Fritjof Capra

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

  While his contemporaries deferred to the authorities of Aristotle and the Church, Leonardo developed and practiced an empirical approach to acquiring independent knowledge, which became known as the scientific method many centuries after him. It involved the systematic and careful observation of natural phenomena, ingenious experiments, the formulation of theoretical models, and many attempts at mathematical generalizations.

  Leonardo used his empirical method—together with his exceptional powers of observation and his “sublime left hand”—to analyze, draw, and paint “with philosophic and subtle speculation…all the qualities of forms.”1 The records he left of his lifelong investigations are superb testimonies of both his art and his science.

  In recent decades, scholars have given us comprehensive analyses of some areas of Leonardo’s science (albeit often from perspectives somewhat different from mine), while other areas remain largely unexplored. Leonardo’s entire corpus of anatomical studies has been analyzed in impressive detail in a magnificent book, Leonardo da Vinci’s Elements of the Science of Man, by the historian of medicine and Leonardo scholar Kenneth Keele.2

  Leonardo’s original contributions to landscape and garden design as well as his outstanding work in botany are discussed in great detail in an insightful volume by botanist William Emboden, Leonardo da Vinci on Plants and Gardens.3 Unfortunately, there is no comparable volume about Leonardo’s voluminous writings on “the motion of the waters,” which include his pioneering studies of fluid flow, as well as his many original thoughts on the ecological dimension of water as the medium and nurturing fluid of life. His related geological observations, centuries ahead of their time, also remain largely unexplored.

  Leonardo’s contributions to mechanics and engineering are discussed extensively in several books, including the beautiful volume on Renaissance Engineers from Brunelleschi to Leonardo da Vinci by science historian Paolo Galluzzi.4 His precise observations and analyses of the flight of birds and his persistent attempts to design workable flying machines are evaluated in a captivating, richly illustrated monograph by science historian Domenico Laurenza, Leonardo on Flight.5 However, no overall assessment of Leonardo’s wide-ranging works in architecture and engineering from the modern perspective of design has been offered so far.6 This would certainly be a fascinating subject.

  Leonardo’s studies of the living forms of nature began with their outward appearance and then turned to methodical investigations of their intrinsic nature. Life’s patterns of organization, its organic structures, and its fundamental processes of metabolism and growth are the unifying conceptual threads that interlink his knowledge of macro-and microcosm. Throughout his life he studied, drew, and painted the rocks and sediments of the Earth, shaped by water; the growth of plants shaped by their metabolism; and the anatomy of the animal body in motion. He used his scientific understanding of the forms of nature as the intellectual underpinning of his art, and he used his drawings and paintings as tools of scientific analysis. Thus Leonardo’s studies of nature’s living forms represent a seamless unity of art and science.

  In the Italian Renaissance, it was not unusual to find painters who were also accomplished sculptors, architects, or engineers. The uomo universale was the great ideal of the time. Nevertheless, Leonardo da Vinci’s synthesis of art and science, and its brilliant applications in numerous fields of design and engineering, were absolutely unique. In subsequent centuries, Leonardo’s scientific concepts and observations were gradually rediscovered, and his vision of a science of organic forms reemerged several times in different epochs. Never again, however, was so much intellectual and artistic genius embodied in a single human being.

  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.7

  For over forty years, Leonardo relentlessly pursued his scientific explorations, driven by his restless and intense intellectual curiosity, his love of nature, and his passion for all living things. His magnificent drawings often reflect that passion with great delicacy and sensitivity. For example, his famous picture of a fetus in utero (Fig. E-1) is accompanied by several smaller sketches that liken the womb to the embryo sac of a flower, picturing the peeled-off layers of the uterine membranes in an arrangement of flower petals. The entire set of drawings vividly shows Leonardo’s tremendous care and respect for all forms of life. They exude a tenderness that is deeply moving.

  Leonardo’s science was a gentle science. He abhorred violence and had a special compassion for animals. He was a vegetarian because he did not want to cause animals pain by killing them for food. He would buy caged birds in the marketplace and set them free, and would observe their flight not only with a sharp observational eye but also with great empathy. Browsing through the Notebooks, one may suddenly get the impression that a single bird has flown right onto the page while Leonardo was discussing something else, followed by a whole flock of fluttering creatures on the subsequent folios.8

  In the designs of his flying machines, Leonardo tried to imitate the flight of birds so closely that he almost gives the impression of wanting to become a bird. He called his flying machine uccello (bird), and when he drew its mechanical wings, he mimicked the anatomical structure of a bird’s wing so accurately and, one almost feels, lovingly, that it is often hard to tell the difference (see Fig. E-2).

  Instead of trying to dominate nature, as Francis Bacon advocated in the seventeenth century, Leonardo’s intent was to learn from her as much as possible. He was in awe of the beauty he saw in the complexity of natural forms, patterns, and processes, and aware that nature’s ingenuity was far superior to human design. “Though human ingenuity in various inventions uses different instruments for the same end,” he declared, “it will never discover an invention more beautiful, easier, or more economical than nature’s, because in her inventions nothing is wanting and nothing is superfluous.”9

  This attitude of seeing nature as a model and mentor is now being rediscovered in the practice of ecological design. Like Leonardo da Vinci five hundred years ago, ecodesigners today study the patterns and flows in the natural world and try to incorporate the underlying principles into their design processes.10 When Leonardo designed villas and palaces, he paid special attention to the movements of people and goods through the buildings, applying the metaphor of metabolic processes to his architectural designs.11 He also considered gardens as parts of buildings, always attempting to integrate architecture and nature. He applied the same principles to his designs of cities, viewing a city as a kind of organism in which people, material goods, food, water, and waste need to flow with ease for the city to be healthy.12

  Figure E-1: The fetus within the womb, c. 1510–12, Anatomical Studies, folio 198r

  Figure E-2: Study for a mechanical wing imitating the wing of a bird, Codex sul volo, folio 7r

  In his extensive projects of hydraulic engineering, Leonardo carefully studied the flow of rivers in order to gently modify their courses by inserting relatively small dams in the right places and at the optimal angles. “A river, to be diverted from one place to another, should be coaxed and not coerced with violence,” he explained.13

  These examples of using natural processes as models for human design, and of working with nature rather than trying to dominate her, show clearly that as a designer, Leonardo worked in the spirit that the ecodesign movement is advocating today. Underlying this attitude of appreciation and respect of nature is a philosophical stance that does not view humans as standing apart from the rest of the living world, but rather as being fundamentally embedded in, and dependent upon, the entire community of li
fe in the biosphere.

  Today this philosophical stance is promoted by a school of thought and cultural movement known as “deep ecology.”14 The distinction between “shallow” and “deep” ecology is now widely accepted as a useful terminology for referring to a major division within contemporary environmental thought. Shallow ecology views humans as above or outside the natural world, as the source of all value, and ascribes only instrumental, or “use,” value to nature. Deep ecology, by contrast, does not separate humans—or anything else—from the natural environment. It sees the living world as being fundamentally interconnected and interdependent and recognizes the intrinsic value of all living beings. Amazingly, Leonardo’s Notebooks contain an explicit articulation of that view: “The virtues of grasses, stones, and trees do not exist because humans know them…. Grasses are noble in themselves without the aid of human languages or letters.”15

  Ultimately, deep ecological awareness is spiritual or religious awareness. When spirituality is understood as a way of being that flows from a deep sense of oneness with all, a sense of belonging to the universe as a whole, it becomes clear that ecological awareness is spiritual in its deepest essence.16 It seems that Leonardo da Vinci’s view of the world had that kind of spiritual dimension. Unlike most of his contemporaries, he hardly ever referred to God’s creation, but preferred to speak of the infinite works and marvelous inventions of nature. The Notebooks are full of passages in which he describes how nature “has ordained” that animals should experience pain, how she has created stones, made the surface of the cornea convex, given movement to animals, and formed their bodies.

  In all of these passages, one senses Leonardo’s great reverence for nature’s boundless creativity and wisdom. They are not couched in religious language, but are deeply spiritual nonetheless.

  During the centuries after Leonardo’s death, while his Notebooks remained hidden, the Scientific Revolution and the Industrial Revolution replaced the organic worldview of the Middle Ages and the Renaissance with the altogether different conception of the world as a machine. The resulting mechanistic paradigm—formulated in scientific language by Galileo, Descartes, Newton, and Locke—has dominated our culture for over three hundred years, during which it has shaped modern Western society and significantly influenced the rest of the world.17

  This paradigm consists of a number of deeply entrenched ideas and values, among them the view of the universe as a mechanical system composed of elementary building blocks, the view of the human body as a machine, the view of life in society as a competitive struggle for existence, and the belief in unlimited material progress to be achieved through economic and technological growth. All of these assumptions have been fatefully challenged by recent events, and a radical revision of them is now occurring.

  As our new century unfolds, it is becoming increasingly apparent that the major problems of our time—whether economic, environmental, technological, social, or political—are systemic problems that cannot be solved within the current fragmented and reductionist framework of our academic disciplines and social institutions. We need a radical shift in our perceptions, thinking, and values. And, indeed, we are now at the beginning of such a fundamental change of world-view in science and society.

  During the last few decades, the mechanistic Cartesian view of the world has begun to give way to a holistic and ecological view not unlike that expressed in the science and art of Leonardo da Vinci. Instead of seeing the universe as a machine composed of elementary building blocks, scientists have discovered that the material world, ultimately, is a network of inseparable patterns of relationships; that the planet as a whole is a living, self-regulating system. The view of the human body as a machine and of the mind as a separate entity is being replaced by one that sees not only the brain, but also the immune system, the bodily tissues, and even each cell as a living, cognitive system. Evolution is no longer seen as a competitive struggle for existence, but rather a cooperative dance in which creativity and the constant emergence of novelty are the driving forces. And with the new emphasis on complexity, networks, and patterns of organization, a new science of quality is slowly emerging.18

  Naturally, this new science is being formulated in a language that is quite different from that of Leonardo’s, as it incorporates the latest achievements of biochemistry, genetics, neuroscience, and other advanced scientific disciplines. However, the underlying conception of the living world as being fundamentally interconnected, highly complex, creative, and imbued with cognitive intelligence is quite similar to Leonardo’s vision. This is why the science and art of this great sage of the Renaissance, with their integrative scope, sublime beauty, and life-affirming ethics, are a tremendous inspiration for our time.

  APPENDIX

  Leonardo’s Geometry of Transformations

  In this appendix, I shall discuss some of the more technical details of Leonardo’s geometry of transformations, which may be of interest to readers familiar with modern mathematics.

  There are three types of curvilinear transformations that Leonardo uses repeatedly in various combinations.1 In the first type, a given figure with one curvilinear side is translated into a new position in such a way that the two figures overlap (see Fig. A-1). Since the two figures are identical, the two parts remaining when the part they have in common (B) is subtracted must have equal areas (A = C). This technique allows Leonardo to transform any area bounded by two identical curves into a rectangular area, that is, to “square” it.

  Figure A-1: Transformation by translation

  However, in accordance with his science of qualities, Leonardo is not interested in calculating areas, only in establishing proportions.2

  The second type of transformation is achieved by cutting out a segment from a given figure, say a triangle, and then reattaching it on the other side (see Fig. A-2). The new curvilinear figure, obviously, has the same area as the original triangle. As Leonardo explains in the accompanying text: “I shall take away portion b from triangle ab, and I will return it at c…. If I give back to a surface what I have taken away from it, the surface returns to its former state.”3 He frequently draws such curvilinear triangles, which he calls falcate (falcates), deriving the term from falce, the Italian word for scythe.

  Figure A-2: Transformation of a triangle into a “falcate”

  Leonardo’s third type of transformation involves gradual deformations rather than movements of rigid figures; for example, the deformation of a rectangle, as shown in Figure A-3. The equality of the two areas can be shown by dividing the rectangle into thin parallel strips and then pushing each strip into a new position, so that the two vertical straight lines are turned into curves.

  Figure A-3: Deformation of a rectangle

  This operation can easily be demonstrated with a deck of cards. However, to rigorously prove the equality of the two areas requires making the strips infinitely thin and using the methods of integral calculus. As Matilde Macagno points out, this example shows again that Leonardo’s way of visualizing these mappings and transformations foreshadows concepts associated with the development of calculus.4

  In addition to these three basic transformations, Leonardo experimented extensively with a geometric theorem involving a triangle and a moon-shaped segment, which is known as the “lunula of Hippocrates” after the Greek mathematician Hippocrates of Chios. To construct this figure, a rectangular isosceles triangle ABC is inscribed in a circle with radius a, and then an arc with radius b is drawn around point C from A to B (see Fig. A-4). The lunula in question is the shaded area bounded by the two circular arcs.

  Figure A-4: The lunula of Hippocrates

  Hippocrates of Chios (not to be confused with the famous physician Hippocrates of Cos) proved in the fifth century B.C. that the area of the lunula is equal to that of the triangle ABC. This surprising equality can easily be verified with elementary geometry, taking into account that the radii of the two arcs are related by the Pythagorean theorem 2a2 =b2. Leonardo appar
ently learned about the lunula of Hippocrates from a mathematical compendium by Giorgio Valla, published in Venice in 1501, and made frequent use of the equality in various forms.5

  On the folio in Codex Madrid II, shown in Fig. 7-6 on Chapter 7, Leonardo sketched a series of transformations involving the three basic types on a single page, as if he had wanted to record a catalog of his basic transformations. In the top two sketches in the right margin of the page, Leonardo demonstrates how a portion of a pyramid can be detached and reattached on the opposite side to create a curvilinear solid. He frequently uses the term “falcate” for such curvilinear pyramids and cones, as he does for curvilinear triangles. These falcates can also be obtained by a continuous process of gradual deformation, or “flow,” which Leonardo demonstrates in the next two sketches with the example of a cone.

  The sketch below the cone shows the bending of a cylinder with a cone inscribed in it. It almost looks like a working sketch for a metal shop, which shows that Leonardo always had real physical objects and phenomena in mind when he worked on his geometric transformations. Indeed, the Codex Atlanticus contains a folio filled with instructions for deforming metal pieces into various shapes. Among many others, these deformations include the bending of a cylinder, as shown here.6