SEVEN
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
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, and 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
CURVILINEAR TRANSFORMATIONS
In today’s mathematical language, the concept of mapping can be applied also to Leonardo’s transformation of a circle into an ellipse, in which the points of one curve are mapped onto those of another together with the mapping of all other corresponding points from the square onto the parallelogram. Alternatively, the operation may be viewed as a continuous transformation—a gradual movement, or “flow,” of one figure into the other—which was how Leonardo understood his “geometry done with motion.” He used this approach in a variety of ways to turn rectilinear into curvilinear figures in such a manner that their areas or volumes are always conserved. These procedures are illustrated and discussed systematically in Codex Madrid II, but there are countless related drawings scattered throughout the Notebooks.41
Leonardo used these curvilinear transformations to experiment with an endless variety of shapes, turning rectilinear planar figures and solid bodies—cones, pyramids, cylinders, etc.—into “equal” curvilinear ones. On an interesting folio in Codex Madrid II, he illustrates his basic techniques by sketching several different transformations on a single page (see Fig. 7-6). In the last paragraph of the text on this folio, he explains that these are examples of “geometry which is demonstrated with motion” (geometria che si prova col moto).42
As Macagno and others have noted, some of these sketches are highly reminiscent of the swirling shapes of substances in rotating liquids (e.g., chocolate syrup in stirred milk), which Leonardo studied extensively. This strongly suggests once again that his ultimate aim was to use his geometry for the analysis of transformations of actual physical forms, in particular in eddies and other turbulent flows.
In these endeavors, Leonardo was greatly helped by his exceptional ability to visualize geometrical forms as physical objects, mold them like clay sculptures in his imagination, and sketch them quickly and accurately. “However abstract the geometrical problem,” writes Martin Kemp, “his sense of its relationship to actual or potential forms in the physical universe was never far away. This accounts for his almost irresistible desire to shade geometric diagrams as if they portrayed existing objects.”43
EARLY FORMS OF TOPOLOGY
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:
Figure 7-6: Leonardo’s catalog of transformations, Codex Madrid II, folio 107r
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 transformation 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 the transformation carries nearby points into nearby points. This latter property is called continuity.44
Leonardo’s geometric transformations of planar figures and solid bodies are clearly examples of topological transformations. Modern topologists call the figures related by such transformations, in which very general geometric properties are preserved, topologically equivalent. These properties do not include area and volume, as topological transformations may arbitrarily stretch, expand, or shrink geometric figures. In contrast, Leonardo concentrated on operations that conserve area or volume, and he called the transformed figures “equal” to the original ones. Even though these represent only a small subset of topological transformations, they exhibit many of the characteristic features of topology in general.
Historians usually give credit for the first topological explorations to the philosopher and mathematician Leibniz who, in the late seventeenth century, tried to identify basic properties of geometric figures in a study he called geometria situ (geometry of place). But topological relationships were not treated systematically until the turn of the nineteenth to the twentieth century, when Henri Poincaré, the leading mathematician of the time, published a series of comprehensive papers on the subject.45 Poincaré is therefore regarded as the founder of topology. The transformations of Leonardo’s “geometry done with motion” are early forms of this important field of mathematics—three hundred years before Leibniz and five hundred years before Poincaré.
One subject that fascinated Leonardo from his early years in Milan was the design of tangled labyrinths of knots. Today this is a special branch of topology. To a mathematician, a knot is a tangled closed loop or path, similar to a knotted rope with its two free ends spliced together, precisely the structures Leonardo studied and drew. In designing such interlaced motifs, he followed a decorative tradition of his time.46 But he far surpassed his contemporaries in this genre, treating his knot designs as objects of theoretical study and drawing a vast quantity of extremely complex interlaced structures.47
Topological thinking—thinking in terms of connectivity, spatial relationships, and continuous transformations—was almost second nature to Leonardo. Many of his architectural studies, especially his designs of radially symmetrical churches and temples, exhibit such characteristics.48 So, too, do many of his numerous diagrams. Leonardo’s topological techniques can also be found in his geographical maps. In the famous map of the Chiana valley (Fig. 7-7), now in the Windsor Collection, he uses a topological approach to distort the scale while providing an accurate picture of the connectivity of the terrain and its intricate waterways.
The central part is enlarged and shows accurate proportions, while the surrounding parts are severely distorted in order to fit the entire system of watercourses into the given format.49
DE LUDO GEOMETRICO
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,50 Leonardo produced a magnificent compendium of topological transformations, titled De ludo geometrico (On the Game of Geometry), on a large double folio in the Codex Atlanticus.51He 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. Previously 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.52 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.53
Figure 7-7: Map of the Chiana valley, 1504, Windsor Collection, Drawings and Miscellaneous Papers, Vol. IV, folio 439v
When he created his double folio of topological equations, Leonardo was over sixty. He continued to explore the geometry of transformations during the last years of his life. But he must have realized that he was still very far from developing it to a point where it could be used to analyze the actual transformations of fluids and other physical forms. Today we know that for such a task, much more sophisticated mathematical tools are needed than those Leonardo had at his disposal. In modern fluid dynamics, for example, we use vector and tensor analysis, rather than geometry, to describe the movements of fluids under the influence of gravity and various shear stresses. However, Leonardo’s fundamental principle of the conservation of mass, known to physicists today as the continuity equation, is an essential part of the equations describing the motions of water and air. As far as the ever-changing forms of fluids are concerned, it is clear that Leonardo’s mathematical intuition was on the right track.
THE NECESSITY OF NATURE’S FORMS
Like Galileo, Newton, and subsequent generations of scientists, Leonardo worked from the basic premise that the physical universe is fundamentally ordered and that its causal relationships can be comprehended by the rational mind and expressed mathematically.54He used the term “necessity” to express the stringent nature of those ordered causal relationships. “Necessity is the theme and inventor of nature, the curb and the rule,” he wrote around 1493, shortly after he began his first studies of mathematics.55
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.56
This new mathematics, naturally, is far more abstract and sophisticated than anything Leonardo could have imagined in the fifteenth and sixteenth centuries. But it 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 the 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.