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
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. Ludwig 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.
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.
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).
9. Ms. A, folio 1v.
10. Ms. A, folio 10r. As noted above, Leonardo, like many of his contemporaries, used the word “pyramid” to also describe cones and other solids that have a single apex; see p. 193, footnote 8.
11. Ms. A, folio 1v.
12. See Keele (1983), p. 46.
13. Ms. Ashburnham II, folio 23r.
14. Ms. A, folio 8v.
15. See Chapter 4.
16. See Chapter 9.
17. See Chapter 2.
18. Arasse (1998), pp. 300–301.
19. See Chapter 4.
20. Ms. A, folio 3r.
21. See Chapter 9.
22. Ms. Ashburnham II, folio 18r.
23. Ibid., folio 25.
24. See Kemp (1981), p. 33.
25. See Chapter 5.
26. See Kemp (1981), p. 35.
27. See Chapter 3.
28. Codex Arundel, folio 70v.
29. See Keele (1983), pp. 55–56.
30. Ms. A, folio 19r.
31. See Keele (1983), p. 141.
32. See Chapter 2.
33. Codex Atlanticus, folio 676r.
34. Ibid.
35. Trattato, chapters 681–82.
36. Clark (1989), p. 129; quoted also on p. 87 in present text.
37. Trattato, chapter 17.
38. See Keele (1983), p. 132; see also p. 226 in present text about Leonardo’s use of the camera obscura.
39. See Chapter 5.
40. Codex Arundel, folio 94v.
41. Ibid.
42. See Keele (1983), pp. 91–92.
43. Ms. F, folio 41v.
44. See Kemp (1981), p. 323; see also p. 155 in present text.
45. Trattato, chapter 25.
46. Codex Atlanticus, folio 372v.
47. Anatomical Studies, folio 118v.
48. Ibid., folio 22v.
49. It would be wrong to read any occult meaning into Leonardo’s frequent use of the term “spiritual.” He defines it clearly to mean simply “invisible and immaterial” and uses it consistently in this sense; see p. 244.
50. The concept of energy was defined precisely only in the seventeenth century. Leonardo uses both potentia and virtù to mean power, or energy.
51. Anatomical Studies, folio 22v.
52. See Capra (1975), p. 61.
53. Ms. Ashburnham II, folio 6v.
54. Ibid.
55. Ms. A, folio 9v.
56. Ibid., folio 61r.
57. More precisely, the water particles move in small circles; see Capra(1975), p. 152.
58. See Chapter 9.
59. Ms. A, folio 61r.
60. Ms. H, folio 67r.
61. There is even some speculation that Huygens may have been aware of Leonardo’s research when he published his famous work on optics, Traité de la lumière, in 1690; see White (2000), p. 177.
62. See Kenneth Keele, “Leonardo da Vinci’s Physiology of the Senses,” in O’Malley (1969).
63. Ms. F, folio 49v.
64. Codex Atlanticus, folio 545v.
65. See White (2000), p. 182.
66. Ibid., p. 183.
67. See Chapter 4.
68. Codex Leicester, folio 4r.
69. See Keele (1983), p. 215.
70. Ms. B, folio 4v.
71. Ms. A, folio 61r.
72. Ibid., folio 22v.
73. See Chapter 8.
74. See Chapter 4.
75. Anatomical Studies, folio 148v.
76. Codex Madrid I, folio 126v.
77. Ibid.
78. See Martin Kemp (1999b), “Leonardo and the Visual Pyramid,” in Farago (1999).
79. Ms. F, folio 34r.
80. Ms. D, folio 4v.
81. See Kemp (1999b).
82. Ms. E, 16v.
CHAPTER 9
1. Codex Atlanticus, folio 949v.
2. Trattato, chapter 28.
3. See Keele (1983), p. 61.
4. Codex Atlanticus, folio 327v.
5. Ms. D, folio 5v.
6. Codex Atlanticus, folio 345r.
7. Ms. F, folio 39v.
8. See Keele (1983), pp. 73–74.
9. Ibid., p. 69.
10. Codex Atlanticus, folio 545r.
11. See Chapter 8.
12. See Chapter 6.
13. Ms. D, folio 1r.
14. See Keele (1983), pp. 74–75.
15. Spectacles were well known in Leonardo’s day. They were of two kinds, those “for the young” (concave lenses) and those “for the old” (convex lenses); see Keele (1983), p. 210.
16. Ibid., p. 204.
17. See Chapter 8.
18. Ms. D, folio 3v.
19. See Keele (1983), p. 201.
20. Anatomical Studies, folio 115r. Leonardo was unaware that central vision actually takes place at the macula on the periphery of the optic disk.
21. Codex Atlanticus, folio 546r.
22. See Chapter 6.
23. Keele (1969), and Keele (1983), p. 60.
24. See Chapter 8.
25. Keele (1969).
26. Keele (1983), p. 63.
27. Ibid., pp. 64–65.
28. The optic chiasma is actually a partial crossing in which each nerve separates into two branches, and the inner branch from each eye crosses over to join the outer branch from the other eye.
29. Codex Atlanticus, folio 832v.
30. The anterior ventricle consists of two almost completely separated lateral wings and is therefore also described as two lateral ventricles.
31. Following Kenneth Keele, I am using Leonardo’s Italian term senso comune for this region of the brain, since the English “common sense” has quite a different meaning; see Keele (1983), p. 62.
32. Leonardo may have coined the term impressiva (or imprensiva) in analogy to related terms like apprensiva and comprensiva, used by medieval scholars; see Farago (1992), pp. 301–2. “Receptor of impressions” is the translation proposed by Martin Kemp; see Kemp (1981), p. 127.
33. Trattato, chapter 28.
34. For more extensive discussions of Leonardo’s studies of the human voice, phonetics, and music, see Giulio Panconcelli-Calzia, “Leonardo’s Work in Phonetics and Linguistics,” and Enrico Magni-Dufflocq, “Da Vinci’s Music,” in Leonardo da Vinci, Reynal, New York, 1956; and especially Keele (1983), p. 215.
35. Keele (1983), p. 219.
36. See Chapter 3.
37. See Chapter 2.
38. See Arasse (1998), p. 222.
39. Anatomical Studies, folio 39r.
40. See Windelband (2001), p. 62; see also p. 145 in present text.
41. See Keele (1983), p. 267.
42. Codex Arundel, 151r, v.
43. See Capra (2002), p. 33.
44. See Chapter 4.
45. See Laurenza (2004b), pp. 86–88.
46. Codex Atlanticus, 434r.
47. Anatomical Studies, folio 114v.
48. See Chapter 5.
49. Codex Atlanticus, folio 166r.
50. See Capra (2002), p. 61.
51. Ms. B, folio 4v.
52. Anatomical Studies, folios 198r and 114v.
53. Codex Atlanticus, folio 680r.
54. See Capra (2002), pp. 40–41.
EPILOGUE
1. See Introduction.
2. Keele (1983); see also Nuland (2000).
3. Emboden (1987).
4. Galluzzi (1996); see also Pedretti (1999); Laurenza, Taddei, and Zanon(2005).
5. Laurenza (2004b).
6. Leonardo’s engineering and architecture are both covered extensively in the beautiful catalog of an exhibit at the Musée des Beaux-Arts de Montréal; see Galluzzi (1987).
7. Codex Madrid I, folio 6r.
8. See, e.g., Ms. E, folios 38ff.
9. Anatomical Studies, folio 114v.
10. See Capra (2002), pp. 229ff.
11. See Chapter 2.
12. See Chapter 2.
13. Codex Leicester, folio 13r; see also folio 32r.
14. See Capra (1996), pp. 6–7.
15. Trattato, chapter 34.
16. See Fritjof Capra and David Steindl-Rast, Belonging to the Universe, HarperSanFrancisco, 1991.
17. See Capra (1982).
18. See Capra (1996), and Capra (2002).
APPENDIX
1. See Macagno (1992b).
2. See Chapter 6.
3. Codex Madrid II, folios 107r and 111v.
4. Macagno (1992b).
5. See Kemp (1981), p. 250.
6. Codex Atlanticus, folio 82r.
7. Macagno (1992b).
8. See Chapter 7.
9. See analysis by Pedretti, and Marinoni; Codex Atlanticus, folio 455.
10. Codex Atlanticus, folio 455.
11. Ibid.
12. Ibid.
13. See Keele (1983), p. 154.
14. Leonardo, apparently, did not feel the need to record the solutions of his topological equations graphically. If he had done so, he would probably have used a different notation, as the equal sign (=) came into common usage only in the seventeenth century; see Kline (1972), p. 260.