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Between Latin, French, English and German: The Language of Transcendence

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And Yet It Is Heard

Part of the book series: Science Networks. Historical Studies ((SNHS,volume 47))

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Abstract

Although the Galileian school still included scholars of the first rank like Evangelista Torricelli  (1608–1647) and Vincenzo Viviani  (1622–1703), the barycentre of European scientific evolution had now shifted to the North. The most important characters for our history had already appeared in the correspondence of Huygens . Through the one with whom our Dutchman had the closest relationship, Leibniz  (1646–1716), we can take up the thread of music which we prefer to follow here. Actually for his model of the solar system, the renowned German philosopher and diplomat had used a term which too few modern astronomers would expect: “circulation harmonique” [“harmonic circulation”].

Un bel air de Musique qui sera tousjour chanté, est comme un beau theoreme de Geometrie rencontré.

[A beautiful musical aria, which will always be sung, is like a fine geometrical theorem found again.]

…when the Chinese will have learnt all they want to know from us, then they will close their doors to us.

Letter to Peter the Great

Aimer est être porté a prendre du plaisir dans la perfection de l’objet aimé.

[To love means being led to take pleasure in the perfection of the object loved.]

Gottfried Wilhelm Leibniz

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Notes

  1. 1.

    Leibniz  1689.

  2. 2.

    Huygens  1901, IX, pp. 367 and 523–527.

  3. 3.

    Leibniz  1960/65, IV, p. 196. Leibniz  1962, IV, p. 243. Bailhache  1992, p. 43. Cf. Serres  1968, pp. 462–464.

  4. 4.

    Tonietti  1988.

  5. 5.

    …organorum …, an ambiguous term which indicates all the instruments, but above all were the organs and harpsichords with a fixed tuning which had to be tempered.

  6. 6.

    The wisecrack that Pythagorean theoreticians could not count beyond five was a joke that I thought I had invented: until I found it in Leibniz . We are all modest commentators on the work of someone else. At the beginning, what would come out is always either the Bible , or the Koran , or the Veda , or the Wujing .

  7. 7.

    Leibniz  1734, pp. 240–242.

  8. 8.

    See above, Sect. 8.3, notes 82–90 and Fig. 8.5.

  9. 9.

    See above, Sect. 9.1.

  10. 10.

    See Sect. 9.3.

  11. 11.

    Bailhache  1992, p. 153.

  12. 12.

    Bailhache  1992, pp. 73–121.

  13. 13.

    Mila  1963, pp. 150–153. Luppi  1989, pp. 21–59 and 150–155.

  14. 14.

    Bailhache  1992, pp. 125–148. Cf. Baas 1976 , pp. 69–72; on the contrary, though attractive and suggestive of some similarities, connections between Leibniz and Johann Sebastian Bach  (1685–1750) are historically unlikely. See below, Sect. 11.3.

  15. 15.

    Bailhache  1992, pp. 89–90 and 98–100. It is curious that the editor and French translator, from the original Latin for the Prussian Academy of Sciences, considered the cubic ratios for the pipes another error among the many that he observed in the work of the German scholar. Instead, as we have already underlined in Sect. 6.7 in note 123, a modern book of acoustics, which takes into consideration the end effect in the pipes would confirm, at least in this case, the correct interpretation of Henfling . Fletcher & Rossing 1991, pp. 474–477.

  16. 16.

    Bailhache  1992, pp. 28–36. The word “Acoustique”, “Acoustics” had already been used by Francis Bacon and Gaspar Schott  (1608–1666). Cf. Gouk 1999 , pp. 108 and 157.

  17. 17.

    See Part I, Chap. 2.

  18. 18.

    Cf. Bailhache  1992, pp. 36–50. Cf. Luppi  1989, pp. 144–154. Cf. Bailhache  1995b. In his article, the Frenchman judges the musical theories of Huygens , Henfling and Leibniz anachronistically by the standard of future justifications based on the harmonics of sound or on the physiology of the ear, like Hermann Helmholtz  (1821–1894); see below, Sect. 12.2. For the division of the octave, he took for correct, true, objective and ‘scientifically’ proven, the so-called “just” ratios of Zarlino. But the belief that these were “just” was an arbitrary dogma, a heritage of the ancient Pythagoreanism, while musicians were successfully taking a completely different direction. In the eighteenth century, Leibniz had understood that the mathematical theory of music would have to adapt to the requirements of composing and playing. In the twentieth century, Bailhache relegated the problems of composers and players to the last position, preferring to leave his own philosophy of mathematics in the first place. For music, even if almost only in this, my essay follows Leibniz . What our French philosopher and mathematician calls, referring to Leibniz , “…sa ‘retraite’ sur le tempérament égal …” [“…his ‘retreat’ over the equable temperament …”] shows, on the contrary, how well the famous German philosopher and mathematician knew the music of his epoch, and how much he desired to harmonise it with his own infinite and infinitesimal numbers. Bailhache  1989; 1995a; 1996. Instead, Haase exaggerated the Pythagoreanism of Leibniz , which is called “…‘säcularisierter’ Pythagoreismus …” [“…‘secularised’ Pythagoreanism …”], almost imprisoning him inside it; Haase  1960 and 1985.

  19. 19.

    Leibniz  1666, pp. 17, 23 and 61. Serres  1968, pp. 186–188 and 468–470. 479,001,600 will be the number of combinations with 12 notes used as a series by Arnold Schönberg ; see Sect. 12.4. Tonietti  2004.

  20. 20.

    Leibniz  1666, p. 34.

  21. 21.

    Leibniz  1666, pp. 68–69.

  22. 22.

    Leibniz  1960/65, IV, pp. 550–551. Bailhache  1992, p. 41.

  23. 23.

    Leibniz  1960/65, VI, p. 605.

  24. 24.

    Leibniz  1960/65, VII, p. 122.

  25. 25.

    Leibniz  1960/65, VII, p. 170. Liebniz 1967, II, p. 747.

  26. 26.

    Leibniz  1967, II, p. 759.

  27. 27.

    Leibniz  1967, II, p. 761.

  28. 28.

    Leibniz  1967, II, p. 764.

  29. 29.

    Leibniz  1962, VII, pp. 9–17. Leibniz  1960/65, VII, p. 32. Leibniz  1966, p. 30. Leibniz  1994, p. 38. Leibniz  2000, II, pp. 381–382. Walker 1972 , pp. 295, 297–298 and 305–306. Tagliagambe  1980, “Chapter I”.

  30. 30.

    Huygens  1901, IX, pp. 450–451.

  31. 31.

    Loxodromics [oblique course] is the route in the sea that intersects the meridians at a constant angle.

  32. 32.

    Leibniz  1962, V, pp. 226–233.

  33. 33.

    Huygens  1905, X, pp. 227–228.

  34. 34.

    Huygens  1905, X, p. 261. In these letters pp. 229–230, 263 and 285–286, Leibniz declared his interest in the division of the octave into 31 parts, and in publishing an article about music by Huygens in the journal Acta Eruditorum. But we already know that the Dutchman would not realise this.

  35. 35.

    Leibniz  1962, V, pp. 306–308.

  36. 36.

    Leibniz  1962, VII, pp. 218–223.

  37. 37.

    Leibniz  1967, II, 163 and 739.

  38. 38.

    Leibniz  1987, pp. 18–21.

  39. 39.

    Leibniz  1987, pp. 182–183. Sabattini & Santangelo  1989, pp. 554–556.

  40. 40.

    See above, Part II, Sect. 9.1.

  41. 41.

    Leibniz  1987, pp. 21–25.

  42. 42.

    Leibniz  1987, pp. 97–98.

  43. 43.

    See above, Part I, Sect. 3.5.

  44. 44.

    Leibniz  1987, pp. 56–61, passim, 98.

  45. 45.

    Leibniz  1987, pp. 105, 115, 118.

  46. 46.

    See above, Sect. 8.3.

  47. 47.

    Leibniz  1987, pp. 121–122, 125–7 and 186.

  48. 48.

    Leibniz  1987, pp. 132–136.

  49. 49.

    Daodejing 1973. Zhuangzi 1982.

  50. 50.

    Ren neng hong dao, fei dao hong ren. Leibniz  1994, p. 90. Confucio 2000, p. 121.

  51. 51.

    Leibniz  1987, pp. 137–141. Leibniz  1967, I, p. 190–191.

  52. 52.

    Leibniz  1987, pp. 143–145. Leibniz  1967, I, pp. 328–329. Leibniz  1967, II, p. 176.

  53. 53.

    Leibniz  1987, pp. 146–147 and 161. Leibniz  1994, pp. 68 and 72.

  54. 54.

    Leibniz  1987, pp. 153–156.

  55. 55.

    Leibniz  1987, p. 168.

  56. 56.

    Leibniz  1967, I, pp. 189–199, 363, 493–494 and passim. Leibniz  1967, II, p. 709 and passim.

  57. 57.

    Leibniz  1987, p. 170.

  58. 58.

    Leibniz  1987, pp. 174–177. Binary calculations were also considered to be useful for music. Leibniz  1966, pp. 279–280. Luppi  1989, pp. 76 and 145. Walker 1972 , p. 297.

  59. 59.

    Leibniz  1962, VII, pp. 223–227. Kreiling 1981 , p. 149. Hofmann 1981 , p. 161. Luppi  1989, p. 71. Walker 1972 , pp. 296–297.

  60. 60.

    Leibniz  1962, VII, p. 321. Here the philosopher of the sufficient reason exchanged the effects for the causes.

  61. 61.

    Leibniz  1987, pp. 179–188. Gernet  1982. Leibniz  1994.

  62. 62.

    Tonietti  2006a.

  63. 63.

    Leibniz  1962, V, p. 357.

  64. 64.

    Leibniz  1967, II, pp. 32, 52–53 and 60.

  65. 65.

    Leibniz  1967, II, pp. 58, 65 and 97. The Philosophia Mosaica had been written by Robert Fludd  (1574–1637), Kepler ’s adversary, already encountered in Sect. 8.3.

  66. 66.

    Leibniz  1967, II, pp. 116–131.

  67. 67.

    Leibniz  1967, II, pp. 689–700.

  68. 68.

    Descartes 1983, p. 44. The Dutch naturalist observed under the microscopy those spermatozoa and bacteria which confirmed the German’s panvitalistic theories.

  69. 69.

    Leibniz  1967, II, pp. 700–706. Leibniz  1967, I, pp. 364–365.

  70. 70.

    Leibniz  1967, I, pp. 300–304, 312–313, 331–334, 347–357, 366–369 and passim. Leibniz  1967, II, pp. 300–304. Serres  1968, pp. 470–472.

  71. 71.

    Leibniz  1948, pp. 365–366. Serres  1968, pp. 461–462. Luppi  1989, pp. 105–112. Baas 1976 , pp. 53–58.

  72. 72.

    Leibniz  1967, I, p. 306. Leibniz  1967, II, pp. 707–714.

  73. 73.

    Leibniz  1960/65, II, p. 95, IV, p. 549, VI, p. 479. Luppi  1989, pp. 130–131 and 140. Serres  1968, p. 473. Baas 1976 , pp. 59–60.

  74. 74.

    Leibniz  1960/65, IV, p. 329. Leibniz  1987, pp. 5–8. Mittelstrass & Aiton  1981, pp. 153–155.

  75. 75.

    Leibniz  1967, II, pp. 724–733. Leibniz  2000, II, p. 256. Koyré  1979. Tonietti  1999a, pp. 199–201. Tonietti  2004, Chap. 16. Walker 1972 , pp. 299–300 and 306–307.

  76. 76.

    Leibniz  1967, II, pp. 733–750. Hofmann 1981 , pp. 164–165. Boyer 1990, pp. 456–475. Kline 1972, p. 380.

  77. 77.

    Leibniz  1967, II, pp. 735–736 and 744. Serres  1968, pp. 420–421.

  78. 78.

    Leibniz  1962, VII, pp. 125–132.

  79. 79.

    Leibniz  1967, II, pp. 158 and 281–285. Mittelstrass & Aiton 1981, pp. 151–152 and 157–158. Hofmann 1981 , p. 165. Boyer  1990, pp. 459–462.

  80. 80.

    Mittelstrass & Aiton 1981, p. 151. Leibniz  1704, pp. 345 and 348. Leibniz  2000, I, pp. 564–569. Leibniz  1967, I, p. 343. Leibniz  1967, II, p. 714. Bailhache  1995b, p. 13. Baas 1976 , pp. 53–59.

  81. 81.

    Huygens  1905, X, p. 261.

  82. 82.

    Writing his name “Jsaacus Neuvtonus”, without any false modesty, he had created the anagram of “Jeova sanctus unus” [“Jehovah holy and one”]. Westfall 1989 , pp. 45 and 304.

  83. 83.

    Newton  1687/1972, p. 577.

  84. 84.

    Casini 1981 and 1984. Tonietti  2000b.

  85. 85.

    McGuire & Rattansi  1966/1989. Gouk 1986 and 1999.

  86. 86.

    Tonietti  2000b.

  87. 87.

    McGuire & Rattansi 1966/1989, pp. 83–84.

  88. 88.

    McGuire & Rattansi  1966/1989. Casini  1981 and 1984. Gouk  1986 and 1999. Tonietti  2000b.

  89. 89.

    Casini 1981. Barone  1989. Westfall 1989 , pp.  272–279. Newton  2006, pp. 215–285.

  90. 90.

    Gouk 1999, pp. 224–228.

  91. 91.

    Gouk 1999 , pp. 153 and 230–233.

  92. 92.

    Gouk 1999 , p. 140.

  93. 93.

    Gouk 1999 , pp. 142–143.

  94. 94.

    Jeans 1980. Gouk 1986, pp. 41–44. Gouk 1999, pp. 233–237. Tonietti  2006b.

  95. 95.

    Newton  1704/1952, pp. 126–128, 225–226 and 346.

  96. 96.

    Gouk 1999 , pp. 242–244. The research of Penelope Gouk was used as a basis by Benjamin Wardhaugh , adding some extra details. However, it sounds very curious to me that the latter completely ignores the musical scholium written by Newton for proposition VIII of the Principia ; see the previous pages. Wardhaugh 2008, pp. 53–56 and 120–125.

  97. 97.

    Newton  2006, pp. 250, 254 and 258. Continuing the series of mutilations, this recent anthology of Newton ’s writings about optics, has ignored the main analogies created with music. The omission of these passages appears to be all the more curious because in the texts, our Englishman also tried to give “mathematical explanations” of the luminous phenomena. But together with the well-known descriptions of the experiments with prisms, the numerical ratios of the intervals between notes were the most mathematical things that he could offer, apart from the geometry of rays.

  98. 98.

    Gouk 1999 , pp. 246–248. Kassler 2004 . Wardhaugh 2008, pp. 125–133.

  99. 99.

    Newton  1687/1972, pp. 510–534. Koyré  1972, pp. 87–126 and 214–217. Dostrovsky  1975, pp. 209–218. Westfall 1989 , pp. 315–318, 394–397, 414, 417, 775–777 and passim. Gouk 1986, p. 85. Gouk 1999 , pp. 248–251.

  100. 100.

    On fluxions, Newton wrote to Oldenburg a letter, so that it were forwarded to Leibniz , and it finally arrived only in 1677. But the fundamental theorem was encoded. “6accdae13eff7i319n404qrr4s8t12ux. …Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et viceversa.” [“…Given an equation including any number of fluent quantities, find the fluxions; and vice versa.”] Newton  1687/1972. Westfall 1989 , “Chapter four”, “Chapter five”, pp. 272–280, 416–417, 441–448, 539–546, 550 and 736–825 and passim. Cf. Koyré  1972, pp. 92–98 and 114–120.

  101. 101.

    Cf. Koyré  1972, pp. 108–114 and 291–303. Cf. Westfall 1989, pp. 475–479. Newton  1687/1972, p. 550.

  102. 102.

    The German astronomer was treated better by Halley , who wrote that Newton ’s propositions were “…found to agree with the Phenomena of the Celestial motions, as discovered by the great Sagacity and Diligence of Kepler .” Gingerich 1970, p. 308.

  103. 103.

    Newton  2006, pp. 216, 220, 232, 237–238, 244–245, 248–249 and passim. Westfall 1989, pp. 254–265 and 791–793. Mamiani  1994a, pp. xxvi–xxviii. Koyré  1972, pp. 31–43.

  104. 104.

    Newton  1994, p. 2.

  105. 105.

    Newton  1994, p. 28.

  106. 106.

    Newton  1994, p. 30.

  107. 107.

    Newton  1994, pp. 80 and 110.

  108. 108.

    Newton  1994, pp. 51, 67 and 73. Mamiani  1994a, pp. xiv, xxxvi–xxxix and 256. Westfall 1989, pp. 341–342.

  109. 109.

    Newton  1994, pp. 120–122.

  110. 110.

    Newton  1994, p. 218.

  111. 111.

    Mamiani  1994a, pp. xxxi–xxxv.

  112. 112.

    Gouk  1986, p. 53. Westfall 1989 , pp. 324–348, 359–371 and 848–873.

  113. 113.

    Mamiani  1994b.

  114. 114.

    Westfall 1981. Westfall 1989 , pp. 254–265, 285–288, 397–407, 419–…and passim.

  115. 115.

    Gouk 1980, p. 586.

  116. 116.

    Gouk 1980, pp. 601 and 605.

  117. 117.

    Gouk  1980, p. 583.

  118. 118.

    Gouk 1999, pp. 201–202 and 212–213.

  119. 119.

    Gouk 1999 , pp. 202–207, 136.

  120. 120.

    Aubrey  1950, p. 165. Gouk 1980, p. 578. Gouk 1999 , pp. 199 and 208. See below, Sect. 12.2. Boyle’s musical interests are described in Wardhaugh 2008, pp. 119–120.

  121. 121.

    Gouk 1980, pp. 586–587 and 589. The experiments with musical sound performed by Hooke are narrated with interesting details in Wardhaugh  2008, pp. 98–118.

  122. 122.

    Aubrey 1950, pp. 166–167. Westfall 1981, pp. 484–486. Gouk 1999 , p. 216 and passim.

  123. 123.

    Westfall 1981, 483. Westfall 1989 , pp. 466–473, 480–483, 486–489 and 536. Shadwell  1927.

  124. 124.

    Westfall 1989, pp. 496–515, 629–635, “Chapter thirteen”, 857–858.

  125. 125.

    Needham 1959, III, pp. 450–454. Martzloff  1988, pp. 25–26.

  126. 126.

    Hooke 1686, p. 37.

  127. 127.

    Hooke 1686, p. 79.

  128. 128.

    Hooke 1686, p. 80.

  129. 129.

    Hooke 1686, pp. 73–75.

  130. 130.

    Hooke 1686, pp. 65 and 67.

  131. 131.

    Westfall 1989, pp. 877–880. The Oxford Companion To Ships and the Sea 1979, p. 376.

  132. 132.

    Westfall 1981. Noble 1994, p. 287. Westfall 1989 , pp. 62, 201, 360–361, 561 e 630–634. Cf. Koyré  1972, pp. 198–203 and 245–289. Arnold  1996.

  133. 133.

    Koyré  1972, pp. 7–8 and 138–147. See here, Part I, Sect. 2.2 and Part II Sect. 10.1. The Englishman Benjamin Wardhaugh is to be recalled, not only for some interesting details about the context of the United Kingdom in the seventeenth century, but also, unfortunately, for some modest omissions. Among other things, these would concern even his famous fellow-countryman, Newton , who had left us the scholium on music, written, in fact, for the Principia ; Wardhaugh 2008, pp. 122–125. Cf. Casini  1981; 1984; 1994; Tonietti  2000b. He finally records, it is true, Newton ’s “…erroneous statement of the relationship between the tension of a string and its pitch.” But he has forgotten to reveal in which text it is found. Nor could the reader, worried about this singular lapsus that afflicted this gigantic English figure, satisfy his curiosity by searching in the secondary literature quoted by him. Thus Newton has unintentionally presented us with another Pythagorean mystery.

  134. 134.

    Private communication of Patricia Radelet de Grave , 29.09.2009. But the opinion of this editress is denied by Daniel Bernoulli in the following Sect. 11.1.

  135. 135.

    Wigner  1967.

  136. 136.

    Berkeley  1971, pp. 77, 81, 85, 87, 97 and 99.

  137. 137.

    Berkeley  1971, pp. 68, 93–95, 99 and 100.

  138. 138.

    Berkeley  1971, pp. 73, 89 and 96.

  139. 139.

    Boyer  1990, pp. 477–506.

  140. 140.

    Koyré  1972, pp. 68–70.

  141. 141.

    Barbour  1951, p. 125. Wardhaugh 2008, pp. 47-, 152–154.

  142. 142.

    Whiteside 1981, p. 310.

  143. 143.

    Section 9.3.

  144. 144.

    Barbour  1951, pp. 114, 122 and 125–128.

  145. 145.

    Dostrovsky  1975, pp. 201–204 and 209–218.

  146. 146.

    See above, Sect. 9.2.

  147. 147.

    Dostrovsky  1975, pp. 204–209; Dostrovsky  1981a, pp. 127–129.

  148. 148.

    Section 9.3. Tannery 1915.

  149. 149.

    Barbour  1951, pp. 193, 123–124 and passim. Tonietti  2006b.

  150. 150.

    Rasch 2004.

  151. 151.

    Crombie  1990, pp. 365, 368, 8, 113, 330. Here, Part I, Sect. 6.7, Chaps. 3–4 and Sect. 5.2.

  152. 152.

    See Part II, Sect. 9.2, at the end.

  153. 153.

    Dostrovsky  1975, pp. 180, 211. McGuire & Rattansi  1966.

  154. 154.

    See above, Sect. 10.2.

  155. 155.

    Kassler 1982, pp. 139, 124 and passim.

  156. 156.

    Gozza 1989, pp. 5–6, 21, 34, 40, 64 and passim.

  157. 157.

    Cohen  1984, passim. Coelho  1992.

  158. 158.

    Kuhn 1985, pp. 42–43.

  159. 159.

    Asinari 1997.

  160. 160.

    Fabbri 2008, pp. 27, 51, 64 and passim.

  161. 161.

    Fabbri 2008, pp. 86–87, 91, 100–103, 116, 123, 156 and passim. See here, Sect. 9.2.

  162. 162.

    Palisca  1989. Palisca 1989a. Palisca  2000. Fabbri 2008, pp. 170–173, 198–199, 220, 222 and passim.

  163. 163.

    Tonietti  1999a.

  164. 164.

    Fabbri 2008, pp. 256–257 and passim.

  165. 165.

    Funkenstein  1996.

  166. 166.

    Funkenstein  1996, p. 86 and passim.

  167. 167.

    Funkenstein  1996, pp. 239, 379 and passim.

  168. 168.

    Funkenstein  1996, pp. 409, 273, 403–404, 230 e passim.

  169. 169.

    See Part I, Sect. 2.4.

  170. 170.

    Funkenstein  1996, pp. 359–361, 267, 169, and passim. However, he admits: “I do not possess the competence.”

  171. 171.

    Funkenstein  1996, pp. 387–388, 433–434 and passim.

  172. 172.

    Funkenstein  1996, pp. 181, 415 and 229.

  173. 173.

    Noble 1994, pp. 12–13, 205 and passim. Donini  1990.

  174. 174.

    Noble 1994, pp. 35–36, 181–185, 287–298 and passim. Abelardo & Eloisa 2008.

  175. 175.

    Noble 1994, pp. 297–298. Though Noble’s was an essay that hardly dealt with the mathematical sciences at all, and too little with natural philosophers, but succeeded well in communicating to us the religious cultural context, an attribute less than unappropriate regarding Newton was inadvertently expressed here: “implicit naturalistic deism”. Instead, the English astronomer had been a heretic Arian theologian, who had self-censured himself out of convenience, and the last of the Christian prophets, as Noble, too, wrote. Noble 1994, pp. 335 and 290.

  176. 176.

    Noble 1994, pp. 74–76, 130, 188 and passim.

  177. 177.

    Kish  1981.

  178. 178.

    Westfall 1989, pp. 201–202 and 422.

  179. 179.

    Newton  1959, v. I, pp. 9–11.

  180. 180.

    Hessen 1971, pp. 151, 171–176 and passim. See above, Sect. 10.2.

  181. 181.

    Hessen 1971, pp. 188, 182, 191, 176 and passim.

  182. 182.

    Hessen 1971, pp. 211, 200 and passim.

  183. 183.

    Hessen 1971, p. 203.

  184. 184.

    Hessen 1971, pp. viii–ix.

  185. 185.

    Koestler 1991, pp. 417–493, 479 and passim. In a hastily added note, the polyglot novelist and essayist reported that the Jesuits had brought “…a Copernican astronomy …” to China. We know from Sect. 8.2 that things went very differently.

  186. 186.

    Koestler 1991, pp. 493, 521, 523–525, 537 and passim.

  187. 187.

    Lovejoy 1966, p. 66 e passim.

  188. 188.

    Lovejoy 1966, pp. 118–127, 137–140, 148, 207, and passim.

  189. 189.

    Lovejoy 1966, pp. 153, 156–160, and passim.

  190. 190.

    Lovejoy 1966, pp. 351–352, 356.

  191. 191.

    Leibniz  1960/65, v. 5, p. 313.

  192. 192.

    Leibniz  1967, v. I, p. 310.

  193. 193.

    Westfall 1989, p. 322.

  194. 194.

    Koyré  1972, p. 172.

  195. 195.

    Koyré  1972, p. 14.

  196. 196.

    Lovejoy 1966, pp. 312–313.

  197. 197.

    Leibniz  1960, v. III, p. 346.

  198. 198.

    Newton  1687/1972, p. 549.

  199. 199.

    Newton  2006, p. 75.

  200. 200.

    Noble 1994, p. 274.

  201. 201.

    Noble 1994, p. 112.

  202. 202.

    Lovejoy 1966, p. 186.

  203. 203.

    Newton  2006, p. 96.

  204. 204.

    Funkenstein  1996, p. 350.

  205. 205.

    Funkenstein  1996, p. 340.

  206. 206.

    Leibniz  1976, v. I, pp. 670, 363 and 372.

  207. 207.

    Newton  1994, pp. 36–37.

  208. 208.

    Newton  1959, I, pp. 94 and 104. Newton  2006, p. 195. Francis Bacon had written: “…examples of a cross, metaphorically taken from the crosses placed at crossroads, to indicate the fork in the road.” Bacon 1965, v. I, p. 422. Koyré  1972, p. 46.

  209. 209.

    Westfall 1989, p. 313.

  210. 210.

    Lovejoy 1966, pp. 57, 99, 104 and passim.

  211. 211.

    Eco 1993, p. 373 and passim.

  212. 212.

    Chomsky  1969.

  213. 213.

    See Part I, Fig. 3.7.

  214. 214.

    Eco 1993, pp. 305–308 and passim. We await with curiosity the so-called quantum computers.

  215. 215.

    Leibniz  1960/65, v. 6, p. 179. Lovejoy 1966, p. 241.

  216. 216.

    Mila  1963, pp. 105–138, 163–178, 177, passim.

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    Tito M. Tonietti

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Tonietti, T.M. (2014). Between Latin, French, English and German: The Language of Transcendence. In: And Yet It Is Heard. Science Networks. Historical Studies, vol 47. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0675-6_4

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