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.
LEONARDO’S NOTEBOOKS
Facsimiles and Transcriptions
ANATOMICAL STUDIES (WINDSOR COLLECTION)
Kenneth Keele and Carlo Pedretti, Leonardo da Vinci: Corpus of the Anatomical Studies in the Collection of Her Majesty the Queen at Windsor Castle, 3 vols., Harcourt Brace Jovanovich, New York, 1978–80.
DRAWINGS AND MISCELLANEOUS PAPERS (WINDSOR COLLECTION)
Carlo Pedretti, The Drawings and Miscellaneous Papers of Leonardo da Vinci in the Collection of Her Majesty the Queen at Windsor Castle, 2 vols., Harcourt Brace Jovanovich, New York, 1982.
Volume I: Landscapes, Plants, and Water Studies
Volume II: Horses and Other Animals
Complete edition to comprise four volumes; volumes 3 and 4 not yet published.
CODEX ARUNDEL
Leonardo da Vinci, Il Codice Arundel 263 nella British Library: edizione in facsimile nel riordinamento cronologico dei suoi fascicoli; a cura di Carlo Pedretti; trascrizioni e note critiche a cura di Carlo Vecce, Giunti, Florence, 1998.
CODEX ATLANTICUS
Leonardo da Vinci, Il codice atlantico della Biblioteca ambrosiana di Milano, trascrizione diplomatica e critica di Augusto Marinoni, Giunti, Florence, 1975–80.
CODEX SUL VOLO
Leonardo da Vinci, The codex on the flight of birds in the Royal Library at Turin, edited by Augusto Marinoni, Johnson Reprint, New York, 1982.
CODICES FORSTER I, II, III
Leonardo da Vinci, I codici Forster del Victoria and Albert Museum di Londra; trascrizione diplomatica e critica di Augusto Marinoni, edizione in facsimile, 3v., Giunti, Florence, 1992.
CODEX LEICESTER (FORMERLY CODEX HAMMER)
Leonardo da Vinci, The Codex Hammer, translated into English and annotated by Carlo Pedretti, Giunti, Florence, 1987.
CODICES MADRID I, II
Leonardo da Vinci, The Madrid Codices, transcribed and translated by Ladislao Reti, McGraw-Hill, New York, 1974.
MANUSCRIPTS AT INSTITUT DE FRANCE
Leonardo da Vinci, I manoscritti dell’Institut de France, edizione in facsimile sotto gli auspici della Commissione nazionale vinciana e dell’Institut de France, trascrizione diplomatica e critica di Augusto Marinoni, Giunti, Florence, 1986–90. (Mss. A, B, C, D, E, F, G, H, I, K, L, M; Ms. A includes as a supplement Ashburnham II, also listed as B.N. 2038; Ms. B includes as a supplement Ashburnham I, also listed as B.N. 2037.)
CODEX TRIVULZIANUS
Leonardo da Vinci, Il codice di Leonardo da Vinci nella Biblioteca trivulziana di Milano, trascrizione diplomatica e critica di Anna Maria Brizio, Giunti, Florence,1980.
TRATTATO DELLA PITTURA (CODEX URBINAS)
Leonardo da Vinci, Libro di pittura, Codice urbinate lat. 1270 nella Biblioteca apostolica vaticana, a cura di Carlo Pedretti, trascrizione critica di Carlo Vecce, Giunti, Florence, 1995.
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Arasse, Daniel. Leonardo da Vinci: The Rhythm of the World. Konecky & Konecky, New York, 1998.
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