previous this endless interplay of geometric motifs was often interpreted as the playful doodling of an aging artist – “a mere intellectual pastime,” in the words of Kenneth Clark. Such assessments were made because art historians were generally not aware of the mathematical significance of Leonardo’s geometry of transformations. Close examination of the double folio shows that its geometric forms, regardless of how complex and fanciful, are all based upon strict topological principles.
67) Since Leonardo’s science was a science of qualities, of organic forms and their movements and transformations, the mathematical “necessity” he saw in nature was not one expressed in quantities and numerical relationships, but one of geometric shapes continually transforming themselves according to rigorous laws and principles. “Mathematical” for Leonardo referred above all to the logic, rigor, and coherence according to which nature has shaped, and is continually reshaping, her organic forms. This meaning of “mathematical” is quite different from the one understood by scientists during the Scientific Revolution and the subsequent three hundred years. However, it is not unlike the understanding of some of the leading mathematicians today. The recent development of complexity theory has generated a new mathematical language in which the dynamics of complex systems – including the turbulent flows and growth patterns of plants studied by Leonardo – are no longer represented by algebraic relationships, but instead by geometric shapes, like the computer-generated strange attractors or fractals, which are analyzed in terms of topological concepts. This new mathematics, naturally, is far more abstract and sophisticated than anything Leonardo could have imagined in the fifteenth and sixteenth centuries. But is used in the same spirit in which he developed his “geometry done with motion” – to show with mathematical rigor how complex natural phenomena are shaped and transformed by the “necessity” of physical forces. The mathematics of complexity has led to a new appreciation of geometry and to the broad realization that mathematics is much more than formulas and equations. Like Leonardo da Vinci five hundred years ago, modern mathematicians today are showing us that understanding of patterns, relationships, and transformations is crucial to understand the living world around us, and that all questions of pattern, order, and coherence are ultimately mathematical.
68) 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 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.
69) 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.
70) Leonardo demonstrated his throughout 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 with great effect. 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.
71) From the 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. 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, 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.
72) 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 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.
73) 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.”
74) 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.” 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 East to the West,” he noted, “and all the other immaterial things have velocities that are by a long way inferior.”
75) 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 continue with rare expression of religious awe: “Certainly none. This is where human discourse turns directly to the contemplation of the divine.” In the Treatise on Painting, Leonardo waxes enthusiastic about the human eye: Don’t you see that the eye embraces the beauty of the whole world? It is the master of astronomy, it practices cosmography, it counsels and corrects all human arts; its transports man to different parts of the world. [The eye] is the prince of mathematics; its sciences are most certain. It has measured the heights and sizes of the stars; it has discovered the elements and their locations…. It has created architecture, perspective, and divine painting…. [The eye] is the window of the human body, through which [the soul] contemplates and enjoys the beauty of the world.
76) “The pupil of the eye,” he concluded, “changes to as many different sizes as there are differences in the degrees of brightness and darkness of the objects which present themselves before it…. Nature has equipped the visual faculty, when irritated by excessive light, with the contraction of the pupil… and here nature works like someone who, having too much light in his house, closes half of a window, and more or less according to necessity.” And then he added: “You can observe that in nocturnal animals such as cats, screech owls, tawny owls and others, which have the pupil small at midday and very large at night.”
77) During the last two decades of the twentieth century, however, a novel conception of the nature of mind and consciousness emerged in the life sciences, which finally overcame the Cartesian division between mind and body. The decisive advance has been to reject the view of mind as a thing; to realize that mind and consciousness are not entities but processes. In the past twenty-five years the study of mind from this new perspective has blossomed into a rich interdisciplinary field known as cognitive science, which transcends the traditional frameworks of biology, psychology, and epistemology. One of the central insights of cognitive science is the identification of cognition, the process of knowing, with the process of life. Accordingly, the interactions of a living organism – plant, animal, or human – with its environment are understood as cognitive interactions. Thus life and cognition become inseparably connected. Mind, – or, more accurately, mental activity – is immanent in matter at all levels of life. This new conception represents a radical expansion of the concept of cognition and, implicitly, the concept of mind. In the new view, cognition involves the entire process of life – including perception, emotion, and behavior – and does not even necessarily require a brain and a nervous system.
78) At the end of life, the reverse process takes place: “While I thought I was learning how to live, I had been learning how to die,” Leonardo wrote movingly late in his life.
79) Nature, as a whole was alive and animated for Leonardo, a world in continual flux and development, in the macrocosm of the Earth as in the microcosm of the human body.
80) Leonardo himself never boasted about his unique talents and skills, and in his thousands of pages of manuscripts he never vaunted the originality of so many of his ideas and discoveries. But he was well aware of his exceptional stature. In the Codex Madrid, in the midst of extensive discussions of the laws of mechanics, we find two lines that can stand as his own definitive epitaph: Read me, O reader, if in my words you find delight, for rarely in the world will one such as I be born again.
Synthesis Ratio :: 274 total of pages read / 23 typewritten pages :: 11.9
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