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Abstract

For many historians, the development of modern science is best explained in terms of the tension between two trends: the search for a mathematical treatment of phenomena, on the one hand, and the demand for mechanical explanations, on the other. Although the importance of Descartes’ Principia philosophiae for natural philosophy is beyond question, this work could have hardly been called Principia mathematica; for quantitative expressions are scarce and nearly no equation is to be found therein. This explains why Descartes is often depicted as the typical mechanical philosopher and contrasted with the founder of the science of mechanics, Galileo.1

Keywords

Incline Plane Heavy Body Simple Machine Mechanical Philosophy Infinite Degree 
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References

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    See the quotations given above, notes 22 to 25. On the medieval roots of these pejorative connotations, see Allard, ”Les Arts mécaniques aux yeux de l’idéologie médiévale On the meaning of ”mechanics“ in the seventeenth century, see Gabbey’s contribution to this volume.Google Scholar
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    In the Discours de la méthode, Descartes insists instead on the negative fact that the mixed sciences make one forget pure mathematics: ”je me plaisais surtout aux mathématiques, [ ... ] mais je ne remarquais point encore leur vrai usage, et pensant qu’elles ne servaient qu’aux arts mécaniques [...],“ Descartes, Oeuvres [Adam e.a.], vI, p. 7.Google Scholar
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    See Séris, ”Descartes et la mécanique,“ pp. 33, 36. According to Gabbey, ”Newton’s MathematicalPrinciples“ and to Gabbey, ”Between Ars and Philosophia Naturalis“ an explicit definition of mechanics emerges in the England of the late 166os thanks to Barrow, Wallis and Boyle. In France I have found it in Poisson and Pardies. Poisson writes: ” [ ... ] il faut prendre garde à ne pas se tromper touchant le mot de mécanique, qui ne signifie pas seulement cette science qui apprend à composer des machines, ou à en connaître les parties, mais sous ce mot on renferme aussi toutes les différentes manières dont un corps se meut pas rapport à certaines lois de la nature qu’on ne peut jamais contester,“ Descartes, Traité de la mécanique [Poisson], p. 18. Pardies, who might have been influenced by Wallis, whose Mechanica he read, writes in the preface of his Discours du mouvement local (1670): ”[ ... ] il n’est pas possible de pénétrer dans les secrets de la physique, ni de réussir dans l’invention et la pratique des arts, sans le secours des mécaniques, c’est-à-dire sans la connaissance des lois du mouvement,“ Pardies, Oeuvres, p. 135.Google Scholar
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    Le Monde, ch. 7, Descartes, Oeuvres [Adam e.a.], xI, p. 47.Google Scholar
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    I use the expression ”pseudo-atomical;” because Descartes refutes atomism. See Roux, ”Descartes atomiste?“ Not all Cartesian explanations are pseudo-atomical. The explanations of lightness and of heaviness, for example, do not rely on corpuscules, but on universal laws of motion.Google Scholar
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    In his Geostatice, Beaugrand claimed that his proposition had received the approbation of Galileo and Castelli (de Beaugrand, Geostatice, p. io), but the reality is somewhat less glorious. Indeed, Beaugrand mentioned his proposition in a letter to Galileo dated 3 November 1635 (Galilei, Le Opere [Favaro], xvi, pp. 336–337 and Mersenne, Correspondance [Tannery e.a.], y, pp. 454–456), but in his answer Galileo did not say a word about geostatics (11 November 1635, Galilei, Le Opere [Favaro], xvi, pp. 340 ff. ). At the end of November 1635, Beaugrand exposed the geostatical proposition to Castelli, who found a demonstration for it and gave to Beaugrand ”a bone to gnaw“, namely the proposition that, if one accepts the supposition that weights converge towards the centre of Earth, there is no centre of gravity in a sphere (see Castelli to Galileo, 3o November 1635, Galilei, Le Opere [Favaro], xvi, pp. 351–352; Cavalieri to Castelli, 19 December 1635, Mersenne, Correspondance [Tannery e.a.], v, p. 548). A frank negative reaction came from Raffaello Magiotti, who thought that Beaugrand’s demonstration implied a ”manifest petition of principle;” Magiotti to [Michelini], 25 January 1636, Galilei, Le Opere [Favaro], xvi, pp. 382–383 and Mersenne, Correspondance [Tannerye.a.], vI pp. 13–14. A few months later, Castelli also admitted that he had lost interest in the geostatical question. See Castelli to [Carcavy], July or August 1636, Mersenne, Correspondance [Tannery e.a.], vI, pp. 128–129.Google Scholar
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    The geostatical question is answered in the fourth proposition of the Geostatice: ”omne grave prope Terrae centrum minus ponderat quam procul, et ejusdem gravis varia pondera eandem habebunt rationem quam a Terrae centro distantiae;” Descartes, Oeuvres [Adam e.a.], II, pp. 645–646.Google Scholar
  50. 49.
    As he said: ”J’aime mieux paraître ignorant en vous répondant mal qu’indiscret en ne vous répondant point du tout;” Fermat to Mersenne, Fermat, Oeuvres [Tannery e.a.], II, p. 3. On Fermat’s arguments, see Costabel, ”Les enseignements“ and the brief ”Appendix I“ of Mahoney, The Mathematical Career, pp. 372–384•Google Scholar
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    I found no clue as to the reason why he did so, neither in Guerlac ”Guy de la Brosse“) nor in Howard ”Guy de la Brosse“). I suspect that it is related to the censorship Beaugrand exercised as Secretary of the Chancellor Séguier (see below, note 53).Google Scholar
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    Desargues’ Brouillon Project, written in 1639 but unpublished at the time, contains an annex called Atteinte aux événements des contrariétés d’entre les actions des puissances ou forces, obviously related to the geostatical controversy. This annex has been first published by Taton, L’Oeuvre mathématique, pp. 181–184, and is briefly commented by Costabel, ”Centre de gravité;” pp. 14–15.Google Scholar
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    As ”secrétaire du Roi“ under the Chancellor Séguier, Beaugrand was in charge of giving ”privilèges“ to scientific books. He received the first printed sheets of the Essais at the beginning of 1637, delayed the publication and, at a time when the Dioptrique had not yet been published, criticized it and sent it to Fermat; after Descartes had pointed out the faults in the Geostatice, Beaugrand charged the Géométrie with plagiarism from Harriot and Viète in three anonymous pamphlets, later published in Tannery, La Correspondance. See Descartes to Mersenne, 22 June 1637, Descartes, Oeuvres [Adam e.a.], I, pp. 390–391; to Mersenne, 1st March 1638, Descartes, Oeuvres [Adam e.a.], II, p. 25; to Mersenne, 31 March 1638, Descartes, Oeuvres [Adam e.a.], II, p. 85; Beaugrand to Mersenne, April 1638, Descartes, Oeuvres [Adam e.a.], v, pp. 504–512. See also the notes of the editors in Descartes, Oeuvres [Adam e.a.], I, pp. 355, 361–362; Descartes, Oeuvres [Adam e.a.], II, pp. 457–459, 269, 326–328, pp. 395–396. This growing inimity goes probably back to the early thirties, when Beaugrand and Descartes competed on mathematical questions. See Mersenne, C0rrest,0ndance [Tannerve.a.] III, pp. 260–262.Google Scholar
  55. 54.
    The notion of relative heaviness is an avatar of the Jordanian gravitas secundum situm (see for example Jordani Liber de ponderibus, ”Proemium;” in Moody e.a. (eds.), The Medieval Science of Weights, pp. 150–151) and the definition of ”moment“ given in Mersenne-Galileo seems to echo it: ”Le moment est l’inclinaison d’ [un] corps, lorsqu’elle n’est pas seulement considérée dans le dit corps, mais conjointement avec la situation qu’il a sur le bras d’un levier ou d’une balance,“ Mersenne, Les Méchaniques de Galilée, p. 444. In the following, I shall quote, whenever possible, the Galilean Le Mecaniche in the translation of Mersenne, for conceptual similarities are more evident when the language is the same. On ”gravitas secundum situm;” see Clavelin, The Natural Philosophy, pp. 151–154 and Galluzzi, Momento, pp. 70–73. On the complexity and fecundity of the Galilean concept of ”momento;” see Clavelin, The Natural Philosophy, pp. 154–155, 168–174 and Galluzzi, Momento, pp. 199 ff.Google Scholar
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    To Mersenne, 13 July 1638: ”Suivant ces deux opinions, dont la première est la plus commune de toutes dans les écoles, et la seconde est la plus reçue entre ceux qui pensent savoir quelque chose de plus que le commun, il est évident que la pesanteur absolue des corps est toujours en eux une même, et qu’elle ne change point du tout à raison de leur diverse distance du centre de la terre;” Descartes, Oeuvres [Adam e.a.], II, pp. 223–224. The first opinion is explicitely presented as the opinion of the Schools, whereas the second one is compatible with the definition of heaviness given in Mersenne, Les Méchaniques de Galilée, p. 443. On the notion of gravity in Galileo’s Mecaniche, see Clavelin, The Natural Philosophy, pp. 154,172–174, and Galluzzi, Momento, pp. 94–95.Google Scholar
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    The third opinion is for example that of Roberval. From it, Descartes deduces that bodies are heavier, the closer they are to the Earth, because he thinks that the activity of a natural agent decreases with its distance from the body it acts upon. A similar idea is expressed in Mersenne, Traité des mouvements, prop. III, quoted in Mersenne, Correspondance [Tannery e.a.], III, p. 632. Contrary to Mersenne and Descartes, Roberval (together with Pascal) argues that if heaviness was an attraction, it should decrease when the body approaches the centre of the Earth (to Fermat, 16 August 1636, § 8, Fermat, Oeuvres [Tannery e.a.], II, pp. 4o-41). The difference between the two arguments may be explained by the fact that Descartes and Mersenne consider the body when it is outside the Earth, Pascal and Roberval when it is inside.Google Scholar
  58. 57.
    For Mersenne, heaviness may be either ”positive et réelle;” or an effect of the pushing air, or an attraction (Mersenne, Traité des mouvements, pp. 630–632); for Roberval and Pascal, heaviness is an attraction, the cause of which may be either in the attracted body, or in the attracting body, or in both (to Fermat, 16 August 1636, §3, Fermat, Oeuvres [Tannery e.a.], II, p. 36).Google Scholar
  59. 58.
    Mersenne, Traité des mouvements, prop. III: ”Encore que l’on ne sache pas la vraie raison de la chute des corps terrestres [ ... ] , l’on peut néanmoins expliquer quelques raisons qui satisferont à plusieurs, soit que la pesanteur des corps les pousse en bas, que l’air les chasse hors de son lieu, que la terre les attire, ou que ces trois causes et plusieurs autres contribuent à cet effet;” quoted in Mersenne, Correspondance [Tannery e.a.], nI, p. 631. However he notices that if bodies were attracted by the Earth, they would accelerate less fast when they were farther from the Earth than when they were closer to it (ibid., p. 632).Google Scholar
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    Roberval and Etienne Pascal to Fermat, 16 August 1636, § 9: ”Puis donc que de ces trois causes possibles de la pesanteur, nous ne savons quelle est la vraie, et que même nous ne sommes pas assurés que ce soit l’une d’icelles, se pouvant faire [que la vraie cause soit composée des deux autres ou] que ce [en] soit une [tout] autre, de laquelle on tirerait des conclusions toutes différentes, il nous semble que nous ne pouvons pas poser d’autres principes [pour raisonner] en cette matière que ceux desquels nous sommes assurés par une expérience continuelle assistée d’un bon jugement,“ Fermat, Oeuvres [Tannery e.a.], II, p. 41.Google Scholar
  61. 60.
    The Cartesian notion of heaviness has both a metaphysical and a physical side. From a metaphysical point of view, Descartes criticizes the usual concepts of heaviness which imply a confusion between substance and accident, body and mind (on this point, see the following letters: to ?, [August 1641], Descartes, Oeuvres [Adam e.a.], III, p. 424; to Arnauld, 29 July 1648, Descartes, Oeuvres [Adam e.a.], v, pp. 222–223; to Elizabeth, 21 May 1643, Descartes, Oeuvres [Adam e.a.], III, pp. 667–668; to Mersenne, 20 April 1646, Descartes, Oeuvres [Adam e.a.], Iv, p. 401). From a physical point of view, he suggests mechanisms that could explain the behaviour of ”heavy“ bodies, without having to suppose that they have in themselves the power to move towards the centre of the earth.Google Scholar
  62. 61.
    Descartes to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 224. Among the authors I have studied, Descartes and Mersenne are the only ones who suggest that one should rely on experiments to answer the geostatical question.Google Scholar
  63. 62.
    Robert Hooke will carry out such experiments in the sixties; see Dugas, La Mécanique, pp. 357–358.Google Scholar
  64. 63.
    The question of why birds prefer to fly high, is discussed, among the others, by Beeckman ([3] -2o October 1624, Journal [De Waard], II p. 306; 2 May-7 July 1634, ibid.,III, p. 348; 24 September 1634, ibid., III, p. 31o; and the references of the editor, ibid., In, p. 253, note 2) and Mersenne, Harmonie universelle,I, p. 207.Google Scholar
  65. 64.
    In April 1634, Descartes asked Mersenne to perform the experiment described in Jean Leurechon’s Récréations mathématiques, problème 86 (to Mersenne, April 1634, Descartes, Oeuvres [Adam e.a.], I, p. 287. The text of this problem is reproduced in Descartes, Oeuvres [Adam e.a.], x, p. 547) . Descartes was dissatisfied with Mersenne’s first experience, performed with an arquebus (to Mersenne, 15 May 1634, Descartes, Oeuvres [Adam e.a.] , I, p. 293). Two years later however, he congratulated Mersenne for having performed it again, apparently in satisfactory conditions (to Mersenne, March 1636, Descartes, Oeuvres [Adam e.a.], I, p. 341). This experience is also alluded to in Mersenne, Harmonie universelle, I, p. 207. It is the subject of an engraving at the beginning of Varignon’s Nouvelles conjectures: in the middle, a gun is pointing towards the zenith; right and left, Marin Mersenne and Pierre Petit tilt their heads backwards and look upwards; below is indicated the question in which they are immersed: ”Retombera-t-il?“Google Scholar
  66. 65.
    Descartes to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 226.Google Scholar
  67. 66.
    Ibid., pp. 226–227. The formulation ”Nous prendrons, s’il vous plaît [ ...] ” shows that Descartes’ supposition is a conventional postulate. In the following, Descartes explains why he introduces all these specifications, that make his determination of heaviness local and empirical.Google Scholar
  68. 67.
    Ibid., p. 227. That Descartes is referring to the two opinions listed at the beginning is clear from his very formulation. Compare the lines quoted in note 55 with the following passage: ”suivant ces deux opinions [ ... ] la pesanteur absolue des corps est toujours en eux une même, et qu’elle ne change point du tout à raison de leur diverse distance du centre de la terre;” ibid., pp. 223–224.Google Scholar
  69. 68.
    Ibid., p. 228. For a good commentary on the meaning of this principle for Cartesian statics, see Séris, Machine, esp. pp. 211–221. The question of why this principle is not one of the laws of nature listed in the Principia philosophiae is addressed in Gabbey, ”Descartes’ Physics?”Google Scholar
  70. 69.
    La Statique, Stevin, Les Oeuvres mathématiques [Girard], pp. 448–449.Google Scholar
  71. 70.
    For an example of this opinion, see Clavelin, The Natural Philosophy, pp.163–164. For formulations of the principle, see Mersenne, Les Méchaniques de Galilée, pp. 439–441,454,464–465,468.Google Scholar
  72. 71.
    Ibid., pp. 445–446.Google Scholar
  73. 72.
    To Huygens, 5 October 1637, Descartes, Oeuvres [Adam e.a.], I, p. 443. That the Cartesian order of reasons runs counter common sense statics is evident from Pardies’ criticism of the Cartesian principle. Although admitting that the principle is “très véritable“ and “indubitable;” Pardies notices that it contains “quelque chose qui ne satisfait pas entièrement l’esprit, qu’il suffise pour faire des démonstrations;” Pardies, La Statique, pp. 102–103.Google Scholar
  74. 73.
    Gabbey (“Descartes’ Physics;” p. 320) believes that the GPS, as he calls it, is not a law of nature because of its empiricity. But in the letter to Mersenne of 12 September 1638, Descartes explains precisely that his principle is not a conclusion reached through experiences: “On s’est imaginé que j’avais apporté les exemples de la poulie, du plan incliné et du levier, afin d’en mieux persuader la vérité, comme si elle eût été douteuse, ou bien que j’eusse si mal raisonné que de vouloir prouver un principe, qui doit de soi être si clair qu’il n’ait besoin d’aucune preuve, par des choses qui sontsi difficiles qu’elles n’avaient peut-être jamais ci devant été bien démontrées par personne. [ ... ] il ne faut que savoir compter jusqu’à deux [...],“ Descartes, Oeuvres [Adam e.a.], II, pp. 358, 36o. Descartes’ reasons for not including the GPs in his set of laws of nature are not epistemological, but exclusively ontological (it is not formulated in terms of matter and motion). This raises however the question of how epistemological evidence can be separated from its ontological reference.Google Scholar
  75. 74.
    To Huygens, 5 October 1637: ”L’effet doit toujours être proportionné à l’action qui est nécessaire pour le produire,“ Descartes, Oeuvres [Adam e.a.], I, p. 438. See also to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 228. Descartes distinguishes in this context between ”action“ (which is the force corresponding exactly to the effect) and ”power“ (which may exceed the effect). In a letter to Mersenne of 15 November 1638, he explains that ”lorsqu’on dit qu’il faut employer moins de force à un effet qu’à un autre, ce n’est pas dire qu’il faille avoir moins de puissance: car encore qu’on en aurait davantage, elle n’y nuit point; mais seulement qu’il y faut moins d’action. [ ... ] Or il n’y a point, ce me semble, d’autre moyen de connaître a priori la quantité d’ [un effet] [ ... ], que de mesurer la quantité de l’action qui cause cet effet, c’est-à-dire de la force qui doit y être employée;” Descartes, Oeuvres [Adam e.a.], II, pp. 432–433. On this point, see also Debeaune to Mersenne, 13 November 1638, Descartes, Oeuvres [Adam e.a.l, v, p. 526.Google Scholar
  76. 75.
    To Mersenne, 12 September 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 356–357. Descartes’ reasoning was criticized by Lamy, who observed that a man who is able to raise a weight of 1 to a height of 10oo is not able to raise a weight of 10oo to a height of 1 (see Lamy, Traités, p. 79). This criticism has been forestalled by Descartes in the letter to Mersenne of 15 November 1638: ”Je ne considérais pas, en cet écrit, la puissance qu’on nomme la force d’un homme, mais seulement l’action qu’on nomme la force par laquelle un poids peut être levé [ ... ] ;” Descartes, Oeuvres [Adam e.a.], II, pp. 432–433.Google Scholar
  77. 76.
    To Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 229. Reiterated discussions with Egidio Festa helped me to understand what is at stake in this passage.Google Scholar
  78. 77.
    As Costabel (”La démonstration;” p. 94) notes, considerations on the direction of forces are of so little import for Descartes that he assimilates the relative heaviness of a body with its opposite force, namely the power to hold it.Google Scholar
  79. 78.
    Guidobaldo del Monte, Mechanicorum liber, ”On the lever;” prop. 4, corollary, in Drabkin e.a. (eds.), Mechanics, p. 300, passim. Galileo, Le Mecaniche: ”E perchè, per fare descendere il peso B, ogni minima gravità accresciutagli è bastante, perô, non tenendo noi conto di questo insensibile, non faremo differenza dal potere un peso sostenere un alto al poterlo movere;” Galilei, Le Opere [Favaro], II, p. 164. Mersenne suppresses any explicit justification and writes only: ”si l’on ajoute quelque chose à l’un [ ...].“ ..]“ On the justification of the principle of virtual speeds, see Galluzzi, Momento, pp. 208–212.Google Scholar
  80. 79.
    To Huygens, 5 October 1637, Descartes, Oeuvres [Adam e.a.], I, p. 438; to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 229.Google Scholar
  81. 80.
    To Mersenne, 12 September 1638: ”J’ai parlé de la force qui sert pour lever un poids à quelque hauteur, laquelle force a toujours deux dimensions, et non de celle qui sert en chaque point pour le soutenir, laquelle n’a jamais qu’une dimension, en sorte que ces deux forces diffèrent autant l’une de l’autre qu’une superficie diffère d’une ligne;” Descartes, Oeuvres [Adam e.a.], II, p. 353. See also ibid., p. 357.Google Scholar
  82. 81.
    This passage should warn us against asserting that Descartes formulated the principle of virtual speeds (Dugas, La Mécanique, p. 155, note 1, goes as far as mentioning ”le caractère différentiel“ of the Cartesian principle of statics); for such a formulation, one needs a somewhat clearer idea about how to manipulate infinitely small quantities.Google Scholar
  83. 82.
    As for the reason why one should take this convergence into account, Descartes says that it is for the sake of mathematical exactitude (to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.] , II, pp. 232–234) . For a similar position, see Fermat to Mersenne, 24 June 1636, Fermat, Oeuvres [Tannery e.a.], II, p. 19; Fermat, Nova in mechanicis theoremata, ibid., p. 23. For an opposite position, see La Statique, Stevin, Les Oeuvres mathématiques [Girard], p. 436.Google Scholar
  84. 83.
    Descartes asserts that every mechanician is familiar with this ratio. It was indeed accepted by Tartaglia, Galileo, Stevin, Roberval and Mersenne, even if they established it through different principles. Guidobaldo, however, preferred to follow Pappus, while Salomon de Caus ignored the problem in the little treatise on simple machines that he inserted in Les Raisons des forces mouvantes, fol. 5v- 9v. As for Descartes, he states it for the first time, without justification, in the letter to Mersenne of 3 May 1632, Descartes, Oeuvres [Adam e.a.], I, p. 247.Google Scholar
  85. 84.
    As Descartes writes, the equation would be true if CB were a part of a circle and CA a part of a spiral (to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 233). The line according to which Pr is a constant is a spiral (figure 2a), because at each point F of the line the angle (MFK) is the same, M being the centre of the Earth, and FK the tangent to the line. The angle (MFK) is constant because Pa on FM is by definition a constant; having fixed the quantity of relative heaviness, you want also Pr on FK constant; consequently, you must have the angle (MFK) constant in the right-angle triangle. This problem had already been solved by Mersenne in the Harmonie universelle, I, pp. 113–12o. Contrary to Mersenne, Descartes affirms however that the spiral reaches the centre of the Earth (to Mersenne, 1 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 39o).Google Scholar
  86. 85.
    To Mersenne,13 July 1638: ”Notez que je dis commencer à descendre, non pas simplement descendre, à cause que ce n’est qu’au commencement de cette descente à laquelle il faut prendre garde,“ Descartes, Oeuvres [Adam e.a.], II, p. 233.Google Scholar
  87. 86.
    As Costabel ”La démonstration;” p. 99), nicely puts it: ”On ne peut engager, au stade de l’élaboration logique où l’on se place, une structure du mouvement que l’on ne connaît pas encore. On ne peut donner des définitions simples (force à deux dimensions) et poser des proportions simples (avec les déplacements) qu’à l’échelon infinitésimal où l’on ”saisit“ le mouvement dans une durée très courte, une ébauche, où le mouvement n’a pas le temps de changer si l’on peut dire. Mais autant, alors, se contenter de considérer les petits déplacements, sans parler du temps ni des vitesses, puisqu’aussi bien ce sont ces petits déplacements qui donnent ”raison des forces à considérer.” I shall come back to the question of why Descartes does not take speed into account.Google Scholar
  88. 87.
    To Mersenne, 11 July 1638: ”La proportion qui est entre la force qui meut ce poids et sa pesanteur, ne se mesure pas par celle qui est entre les deux diamètres de ces cercles, ou entre leurs deux circonférences, mais plutôt par celle qui est entre la circonférence du premier et le diamètre du second;” Descartes, Oeuvres [Adam e.a.], II, p. 235.Google Scholar
  89. 88.
    ”Pour mesurer exactement quelle doit être cette force en chaque point de la ligne courbe ABCDE, il faut penser qu’elle y agit tout de même que si elle trainait ce poids sur un plan circulairement incliné, et l’inclinaison de chacun des points de ce plan circulaire, ou sphérique, se doit mesurer par celle de la ligne droite qui touche le cercle en ce point-là,“ ibid., p. 236.Google Scholar
  90. 89.
    In Le Mecaniche we read: ”Il considerare questo grave discendente, e sostenuto dalli semidiametri BF, BL ora meno e ora più, e constretto a camminare per la circonferenza CGL, non è diverso da quello che sarà imaginarsi la medesima circonferenza CFLI esser una superficie cosi piegata, e sottoposta al medesimo mobile, si che, appoggiandovisi egli sopra, fosse constretto a descendere in essa; perché se nell’uno e nell’altro modo disegna il mobile il medesimo viaggio, niente importerà s’egli sia sospeso dal centro B e sostenuto dal semidiametro del cerchio, o pure se, levato tale sostegno, s’appoggi e cammini su la circonferenza CFLI;” Galilei, Le Opere [Favaro], II, p.184. There is no equivalent of this passage is Mersenne’s translation, whereas the other passage is translated as follows: ”Le point d’inclination F de la circonférence cI ne diffère point de l’inclination de la tangente CFH, que par l’angle insensible de contact;” Mersenne, Les Méchaniques de Galilée, p. 485.Google Scholar
  91. 90.
    The first distinction corresponds to the first criticism addressed to Beaugrand (to Mersenne, 29 June 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 184–185). The distinction between solid and liquid bodies, which does not appear in the other writings on geostatics, is fundamental in Descartes’ physics. Suffice it to say here that in Le Monde, Descartes notes that the first difference to be noticed is the one between hard and liquid bodies (chap. 3, Descartes, Oeuvres [Adam e.a.], vI, p. 12). To give a mechanical explanation of liquidity and hardness does not mean to get rid of them. My (somewhat paradoxical) impression is that, besides his famous identification of matter and motion, Descartes had a great sensibility for concrete differences between materials.Google Scholar
  92. 91.
    This assimilation may remind us of the Galilean assimilation of a body sustained by a circularly inclined plane with a body suspended from the centre of a lever. What makes the difference however (and the physically absurd character of Descartes’ suggestion) is that, contrary to the Galilean body, the Cartesian body is not suspended from any fixed part; consequently, there is no equivalent to the constraint represented by the inclined plane.Google Scholar
  93. 92.
    To Mersenne, 11 July 1638, Descartes, Oeuvres [Adam e.a.1, II, pp. 238–239.Google Scholar
  94. 93.
    According to the editors of Mersenne’s Correspondance, this observation goes back to the Middle Ages and has been summarized in the unpublished continuation of Mersenne’s Quaestiones in Genesim. See Mersenne, Correspondance [Tannery e.a.], vII, pp. 372–373.Google Scholar
  95. 94.
    To Mersenne, 11 July 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 24o-241.Google Scholar
  96. 95.
    On this point, see Costabel, ”La démonstration“ pp. 95–96.Google Scholar
  97. 96.
    Descartes had already formulated this conclusion in a letter to Huygens dated 5 October 1637 (Descartes, Oeuvres [Adam e.a.], I, pp. 446–447). Although he claimed to be the first to make this point, Stevin (see above, note 47), Castelli (see above, note 44) and Fermat (Nova in mechanicis theoremata, Fermat, Oeuvres [Tannery e.a.], II, pp. 25–26) had already had similar thoughts.Google Scholar
  98. 97.
    On this demonstration, see Costabel, ”La démonstration;” pp. 96–98 and ”Centre de gravité,“ pp. 11–13. Costabel points out that the determination of the new centre of gravity suggested here is contradictory, because it relies on the supposition that the old centre of gravity still holds; curiously enough, Descartes had pointed out a similar contradiction in his critique of Beaugrand’s book (to Mersenne, 29 June 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 185–186).Google Scholar
  99. 98.
    In the very last paragraph of the letter, Descartes suggested a way of determining the position of the centre of gravity of a sphere, which he later on declared false (to Mersenne, 15 November 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 451–452). As I shall report below, this led Duhem to conclude that Descartes’ contribution to the history of mechanics was to show that the notion of centre of gravity is not compatible with the convergence of verticals. It is certainly true that the hypothesis of convergent verticals leads to the paradoxical conclusion that the usual notion of centre of gravity does not hold; however, Descartes does not express what one should conclude from this paradox. Does it mean that we should confine ourselves to practical procedures when we want to determine the centre of gravity of a body (e.g. suspending the body and considering that the centre of gravity is the intersection of the verticals thus obtained)? That we should neglect the convergence of verticals? That we should forge another mathematical notion of centre of gravity? Or that we should stop altogether pretending that there is such a thing as a science of weights?Google Scholar
  100. 99.
    Descartes, Oeuvres [Adam e.a.], Ix, pp. 14, 17. On the relationship between this statement and the claim that physics and mechanics are identical, see Gabbev. ”Descartes’ Phvsics“ nn 315 3200–3,1Google Scholar
  101. 100.
    Descartes gave to Huygens the permission to circulate his Explication des engins. See the letters to Huygens, 4 December 1637 [wrongly given in Descartes, Oeuvres [Adam e.a.], I, p. 507 as a letter from January, 25, 1638]; to Pollot, 12 February 1638, Descartes, Oeuvres [Adam e.a.], I, pp. 518519; to Huygens, [March 1638], Descartes, Oeuvres [Adam e.a.], II, p. 51. The treatise was finally published by Nicolas Poisson in 1668. Consistent with his wish to preserve his anonymity and with the current norms of ”honnêteté;” Descartes let Mersenne publish the letter of June 1638 on condition that his own name and certain insulting words concerning Beaugrand should be omitted (to Mersenne, 27 July 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 271–272). As for the Examen de la question géostatique, Descartes noted that it is not ”assez complet ni achevé pour aller seul“ and suggested its publication in a collection of objections ”car aussi bien ne sera-ce qu’un ramas de toutes sortes de matières;” to Mersenne, 11 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 392. This collection was never realized, but in 1644 Mersenne published some extracts of the Examen in his Cogitata physico-mathematica, after having once again asked for Descartes’ authorization (see his letter to Mersenne, 3 February 1643, Descartes, Oeuvres [Adam e.a.], III, p. 613).Google Scholar
  102. l01.
    Garber, ”A Different Descartes;” pp. 207–208, 211, focuses on the question of why Descartes has not published anything on mechanics.Google Scholar
  103. 102.
    To Mersenne, 12 September 1638: ”cette question de nul usage;” Descartes, Oeuvres [Adam e.a.], II, p. 358.Google Scholar
  104. 103.
    To Huygens, 5 October 1637: ”Les poils blancs qui se hâtent de me venir m’avertIssent que je ne doIs plus étudier à autre chose qu’aux moyens de les retarder. C’est maintenant à quoi je m’occupe;” Descartes, Oeuvres [Adam e.a.], I, pp. 434–435; to Huygens, 4 December 1637 (given in Descartes, Oeuvres [Adam e.a.], I, pp. 506–507 as 25 January 1638): ”Je travaille maintenant à composer un abrégé de médecine:”Google Scholar
  105. 104.
    The story of the notion of centre of gravity is part of another story, which is the core of Les Origines de la statique, vol. II, namely that of the Torricelli’s principle, which states that two heavy bodies linked together cannot move unless their common centre of gravity goes down.Google Scholar
  106. 105.
    Duhem, Les Origines, II, pp. 6–31, 99–104.Google Scholar
  107. 106.
    Ibid, pp. 156–185. I have expressed my doubts about this conclusion above, note 98.Google Scholar
  108. 107.
    Costabel’s ”Centre de gravité“ is a broad presentation of the history of the notion of centre of gravity from Pappus to d’Alembert; ”La démonstration“ is about its treatment by Descartes; ”Les enseignements“ about its treatment by Fermat.Google Scholar
  109. 108.
    See for example Costabel, ”Centre de gravité,“ p. 13; ”La démonstration;” p. 98. The Duhemian inspiration is especially clear in ”Centre de gravité,“ where Costabel, after an a-historical presentation of the modern notion, exposes the problem as discussed by Pappus, Albert of Saxony, Guidobaldo and Benedetti, just as Duhem did.Google Scholar
  110. 109.
    Costabel distributes blame and praise in continuation, e.g.: ”Il faut reconnaître à Descartes le mérite de l’avoir compris;” Costabel, ”Centre de gravité;” p. 10; ”On ne peut qu’admirer ces précautions d’un esprit épris de rigueur [ ... ] . On ne saurait exiger plus de clarté [ ... ] . Une méthodologie dont la valeur mérite d’être reconnue;” Costabel, ”La démonstration;’ pp. 90, 94, 100; ”Le mérite d’avoir bien compris cette conclusion désastreuse revient à Fermat;” Costabel, ”Les enseignements;“ p. 117.Google Scholar
  111. 110.
    In the case of Descartes, Costabel does not say a word about the physical answer; in the mathematical answer, he concentrates exclusively on the explanation of the simple machines and on the proposition that there is no centre of gravity. In the case of Fermat, he ignores that most of Fermat’s writings on mechanics in 1636 are devoted to discussing a case that makes little sense in a modern perspective, namely that of a ”lever“ in which weights act towards the fulcrum.Google Scholar
  112. 111.
    To Mersenne, i1 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 225.Google Scholar
  113. 112.
    Carla Rita Palmerino has suggested that this contradiction may be the result of a kind of reductioad absurdum, by which Descartes would have shown that the assumption of an absolute heaviness is false. Although interesting, this interpretation is, in my opinion, not really suited to Descartes’ scientific psychology. In general, Descartes is very explicit about what he wants to prove and thus, instead of reductiones ad absurdum, he prefers direct demonstrations, which go from the obvious to the obvious.Google Scholar
  114. 113.
    Descartes compares his way of proceeding to the supposition of the equality of median, which eases calculations for astronomers. Such a comparison is not totally convincing: the astronomers make a mean between different measures; Descartes assumes in his letter to Mersenne that a local and conventional definition of heaviness can be extended everywhere, but his own notion of heaviness stands in the way of this very extension.Google Scholar
  115. 114.
    See Gabbey, ”Descartes’ Physics“; Garber, ”A Different Descartes”Google Scholar
  116. 115.
    I have argued above, note 73, that, although the principle of statics involves the ontologically obscure notion of gravity, Descartes does not think it can be known by experiment.Google Scholar
  117. 116.
    Garber, ”A Different Descartes;” pp. 196–198, described the Galilean paradigm as a combination of mathematics and physics aimed at the description of the behaviour of heavy bodies. I have at least two objections to such a paradigm: it is much too gross to grasp the singularity of Galilean science, and it would be valid for all mechanicians of the sixteenth century; the very reception of Galileo shows that there is no established paradigm, but a complex tradition with different trends.Google Scholar
  118. 117.
    Ibid., pp. 214–216. I do not see what kind of workable experiments Garber is thinking of, and I show in 3.3. the many conceptual diffIculties that Descartes encountered while working out his notion of heaviness.Google Scholar
  119. 118.
    In that respect, Gabbey’s rhetoric is striking: he begins by picking up ”anomalies;” ”intriguing remarks“ and ”inconsistencies“; next he ”disentangles“ them; last he ”concludes“ that they are only apparent. Gabbey, ”Descartes’ Physics,“ pp. 311–314, 315–320.Google Scholar
  120. 119.
    See the letters to Mersenne, 12 September 1638, Descartes, Oeuvres [Adam e.a.], II, p. 355; 11 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 391; 15 November 1638, Descartes, Oeuvres [Adam e.a.], II, p. 433 and the references given below, note 125.Google Scholar
  121. 120.
    This is, by the way, perfectly legitimate in a simple machine like the lever: because both ends are linked, the time which one end needs to go up is the same as the time which the other end needs to go down. Hence it does not matter whether one speaks of spaces or of speeds.Google Scholar
  122. 121.
    See for example Guidobaldo, Mechanicorum liber, ”De trochlea,“ in Drabkin e.a. (eds.), Mechanics, p. 317; Hérigone, Cursus mathematicus, vol. 3 (1634), p. 291; Mersenne, Les Méchaniques de Galilée, pp. 439–441.Google Scholar
  123. 122.
    Roberval, Traité de méchaniques, p. 1. The inference from spaces to times is explicitly justified by the proposition that ”le temps est en raison des chemins“ (ibid., p. 12).Google Scholar
  124. 123.
    To Huygens, 4 December 1637, Descartes, Oeuvres [Adam e.a.], I, p. 648.Google Scholar
  125. 124.
    See for example Koyré, Études galiléennes, pp. 131, 331–332, 341; Séris, Machine, pp. 216–218.Google Scholar
  126. 125.
    To Mersenne, 15 November 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 433–434; 2 February 1643, Descartes, Oeuvres [Adam e.a.], III, p. 614; [to Boswell, 1646?], Descartes, Oeuvres [Adam e.a.], Iv, p. 685; [to Boswell, 1646?], Descartes, Oeuvres [Adam e.a.], Iv, p. 696.Google Scholar
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    Aristotle, First Analytics, 53b5 ff. and 57a38 ff. This principle has been often commented upon by astronomers, in the context of a comparison between the Ptolemaic and the Copernican hypothesis.Google Scholar
  128. 127.
    Mersenne, Traité de l’harmonie universelle, pp. 395, 399, 404. See also the texts quoted by Duhem, Les Origines, I, appendix I, pp. 291–296. In that sense, Duhem may be right in arguing that Descartes’ target was the Aristotelian axiom that speeds are proportional to motive actions. He is however over-interpreting when he asserts that, having realized that there was no simple ratio between force and speed, Descartes aimed at making statics independent from the false Aristotelian dynamics, waiting for the time when a true dynamics would be elaborated (ibid, pp. 342–348) — not to speak of the absurd distinction Duhem makes between Descartes’ allegedly visionary restriction and the dullness of Galileo, whom he accuses of having been stuck with the Aristotelian axiom and of not deserving the honour of being called the father of modern dynamics (ibid., pp. 247–248, 253, 255, 260–261).Google Scholar
  129. 128.
    It is interesting to note that the meaning of this argument was already unclear for the first editor and commentator of Descartes’ mechanics, the oratorian Nicolas Poisson. Not only does Poisson claim that there is no difference between considering speeds and displacements (Descartes, Traité de la mécanique [Poisson], p. 22), but he also criticizes Descartes for having laughed at Galileo: ”On pourrait lui répondre [to Descartes] : Quid rides? Mutato nomine de te fabula narratur,“ ibid., P. 45.Google Scholar
  130. 129.
    Previous to Descartes, this argument against Aristotle has been used by Benedetti, Diversarum speculationum mathematicarum, et physicarum liber, cap. 11, in Drabkin e.a. (eds.), Mechanics, pp. 180–181 and by Baldi, In mechanica exercitationes, p. 36. The formulation found in Stevin’s La statique is particularly clear: ”Là où il n’y a pas de circonférence, elle ne sera pas cause de ce qui advient, ainsi donc la circonférence ne sera pas la cause de l’équilibration. [ ... ] Le mouvement et la description des circonférences n’advient là que par accident;” Stevin, Les Oeuvres mathématiques [Girard], p. 5o1. In this context, it is interesting to note that Stevin mentions, at the beginning of his explanation of pulleys, a general rule, apparently common at the time, which does not involve time, but which is too general to be a workable principle: ”On tient pour règle générale en mathématique, que Comme l’espace de l’agent, à l’espace du patient: Ainsi la puissance du patient, à la puissance de l’agent;” ibid., p. 509. After Descartes, the anti-Aristotelian argument appears in Lamy. Traités. p. 74• Google Scholar
  131. 130.
    According to the editors of Mersenne’ s correspondence the fragments date from as early as 1630, but the editors of Descartes’ works argue that they were most probably written between 1635 and 1637 (Descartes, Oeuvres [Adam e.a.], Iv, pp. 814–818).Google Scholar
  132. 131.
    To Mersenne, 12 September 1638, Descartes, Oeuvres [Adam e.a.], II, p. 354; 15 November 1638, Descartes, Oeuvres [Adam e.a.], II, pp. 433–434; 2 February1643, Descartes, Oeuvres [Adam e.a.], III, p. 614; [to Boswell, 1646?], Descartes, Oeuvres [Adam e.a.], Iv, p. 685; [to Boswell, 1646?], Descartes, Oeuvres [Adam e.a.], Iv, pp. 694–696.Google Scholar
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    ”La statique enseigne seulement à mettre en équilibre le mouvant avec l’ému. Et [ ... ] touchant la pesanteur ou la puissance, que le mouvant a besoin d’avoir encore davantage, pour faire que l’ému se puisse mouvoir [ ... ], la statique ne montre pas la manière de trouver telle pesanteur ou puissance mathématiquement, pour ce que l’ému et ses empêchements n’ont aucune proportion avec un autre ému et ses empêchements,“ La Statique, Stevin, Les Oeuvres mathématiques [Girard], P. 469.Google Scholar
  134. 133.
    His earlier position on the resistance of the air is somewhat different. To Mersenne, 18 December 1629: ”Pour le quantum [de l’empêchement de l’air par les mouvements], je l’ignore, et encore qu’il se pût faire mille expériences pour le trouver à plus près, toutefois, pour ce qu’elles ne se peuvent justifier par raison, au moins que je puis atteindre, je ne crois pas qu’on doive prendre la peine de les faire,“ Descartes, Oeuvres [Adam e.a. ] , I, vv. 99–100.Google Scholar
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    To Mersenne, 2 February 1643, Descartes, Oeuvres [Adam e.a.], III, p. 614.Google Scholar
  136. 135.
    To Mersenne, 12 September 1638: ”Encore qu’il n’y ait aucun mouvement qui n’ait quelque vitesse, toutefois il n’y a que les augmentations ou diminutions de cette vitesse qui sont considérables, et lorsque, parlant du mouvement d’un corps, on suppose qu’il se fait selon la vitesse qui lui est plus naturelle, c’est le même que si on ne la considérait pas du tout,“ Descartes, Oeuvres [Adam e.a.], II, P. 355.Google Scholar
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    The more complete text on this question, although written in the somewhat different context of the controversy on the centres of oscillation, is the letter to Mersenne, 7 September 1646, Descartes, Oeuvres [Adam e.a.], Iv, p. 5oo. See also, to Mersenne, 3o March 1646, Descartes, Oeuvres [Adam e.a.], Iv, pp. 385–386; 20 April 1646, Descartes, Oeuvres [Adam e.a.], Iv, pp. 391–392; 15 May 1646, Descartes, Oeuvres [Adam e.a.], Iv, p. 417; 2 November 1646, Descartes, Oeuvres [Adam e.a.], Iv, p. 547.Google Scholar
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    Principia Philosophiae, Descartes, Oeuvres [Adam e.a.], VIII-1, pp. 71–75. For a general presentation of the Cartesian concepts implied here, see Roux, La philosophie mécanique, pp. 408–433.Google Scholar
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    To Huygens, March 1638, Descartes, Oeuvres [Adam e.a.], II, p. 5o; to Mersenne, 12 September 1638, Descartes, Oeuvres [Adam e.a.], II, p. 355; to Debeaune, 3o April 1639, Descartes, Oeuvres [Adam e.a.], II, p. 544.Google Scholar
  140. 139.
    Galileo is explicitely criticized in the following letters: to Mersenne, 3o August 1637, Descartes, Oeuvres [Adam e.a.], I, p. 392; to Mersenne, 11 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 385; and to Mersenne, 29 January 1640, Descartes, Oeuvres [Adam e.a.], III, pp. 9–11, passim. On this criticism, see for example Koyré, Études galiléennes, pp. 131–134. For a rich and detailed account of the reception of Galileo’s law of fall not only in Descartes, but in the French community, see Palmerino, “Infinite Degrees of Speed.“Google Scholar
  141. 140.
    The remarks made by Carla Rita Palmerino on a former version of this paper, as well as her paper ”Infinite Degrees of Speed“ have helped me greatly in clarifying the following points.Google Scholar
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    See Le Monde, Descartes, Oeuvres [Adam e.a.], xI, pp. 74–76; Principia philosophiae, Descartes, Oeuvres [Adam e.a.], vIII-1, pp. 213–215. Another explanation is suggested in the Principia, but immediately forgotten; on these two explanations, see Roux, La philosophie mécanique, esp. pp. 534–544.Google Scholar
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    See for example the letters to Mersenne of 29 January 1640, Descartes, Oeuvres [Adam e.a.], III, pp. 9–10; 11 March 1640, Descartes, Oeuvres [Adam e.a.], In, pp. 37–38; and November 1640, Descartes, Oeuvres [Adam e.a.], III, p. 593•Google Scholar
  144. 143.
    To Mersenne, 11 March 1640: ”Je ne puis déterminer la vitesse dont chaque corps descend au commencement, car c’est une question purement de fait, et cela dépend de la vitesse de la matière subtile;” Descartes, Oeuvres [Adam e.a.], III, p. 36.Google Scholar
  145. 144.
    In a letter to Mersenne of October-November 1631, Descartes notices that the supposition ”que la force qui faisait mouvoir cette pierre, agissait toujours également [ ... ] répugne apertement aux lois de la nature: car toutes les puissances naturelles agissent plus ou moins, selon que le sujet est plusou moins disposé à recevoir leur action; et il est certain qu’une pierre n’est pas également disposée à recevoir un nouveau mouvement ou une nouvelle augmentation de vitesse, lorsqu’elle se meut déjà fort vite et lorsqu’elle se meut fort lentement;” Descartes, Oeuvres [Adam e.a.], I, p. 230. See also Descartes’ letter to Mersenne, 15 November 1638, Descartes, Oeuvres [Adam e.a.], II, p. 443; to Debeaune, 3o April 1639, Descartes, Oeuvres [Adam e.a.], II, p. 544; to Mersenne, 27 August 1639, Descartes, Oeuvres [Adam e.a.], II, p. 571; to Mersenne, 11 June 1640, Descartes, Oeuvres [Adam e.a.], III, p. 79; to Mersenne, 3o August 1640, Descartes, Oeuvres [Adam e.a.], III, p. 164.Google Scholar
  146. 145.
    To Mersenne, October-November 1632, Descartes, Oeuvres [Adam e.a.], I, p. 261.Google Scholar
  147. 146.
    The link is explicit in the Principia philosophiae, Descartes, Oeuvres [Adam e.a.], vIII-1, p. 214. Palmerino, ”Infinite Degrees of Speed,“ p. 285, comments on the letter of 1632.Google Scholar
  148. 147.
    Descartes’ more explicit attempt to give an evaluation of the law of heaviness appears in a manuscript, conserved by Leibniz and published in Descartes, Oeuvres [Adam e.a.], xI, pp. 629–630. Descartes first notices that the law he formulated when he met Beeckman, and which states that if the body travels 1 unit of space in the first moment, it will travel 4/3 unit of space in the second, would be true only for a body impelled in the void by a uniform force (that would be a force imparted by the mind). But given that there is no void, but only a full space the resistance of which augments in geometrical proportion to the speed of the body, and that the impulse of the force imparted upon the body decreases in geometrical proportion, Descartes concludes that the ”true“ law should be in geometrical proportion. Then he considers different hypotheses and notes in particular that a geometrical augmentation and a geometrical diminution should be combined in order for the arithmetical law in 4/3 to remain valid.Google Scholar
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    On this point, see Séris, Machine, pp. 219–221.Google Scholar
  150. 149.
    For Guidobaldo, see [to Boswell, 1646?], Descartes, Oeuvres [Adam e.a.], Iv, p. 696. For Stevin, see to Huygens, 1st November 1635, Descartes, Oeuvres [Adam e.a.], I, p. 331; to Mersenne, 13 July 1638, Descartes, Oeuvres [Adam e.a.], II, p. 247; to Mersenne, 11 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 391.Google Scholar
  151. 150.
    To Mersenne, 11 October 1638, Descartes, Oeuvres [Adam e.a.], II, p. 391.Google Scholar
  152. 151.
    Huygens to Descartes, 8 September 1637, Descartes, Oeuvres [Adam e.a.], I, p. 397. My opinion is that Descartes did not read Mersenne’s translation, because otherwise, he would have explained at this point why Galileo’s way of proceeding was not as good as his own. This does not exclude however the possibility that he may have had an earlier glimpse of one of the manuscripts of Le Mecaniche circulating in Paris in the twenties. There is good evidence for this circulation. In April 1635, Diodati wrote that a manuscript of Le Mecaniche had been brought to France some 16 or 18 years before (Galilei, Le Opere [Favaro], xvI, p. 255); the two existing Parisian manuscripts date from 1623 and from 1627; in his first letter to Galileo, dated 1st February 1629, Mersenne notes: ”vidimus etiam tractatum Mechanicorum, quem e tua manu putant ortum;” Mersenne, Correspondance [Tannery e.a.], II, p. 175.Google Scholar
  153. 152.
    Luca Valerio’s De Centro Gravitatis, as the homonymous later treatise by Guldin, determines in an Archimedean fashion the centre of gravity of various figures and solides. On Mersenne’s Synopsis and its sources, see Duhem, Les Origines, I, pp. 295–299, II, pp. 123–129, passim. Google Scholar
  154. 153.
    Roberval, Traité de méchaniques, pp. 1, 11, 15, 21.Google Scholar
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    Nova in mechanicis theoremata, Fermat, Oeuvres [Tannery e.a.], II, p. 26.Google Scholar

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  • Sophie Roux

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