Fritjof Capra (27 page)

SEVEN

Geometry Done with Motion

L
eonardo 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.”
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In his Anatomical Studies, he proclaimed, in evident homage to Plato, “Let no man who is not a mathematician read my principles.”
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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.
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Geometry was considered more fundamental, especially among Renaissance artists, for whom it represented the foundation of perspective, and thus the mathematical underpinning of painting.
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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.”
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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.”
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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.
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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.”
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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.”
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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.”
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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.”
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In other words, Leonardo is establishing the mathematical law that for freely falling bodies there is a linear relationship between velocity and time.
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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.
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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.
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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.
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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.
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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.
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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