APPENDIX
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 apparently 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
The last two sketches in the right margin represent examples of so-called parallel shear, which are extended to circular shears in the three sketches in the center of the page. In these examples, rectilinear figures are transformed into spirals, and the conservation of area is far from obvious. To Leonardo, the circular operations evidently looked like legitimate extensions of his linear deformations. Indeed, as Matilde Macagno has shown with the help of elementary calculus, Leonardo’s intuition was absolutely correct.7
The two sketches below the circular shears, finally, show examples of the “squaring” of surfaces bounded by two parallel curves. There is a striking similarity between these surfaces and those in the three sketches just above, which suggests that Leonardo probably thought of the two techniques as alternative methods for squaring surfaces bounded by parallel curves.
As I have discussed, Leonardo’s geometric transformations of planar figures and solid bodies may be seen as early forms of topological transformations.8 Leonardo restricted them to transformations in which area or volume are conserved, and he called the transformed figures “equal” to the original ones. Topologists call the figures related by such transformations, in which very general geometric properties are preserved, topologically equivalent.
Modern topology has two main branches, which overlap considerably. In the first, known as point-set topology, geometric figures are regarded as collections of points, and topological transformations are seen as continuous mappings of those points. The second branch, called combinatorial topology, treats geometric figures as combinations of simpler figures, joined together in an orderly manner.
Leonardo experimented with both of these approaches. The operations shown in Figure 7-6 can all be seen as continuous deformations or, alternatively, as continuous mappings. On the other hand, his ingenious transformation of a dodecahedron into a cube (Fig. 7-5 on Chapter 7) is a beautiful and elaborate example of combinatorial topology.
The concept of continuity, which is central to all topological transformations, has to do, ultimately, with very basic properties of space and time. Hence topology is seen today as a general foundation of mathematics and a unifying conceptual framework for its many branches. In the early sixteenth century, Leonardo da Vinci saw his geometry of continuous transformations in a similar vein—as a fundamental mathematical language that would allow him to capture the essence of nature’s ever-changing forms.
The double folio in the Codex Atlanticus (see Chapter 7) represents the culmination of Leonardo’s explorations of topological transformations. These drawings were intended for a comprehensive treatise, for which he proposed several titles—Treatise on Continuous Quantity, Book of Equations, and De ludo geometrico (On the Game of Geometry).
The diagrams shown on the two sheets display a bewildering variety of geometric forms built from intersecting circles, triangles, and squares, which look like playful variations of floral patterns and other aesthetically pleasing motifs, but turn out to be rigorous “geometric equations” based upon topological principles.
The double folio is divided equally by nine horizontal lines on which Leonardo has placed a regular array of semicircles (and, in the last now, some circles), filled with his geometric designs.9 The starting point for each diagram is always a circle with an inscribed square. Depending on how the circle is cut in half, two equivalent basic diagrams are obtained (see Fig. A-5), one with a rectangle and the other with a triangle inside the semicircle.
Since the white areas in the two diagrams are equal, both representing half of the inscribed square, the shaded areas must also be equal. As Leonardo explains in the accompanying text, “If one removes equal parts from equal figures, the remainder must be equal.”10
Figure A-5: The two basic diagrams, from Codex Atlanticus, folio 455, row 3
The two figures are then filled with shaded segments of circles, bisangoli (“double angles” shaped like olive leaves), and falcates (curvilinear triangles) in a dazzling variety of designs. In all of them, the ratio between the shaded areas (also called “empty”) and the white areas (also called “full”) is always the same, because the white areas—no matter how fragmented they may be—are always equal to the original inscribed half square (rectangle or triangle), and the shaded areas are equal to the original shaded areas outside the half square.
These equalities are by no means obvious, but the text underneath each diagram specifies how parts of the figure can successively be “filled in” (i.e., how shaded and white parts can be interchanged) until the original rectilinear half square is recovered and the figure has thus been “squared.” The same principle is always repeated: “To square [the figure], fill in the empty parts.”11
In Figure A-6, I have selected a specific diagram from the double folio to illustrate Leonardo’s technique. The text under the diagram reads: “To square, fill in the triangle with the four falcates outside.”12 I have redrawn the diagram in Figures A-7 a and b so as to make its geometry explicit. Inside the large half circle with radius R, Leonardo has generated eight shaded segments B by drawing four smaller half circles with half the radius, r = R/2 (see Fig. A-7 a). The falcates he mentions are the white areas marked F.
Figure A-6: Sample diagram (number 7 in row 7, Codex Atlanticus, folio 455)
By specifying that the four “empty” (shaded) areas inside the triangle are to be “filled in” with the four falcates, Leonardo indicates that the areas F and B are equal. Here is how he might have reasoned. Since he knew that the area of a circle is proportional to the square of its radius,13 he could show that the area of the large half circle is four times that of each small half circle, and that consequently the area of the large segment A is four times that of the small segment B (see Fig. A-7 b). This means that, if two small segments are subtracted from the large segment, the area of the remaining curved figure (composed of two falcates) will be equal to the area subtracted, and hence the area of the falcate F is equal to that of the small segment B.
Figure A-7: Geometry of the sample diagram
For the other figures, the squaring procedure can be more elaborate, but eventually the original diagrams are always recovered. This is Leonardo’s “game of geometry.” Each diagram represents a geometric—or, rather, topological—equation, and the accompanying instruction describes how the equation is to be solved to square the curvilinear figure. This is why Leonardo proposed Book of Equations as an alternative title for his treatise. The successive steps of solving the equations can be depicted geometrically, as shown (for example) in Figure A-8.14
Figure A-8: Squaring the sample diagram
Leonardo delighted in drawing endless varieties of these topological equations, just as Arab mathematicians in previous centuries had been fascinated by exploring wide varieties of algebraic equations. Occasionally he was carried away by the aesthetic pleasure of sketching fanciful geometric figures. But the deeper significance of his game of geometry was never far from his mind. The infinite variations of geometric forms in which area or volume were always conserved were meant to mirror the inexhaustible transmutations in the living forms of nature within limited and unchanging quantities of matter.