Fritjof Capra (37 page)

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
, chapters 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