The Science of Leonardo: Inside the Mind of the Great Genius of the Renaissance
Page 29
CHAPTER 3
1. See, e.g., Ludwig H. Heydenreich, Leonardo da Vinci, 2 vols., Macmillan, New York, 1954; Clark (1989); Bramly (1991).
2. “Ser” was the traditional title of a notary.
3. Codex Atlanticus, folio 888r.
4. See Arasse (1998), pp. 108–9.
5. Ibid., p. 39.
6. Codex Atlanticus, folio 327v.
7. See Chapter 4.
8. See Bramly (1991), p. 53.
9. See Arasse (1998), p. 502 n. 71.
10. Vasari probably exaggerated when he called Verrocchio “a close friend” of Ser Piero, but it is quite likely that the notary knew the artist, since many of his clients were patrons of the arts.
11. Bramly (1991), pp. 65–66.
12. Ibid., pp. 67–69.
13. Domenico Laurenza, “Leonardo: La scienza trasfigurata in arte,” Le Scienze, Rome, maggio 2004a, p. 6.
14. See Arasse (1998), p. 54.
15. See Bramly (1991), pp. 71–72; Carlo Pedretti, Leonardo: The Machines, Giunti, Florence, 1999, p. 16.
16. See Laurenza (2004a), p. 7.
17. Ms. G, folio 84v.
18. See Jane Roberts, “The Life of Leonardo,” in Martin Kemp and Jane Roberts, eds., Leonardo da Vinci: Artist, Scientist, Inventor, Catalogue of Exhibition at Hayward Gallery, Yale University Press, 1989.
19. See Keele (1983), p. 9.
20. See Domenico Laurenza, Leonardo on Flight, Giunti, Florence, 2004b, p. 16.
21. See Keele (1983), p. 9.
22. See Martin Kemp, Leonardo, Oxford University Press, 2004, pp. 13–14.
23. See Bramly (1991), p. 144.
24. Clark (1989), p. 59.
25. See Bramly (1991), p. 156.
26. Ibid., p. 91.
27. Ibid., p. 157; see also White (2000), p. 83.
28. See White (2000), p. 81.
29. See Pedretti (1999), p. 16.
30. See Keele (1983), pp. 9–11.
31. See Arasse (1998), pp. 350–61.
32. Clark (1989), pp. 74–75.
33. See Chapter 1.
34. See Clark (1989), p. 78.
35. Roberts (1989).
36. Arasse (1998), p. 361.
37. Anonimo Gaddiano (1542).
38. Ludovico’s official title was Duke of Milan and, like other powerful rulers of the Renaissance, he was often referred to as a prince.
39. See Pedretti (1999), p. 32.
40. See Bramly (1991), p. 158.
41. See Chapter 1.
42. Codex Atlanticus, folio 1082r.
43. See Clark (1989), p. 85.
44. Keele (1983), p. 11.
45. See Bramly (1991), pp. 183–84.
46. See Chapter 2.
47. See Kemp (1981), pp. 93–94.
48. For the comparison of the two paintings, based on the analysis of Leonardo’s geology, see Pizzorusso (1996); for the corresponding botanical analysis, see Emboden (1987), p. 125.
49. Clark (1989), p. 93.
50. See Chapter 2.
51. Laurenza (2004a), p. 23.
52. See Arasse (1998), p. 43.
53. See Keele (1983), p. 20.
54. See Laurenza (2004a), p. 24.
55. Codex Atlanticus, folio 888r.
56. See Arasse (1998), p. 37.
57. See Emboden (1987), p. 21; Kemp (2004), pp. 165–66.
58. Clark (1989), p. 129.
59. See Chapter 9.
60. Laurenza (2004a), p. 27.
61. See Guillaume (1987).
62. See Chapter 2.
63. See Bramly (1991), p. 192.
64. See Heydenreich (1969).
65. See Arasse (1998), p. 397.
66. See Chapter 2.
67. See White (2000), pp. 127–28.
CHAPTER 4
1. See Kemp (2004), pp. 38–40.
2. See Keele (1983), p. 22.
3. Ibid., p. 22.
4. See Keele (1983), p. 22; Fazio Cardano was the father of the famous mathematician Girolamo Cardano, the founder of probability theory.
5. See Laurenza (2004b), p. 40.
6. See Ladislao Reti, ed., The Unknown Leonardo, McGraw-Hill, New York, 1974, pp. 272–73.
7. See Kemp (1981), p. 194.
8. Quoted in Richter (1952), p. 322.
9. See Chapter 7.
10. For a discussion of the golden section and its connection with the Platonic solids, see Mario Livio, The Golden Ratio, Broadway Books, New York, 2002.
11. Luca Pacioli, De divina proportione, Paganinum de Paganinis, Venice, 1509; fascsimile edition of the ms. in the Biblioteca Ambrosiana di Milano published by Fontes Ambrosiani XXXi, G. Biggiogero and F. Riva, eds., Milan, 1966.
12. See Chapter 1.
13. See Bramly (1991), pp. 294–95.
14. Clark (1989), p. 146.
15. See Hope (2001).
16. Clark (1989), p. 149.
17. The portrait, now in the Louvre, is also known as La Belle Ferronière.
18. See Chapter 2.
19. See Codex Leicester, folio 4r.
20. See Chapter 1.
21. See Bramly (1991), p. 308.
22. See Chapter 2.
23. Ludovico briefly regained possession of Milan in 1500 before being captured and taken to France as prisoner, where he remained until his death in 1508.
24. See Keele (1983), p. 25.
25. See Bramly (1991), p. 307.
26. See Arasse (1998), p. 210.
27. Ibid., p. 417.
28. Kemp (1981), p. 218.
29. See Bramly (1991), p. 310.
30. The drawing is now in the Louvre; see Arasse (1998), p. 398.
31. See Codex Atlanticus, folio 638vd.
32. See Keele (1983), pp. 28–29.
33. See also Chapter 2.
34. Codex Leicester, folio 22v.
35. See Keele (1983), p. 28.
36. These rooms, with fading frescoes on their walls, may have been identified recently in a building in central Florence; see International Herald Tribune, January 19, 2005.
37. See Arasse (1998), p. 448.
38. See White (2000), pp. 208–9.
39. See Keele (1983), pp. 30–32.
40. See Bramly (1991), pp. 330–31.
41. Ibid., p. 332.
42. Codex Arundel, folio 272r.
43. See Chapter 1.
44. Codex Forster I, folio 3r.
45. See Chapter 7.
46. See Laurenza (2004b).
47. Ibid., p. 96.
48. See Bramly (1991), pp. 348–49.
49. See Emboden (1987), pp. 62–65.
50. See Chapter 1.
51. See Kemp (1981), p. 270.
52. See Bramly (1991), pp. 356–58.
53. This bronze group, Saint John the Baptist Preaching to a Levite and a Pharisee, can still be seen above the Baptistery’s north door. The life-size statues do seem to exhibit Leonardesque features.
54. Codex Arundel, folio 1r.
55. Anatomical Studies, folio 154r.
56. Ibid., folio 113r.
57. Ibid., folio 69v.
58. See Keele (1983), pp. 321–22.
59. See Farago (2003).
60. See Emboden (1987), p. 24.
61. See Bramly (1991), pp. 370–71.
62. See Laurenza (2004a), p. 87.
63. See Emboden (1987), pp. 65–68.
64. See Chapter 1.
65. See Bramly (1991), pp. 385–86.
66. See Chapter 9.
67. Historians long believed that the dissections themselves got Leonardo into trouble with the pope. However, Domenico Laurenza has documented that there were no religious or ethical objections to dissections in Italy at the time. According to Laurenza, it was the clash between Leonardo’s Aristotelian view of the soul and Leo X’s Thomistic view that was at the root of the pope’s ban; see Domenico Laurenza, “Leonardo nella Roma di Leone X,” XLIII Lettura Vinciana, Biblioteca Leonardiana, Vinci, 2003.
68. See Bramly (1991), pp.
384–85.
69. See Chapter 1.
70. Drawings and Miscellaneous Papers, vol. I, folio 67r.
71. The Leda was lost or destroyed in the early eighteenth century; see Bramly (1991), p. 465 n. 49.
72. Arasse (1998), p. 462.
73. Trattato, chapter 25.
74. See Bramly (1991), p. 397.
75. See Arasse (1998), p. 152.
76. See Bramly (1991), p. 398.
77. Ibid., p. 399.
78. Quoted by Kemp (1981), p. 349.
79. Quoted by Bramly (1991), p. 400.
80. See Chapter 2.
81. Quoted by Bramly (1991), p. 400.
82. See Keele (1983), p. 41.
83. See Introduction.
84. See Chapter 7.
85. Anatomical Studies, folio 113 r.
86. Codex Atlanticus, folio 673 r.
87. See Chapter 2.
88. See Keele (1983), p. 40.
89. See Chapter 2.
90. Codex Trivulzianus, folio 27r.
91. See Bramly (1991), pp. 406–7.
92. Quoted by Bramly (1991), pp. 411–12.
93. See Carlo Pedretti and Marco Cianchi, Leonardo: I codici, Giunti, Florence, 1995; see also Bramly (1991), p. 417.
94. See Reti (1974).
CHAPTER 5
1. Thomas S. Kuhn, The Structure of Scientific Revolutions, University of Chicago Press, 1962; see also Capra (1996), p. 5.
2. See, e.g., George Sarton, The Appreciation of Ancient and Medieval Science during the Renaissance, University of Pennsylvania Press, Philadelphia, 1955; Marie Boas, The Scientific Renaissance, Harper & Brothers, New York, 1962.
3. “Byzantine Empire” is the term commonly used to refer to the Greek-speaking Eastern Roman Empire during the Middle Ages. Its capital was Constantinople, today’s Istanbul.
4. See Karen Armstrong, Islam: A Short History, Modern Library, New York, 2000, pp. 5–6.
5. See Chapter 4.
6. See Sarton (1955), p. 4.
7. See Pedretti (1999), p. 83.
8. Ibid., p. 91.
9. See Chapter 39.
10. Anatomical Studies, folio 139v.
11. See George Sarton, “The Quest for Truth: A Brief Account of Scientific Progress during the Renaissance,” in Robert M. Palter, ed., Toward Modern Science, vol. 2, Noonday Press, New York, 1961.
12. See Chapter 4.
13. See Kemp (1981), pp. 159–60.
14. See Fritjof Capra, The Tao of Physics, Shambhala, Berkeley, 1975; 25th Anniversary Edition by Shambhala, Boston, 2000, pp. 55–56.
15. See Capra (1996), p. 18.
16. See Wilhelm Windelband, A History of Philosophy, published originally in 1901 by Macmillan; reprinted by The Paper Tiger, Cresskill, N.J., 2001, p. 149.
17. See Chapter 9.
18. See Chapter 8.
19. See, Chapter 7.
20. Sarton (1955), p. 171.
21. Irrational numbers, e.g., square roots, cannot be expressed as ratios, or quotients, of integers.
22. Al jabr refers to the process of reducing the number of unknown mathematical quantities by binding them together in equations.
23. See Capra (1996), p. 114.
24. See Sarton (1955), p. 52.
25. See Capra (1982), p. 306.
26. Ibid., p. 311.
27. See Sarton (1955), p. 7.
28. Ibid., pp. 169–70.
29. Anatomical Studies, folio 136r.
30. See Boas (1962), p. 131.
31. See Chapter 3.
32. See Chapter 4.
33. See Kemp (1981), p. 323.
34. See Emboden (1987), p. 141.
35. See Chapter 4.
CHAPTER 6
1. See, for example, Kuhn (1962).
2. Quoted in Capra (1982), p. 101.
3. For the classical work on Leonardian paleography, see Gerolamo Calvi, I manoscritti di Leonardo da Vinci dal punto di vista cronologico storico e biografico, Bramante, Busto Arsizio, 1982; first published in 1925, republished in 1982 with a foreword by Augusto Marinoni.
4. A list of the scholarly editions of Leonardo’s Notebooks is given in the Bibliography on pp. 299–301.
5. Codex Trivulzianus, folio 20v.
6. Codex Forster III, folio 14r.
7. Trattato, chapter 33.
8. Codex Atlanticus, folio 323r.
9. Ibid., folio 534v.
10. Ms. E, folio 55r.
11. See Introduction and Chapter 2.
12. Clark (1989), p. 255.
13. E. H. Gombrich, preface to Leonardo da Vinci: Artist, Scientist, Inventor, Catalogue of Exhibition at Hayward Gallery, Yale University Press, 1989.
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 = 2x + 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 = 2x + 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.