by The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance
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.
NOTES
Citations of Leonardo’s manuscripts refer to the scholarly editions listed in the Bibliography. I have retranslated some of the passages by staying closer to the original texts, so as to preserve their Leonardesque flavor.
PREFACE
1. Kenneth Clark, Leonardo da Vinci, Penguin, 1989, p. 258.
2. Ibid., p. 255.
3. Martin Kemp, “Leonardo Then and Now,” in Kemp and Jane Roberts, eds., Leonardo da Vinci: Artist, Scientist, Inventor, Catalogue of Exhibition at Hayward Gallery, Yale University Press, 1989.
INTRODUCTION
1. See Chapter 4.
2. Trattato, chapter 19; “sensory awareness” is my translation of Leonardo’s term senso comune, which I discuss on Chapter 9.
3. Ms. Ashburnham II, folio 19v.
4. Trattato, chapter
s 6 and 12.
5. Codex Leicester, folio 34r.
6. Daniel Arasse, Leonardo da Vinci: The Rhythm of the World, Konecky & Konecky, New York, 1998, p. 80.
7. Fritjof Capra, The Web of Life, Doubleday, New York, 1996, p. 100.
8. For a detailed account of the history and characteristics of systemic thinking, see Capra (1996).
9. Ibid., p. 112.
10. See Chapter 7.
11. See Chapter 7.
12. See Chapter 9.
13. See Chapter 9.
14. Ms. A, folio 3r.
15. Arasse (1998), p. 311.
16. See Chapter 8.
17. Arasse (1998), p. 20.
18. Trattato, chapter 367.
19. Irma Richter, ed., The Notebooks of Leonardo da Vinci, Oxford University Press, New York, 1952, p. 175.
20. See Chapter 9.
21. Anatomical Studies, folio 153r.
22. See Chapter 9.
23. See Epilogue.
24. Anatomical Studies, folio 173r.
CHAPTER 1
1. Giorgio Vasari, Lives of the Artists, published originally in 1550; trans. George Bull, 1965; reprinted as Lives of the Artists, vol. 1, Penguin, 1987.
2. Paolo Giovio, “The Life of Leonardo da Vinci,” written around 1527, first published in 1796; translation from the original Latin by J. P. Richter, 1939; reprinted in Ludwig Goldscheider, Leonardo da Vinci, Phaidon, London, 1964, p. 29.
3. Vasari (1550), pp. 13–14.
4. Serge Bramly, Leonardo, HarperCollins, New York, 1991, p. 6.
5. Anonimo Gaddiano, “Leonardo da Vinci,” written around 1542; trans. Kate Steinitz and Ebria Feigenblatt, 1949; reprinted in Goldscheider(1964), pp. 30–32. This manuscript, now in the Biblioteca Nazionale in Florence, was formerly housed in the Biblioteca Gaddiana, the private library of the Florentine Gaddi family.
6. Trattato, chapter 36.
7. Giorgio Nicodemi, “The Portrait of Leonardo,” in Leonardo da Vinci, Reynal, New York, 1956.
8. Ibid.
9. Clark (1989), p. 255.
10. Trattato, chapter 50.
11. Ms. H, folio 60r.
12. Bramly (1991), p. 342.
13. Ms. Ashburnham II, folios 31r and 30v.
14. See Arasse (1998), p. 430.
15. See Martin Kemp, Leonardo da Vinci: The Marvellous Works of Nature and Man, Harvard University Press, Cambridge, Mass., 1981, p. 152.
16. Bramly (1991), p. 115.
17. Vasari (1550); translation of this passage by Daniel Arasse; see Arasse(1998), p. 477.
18. See, e.g., Bramly (1991), p. 119.
19. See Michael White, Leonardo: The First Scientist, St. Martin’s/Griffin, New York, 2000, pp. 132–33.
20. See Bramly (1991), p. 241.
21. Ibid., p. 133.
22. Charles Hope, “The Last ‘Last Supper’,” New York Review of Books, August 9, 2001.
23. Kenneth Keele, Leonardo da Vinci’s Elements of the Science of Man, Academic Press, New York, 1983 p. 365.
24. See Penelope Murray, ed., Genius: The History of an Idea, Basil Blackwell, New York, 1989.
25. Quoted by Wilfrid Mellers, “What is Musical Genius?,” Murray(1989), p. 167.
26. See Andrew Steptoe, ed., Genius and the Mind, Oxford University Press, 1998.
27. See David Lykken, “The Genetics of Genius,” in Steptoe (1998).
28. Kenneth Clark, quoted by Sherwin B. Nuland, Leonardo da Vinci, Viking Penguin, New York, 2000, p. 4.
29. Quoted by David Lykken in Steptoe (1998).
30. Quoted by Bramly (1991), p. 281.
31. Quoted by Richter (1952), p. 306.
32. Murray (1989), p. 1.
CHAPTER 2
1. See Jacob Burckhardt, The Civilization of the Renaissance in Italy, original German edition published in 1860, Modern Library, New York, 2002.
2. See Bramly (1991), p. 100.
3. Trattato, chapter 79.
4. See Robert Richards, The Romantic Conception of Life, University of Chicago Press, 2002.
5. See Fritjof Capra, Uncommon Wisdom, Simon & Schuster, New York, 1988, p. 71.
6. Trattato, chapter 42.
7. Ibid., chapter 33.
8. Ibid., chapter 13.
9. Kemp (1981), p. 161.
10. Trattato, chapter 68.
11. Anatomical Studies, folio 50v.
12. See Fritjof Capra, The Hidden Connections, Doubleday, New York, 2002, p. 119.
13. See Penny Sparke, Design and Culture in the Twentieth Century, Allen & Unwin, London, 1986.
14. Codex Atlanticus, folio 323r.
15. Trattato, chapter 23.
16. Clark (1989), p. 63.
17. See Arasse (1998), p. 274.
18. Quoted by Arasse (1998), p. 275.
19. Anatomical Studies, folio 139v.
20. Arasse (1998), p. 202.
21. Ibid., p. 283.
22. Kemp (1981), p. 56.
23. Arasse (1998), p. 284.
24. See Capra (2002), pp. 13–14 and p. 116.
25. See Claire Farago, “How Leonardo Da Vinci’s Editors Organized His Treatise on Painting and How Leonardo Would Have Done It Differently,” in Lyle Massey, ed., The Treatise on Perspective: Published and Unpublished, National Gallery of Art, Washington, distributed by Yale University Press, 2003.
26. See Claire Farago, Leonardo da Vinci’s Paragone: A Critical Interpretation with a New Edition of the Text in the Codex Urbinas, E. J. Brill, Leiden, 1992.
27. Trattato, chapters 14, 19.
28. Ibid., chapter 29.
29. Ibid., chapters 38, 41.
30. See Bram Kempers, Painting, Power and Patronage, Allen Lane, London, 1992; Steptoe (1998), p. 255.
31. See Arasse (1998), p. 293.
32. See Introduction.
33. Clark (1989), p. 167.
34. Kemp (1981), p. 97.
35. Trattato, chapter 412.
36. Clark (1989), p. 129. In the history of Italian art, the fifteenth century is known as the quattrocento (four hundred), the sixteenth century is called the cinquecento (five hundred), and so on.
37. Trattato, chapter 124.
38. See Introduction.
39. See Chapter 8.
40. These notes are in Manuscripts C and Ashburnham II.
41. Kemp (1981), p. 98.
42. See Bramly (1991), pp. 101–2.
43. Ibid., p. 106.
44. See Kemp (1981), pp. 94–96.
45. See Ann Pizzorusso, “Leonardo’s Geology: The Authenticity of the Virgin of the Rocks,” Leonardo, vol. 29, no. 3, pp. 197–200, MIT Press, 1996.
46. See William Emboden, Leonardo da Vinci on Plants and Gardens, Dioscorides Press, Portland, Ore., 1987, p. 125.
47. Trattato, chapter 38.
48. Bramly (1991), p. 228.
49. This Roman equestrian statue of the Gothic king Odoacer no longer exists. It was destroyed in the eighteenth century; see Bramly (1991), p. 232.
50. Codex Atlanticus, folio 399r.
51. This treatise, mentioned by Vasari and by Lomazzo, has been lost.
52. Kemp (1981), p. 205.
53. Codex Madrid II, folio 157v.
54. See Bramly (1991), pp. 234–35.
55. Codex Atlanticus, folio 914ar.
56. Bramly (1991), p. 250.
57. See Chapter 2.
58. I am grateful to ecodesigner Magdalena E. Corvin for illuminating discussions and correspondence on the nature of design.
59. See Bramly (1991), p. 232.
60. See Clark (1989), p. 139.
61. See Bramly (1991), p. 219.
62. Codex Atlanticus, folio 21r.
63. See Bramly (1991), p. 272.
64. See Pierre Sergescu, “Léonard de Vinci et les mathématiques,” cited in Arasse (1998), p. 65.
65. See Kemp (1981), p. 88.
66. See Paolo Galluzzi, Renaissance Engineers, Giunti, Florence, 1996, p. 187.
67. Clark (1989), p. 110.
68. L
udwig H. Heydenreich, “Leonardo and Bramante: Genius in Architecture,” in C. D. O’Malley, ed., Leonardo’s Legacy: An International Symposium, University of California Press, Berkeley and Los Angeles, 1969; Carlo Pedretti, Leonardo, Architect, Rizzoli, New York, 1985.
69. See Pedretti (1985) for a full account of Leonardo’s architectural work; see also Jean Guillaume, “Léonard et l’architecture,” in Paolo Galluzzi and Jean Guillaume, eds., Léonard de Vinci: ingénieur et architecte, Musée des Beaux-Arts de Montréal, 1987.
70. Arasse (1998), p. 173.
71. Heydenreich (1969).
72. Arasse (1998), pp. 179–80. Mannerism is a style of art and architecture developed in the late sixteenth century and characterized by spatial incongruity and elegant, elongated figures.
73. Kemp (1981), p. 110.
74. See, for example, White (2000), p. 124.
75. Codex Atlanticus, folio 730r.
76. See Guillaume (1987); see also Arasse (1998), pp. 165–68.
77. Anatomical Studies, folio 97r.
78. Emboden (1987).
79. Ms. B, folio 38r.
80. Codex Atlanticus, folio 184v.
81. Ms. B, folio 16r; see also Guillaume (1987).
82. Nuland (2000), p. 53.
83. See Bramly (1991), pp. 402–3.
84. See International Healthy Cities Foundation, www.healthycities.org.
85. Arasse (1998), p. 152.
86. Ibid., p. 233.
87. Ibid., pp. 239–40.
88. See Kate Steinitz, “Le dessin de Léonard de Vinci pour la représentation de la Danaé de Baldassare Taccone,” in Jean Jacquot, ed., Le Lieu théâtral à la Renaissance, Paris, 1968.
89. See Arasse (1998), p. 239.
90. See Bramly (1991), p. 301.
91. See Kemp (1981), p. 182., for a detailed description of the vaulted architecture and Leonardo’s matching design.
92. See Arasse (1998), p. 138.
93. Quoted in Fritjof Capra, The Turning Point, Simon & Schuster, New York, 1982, p. 68.
