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Hobbes’s model of refraction and derivation of the sine law

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This paper aims both to tackle the technical issue of deciphering Hobbes’s derivation of the sine law of refraction and to throw some light to the broader issue of Hobbes’s mechanical philosophy. I start by recapitulating the polemics between Hobbes and Descartes concerning Descartes’ optics. I argue that, first, Hobbes’s criticisms do expose certain shortcomings of Descartes’ optics which presupposes a twofold distinction between real motion and inclination to motion, and between motion itself and determination of motion; second, Hobbes’s optical theory presented in Tractatus Opticus I constitutes a more economical alternative, which eliminates the twofold distinction and only admits actual local motion, and Hobbes’s derivation of the sine law presented therein, which I call “the early model” and which was retained in Tractatus Opticus II and First Draught, is mathematically consistent and physically meaningful. These two points give Hobbes’s early optics some theoretical advantage over that of Descartes. However, an issue that has baffled commentators is that, in De Corpore Hobbes’s derivation of the sine law seems to be completely different from that presented in his earlier works, furthermore, it does not make any intuitive sense. I argue that the derivation of the sine law in De Corpore does make sense mathematically if we read it as a simplification of the early model, and Hobbes has already hinted toward it in the last proposition of Tractatus Opticus I. But now the question becomes, why does Hobbes take himself to be entitled to present this simplified, seemingly question-begging form without having presented all the previous results? My conjecture is that the switch from the early model to the late model is symptomatic of Hobbes’s changing views on the relation between physics and mathematics.

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Notes

  1. In what follows, I will cite the titles of the texts as indicated in this chart. A detailed bibliography is provided at the end of this article.

  2. Although Hobbes’s original letter has been lost, Descartes’ quotations in his reply correspond very closely to the published TO I (Brandt 1928, pp. 94–96).

  3. For a summary of views, see Giudice (1999, pp. 12–13), note 45.

  4. Raylor (2005) and Malcolm (2005). Thanks to the reviewer for pointing me to these articles.

  5. I think there is also another reason to date TO II before Anti-White which no commentators have hitherto noticed, that is, inTO II Hobbes falsely posited that under the diastole-systole model of the generation of light, the velocity of the propagation of light should decrease with the distance to the center of the lucid body in a relation based on square roots rather than cubic roots, a glitch that was later fixed in Anti-White. Cf. TO II, I. 6–7, p. 149; Anti-White, IX. 4, pp. 162–163.

  6. See Giudice (1999) for a discussion on the connection between optics and physics/physiology, and Malet (2001) for a discussion on the place of vision in Hobbes’s philosophical system.

  7. For an interpretation of some of the early texts, see Sabra (1981) and Schuster (2000).

  8. AT X, p. 336. Here Descartes has not explained exactly how this analogy with the balance can help to determine the relation between the incident angle and the refracted angle. An earlier text that deals with refraction was written around 1619–1621, but there Descartes only gives the rough estimate that when the light travels from a rarer to a denser medium, it would be refracted toward the normal and vice versa (AT X, pp. 242–243). Throughout this paper, AT refers to Descartes (1964–1974) and the Roman numeral refers to its volume number.

  9. “Introduction,” Oeuvres completes III (Descartes 2009), pp. 18–19.

  10. AT VI, p. 100.

  11. AT VI, pp. 87–88; also see chap. 13 of Le monde for a detailed description of light as the centrifugal tendency of the second element. (AT XI, pp. 84–97; esp. p. 86 for the explanation of the centrifugal tendency).

  12. AT VI, pp. 99–100.

  13. Sabra (1981, pp. 109–111).

  14. “Tendency” corresponds to the Latin verb “tendere” (in French, tendre), which is equivalent to conatus (cf. Principia, III. 57, AT VIII, pp. 108–109), or “endeavor.” Conatus was also translated into French as effort, or inclination (Principia III. 56–57, AT IX B, p. 131; cf. Le monde XIII, AT XI, p. 84).

  15. Principia III. 55, AT VIII, p. 108. For a lucid exposition of Descartes’ theory of light in comparison with the modern motion of pressure, see Shapiro (1974). Descartes’ theory of light is perhaps proposed to accommodate light, whose transmission is instantaneous and in straight lines, in a plenum universe where actual motions usually occur in circles. Thanks to the reviewer for pointing out the importance of this issue and its relevance to Hobbes’s polemics with Descartes.

  16. AT VI, p. 89.

  17. AT I, p. 450. This is Descartes’ quotation of Fermat’s original letter.

  18. AT I, p. 451.

  19. Costabel (1967, p. 237).

  20. Descartes to Clerselier, February 17, 1645 (AT IV, p. 185). Also see Descartes to Mersenne, April 26, 1643 (AT III, p. 650).

  21. On the use of French word motion, see Gabbey (1980, pp. 311–312), note. 125.

  22. On the objection of Fermat, see Sabra (1981, pp. 116–136); on the objection of Roberval, see Garber (1992, p. 186).

  23. Gabbey (1980, p. 258); Knudsen and Pedersen (1969, p. 185). Also see Garber (1992, pp. 188–193).

  24. TO I, Hypothesis I, OL V, p. 217.

  25. TO II, I. 8, p. 150.

  26. TO II, I. 10, p. 152. On Hobbes’s critique of Descartes in TO II, especially concerning the rejection of inclination, see Bernhardt (1979, pp. 435–437).

  27. This is the second analogy in the First Discourse of Dioptrique.

  28. TO II, IV. 13, p. 207.

  29. Bernhardt (1977, p. 16), note. 34.

  30. The following passage from TO II is interesting: “Descartes himself, in order to explain the action of light, always employs words signifying actual motion and he talks about (in refraction) the very [here I read easque rather than eamque] laws of the motion of the ball.” TO II, I. 10, p. 152. Thanks to the reviewer for directing me to relevant passages.

  31. For a discussion on the Cartesian conatus as merely counterfactual, see Garber (1992, pp. 219–223, 1994).

  32. For Hobbes, see De Corpore, XV. 1, OL I, p. 175; for Descartes, see Garber’s discussion in Garber (1992, pp. 182–185).

  33. Correspondence, Letter 34, p. 103; translation of Noel Malcolm, pp. 108–109.

  34. Correspondence, Letter 36, p. 117.

  35. Gabbey (1980, p. 259).

  36. Correspondence, Letter 34, p. 103; translation of Noel Malcolm, p. 108.

  37. Correspondence, Letter 30, p. 69.

  38. Ibid.

  39. For general backgrounds of Hobbes’s natural philosophy, see Garber (2016).

  40. Sabra (1981, p. 117).

  41. Schuster (2000, p. 267). Schuster later in the same article argues that Descartes in a way reverse-engineered the physical explanation from an already obtain mathematical result.

  42. TO I, OL V, pp. 221–223; TO II, II. 2, p. 160; FD, pp. 122–124. The description of line of light as a wave front is due to Shapiro (1973), in which he argues that Hobbes’s theory is a forerunner of the subsequent wave theory of light.

  43. Hobbes himself did not set out these rules in a systematic way. I summarized them from his scattered remarks as well as principles at work during his actual derivation.

  44. TO I, Postulatum, OL V, p. 226.

  45. TO II, II. 6, p. 163.

  46. TO I, ibid. This explanation can be traced back to Alhazen, who thinks that the light perpendicular to the surface has the greatest power hence could not be refracted, and adduces several analogies to explain this, e.g., a sword striking an armor perpendicularly can often break through the armor, but when striking obliquely often fails to do so. Cf. Lindberg (1968, pp. 26–27).

  47. TO I, OL V, p. 223.

  48. TO II, II. 3 & II. 7, pp. 161, 163.

  49. FD, pt. 1, chap. 4, p. 124.

  50. TO II, II. 10, pp. 165–166; TO I, Prop. VI, OL V, p. 227. This is in fact a more traditional view compared to that of Descartes, cf. Lindberg (1968) and Sabra (1981, pp. 98–99).

  51. In the Elements there are two kinds of propositions, theorem and problem. The former takes the form of an indicative statement, while the latter is expressed in infinitive with imperative meaning. The goal of problem is to construct a figure that satisfies certain conditions. Cf. Beere and Morison, “A Mathematical Form of Knowing How: The Nature of Problems in Euclid’s Geometry.”

  52. My reconstruction is largely indebted to Shapiro’s (Shapiro 1973, pp. 258–263), though he did not explicitly point out the three rules assumed by Hobbes’s model and Hobbes’s justifications for them.

  53. TO I, Prop VI, OL V, p. 227; TO II, II. 10, p. 166.

  54. Ibid.

  55. TO II, II. 10, p. 166; FD, p. 126a.

  56. Cf. TO I, OL V, p. 223.

  57. Hobbes calls “angle refracted [angulus refractus]” what we usually understand to be the angle of refraction, i.e., the angle formed by the refracted light and the normal of the surface (cf. De Corpore, XXIV, Def. V, OL I, pp. 305–306). But this is simply a terminological issue, and I shall stick to what is more in keeping with our usage.

  58. Hobbes only used this form of the model to demonstrate the much weaker claim that the angle of incidence or refraction is smaller on the side of denser medium (TO I, OL V, p. 224; FD, pp. 125–126a), but did not demonstrate the sine law based on it. However, as we have seen, it is not hard to prove the sine law under this form of the model, and the form of the model that Hobbes actually used to prove to sine law was only a simplified form based on the face that LT = IS. Cf. TO II, II. 10, pp. 166–167 and Shapiro’s reconstruction of it in Shapiro (1973, p. 260), note 410.

  59. Shapiro (1973, p. 172); Horstmann (2000, pp. 437–439).

  60. De Corpore, XV. 2, OL I, p. 177.

  61. Hobbes stipulates that the surface itself belong to the lower medium.

  62. De Corpore, XXIV. 4, OL I, pp. 309–310. Translation based on Concerning Body, EW I, pp. 379–380.

  63. De Corpore, XVI. 1, OL I, p. 185. Examples can be found throughout Part III, but one can get a taste of it by taking a look at the following article, XVI. 2, where Hobbes discusses uniform motion.

  64. De Corpore, VII. 3, OL I, p. 85.

  65. De Corpore, XV. 1, OL I, p. 176.

  66. De Corpore, VIII. 16, OL I, p. 101.

  67. Such as the motion of a horologe, ibid.

  68. See note 58.

  69. TO I, OL V, p. 225.

  70. TO I, OL V, p. 225 and FD, pp. 126j–126m; TO II, II. 10, p. 166.

  71. TO I, OL V, p. 239, last line and p. 240, l. 2, 14, etc.

  72. TO I, OL V, p. 240, l. 21. It should be noted that here the length of the line of light is not fixed, since if A were at another place, AB would have been different. Therefore, here the line of light is only a temporary device to determine the direction of CK and is not an actual part of the incident and refracted rays. Shapiro has taken the line of light as a given and hence has not noticed that here only AC and CK are the rays proper (Shapiro 1973, p. 262).

  73. For some statements about mixed mathematics in Aristotle, see Posterior Analytics, I.7, 75b13–20; I.9, 76a10–15; I.13, 78b34–79a16.

  74. For a clear exposition of the difference between physics and the mathematical sciences in Aristotle, see Lennox (1986) and Biener (2008), chap. 2.

  75. For an overview on Galileo’s influence on Hobbes, see Giudice (2016) and Baldin (2020).

  76. Opere VI, p. 232. Opere refers to Galilei (1890-1909) and the Roman numeral refers to its volume number.

  77. See, e.g., Salviati’s claim that the cause of gravity cannot be known for certain in the Second Day of Dialogo, Opere VII, pp. 260–261, and Salviati’s similar claim in the Third Day of Discorsi that the causes of accelerated motion are largely irrelevant to the mathematical investigations on the properties (passiones) of such motion, Opere VIII, p. 202. Galileo also said in two letters that he only wanted to investigate the sintomi of motion (Opere X, pp. 351–352; XVIII, p. 12), and sintomi here roughly denote the characteristics and properties of motion in contrast with its generation and true nature.

  78. Giudice (2016), sec. 4.

  79. See the opening paragraph of TO II, p. 147. There Hobbes contrasts a posteriori physics with a priori mathematics, but he does not express the view that the hypotheses that physics must make are made from mathematics.

  80. On this point see Jesseph (1999), chap. 3.

  81. Thanks to the reviewer for making me reconsider the issue.

  82. De Corpore, VI. 6, OL I, pp. 62–63. Besides, the chapters on the classification of the sciences in the English Leviathan (1651) and its Latin translation (1668) also differ with respect to the status of geometry. The Latin Leviathan is more in accord with the view in De Corpore, while in the English Leviathan we do not see a definitive statement of the prominent status of geometry in Hobbes’s system (cf. Curley’s edition, pp. 47–50).

  83. Notes on an early draft of De Corpore are published by the editors of Anti-White as Appendix III of the volume. The early version of the methodological Chapter VI can be found in pp. 472–473, which curiously does not mention geometry or mathematics. As the editors of the volume remark, these notes do not seem to have been written continuously, but rather roughly span the period between 1645 and 1649 (pp. 83–84).

  84. We could say that Hobbesian geometry (or geometrical physics) is arbitrary because it does not need to rely on any physical premise about how bodies are supposed to behave but is only concerned with whether our geometrical construction could be carried through. Arbitrariness in this sense is different from the arbitrariness of conventional definitions, which is also a much-discussed topic in Hobbesian studies. As for a refutation of the view that the Hobbesian science is not arbitrary in the latter sense, see Jesseph (2010).

  85. Giudice has observed this fact in Giudice (2016, p. 102).

  86. Hobbes’s derivation of the quantitative relation between the quickness of light and the distance from its lucid origin on the basis of his early model of the generation of light also constitutes an example for this more traditional kind of mixed mathematics: one first assumes a specific physical process—in this case the diastole and systole of the star, and then delineates and derives its quantitative properties (for texts in TO II and Anti-White, see note 5). Galileo’s studies of various natural phenomena, e.g., his study of uniformly accelerated motion in the Third Day of Discorsi, all roughly follow this approach. As mentioned above, Galileo deliberately avoids entering into discussions of the real causes of these phenomena, but rests content with attaining quantitative results that best fit with our experiments.

  87. De Corpore, VI. 6. And in the earlier draft of De Corpore cited in note 83 we do not find such a statement.

  88. Anti-White, XXIII. 18, pp. 283–284.

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Acknowledgements

I would like to thank Professor Dan Garber for his cordial help in getting this paper in its proper shape. This paper was presented in the 2019 Princeton-Humboldt Graduate Conference, and I thank the audience and Éve-Lyne Perron who commented on it. I also thank the reviewer and the editor for their helpful comments.

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    Hao Dong

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Dong, H. Hobbes’s model of refraction and derivation of the sine law. Arch. Hist. Exact Sci. 75, 323–348 (2021). https://doi.org/10.1007/s00407-020-00265-w

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