Fritjof Capra (40 page)

14
. Ms. A, folio 47r, and Ms. M, folio 57r; see also Keele (1983), pp. 132–33.

15
. See Keele (1983), pp. 136–37.

16
. Ibid., p. 141.

17
. See Codex Atlanticus, folio 1b.

18
. See Keele (1983), p. 135.

19
. Anatomical Studies, folio 104r.

20
. Nuland (2000), p. 131.

21
. Keele (1983), pp. 244–45.

22
. Ibid., p. 301.

23
. See Enzo Macagno, “Lagrangian and Eulerian Descriptions in the Flow Studies of Leonardo da Vinci,”
Raccolta Vinciana
, Fasc. XXIV, 1992a.

24
. See Chapter 3.

25
. See Augusto Marinoni, introduction to Leonardo da Vinci,
II codice atlantico della Biblioteca ambrosiana di Milano
, vol. 1, pp. 18–25, Giunti, Florence, 1975.

26
. See Capra (1996), p. 18.

27
. Ibid., p. 22.

28
. Codex Atlanticus, folio 1067.

29
. See Capra (1982).

30
. See Frank Zöllner and Johannes Nathan,
Leonardo da Vinci: The Complete Paintings and Drawings
, Taschen, 2003, pp. 384–99.

31
. See Keele (1983), p. 142.

32
.
Trattato
, chapter 501.

33
. See Bramly (1991), p. 257.

34
. Anatomical Studies, folio 69v.

35
. See, for example, Martin Kemp (1999a), “Analogy and Observation in the Codex Hammer,” in Claire Farago, ed.,
Leonardo’s Science and Technology
, Garland Publishing, New York, 1999; Arasse (1998), p. 74.

36
. Arasse (1998), p. 19.

37
. Ms. C, folio 26v.

38
. See Capra (1996), p. 169.

39
. Codex Atlanticus, folio 813.

40
. Ibid., folio 508v.

41
. See p. 48; see also Stephen Jay Gould, “The Upwardly Mobile Fossils of Leonardo’s Living Earth,” in Stephen Jay Gould,
Leonardo’s Mountain of Clams and the Diet of Worms
, Harmony Books, New York, 1998.

42
. Codex Arundel, folio 172v.

43
. See Chapter 2.

44
. See Emboden (1987), p. 163.

45
. See Keele (1983), p. 316.

46
. See Emboden (1987), p. 171.

47
.
Trattato
, chapter 21.

48
. See Introduction.

49
. Anatomical Studies, folio 153r.

50
. Codex
sul volo
, folio 3r.

51
. See Marshall Clagett, “Leonardo da Vinci: Mechanics,” in Farago (1999).

52
. Codex Atlanticus, folio 481.

53
. See Clagett (1999).

54
. See Chapter 2.

55
. See Pedretti (1999); also see Domenico Laurenza, Mario Taddei, and Edoardo Zanon,
Le Macchine di Leonardo
, Giunti, Florence, 2005.

56
. See, for example, Kemp and Roberts (1989), pp. 218–41.

57
. See Chapter 4.

58
. For a detailed description of the purpose and functioning of this machine, see Bern Dibner, “Leonardo: Prophet of Automation,” in O’Malley (1969).

59
. See, for example, Kemp (1989), p. 227.

60
. For a detailed description of this mechanism, see Dibner (1969).

61
. Codex Forster II, folios 86r and 87r.

62
. Codex Madrid I, cover.

63
. Ibid., folio 95r.

64
. Codex Leicester, folio 25r.

65
. Ms. E, folio 54r.

66
. For a comprehensive account of Leonardo’s studies on flight, see Laurenza (2004b).

67
. See Chapter 4.

68
. Codex Atlanticus, folio 1058v.

69
. In Newton’s formulation, the law reads: “For any action there is an equal and opposite reaction.”

70
. Laurenza (2004b), p. 44.

71
. See Kemp and Roberts (1989), p. 236.

72
. See Chapter 4.

73
. Codex
sul volo
, folio 16r.

74
. See Kemp (2004), pp. 127–29.

75
. Kemp (1989), p. 239.

76
. Kenneth Keele,
Leonardo da Vinci on Movement of the Heart and Blood,
Lippincott, Philadelphia, 1952, p. 122.

77
. Anatomical Studies, folio 81v.

78
. Ibid., folio 198v.

79
. Nuland (2000), p. 161.

80
. Ms. I, folio 18r.

81
. Clark (1989), p. 250.

         
CHAPTER
7         

1
. Ms. G, folio 96v.

2
. Anatomical Studies, folio 116r.

3
. See Chapter 5.

4
. See Chapter 2.

5
. See Chapter 9.

6
. Quoted in Capra (1982), p. 55.

7
. An arithmetic progression is a sequence of numbers such that the difference between successive terms is a constant. For example, the sequence 1, 3, 5, 7,…is an arithmetic progression with common difference 2. Functions are relationships between unknown variable numbers, or “variables,” denoted by letters. For example, in the equation
y
= 2
x
+ 1, the variable
y
is said to be a function of
x
. In linear functions, such as in this example, the variables are raised only to the first power. The graphs corresponding to these functions are straight lines; hence the term “linear.” Arithmetic progressions are special cases of linear functions in which the variables are discrete numbers. Thus, in the example above, the equation
y
= 2
x
+ 1 turns into the sequence 1, 3, 5, 7,…if
x
is restricted to positive integers.

8
. Ms. A, folio 10r; see also p. 214. It should be noted that, like many medieval and Renaissance writers, Leonardo uses the word “pyramid” to describe all solids that have regular or irregular bases and one apex, including cones; see Keele (1983), p. 153.

9
. Ms. M, folio 59v.

10
. Ibid.

11
. Ms. M, folio 45r.

12
. See Keele (1983), pp. 113–14.

13
. See Morris Kline,
Mathematical Thought from Ancient to Modern Times
, Oxford University Press, New York, 1972, p. 338.

14
. Keele (1983), p. 157.

15
. See E. H. Gombrich, “The Form of Movement in Water and Air,” in O’Malley (1969).

16
. See Chapter 2.

17
. Arasse (1998), p. 271.

18
. See Chapter 4.

19
. Clark (1989), p. 38.

20
. Codex Madrid II, folio 67r.

21
. The theory of functions deals with relationships among continuous variable numbers, or variables. Differential calculus is a branch of modern mathematics used to calculate the rate of change of a function with respect to the variable on which it depends.

22
. Codex Arundel, folios 190v and 266r.

23
. From The Notebooks of Paul Klee (1961), quoted in Francis Ching,
Architecture: Form, Space, and Order
, 2nd ed., John Wiley, New York, 1996, p. 1.

24
. Codex Arundel, folio 190v.

25
. Matilde Macagno, “Geometry in Motion in the Manuscripts of Leonardo da Vinci,”
Raccolta Vinciana
, Fasc. XXIV, 1992b, and “Transformation Geometry in the Manuscripts of Leonardo da Vinci,”
Raccolta Vinciana
, Fasc. XXVI, 1995.

26
. Codex Madrid II, folio 107r.

27
. See Kline (1972), p. 340.

28
. Ms. M, folio 66v.

29
. See Keele (1983), p. 276.

30
. Codex Atlanticus, folio 781ar.

31
. See Chapter 4.

32
. Codex Forster I, folio 3r.

33
. Codex Madrid II, folio 72r.

34
. Ibid., folio 112r.

35
. Anatomical Studies, folio 121r.

36
. Codex Atlanticus, folio 124v.

37
. Ms. G, folio 96r.

38
. See Macagno (1995).

39
. Ibid.

40
. See Kline (1972), p. 349.

41
. For detailed discussions of Leonardo’s three basic types of curvilinear transformations, see appendix, pp. 267–74.

42
. For a more detailed discussion of the transformations sketched on this folio, see appendix, pp. 269–71.

43
. Kemp (1981), p. 253.

44
. Kline (1972), p. 1158.

45
. Ibid., p. 1170.

46
. See Chapter 2.

47
. See Pedretti (1985), p. 296.

48
. See Chapter 2.

49
. See Arassse (1998), p. 212.

50
. See Chapter 4.

51
. Codex Atlanticus, folio 455; see also Arasse (1998), pp. 122–23.

52
. Clark (1989), p. 39.

53
. See appendix, pp. 271–74.

54
. Today we would qualify this assertion by saying that the causal relationships in nature can be represented by
approximate
mathematical models.

55
. Codex Forster III, folio 43v.

56
. See Capra (1996), p. 128.

         
CHAPTER
8         

1
. See Chapter 6.

2
. See George Lakoff and Mark Johnson,
Philosophy in the Flesh
, Basic Books, New York, 1999, p. 94; see also p. 252 in present text.

3
. Codex Trivulzianus, folio 20v.

4
. See Chapter 3.

5
. See Chapter 4.

6
. The treatise on perspective is contained in Ms. A, folios 36–42; the diagrams of geometrical optics are in Manuscripts A and C.

7
. See Chapter 2.

8
. James Ackerman, “Science and Art in the Work of Leonardo,” in O’Malley (1969).