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New Forms of Natural Philosophy and Mixed Mathematics

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Book cover The Path to Post-Galilean Epistemology

Part of the book series: History of Mechanism and Machine Science ((HMMS,volume 34))

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Abstract

Between the sixteenth and seventeenth centuries natural philosophy underwent great changes. Formal and final causes were replaced by efficient causes and the world became a huge machine. Greek philosophy turned toward mechanistic philosophy, a kind of philosophy more easily digestible by nonprofessional philosophers, mathematicians in particular. This change occurred in a period where specializations were not pushed away as today. Many of those who were called mathematicians could actually have a solid background in the philosophy of nature and sometimes even in metaphysics and theology. On the other hand, those who were called philosophers also had in general a mathematical preparation, not always deep but in many cases not negligible. The dichotomy, mathematician (philosopher)-philosopher (mathematician), does not cover the entire industry that today is associated with scientific knowledge. There were also physicians, alchemists, natural magicians, and educated technicians. However, they also had a nonspecialized culture and moved between natural philosophy and mathematics. All of them could so easily take possession of themes of the new philosophy of nature and integrate it into the already waiting conceptual framework of the mixed mathematics.

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Notes

  1. 1.

    p. 84.

  2. 2.

    Regarding the possibility of using a unique label (Aristotelians) to identify philosophers variously connected to Aristotle see [83, 224, 225].

  3. 3.

    pp. 135–138.

  4. 4.

    p. 210

  5. 5.

    p. 83.

  6. 6.

    p. 60.

  7. 7.

    p. 392.

  8. 8.

    p. 106.

  9. 9.

    p. 32.

  10. 10.

    p. 120.

  11. 11.

    f. 108 v.

  12. 12.

    p. 605.

  13. 13.

    p. 101.

  14. 14.

    pp. 96–97

  15. 15.

    p. 98.

  16. 16.

    p. 177.

  17. 17.

    pp. 238–239.

  18. 18.

    Liber II, Lectio 3, pp. 282, 284.

  19. 19.

    pp. 46–47.

  20. 20.

    pp. 21–22.

  21. 21.

    pp. 187–242.

  22. 22.

    p. 61.

  23. 23.

    p. 9.

  24. 24.

    pp. 331–332.

  25. 25.

    vol. 1, p. 275.

  26. 26.

    p. 330.

  27. 27.

    One possible interpretation, using a modern notation, is shown below:

    $$ \begin{aligned} (a:b) \,{:}{:}\, (c:d) \leftrightarrow \\ \forall m \forall n( ma\succ nb\rightarrow m c \succ nd \& \\ ma\simeq nb \rightarrow m c \simeq nd \& \\ ma\prec nb \rightarrow m c \prec nd) \end{aligned}$$

    where \(\succ , \prec , \simeq \), generalize the concept of \(>, <, =\) respectively, from numbers to geometrical magnitudes.

  28. 28.

    p. 107. Translation in [180].

  29. 29.

    p. 107. Comment to Def. 5.

  30. 30.

    p. 5.

  31. 31.

    pp. 24–28

  32. 32.

    pp. 55–56.

  33. 33.

    pp. 1–4,

  34. 34.

    pp. 61–78.

  35. 35.

    p. 352.

  36. 36.

    p. 362.

  37. 37.

    p. 360.

  38. 38.

    p. 191.

  39. 39.

    pp. 191–192.

  40. 40.

    p. 194.

  41. 41.

    p. 175.

  42. 42.

    p. 154.

  43. 43.

    pp. 81–84.

  44. 44.

    pp. 81–84.

  45. 45.

    vol. 1, p. 226.

  46. 46.

    pp. 108–109.

  47. 47.

    p. 1.

  48. 48.

    p. 1.

  49. 49.

    p. 51.

  50. 50.

    Preface. Translation in [77].

  51. 51.

    Letter of October 9th 1580. There is a reference to this experience also in [59], p. 29 r.

  52. 52.

    Letter of December 18th, 1580.

  53. 53.

    pp. 1808–115.

  54. 54.

    p. 236. Transcribed in [150], vol. IV, pp. 387–398.

  55. 55.

    p. 25v.

  56. 56.

    p. 78v–79r. Translation in [77].

  57. 57.

    pp. 5–6.

  58. 58.

    Uniformly heavy body is defined by Tartaglia as a bodies that is sufficiently compact to suffer only a negligible air resistance. In this way the heaviness of a body, intended not in the absolute sense but as characterized by a downward tendency, remains uniform (in time) along the trajectory. Indeed a science of motion can only exist if inessential accidents are not accounted for, such as the irregular variation of air resistance. However, Tartaglia did not limit himself to a definition. He also stated that uniformly heavy bodies exist (this is an implicit supposition) and furnishes the characteristics they should have: made up of matter that is sufficiently compact (such as stone or iron) and have a spherical shape [243], p. 1r.

  59. 59.

    p. 3r.

  60. 60.

    p. 3v. Translation in [77].

  61. 61.

    pp. 557.

  62. 62.

    pp. 78–79.

  63. 63.

    pp. 3r–3v. Translation in [77].

  64. 64.

    pp. 80–81.

  65. 65.

    pp. 11r–11v.

  66. 66.

    p. 16r.

  67. 67.

    pp. 18v–19r.

  68. 68.

    p. 28v.

  69. 69.

    p. 92.

  70. 70.

    pp. 247–248.

  71. 71.

    p. 238.

  72. 72.

    p. 552.

  73. 73.

    p. 316.

  74. 74.

    p. 165.

  75. 75.

    p. 495.

  76. 76.

    p. 482.

  77. 77.

    p. 7.

  78. 78.

    In music the sequence of three numbers \(n_1, n_2, n_3\) is called harmonic if \(n_2\) is the harmonic mean of \(n_1\) and \(n_3\), that is, if \(\frac{1}{n_2}= \frac{1}{2}\left( \frac{1}{n_1}+\frac{1}{n_3} \right) \).

  79. 79.

    p. 547.

  80. 80.

    p. 481.

  81. 81.

    487.

  82. 82.

    Many ancient scholars referred to different account about Pythagoras’ attributions. Among them Nicomachus (first century AD), Theon of Smyrna (second century AD), Boethius (c 480–524) [74], pp. 172–173. Most of the observations reported by them, as the tale of the hammers, could not have been made, and Pythagoras’s discoveries sound like a legend. What is however valuable about ancient witnesses is the importance of ratios and arithmetics for music.

  83. 83.

    pp. 103–105.

  84. 84.

    Here there are problems of interpretation. Indeed the variation of volume could be due either to variation in diameter and/or in length. But the pitches of pipes do not depend on the diameter.

  85. 85.

    p. 145.

  86. 86.

    p. 147.

  87. 87.

    De intervalli musicis, pp. 277–283.

  88. 88.

    p. 283. Translation in [191], adapted.

  89. 89.

    It might be thought that he had been influenced by the Problemata of Aristotle , where Problem 35b of Book 19 made considerations on the frequency of vibration of strings [12], vol. 1, 920b.

  90. 90.

    pp. 76–77.

  91. 91.

    p. 283. Translation in [191].

  92. 92.

    p. 26.

  93. 93.

    pp. 157–158.

  94. 94.

    vol. 1, pp. 54–55.

  95. 95.

    vol. 1, p. 249. Translation in [249], adapted.

  96. 96.

    vol. 1, p. 92.

  97. 97.

    pp. 126–127.

  98. 98.

    vol. 1, p. 160.

  99. 99.

    vol. 4, pp. 206–207. Translation in [48], adapted.

  100. 100.

    p. 22. Clifford Ambrose Truesdell states that Fracastoro’s writings do not read like a work of an originator, and conjecture that future studies of medieval sources could reveal a considerable knowledge of acoustics. Moreover, Truesdell claims that Fracastoro had the idea that sound was a vibratory motion of definite frequency [249], pp. 22–23; but on this point Truesdell most probably attributed too much to Fracastoro.

  101. 101.

    vol. 4, pp. 206. Translation in [48].

  102. 102.

    This is a result of the modern theory of vibrating strings, considering that some form of damping is always present.

  103. 103.

    Second partie. Livre troisiesme des instrumens à chordes, pp. 123–125.

  104. 104.

    Second partie. Livre troisiesme des instrumens à chordes, p. 123. Mersenne’s rules translated into modern algebraic language are equivalent to the relation:

    $$ f\propto \frac{1}{l} \sqrt{\frac{F}{A}}$$

    where f the frequency of vibration, L the length, F the force of traction (the weight attached to the string according to Mersenne language), and A the cross-section area of the string. The ratio F / A is the stress (modern meaning) of the string. Note that the frequency is defined less a constant of proportionality.

  105. 105.

    Second partie. Livre troisiesme des instrumens à chordes, pp. 123–125.

  106. 106.

    Using the rule of the parallelogram of forces it is not difficult to show – this is true for a modern student but was also true for Mersenne’s contemporaries – that for strings of equal lengths and forces of tension, the transverse force necessary to display their middle points is directly proportional to the displacement (actually only when angles in a and b (see Fig. 3.7) can be assimilated to sines).

  107. 107.

    part 1, Livre troisieme du mouvement, p. 157.

  108. 108.

    part 1, Livre troisieme du mouvement, p. 158.

  109. 109.

    Premiere partie, Livre troisieme du mouvement, p. 158.

  110. 110.

    p. 145. Translation in [74].

  111. 111.

    1 pound is 16 ounces.

  112. 112.

    Second partie. Livre troisiesme des instrumens à chordes, p. 123.

  113. 113.

    p. 429.

  114. 114.

    p. 187.

  115. 115.

    Premiere partie. Livre troisieme du mouvement, p. 169.

  116. 116.

    Premiere partie. Livre troisieme du mouvement, p. 172.

  117. 117.

    XIX, 918b.

  118. 118.

    Second partie. Livre quatriesme des instrumens, p. 210.

  119. 119.

    p. 129

  120. 120.

    pp. 118–120

  121. 121.

    p. 299.

  122. 122.

    p. 349.

  123. 123.

    p. 351–352.

  124. 124.

    Second partie. Livre sexieme des orgues, p. 362.

  125. 125.

    pp. 205–206.

  126. 126.

    pp. 119–140.

  127. 127.

    p. 3.

  128. 128.

    p. 120.

  129. 129.

    Grosseteste also wrote some technical works on optics, such as De iride, De lineis angulis et figuris.

  130. 130.

    p. 214.

  131. 131.

    pp. 1–167.

  132. 132.

    pp. 16–17.

  133. 133.

    p. 61. Translation in [257].

  134. 134.

    pp. 61–65. Translation in [257].

  135. 135.

    p. 63. Translation in [257].

  136. 136.

    p. 66. Translation in [257].

  137. 137.

    p. 66.

  138. 138.

    p. 91.

  139. 139.

    The previous description is nearly verbatim derived from [235], pp. 186–187.

  140. 140.

    pp. 92, 93. Translation in [257].

  141. 141.

    pp. 215–218.

  142. 142.

    p. 94.

  143. 143.

    pp. 323–330.

  144. 144.

    p. 5. Translation in [135].

  145. 145.

    Not numbered page. Summary of part III, Chap. 32.

  146. 146.

    p. 6.

  147. 147.

    p. 80.

  148. 148.

    pp. 19–20.

  149. 149.

    p 56.

  150. 150.

    p. 43. Translation in [135].

  151. 151.

    p. 45. Translation in [135].

  152. 152.

    p. 48. Translation in [135].

  153. 153.

    pp. 171–172.

  154. 154.

    p. 180.

  155. 155.

    p. 193. Translation in [135].

  156. 156.

    p. 191.

  157. 157.

    p. 213.

  158. 158.

    p. 196. Translation in [135].

  159. 159.

    pp. 196–197.

  160. 160.

    p. 199.

  161. 161.

    p. 349.

  162. 162.

    p. 3.

  163. 163.

    p. 53.

  164. 164.

    p. 56.

  165. 165.

    p. 79.

  166. 166.

    vol. 1, p. 376.

  167. 167.

    vol. 15, p. 146.

  168. 168.

    p. 12, note 2.

  169. 169.

    p. 197.

  170. 170.

    vol. 1, p. 356.

  171. 171.

    p. 74.

  172. 172.

    vol. 3, pp. 15–16.

  173. 173.

    vol. 3, p. 16; p. 29.

  174. 174.

    vol. 1, pp. 355–356.

  175. 175.

    A longer, modern, list could also include, before Boyle : Bacon , Beeckman, Galileo , Mersenne , Charlton, Digby, More , Hobbes , Cordemoy, La Forge, Malebranche, Regis, Rohault, Huygens , and Spinoza . After Boyle , Locke , Hooke , Leibniz , and Newton [119], p. 71.

  176. 176.

    p. 10.

  177. 177.

    Aristotele Physica, 1.4, 187b; Aristotele De generatione et corruptione, 327b.

  178. 178.

    pp. 155–159.

  179. 179.

    p. 570.

  180. 180.

    vol. 1, pp. 102–172.

  181. 181.

    vol. 1, p. 151.

  182. 182.

    vol. 1, p. 154. Translation in [169].

  183. 183.

    vol. 1, p. 157. Translation in [169].

  184. 184.

    p. 33.

  185. 185.

    vol. 1, p. 276. Translation in [159].

  186. 186.

    vol 1, p 268.

  187. 187.

    vol. 1, p. 366. Translation in [159].

  188. 188.

    vol 1, p 372 cols 1,2

  189. 189.

    p. 157.

  190. 190.

    vol. 1, pp. 229–282.

  191. 191.

    pp. 65–66.

  192. 192.

    vol 3, pp. 478–563.

  193. 193.

    vol 3, p. 497. Translation in [106].

  194. 194.

    vol 3, p 495.

  195. 195.

    p. 174.

  196. 196.

    p. 157.

  197. 197.

    vol 1, p 338. Translation in [106].

  198. 198.

    vol. 4, p. 41

  199. 199.

    p. 119.

  200. 200.

    vol. 1, p. 244.

  201. 201.

    vol. 4. pp. 52–55. See also [107].

  202. 202.

    p. 169.

  203. 203.

    p. 11.

  204. 204.

    pp. 80–81.

  205. 205.

    p. 88.

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  1. Ingegneria Strutturale e Geotecnica, Sapienza University of Rome, Rome, Italy

    Danilo Capecchi

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Capecchi, D. (2018). New Forms of Natural Philosophy and Mixed Mathematics. In: The Path to Post-Galilean Epistemology. History of Mechanism and Machine Science, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-58310-5_3

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