by The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance
Leonardo uses this particular diagram to illustrate the increase of the velocity of falling bodies with time. “The natural motion of heavy things,” he explains, “at each degree of its descent acquires a degree of velocity. And for this reason, such motion, as it acquires power, is represented by the figure of a pyramid.”10 We know that the phrase “each degree of its descent” refers to units of time, because on an earlier page of the same Notebook he writes: “Gravity that descends freely in every degree of time acquires…a degree of velocity.”11 In other words, Leonardo is establishing the mathematical law that for freely falling bodies there is a linear relationship between velocity and time.12
In today’s mathematical language, we say that the velocity of a falling body is a linear function of time, and we write it symbolically as v = gt, where g denotes the constant gravitational acceleration. This language was not available to Leonardo. The concept of a function as a relation between variables was developed only in the late seventeenth century. Even Galileo described the functional relationship between velocity and time for a falling body in words and in the language of proportion, as did Leonardo 140 years before him.13
For most of his life, Leonardo believed that his “pyramidal” progression was a universal mathematical law describing all quantitative relationships between physical variables. He discovered only late in life that there are other kinds of functional relationships between physical variables, and that some of those, too, could be represented by pyramids. For example, he realized that a quantity could vary with the square of another variable, and that this relationship, too, was embodied in the geometry of pyramids. In a sequence of square pyramids with a common apex, the areas of the bases are proportional to the squares of their distances from the apex. As Kenneth Keele noted, there can be no doubt that with time Leonardo would have revised and extended many applications of his pyramidal law in the light of his new insights.14 But as we shall see, Leonardo preferred to explore a different kind of mathematics during the last years of his life.
DRAWINGS AS DIAGRAMS
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” (Fig. 7-2), 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.15
Similarly, Leonardo’s anatomical drawings, which he called “demonstrations,” are not always faithful pictures of what one would see in an actual dissection. Often, they are diagrammatic representations of the functional relationships between various parts of the body.16
For example, in a series of drawings of the deep structures of the shoulder (Fig. 7-3), Leonardo combines different graphical techniques—individual parts shown separated from the whole, muscles cut away to expose the bones, parts labeled with a series of letters, cord diagrams showing lines of forces, among others—to demonstrate the spatial extensions and mutual functional relationships of anatomical forms. These drawings clearly display characteristics of mathematical diagrams, used in the discipline of anatomy.
Leonardo’s scientific drawings—whether they depict elements of machines, anatomical structures, geological formations, turbulent flows of water, or botanical details of plants—were never realistic representations of a single observation. Rather, they are syntheses of repeated observations, crafted in the form of theoretical models. Daniel 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.17
Figure 7-2: “Water falling upon water,” c. 1508–9, Windsor Collection, Landscapes, Plants, and Water Studies, folio 42r
Figure 7-3: Deep structures of the shoulder, c. 1509, Anatomical Studies, folio 136r
GEOMETRY IN MOTION
In addition to using his phenomenal drawing skills, Leonardo also pursued a more formal mathematical approach to represent nature’s forms. He became seriously interested in mathematics when he was in his late thirties, after his visit to the library of Pavia. He furthered his studies of Euclidean geometry a few years later with the help of mathematician and friend Luca Pacioli.18 For about eight years he diligently went through the volumes of Euclid’s Elements and studied several works of Archimedes. But he went beyond Euclid in his own drawings and notes. As Kenneth Clark observed, “Euclidean order could not satisfy Leonardo for long, for it conflicted with his sense of life.”19
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.”20 It was evident to Leonardo that a mathematics 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.21 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 fores had-owed subsequent developments.
In contrast to Euclid’s geometry of rigid static figures, Leonardo’s conception of geometric relationships is inherently dynamic. This is evident even from his definitions of the basic geometric elements. “The line is made with the movement of the point,” he declares. “The surface is made by the transverse movement of the line;…the body is made by the movement of the extension of the surface.”22 In the twentieth century, the painter and art theorist Paul Klee used almost identical words to define line, plane, and body in a passage that is still used today to teach design students the primary elements of architectural design:
The point moves…and the line comes into being—the first dimension. If the line shifts to form a plane, we obtain a two-dimensional element. In the movement from plane to spaces, the clash of planes gives rise to body.23
Leonardo also drew analogies between a segment of a line and a duration of time: “The line is similar to a length of time, and as the points are the beginning and end of the line, so the instants are the endpoints of any given extension of time.”24 Two centuries later this analogy became the foundation of the concept of time as a coordinate in Descartes’ analytic geometry and in Newton’s calculus.
Figure 7-4: Family of water jets flowing out of a pressurized bag, Ms. C, folio 7r (sides have been reversed to make the similarity with modern diagrams of geometric curves more evident)
As mathematician Matilde Macagno points out,25 on the one hand, Leonardo uses geometry to study trajectories and various kinds of complex motions in natural phenomena; on the other hand, he uses motion as a tool to demonstrate geometrical theorems. He called his approach “geometry which is demonstrated with motion” (geometria che si prova col moto), or “done with motion” (che si fa col moto).26
Leonardo’s Notebooks contain a large number of drawings and discussions of trajectories of all kinds, including flight paths of projectiles, balls rebounding from walls, water jets descending through the air and falling into ponds, jets ricocheting across a water tank, and the propagation of sound and its reverberation as an echo. In all these cases, Leonardo pays careful attention to the geometries of the trajectories, their curves, angles of incidence and reflection, an
d so on. Of special significance are drawings of families of path-lines that depend on a single parameter; for example, a family of water jets flowing out of a pressurized bag, generated by different inclinations of a nozzle (see Fig. 7-4). These drawings can be seen as geometric precursors of the concept of a function of continuous variables, dependent on a parameter.
The concepts of functions, variables, and parameters were developed gradually in the seventeenth century from the study of geometric curves representing trajectories, and were clearly formulated only in the eighteenth century by the great mathematician and philosopher Gottfried Wilhelm Leibniz.27
The second, highly original branch of Leonardo’s geometry is a geometry of continuous transformations of rectilinear and curvilinear shapes, which occupied him intensely during the last twelve years of his life. The central idea underlying this new type of geometry is Leonardo’s conception of both movement and transformation as processes of continual transition, in which bodies leave one area in space and occupy another. “Of everything that moves,” he explains, “the space which it acquires is as great as that which it leaves.”28
Leonardo saw this conservation of volume as a general principle governing all changes and transformations of natural forms, whether solid bodies moving in space or pliable bodies changing their shapes. He applied it to the analysis of various movements of the human body, including in particular the contraction of muscles,29 as well as to the flow of water and other liquids. Here is how he writes about the flow of a river: “If the water does not increase, nor diminish, in a river, which may be of varying tortuosities, breadths and depths, the water will pass in equal quantities in equal times through every degree of the length of that river.”30
The realization that the same volume of water can take on an infinite number of shapes may well have inspired Leonardo to search for a new, dynamic geometry of transformations. It is striking that his first explorations of such a geometry in the Codex Forster coincide with increased studies of the shapes of waves and eddies in flowing water.31 Leonardo evidently thought that, by developing a “geometry done with motion,” based on the conservation of volume, he might be able to describe the continual movements and transformations of water and other natural forms with mathematical precision. He methodically set out to develop such a geometry, and in doing so anticipated some important developments in mathematical thought that would not occur until several centuries later.
“ON TRANSFORMATION”
Leonardo’s ultimate aim was to apply his geometry of transformations to the movements and changes of the curvilinear forms of water and other pliable bodies. But in order to develop his techniques, he began with transformations of rectilinear figures where the conservation of areas and volumes can easily be proven with elementary Euclidean geometry. In so doing, he pioneered a method that would become standard practice in science during the subsequent centuries—to develop mathematical frameworks with the help of simplified unrealistic models before applying them to the actual phenomena under study.
Many of Leonardo’s examples of rectilinear transformations are contained in the first forty folios of Codex Forster I under the heading “A book entitled ‘On Transformation,’ that is, of one body into another without diminution or increase of matter.”32 This sounds like conservation of mass, but in fact Leonardo’s drawings in these folios all have to do with conservation of area or volume. For solid bodies and incompressible liquids, conservation of volume does imply conservation of mass, and the wording of his title shows us that Leonardo’s geometrical explorations were clearly intended for the study of such material bodies.
He begins with transformations of triangles, rectangles (which he calls “table tops”), and parallelograms. He knows from Euclidean geometry that two triangles or parallelograms with the same base and height have the same area, even when their shapes are quite different. He then extends this reasoning to transformations in three dimensions, changing cubes into rectangular prisms and comparing the volumes of upright and inclined pyramids.
In his most sophisticated example, Leonardo transforms a dodecahedron—a regular solid with 12 pentagonal faces—into a cube of equal volume. He does so in four clearly illustrated steps (see Fig. 7-5): First, he cuts up the dodecahedron into 12 equal pyramids with pentagons as bases; then he cuts each of these pyramids into 5 smaller pyramids with triangular bases, so that the dodecahedron has now been cut into 60 equal pyramids; then he transforms the triangular base of each pyramid into a rectangle of equal area, thereby conserving the pyramid’s volume; and in the last step, he ingeniously stacks the 60 rectangular pyramids into a cube, which evidently has the same volume as the original dodecahedron.
In a final flourish, Leonardo then reverses the steps of the whole procedure, beginning with a cube and ending up with a dodecahedron of equal volume. Needless to say, this set of transformations shows great imagination and considerable powers of visualization.
MAPPINGS OF CURVES AND CURVED SURFACES
As soon as Leonardo achieved sufficient confidence and facility with transformations of rectilinear figures, he turned to the main topic of his mathematical explorations—the transformations of curvilinear figures. In an interesting “transitional” example, he draws a square with an inscribed circle and then transforms the square into a parallelogram, thereby turning the circle into an ellipse. On the same folio, he transforms the square into a rectangle, which elongates the circle into a different ellipse. Leonardo explains that the relationship of the figura ovale (ellipse) with respect to the parallelogram is the same as that of the circle with respect to the square, and he asserts that the area of an ellipse can easily be obtained if the right equivalent circle is found.33
Figure 7-5: Transforming a dodecahedron into a cube, Codex Forster I, folio 7r
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 to 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.”34 However, as the day progressed, he came to the realization that this attempt, too, was futile.
Even though Leonardo could not succeed in solving the classical problem of squaring the circle, he did come up with two ingenious and unorthodox solutions, both of which are revealing about his mathematical thinking. He divided the circle into a number of sectors, which in turn are subdivided into a triangle and a small circular segment. These sectors are then rearranged in such a way that they form an approximate rectangle in which the short side is equal to the circle’s radius (r) and the long side is equal to half the circumference (C/2). As this procedure is carried out with larger and larger numbers of triangles, the figure will tend toward a true rectangle with an area equal to that of the circle. Today, we would write the formula for the area as A = r (C/2) = r2π.
The last step in this process involves the subtle concept of approaching the limit of an infinite number of infinitely small triangles, which was understood only in the seventeenth century with the development of calculus. The Greek mathematicians all shied away from infinite numbers and processes, and thus were unable to formulate the mathematical concept of a limit. It is interesting, however, that Leonardo seems to have had at least an intuitive grasp of it. “I square the circle minus the smallest portion of it that the intellect can imagine,” he wrote in the Windsor manuscripts, “that is, the
smallest perceptible point.”35 In the Codex Atlanticus he stated: “[I have] completed here various ways of squaring the circles…and given the rules for proceeding to infinity.”36
Leonardo’s second method of squaring the circle is much more pragmatic. Again, he divides the circle into many small sectors, but then—perhaps encouraged by his intuitive grasp of the limiting process in the first method—he simply rolls half of the circumference on a line and constructs the rectangle accordingly, its short side being equal to the radius. Thus he arrives again at the correct formula, which he properly attributes to Archimedes.37
Leonardo’s second method, which greatly appealed to his practical mind, involves what we now call the mapping of a curve onto a straight line. He compared it to measuring distances with a rolling wheel, and he also extended the process to two dimensions, mapping various curved surfaces onto planes.38 On several folios of Manuscript G, he described procedures for rolling cylinders, cones, and spheres on plane surfaces to find their surface areas. He realized that cylinders and cones can be mapped onto a plane, line by line, without any distortion, while this is not possible for spheres. But he experimented with several methods of approximately mapping a sphere onto a plane, which corresponds to the cartographer’s problem of finding accurate plane maps of the surface of the Earth.
One of Leonardo’s methods involved drawing parallel circles on a portion of the sphere, thereby marking off a series of small strips, and then rolling the strips one by one, so that an approximate triangle is generated on the plane. The strips were probably freshly painted so that they left an imprint on the paper. As Macagno points out, this technique foreshadows the development of integral calculus, which began in the seventeenth century with various attempts to calculate the lengths of curves, areas of circles, and volumes of spheres.39 Indeed, some of these efforts involved dividing curved surfaces into small segments by drawing a series of parallel lines, as Leonardo had done two centuries earlier.40
