Leonardo’s conclusions are fully borne out by modern mechanics. Today the force of friction is defined as the product of the frictional coefficient (measuring the roughness of the surfaces) and the force perpendicular to the contact surface (which depends both on the object’s weight and the slope of the surface).
Leonardo’s studies of power transmission led him to investigate the medieval belief that power could be harnessed through perpetual motion machines. At first he accepted this idea. He designed a host of complex mechanisms to keep water in perpetual motion by means of various feedback systems. But eventually he realized that any mechanical system will gradually lose its power because of friction. In the end, Leonardo scoffed at attempts to build perpetual motion machines. “I have found among the excessive and impossible delusions of men,” he wrote in the Codex Madrid, “the search for continuous motion, which is called by some the perpetual wheel.”62
Leonardo extended his keen interest in friction to his extensive studies of fluid flows. The Codex Madrid contains meticulous records of his investigations and analyses of the resistance of water and air to moving solid bodies, as well as of water and fire moving in air.63 Well aware of the internal friction of fluids, known as viscosity, Leonardo dedicated numerous pages in the Notebooks to analyzing its effects on fluid flow. “Water has always a cohesion in itself,” he wrote in the Codex Leicester, “and this is the more potent as the water is more viscous.”64
Air resistance was of special interest to Leonardo, because it played an important role in one of his great passions—the flight of birds and the design of flying machines. “In order to give the true science of the movement of birds in the air,” he declared, “it is necessary first to give the science of the winds.”65
THE DREAM OF FLYING
The dream of flying like a bird is as old as humanity itself. But nobody pursued it with more intensity, perseverance, and commitment to meticulous research than Leonardo da Vinci. His “science of flight” involved numerous disciplines—from fluid dynamics to human anatomy, mechanics, the anatomy of birds, and mechanical engineering. He diligently pursued these studies throughout most of his life, from the early years of his apprenticeship in Florence to his old age in Rome.66
The first intense period of research on flying machines began in the early 1490s, about a decade after Leonardo’s arrival in Milan.67 His experiments during this period combined mechanics and the anatomy of the human body. He carefully investigated and measured the body’s ability to generate various amounts of force in order to find out how a human pilot might be able to lift a flying machine off the ground by flapping its mechanical wings.
Leonardo realized that the air under a bird’s wing is compressed by the downstroke. “See how the wings, striking against the air, sustain the heavy eagle in the thin air on high,” he noted in the Codex Atlanticus, and then he added a remarkable observation: “As much force is exerted by the object against the air as by the air against the object.”68 Leonardo’s observation was restated by Isaac Newton two hundred years later and has since been known as Newton’s third law of motion.69
Figure 6-6: Leonardo’s “flying ship,” Ms. B, folio 80r
The result of these investigations was Leonardo’s so-called flying ship, his first design of a flying machine (see Fig. 6-6). From the human point of view, the design is rather strange. Crouched down in the center of the craft, the pilot generates the necessary force by pushing two pedals with his feet while simultaneously turning two handles with his hands. As historian Domenico Laurenza points out, “There is no note, no mention to be found…of how the pilot will steer the machine in flight; he becomes almost an automatic pilot: he simply has to generate the force to lift off the ground.”70
During these years, Leonardo designed a series of much more realistic flying machines in which the pilot is placed horizontally (see Fig. 6-7). These designs involve more varied and subtle movements. Human arms and legs are used to make the wings flap. Other movements turn the wings, angling them in the upstroke and opening them to the air in the downstroke, as birds do when flapping in flight. Yet other movements are used to maintain balance and change direction.
These drawings (in Manuscript B and the Codex Atlanticus) represent Leonardo’s most sophisticated designs of flying machines. They became the basis of several models built by modern engineers.71 Figure 6-8 shows one of these models, built from materials that were available in the Renaissance. Unfortunately, the limitations of these materials—wooden struts, leather joints and thongs, and skin of strong cloth—make it evident why Leonardo could not create a viable model of his flying machines, even though they were based on sound aerodynamic principles. The combined weight of the machine and its pilot was simply far too heavy to be lifted by human muscle power.
Eventually, Leonardo became aware that he could not achieve the required power-to-weight ratio for successful flight. Ten years after his experiments with flying machines in Milan, he entered into another intense period of research in Florence, which involved his making careful and methodical observations of birds in flight, down to the finest anatomical and aerodynamic details.72
Figure 6-7: Design for flying machine, Ms. B, folio 74v
In the resulting Notebook, Codex on the Flight of Birds, Leonardo concludes that human flight with mechanical wings might not be possible because of the limitations of our anatomy. Birds have powerful pectoral muscles, he notes, that allow them to flee rapidly from predators, or to carry heavy prey, but they need only a fraction of that force to sustain themselves in the air during normal flight.73
His observations led Leonardo to speculate that, even though human beings would not be able to fly by flapping mechanical wings, “soaring flight,” or gliding, might be possible, since this required much less force. During his last years in Florence he began to experiment with designs of flying machines that had fixed wings, not unlike a modern hang glider.
Based on these designs, British engineers recently built a glider and tested it successfully in a flight from the chalk cliffs in southeast England known as the Sussex Downs. This maiden flight of “Leonardo’s glider,” reportedly, exceeded the first attempts by the Wright brothers in 1900.74
Figure 6-8: Working model of the flying machine, Museum of the History of Science, Florence
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 of 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.”75
THE MYSTERY OF HUMAN LIFE
The third grand theme in Leonardo’s anatomical research (in addition to the themes of harmony and proportion, and the body in motion) is his persistent quest for understanding the nature of life. It is the leitmotiv of his anatomies of the body’s internal organs, and in particular of his investigations of the heart—the bodily organ that has served as the foremost symbol of human existence and emotional life throughout the ages.
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.
Because he did not see the body as a machine, Leonardo’s main concern was not the mechanical transportation of blood, but the twin problems, as he saw them, of how the actions of the heart maintained the blood at body temperature and how they produced the “vital spirits” that keep us alive. He accepted the ancient notion that these vital sp
irits arise from a mixture of blood and air—which is essentially correct, if we identify them with oxygenated blood—and he developed an ingenious theory to solve both problems.
In the absence of any knowledge of chemistry, Leonardo used his extensive understanding of turbulent flows of water and air, and of the role of friction, in his attempt to explain the origin of both the blood-air mixture and the body temperature. This included a meticulous description of many subtle features of blood flow—including the coordinated actions of the heart’s four chambers (when all his contemporaries knew only of two), and the corresponding synchronized actions of the coronary valves—which he pictured in a series of superb drawings. According to the eminent physician and Leonardo scholar Kenneth Keele:
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.76
Leonardo missed some crucial details about the mechanics of blood circulation, which were discovered by William Harvey a hundred years later, and without chemistry he could not explain the oxygen exchange between the blood and the tissues of the lungs and body. But amazingly, he recognized many subtle features of cellular metabolism without even knowing about cells—for example, that heat energy supports the metabolic processes, that oxygen (the “vital spirits”) sustains them, that there is a constant flow of oxygen from the heart to the body’s periphery, and that the blood returns with waste products from the tissue metabolism. In other words, Leonardo developed a theory of the functioning of the heart and the flow of blood that allowed him to understand some of the essential features of biological life.
During the last decade of his life, while he was engaged in his most advanced studies of the human heart, Leonardo also became intensely interested in another aspect of the mystery of life—its origin in the processes of reproduction and embryonic development. That he had always considered embryology as an integral part of his studies of the human body is evident from the grandiose outline of a planned (but never assembled) treatise on the movements of the body, written about twenty years earlier. This long and detailed outline begins with the following sweeping declaration:
This work should begin with the conception of man, and should describe the nature of the womb, and how the child lives in it, and to what stage it resides in it, and in what way it acquires life and food, and its growth, and what interval there is between one degree of growth and another, and what it is that pushes it out of the body of the mother.77
Leonardo’s embryological studies, based largely on dissections of cows and sheep, included most of the topics he had listed and led him to remarkable observations and conclusions. While most authorities in his day believed that all inherited characteristics derived from the father, he asserted unequivocally: “The seed of the mother has equal power in the embryo to the seed of the father.”78
He described the life processes of the fetus in the womb, including its nourishment through the umbilical cord, in astonishing detail, and he also made a series of measurements on animal fetuses to determine their rates of growth. Leonardo’s embryological drawings are graceful and touching revelations of the mysteries surrounding the origins of human life (see Fig. E-1 on p. 261). 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 scientific perception as well.79
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,”80 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.”81
SEVEN
Geometry Done with Motion
Leonardo was well aware of the critical role of mathematics in the formulation of scientific ideas and in the recording and evaluation of experiments. “There is no certainty,” he wrote in his Notebooks, “where one can not apply any of the mathematical sciences, nor those which are connected with the mathematical sciences.”1 In his Anatomical Studies, he proclaimed, in evident homage to Plato, “Let no man who is not a mathematician read my principles.”2
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 the fields of integral calculus and topology, as I shall discuss below.
Leonardo’s mathematical diagrams and notes are scattered throughout his Notebooks. Many of them have not yet been fully evaluated. While we have illuminating books by physicians on his anatomical studies and detailed analyses of his botanical drawings by botanists, a comprehensive volume on his mathematical works by a professional mathematician still needs to be written. Here, I can give only a brief summary of this fascinating side of Leonardo’s genius.
GEOMETRY AND ALGEBRA
In the Renaissance, as we have seen, mathematics consisted of two main branches, geometry and algebra, the former inherited from the Greeks, while the latter had been developed mainly by Arab mathematicians.3 Geometry was considered more fundamental, especially among Renaissance artists, for whom it represented the foundation of perspective, and thus the mathematical underpinning of painting.4 Leonardo fully shared this view. And since his approach to science was largely visual, it is not surprising that his entire mathematical thinking was geometric. He never got very far with algebra, and indeed he frequently made careless errors in simple arithmetical calculations. The really important mathematics for him was geometry, which is evident from his praise of the eye as “the prince of mathematics.”5
In this he was hardly alone. Even for Galileo, one hundred years after Leonardo, mathematical language essentially meant the language of geometry. “Philosophy is written in that great book which ever lies before our eyes,” Galileo wrote in a much quoted passage. “But we cannot understand it if we do not first learn the language and characters in which it is written. This language is mathematics, and the characters are triangles, circles, and other geometrical figures.”6
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.7 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.”8
Figure 7-1: The “pyramidal law,” Ms. M, folio 59v
In his notes, Leonardo often represented such a pyramid, or cone, in a vertical section, that is, simply as a triangle, where the triangle’s base represents the edge of the object and its apex a point in the eye. Leonardo then used this geometric figure—the isosceles triangle (i.e., a triangle with two equal sides
)—to represent arithmetic progressions and linear algebraic relationships, thus establishing a visual link between the proportions of perspective and quantitative relationships in many fields of science, for example, the increase of the velocity of falling bodies with time, discussed below.
He knew from Euclidean geometry that in a sequence of isosceles triangles with bases at equal distances from the apex, the lengths of these bases, as well as the distances of their endpoints from the apex, form arithmetic progressions. He called such triangles “pyramids” and accordingly referred to an arithmetic progression as “pyramidal.”
Leonardo repeatedly illustrates this technique in his Notebooks. For example, in Manuscript M he draws a “pyramid” (isosceles triangle) with a sequence of bases, labeled with small circles and numbers running from 1 to 8 (see Fig. 7-1). Inside the triangle, he also indicates the progressively increasing lengths of the bases with numbers from 1 to 8. In the accompanying text, he gives a clear definition of arithmetic progression: “The pyramid…acquires in each degree of its length a degree of breadth, and such proportional acquisition is found in the arithmetic proportion, because the parts that exceed are always equal.”9
The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance Page 20