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Abstract
Descartes' allusions, in the Meditations and the Principles, to the individual moments of duration, has for some years stirred controversy over whether this commits him to a kind of time atomism. The origins of Descartes' way of treating moments as least intervals of duration can be traced back to his early collaboration with Isaac Beeckman. Where Beeckman (in 1618) conceived of moments as (mathematically divisible) physical indivisibles, corresponding to the durations of uniform motions between successive impacts on a body by microscopic particles, Descartes was able to give a mathematical treatment of the problem of fall in which moments were rendered mathematical minima of motion that were necessarily devoid of extension. This achievement, coupled with his innovation of conceiving force as instantaneous tendency to motion, subsequently led him to disdain Beeckman's discretist physics with its extended indivisible moments. Nevertheless, he was not able to eradicate a fundamental tension in his philosophy between force as a quantity of motion, and force as an instantaneous tendency to motion. For by his principles, action, motion, quantity of motion, and indeed existence, all require some minimal interval of duration. This explains his need to refer to moments as least conceivable parts of duration, and this is what has given rise to the impression that he supposed duration to be composed of such parts, contrary to his commitment to continuous creation.
Notes
2 ‘Neque vim harum rationum effugio, si supponam me forte semper fuisse ut nunc sum, tanquam si inde sequeretur, nullum existentiae meae authorem esse quaerendum. Quoniam enim omne tempus vitae in partes innumeras dividi potest, quarum singulae a reliquis nullo modo dependent, ex eo quod paulo ante suerim, non sequitur me nunc debere esse, nisi aliqua causa me quasi rursus creet ad hoc momentum, hoc est me conservet. Perspicuum enim est attendenti ad temporis naturam, eâdem plane vi & actione opus esse ad rem quamlibet singulis momentis quibus durat conservandam, quâ opus esset ad eandem de novo creandam, si nondum existeret; adeo ut conservationem solà ratione a creatione differre, sit etiam unum ex iis quae lumine naturali manifesta sunt’, R. Descartes, ‘Meditations 3’, in Oeuvres De Descartes, edited by C. Adam and P. Tannery, 11 vols (Paris: Librairie Philosophique J. Vrin, 1983 (abbreviated AT VII 48–49, etc.), 48–49. All translations of primary sources here are my own; but I also reference the standard English translations, primarily from The Philosophical Writings Of Descartes, translated and edited by J. Cottingham, R. Stoothoff and D. Murdoch, 2 vols, (Cambridge: Cambridge University Press, 1988), abbreviated CSM II 28–29; and The Philosophical Writings of Descartes: The Correspondence, edited by J. Cottingham, R. Stoothoff, D. Murdoch and A. Kenny (Cambridge: Cambridge University Press, 1991), abbreviated CSM-K 28–29, etc.
1 See, for example, J.-M. Beyssade, La Philosophie Première de Descartes (Paris: Flammarion, 1979); M. Gueroult, Descartes's Philosophy Interpreted According to the Order of Reasons, edited and translated by R. Ariew, 2 vols (Minneapolis, MN: University of Minnesota Press, 1984); R.T.W. Arthur, ‘Continuous creation, continuous time: A refutation of the alleged discontinuity of Cartesian time’, Journal of the History of Philosophy, 26:3 (1988), 349–375; D. Garber, Descartes' Metaphysical Physics (Chicago, IL: University of Chicago Press, 1992), J.E.K. Secada, ‘Descartes on time and causality’, Philosophical Review, 99 (1990), 45–72; H. Frankfurt, ‘Continuous creation, ontological inertia, and the discontinuity of time’, in Necessity, Volition, and Love, edited by H. Frankfurt (Cambridge: Cambridge University Press, 1999); C. Bonnen and D. Flage, ‘Descartes: The matter of time’, International Studies in Philosophy, 32 (2000), 1–11; G. Gorham, ‘Cartesian causation: continuous, instantaneous, overdetermined’, Journal of the History of Philosophy, 42 (2004), 389–423; G. Gorham, ‘Descartes on Time and Duration’, Early Science and Medicine, 12 (2007), 28–54; G. Gorham, ‘Cartesian Temporal Atomism: A New Defence, A New Refutation’, British Journal for the History of Philosophy, 16:3 (2008), 625–637; K. Levy, ‘Is Descartes a temporal atomist?’, British Journal for the History of Philosophy, 13 (2005), 627–674.
3 See in particular Secada, ‘Descartes on time and causality’, and R. Sorabji, Time, Creation and the Continuum: Theories in Antiquity and the Early Middle Ages (Ithaca, NY: Cornell University Press, 1983).
4 I am setting aside the case of Diodorus Cronus, whose time atomism has only recently been established by the fine work of Sorabji (Sorabji, Time, Creation and the Continuum).
6 ‘Eodem modo etiam dico implicare contradictionem, vt aliquae dentur atomi, quae concipiantur extensae ac simul indivisibiles; quia, quamvis Deus eas tales efficere poterit, ut a nulla creatura dividantur, certe non possumus intelligere ipsum se facultate eas dividendi privare potuisse.’ (Descartes to More, 5th February 1649; AT V 273; CSM-K 363).
5 ‘Je n'approuve point non plus ses indivisibles […] un cors […] ne peut estre composé d'indivisibles, à cause qu'un indivisible ne peut avoir aucune longueur, largeur, ny profondeur.’ (Descartes to Lacombe; AT III 213–14; CSMK 155). I am indebted to Geoffrey Gorham for reminding me of this passage.
7 ‘Il fait considerer une ligne droite, descrite par le mouvement d'un cercle, pour prouver qu'elle est composée d'une infinité de poins actu, ce qui n'est qu'une imagination toute pure.’ (Descartes to Mersenne, 11th October 1638; AT II 383)
8 See Garber, Descartes' Metaphysical Physics, 273: ‘I will show that the issue of the continuity of time is irrelevant to Descartes’ arguments'. In fact, though, Garber argues this a few pages earlier in the book, concluding that ‘there is no strong reason for attributing either view [temporal atomism or its antithesis] to Descartes’. (269)
9 Arthur, ‘Continuous Creation’, 374.
10 AT VIIIA 26; CSM 211.
11 AT VIIIA 30; CSM 1 214.
12 See Arthur ‘Continuous Creation’, 353–355. As I also argue there (363–367), the supposition that parts of duration could be merely contiguous without being continuous also seems incompatible with Cartesian principles. Just as the lack of a real distinction between body and extension makes it impossible, when two bodies relatively at rest are adjacent with one another, for there to be two distinct endpoints of the bodies at one point of extension, so there would be no real distinction between the endpoints of contiguous parts of a thing's duration, so that all such contiguous parts would form a continuous stretch.
13 Gorham, ‘Cartesian Temporal Atomism’, 626–627.
14 In his reply to Gassendi, Descartes writes: ‘And this is clearly demonstrated by what I explained about the independence of the parts of time, which you try in vain to evade by proposing “the necessity of the sequence which exists among the parts of time” considered in the abstract. It is not this that is at issue here, but rather the time or duration of the enduring thing, and you will not deny that the individual moments of this time could be separated from those next to them, that is, that the enduring thing could at any single moment cease to exist. [Hocque apertè demonstratur ex eo quod explicui de partium temporis independentiâ, quodque frustra conaris eludere, proponendo necessitatem consecutionis quae est inter partes temporis in abstracto considerati; de quo hîc non est quaestio, sed de tempore, seu duratione rei durantis, cujus non negas singula momenta posse a vicinis separari, hoc est rem durantem singulis momentis desinere esse.]’, AT VII 369–370; CSM II 254–255).
15 Gorham ‘Cartesian Temporal Atomism’, 635; compare with Descartes' ‘It is a manifest contradiction for them [bodies] to be apart, or to have a distance between them, when the distance in question is nothing [ac manifestè repugnat ut distent, sive ut inter ipsa sit distantia, & tamen ut ista distantia sit nihil.]’, AT VIIIA 50; CSM I 231.
16 Sorabji, Time, Creation and the Continuum, 297–298. In his discussion of Augustine's and Malebranche's shared doctrine that continuance of any thing in existence depends on God, Sorabji also attributes the crucial difference between Malebranche's occasionalism and Augustine's non-occasionalist view to the fact that ‘what [Malebranche] adds, and what is missing, so far as I know, from Augustine, is the idea that continuation depends on continuous re-creation’ (304). Continuous re-creation is continuous creation interpreted as involving discrete time, so again we see continuous creation, occasionalism, and temporal atomism understood as a package.
17 Compare with Sukjae Lee's comment: ‘Given how many prominent Cartesians were indeed occasionalists, it is not surprising that Descartes has been suggested as the source of this new wave of occasionalism that swept through the European continent in the second half of the seventeenth century’, S. Lee, ‘Occasionalism’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/occasionalism/ (2008). But, in fact, many qualifications have to be made. As Nadler has argued persuasively, La Forge was ‘an occasionalist with respect to bodies, to matter in motion’, but ‘not an occasionalist when it comes to the mind’, S. Nadler, ‘Louis de La Forge and the Development of Occasionalism: Continuous Creation and the Activity of the Soul,’ Journal of the History of Philosophy, 36:2 (1998), 215–231 (224); similarly Clerselier. But I will not dwell on such niceties here, since all these thinkers denied body-body causation, which I think is sufficient to distinguish them from Descartes.
18 For this conclusion, see especially the arguments of D. Garber, ‘How God Causes Motion: Descartes, Divine Sustenance and Occasionalism’, The Journal of Philosophy, 84 (1986), 567–580; Garber, Descartes' Metaphysical Physics; and of S. Nadler, ‘Louis de la Forge’; S. Nadler, ‘The Occasionalism of Louis de La Forge’ in Causation in Early Modern Philosophy, edited by S. Nadler (University Park, PA: Pennsylvania State University Press, 1993) 57–73; S. Nadler, ‘Cordemoy and Occasionalism’, Journal of the History of Philosophy, 43:1 (2005), 37–54.
19 R.T.W. Arthur, ‘Time Atomism and Ash'arite Origins for Cartesian Occasionalism, Revisited’ in Asia-Europe Dialogue and the Making of Modern Science, edited by A. Bala (Palgrave Macmillan, forthcoming).
20 See Sorabji's excellent discussion in his Time, Creation and the Continuum, chapter 24: Atoms and Time-Atoms after Aristotle, 365–383, and chapter 25: Atoms and Divisible Leaps in Islamic Thought, 384–402.
21 Here I have relied on M. Fakhry, Islamic Occasionalism and its Critique by Averröes and Aquinas (London: George Allen and Unwin, 1958). Although al-Nazzam held that substances and their accidents revert back to nothingness before being recreated each moment, the attribution of this view to the Mûtakallimûn has been disputed by A. Dhanani, The Physical Theory of Kalam (Leiden: EJ Brill, 1994), 43–47. See Nadler, ‘Louis de La Forge’, 17, 217.
22 Nadler, ‘Louis de La Forge’, 219.
23 Malebranche, Entretiens, VII, §7; OC 12:157–58; quoted in translation from Nadler's illuminating article ‘Louis de La Forge’, 217.
24 Augustine, de. Gen. ad Lit. 4.12.22; quoted in translation from Sorabji, Time, Creation and the Continuum, 304. The same doctrine is repeated by St. Thomas Aquinas: ‘Creatures are conserved in being by God […] For the being of each creature depends on God in such a way that, unless creatures are conserved in being by the operation of the divine power, they could not subsist for a moment but would be reduced to nothing [Creaturae conservantur in esse a Deo […] dependet enim esse cujuslibet creaturae a Deo ita quod nec ad momentum subsistere possent sed in nihilum redigerentur, nisi operatione divinae virtutus conservantur in esse]’, Summa Theologiae, Ia., q. l o 4, art. 1, responsio; quoted from Nadler, ‘Louis de La Forge’, 216.
25 ‘Cùm negas nos continua causae primae influxu in digère, ut conservemur, negas rem quam Metaphysici omnes ut manifestam affirmant, sed de quâ saepe illiterati non cogitant, quia tantùm ad causas secundum fieri, non autem secundum esse, attendunt. Sic Architectus est causa domûs & pater filii secundum fieri tantùm, ideoque, cùm opus absolutum est, potest absque istiusmodi causâ remanere; sed sol est causa lucis ab ipso procedentis, & Deus est causa rerum creatarum, non modo secundum fieri, sed etiam secundum esse, ideoque débet semper eodem modo influere in essendum, ut eundem conservet’, AT VII 369–370; CSM II 254–255.
26 Sorabji, Time, Creation and the Continuum, 302–303.
27 Sorabji, Time, Creation and the Continuum, 302.
28 I therefore dissent from Sukjae Lee's conclusion of his SEP article on occasionalism that ‘If we bracket the issue of how to render a full-blown occasionalism with a sufficiently robust account of free will, then, as we have hopefully seen, the CCC argument is a fairly powerful argument as an argument for global occasionalism’, Lee, ‘Occasionalism’.
29 For a defence of the view that Descartes really ascribes a force of a body's motion to a moving body, see R.T.W. Arthur, ‘Beeckman, Descartes and the Force of Motion’, Journal for the History of Philosophy, 45:1 (2007), 1–28. I think force is in body only so long as God sustains it by his creative action. I am not persuaded by Dan Garber's arguments that ‘Cartesian force is nowhere at all’, Garber, Descartes' Metaphysical Physics, 297.
30 I do not mean to deny that, for Descartes, God is the ultimate cause of everything in the universe that does not depend on human free will. As he writes to Elizabeth, ‘he would not be supremely perfect if anything could happen in the world without coming entirely from him [& il ne seroit pas souuerainement parfait, s'il pouuoit arriuer quelque chose dans le monde, qui ne vint pas entièrement de luy]’, AT IV 314; CSMK 272. But this does not entail that he is the immediate cause, as in occasionalism. On my reading, bodies cause things to happen provided they are sustained in existence and in motion by God. For a different reading, see Gorham, ‘Cartesian causation’, especially 410 ff.
31 Compare with Geoffrey Gorham's argument that the independence of the parts of duration for Descartes is a consequence of their causal non-simultaneity, not their discreteness: ‘Descartes derives the key premise of his argument—the independence of the parts of time—from the very simple assumption that causes and effects are necessarily simultaneous’, Gorham, ‘Cartesian causation’, 390.
32 ‘Neque enim illum conservat nisi praecisè qualis est eo ipso temporis momento quo conservat, nullâ habitâ ratione ejus qui forte fuit paulô antè. Ac quamvis nullus motus fiat in instanti, manifestum tamen est omne id quod movetur, in singulis instantibus quae possunt designari dum movetur, determinatum esse ad motum suum continuandum versus aliquam partem, secundùm lineam rectam, non autem unquam secundùm ullam lineam curvam’, AT VIIIA 63–64; CSM I 242.
33 ‘Que, lors q'un corps se meut, encore que son mouvement se fasse le plus souvent en ligne coube, & qu'il ne s'en puisse jamais faire aucun, qui ne soit en quelque façon circulaire, ainsi qu'il a esté dit cy-dessus, toutesfois chacune de ses parties en particulier tend tousjours à continuer le sien en ligne droite. Et ainsi leur action, c'est à dire l'inclination qu'elles ont à se mouvoir, est differente de leur mouvement’, AT XI 43–44; CSM I 96.
34 ‘Cette Regle est appuyé sur le mesme fondement que les deux autres, & ne dépend que de ce que Dieu conserve chaque chose par une action continuë, & par consequent, qu'il ne la conserve point telle qu'elle peut avoir esté quelque temps auparavant, mais précisément telle qu'elle est au mesme instant qu'il la conserve’, AT XI 44; CSM I 96.
35 ‘supposant qu'il a mis certaine quantité de mouvemens dans toute la matiere en general, dés le premier instant qu'il l'a creée, il faut avouer qu'il y en conserve toujours autant, ou ne pas croire qu'il agisse toujours en mesme sorte’, AT XI 43; CSM I 96.
36 G.W. Leibniz, Philosophical Texts, translated and edited by R.S. Woolhouse and R. Francks (Oxford: Oxford University Press, 1998), 157.
37 Leibniz, Philosophical Texts, 157.
38 This has already been observed by Ed Slowik in his perceptive analysis of Cartesian physics: ‘In one sense, of course, [quantity of motion] is non-instantaneous; i.e., it includes the Cartesian concept of speed, which is only manifest over a non-instantaneous temporal period […] Yet, [quantity of motion] is also closely tied to the instantaneous property which we can loosely entitle, ‘tendency’ […]', E. Slowik, Cartesian Spacetime (Dordrecht/Boston/London: Kluwer Academic Publishers, 2002), 113.
41 JIB I 261; AT X 58.
39 ‘Haec ita demonftravit Mr. Peron, cum ci anfam praebuissem, rogando an possit quis scire quantum spacium res cadendo conficeret unicà horà, cùm scitur quantum conficiat duabus horis, fecundùm mea fundamcnta, viz. quod semel movetur, femper movetur, in vacuo, & supponendo inter terram & lapidem cadentem esse vacuum’ (23rd November–26th December 1618), AT X 60; and in C. De Waard, Journal tenu par Isaac Beeckman de 1604 à 1634, 4 vols (The Hague: M. Nijhoff, 1939–1953), hereafter JIB, with volume and page (JIB I 263).
40 JIB I 264.
42 ‘Sic termini hi octo ad 16 se habent ut 36 ad 136, quod nondum est ut 1 ad 4. Si igitur descensus lapidis fiât per distincta intervalla, trahente terra per corporeos spiritus, erunt tamen haec intervalla seu momenta tam exigua, ut proportio eorum arithmeticà ob multitudinem particularum, non sensibiliter fuerit minor quàm 1 ad 4’, AT X 61; JIB I 263; 23rd November–26th December 1618.
43 ‘In propositâ quaestione, ubi imaginatur singulis temporibus novam addi vim quâ corpus grave tendat deorsum, dico vim illam eodem pacto augeri, quo augentur lineae transversae de, fg, hi, & aliae infinitae transversae, quae inter illas possunt imaginari. Quod vt demonstrem, assumam pro primo minimo vel puncto motûs, quod causatur à prima quae imaginari potest attractivâ vi terrae, quadratum alde. Pro secundo minimo motûs, habebimus duplum, nempe dmgf: pergit enim ea vis quae erat in primo minimo, & alia nova accedit illi aequalis. Item in tertio minimo motûs […]’, JIB IV 49; AT X 75–76.
45 ‘Quôd si denique pro illo minimo assumam verum minimum, nempe punctum, tum illae partes protuberantes nullae erunt, quia non possunt esse totum punctum, ut patet, sed tantùm média pars minimi alde; atqui puncti media pars nulla est’, AT X 77.
44 JIB IV 50.
46 AT X 60; JIB I 262.
47 AT X 59; JIB I 262.
48 ‘Si verô momentum minimum spatii sit alicujus quantitatis, erit arithmetica progressio. Nec poterit sciri ex uno casu, quantum singulis horis perficiat; sed opus erit duobus casibus,ut inde sciamus quantitatem primi momenti. Ita autem ego supposueram; at, quia magis placet suppositio momenti indivisibilis, hœc non explicabo fusius’, AT X 61; JIB I 263.
49 ‘Placuit quidem autem nobis triangularis haec proportio, non quod reverâ non foret aliquod <minimum> physicum mathematicé divisibile spacium, per quod minimum physica vis attractiva rem movet (vis enim haec non est revera continua, sed dicreta est, ut beligicè loquar, sy trect met clyne hrtkens, ac propterea constant augmenta praedita, ex verâ arithemticâ progressione); sed placuit, inquam, quia hoc <minimum> est tam parvum et insensibile, ut propter multitudinem terminorum progrsssionis, proportio numerorum non sensibiliter differat à proportione triangulari continua’, AT X; JIB I 264.
50 Alexandre Koyré, for example, in his masterful work Galileo Studies, writes: ‘However, it should not be forgotten that Beeckman, while he was certainly a good physicist, was a rather mediocre mathematician’, A. Koyré, Galileo Studies (translation of Études Galiléennes (1939) by John Mepham) (Atlantic Highlands, NJ: Humanities Press, 1978), 86.
51 ‘Hinc autem sic concludit: cum vis celeritatem faciens crescat semper aequaliter, nempe singulis momentis unitate, resistentia vero aeris celeritatem impediens semper inaequaliter, nempe 1° momento sit quidem minor unitate, sed aliquantulum augeatur secundo momento et sequentibus, necessario, inquit, eo usque perveniet ut ista resistientia sit aequalis impulsui gravitatis, tantumque detrahat ex celeritate quantum vis gravitatis adiungit. Eo autem momento quo id contingit, certum est, inquit, pondus celerius non descendere quam momento proxime praecedenti; sed neque sequentibus momentis celeritas augebitur vel minuetur, quia deinceps aeris resistentia manet aequalis […]’, AT I 91.
52 JIB IV 172; AT I 92.
53 JIB IV 172; AT I 92.
54 ‘ainsy de suitte aus autres momans l'empeschemant de l'aer sera, 15/16, 31/32, 63/64, 127/128, 255/256 et sic in infinitum […] Ac proinde nunquam tantum detrahitur de celeritate per resistentiam aeris quantum ei accrescit per gravitatem, quae nempe singulis momentis illam auget unitate’, JIB 172; AT I 93–94.
55 G. Galilei, Dialogues Concerning Two New Sciences, translated by H. Crew and A. de Salvio (New York, NY: Dover Galileim 1954), 62–63.
56 K. Van Berkel, ‘Descartes’ Debt to Beeckman', in Descartes' Natural Philosophy, edited by S. Gaukroger, J. Schuster and J. Sutton (London/New York: Routledge, 2000), 46–59.
57 ‘Il suppose que la vitesse des poids qui descendent, s'augmente tousiours esgalement, ce que i'ay autrefois creu comme luy; mais ie croy maintenant sçauoir par demonstration qu'il n'est pas vray’, Descartes to Mersenne, AT II 386; CSMK 126.
58 AT II 399; CSMK 128. Indeed, according to the fine analysis of Carla Palmerino (C.R. Palmerino, ‘Infinite Degrees of Speed: Marin Mersenne and the Debate over Galileo's Law of Free Fall’, Early Science and Medicine, 4:4 (1999), 269–328), Descartes was forced, because of the principle of the conservation of the quantity of motion, ‘to assume that all heavy bodies received a determined degree of speed at the very moment at which they collided with the subtle matter and that this speed increased with each successive collision, the limit of acceleration being given by the very speed of the pushing medium. The motion of fall was therefore not uniform. Instead, one had to assume that the higher the speed of the falling body, the less the effect exercised on it by the little pushes of the subtle matter’ (294–295). This, of course, only heightens the irony of Descartes' criticisms of Beeckman. (My thanks to Geoff Gorham for bringing this article to my attention.)
59 ‘The things which are perceivable by the senses are helpful in enabling us to conceive Olympian matters. The wind signifies spirit; movement with the passage of time signifies life; light signifies knowledge; heat signifies love; and instantaneous activity signifies creation [Sensibilia apta concipiendis Olympicis: ventus spiritum significat, motus cum tempore vitam, lumen cognitionem, calor amorem, activitas instantanea creationem]’, AT X 218; CSM I 5.
60 AT XI 44; CSM I 96.
61 AT VIIIA 63; CSM I 242.
62 AT VIIIA 66; CSM I 243.
63 S. Gaukroger, Descartes: An Intellectual Biography (Oxford: Oxford University Press, 1995), 89; compare with J.A. Schuster, Descartes and the Scientific Revolution, 1618-1634, 2 vols (Ann Arbor, MI, 1977), I, 99.
64 Thus, while I am in complete accord with Gaukroger that ‘Descartes is committed to the doctrine of instantaneous action from early 1619 onwards’ (Gaukroger, Descartes, 13), I reject his claims that ‘the metaphysical doctrine of the instantaneous nature of divine action first appears in 1640’, and that it is ‘far more likely to be a metaphysical legitimation of a long-held physical theory’ (Gaukroger, Descartes, 13). As we have seen, it is in Le Monde as well as the Olympica, and appears to have the same metaphysical import from the beginning.
65 ‘Ex quibus patet, si imaginetur, verbi gratiâ, lapis ex a ad b trahi à terrâ in vacuo per vim quae aequaliter ab illâ semper fluat, priori remanente, motum primum in a se habere ad ultimum qui est in b, ut punctum a se habet ad lineam bc; mediam verò partem gb triplo celerius pertransiri à lapide, quàm alia media pars ag, quia triplo majori vi à terrâ trahitur: spatium enim fgbc triplum est spatii afg, ut facilè probatur; & sic proportione dicendum de caeteris partibus’, AT X 77.
66 ‘Unde fit triangulus ACD qui repraesentat augmentum celeritatis motus in descenfu ponderis ab A usque ad C, & ABE qui repraesentat augmentum celeritatis in priori media parte spatii quod pondus percurrit, & trapezium BCDE quod repraesentat augmentum celeritatis in posteriori media parte spatii quod pondus percurrit, nempe BC. Et cum trapezium BCDE fit triplo maius triangulo ABE, ut patet, inde sequitur pondus triplo celerius descensurum a B ad C quam ab A ad B’, AT I 72–73; CSMK 9.
68 Aristotle, Physics vi.2, 232a 24–27
69 That Descartes applied this rule is also noted by Palmerino: ‘By applying the medieval theory of proportions according to which “for equal spaces velocities are inversely proportional to times elapsed,” Descartes arrives at the conclusion that the time required to cross the distance AB is three times longer than the time required for BC’, Palmerino, ‘Infinite Degrees of Speed’, 284.
67 R.T.W. Arthur, ‘On the mathematization of motion before instantaneous velocity’, unpublished workshop paper for On the Contested Expanding Rôle of Applied Mathematics from the Renaissance to the Enlightenment, Pisa, 13 September 2010.
70 See A. Le Tenneur, De Motu Naturaliter Accelerato Tractatus Physico-Mathematicus (Paris: Ludovic Boullenger, 1649); and P. Gassendi, ‘Epistolae tres de Proportione qua gravia decidentia accelerantur’, in Opera Omnia, 6 vols (Lyon, 1658) (reprinted in Stuttgart-Bad Canstatt: Friedrich Frommann, 1964); III, 564ff.
71 ‘nempe prima linea dénotât vim celeritatis impressam 1 o momento, 2 a linea vim impressam 2° momento, 3 a vim 3 o inditam, & sic consequenter’, AT I 72; CSMK 8.
72 See Arthur, ‘Beeckman, Descartes and the Force of Motion’.
73 ‘Mais le conatus, ainsi acquis et accru dans la durée, au moins dans la durée minimale d'un moment, peut être attribué au mobile pris à un instant de son mouvement et en un point de sa trajectoire’; J.-M. Beyssade, La Philosophie Première de Descartes (Paris: Flammarion, 1979), 141.
74 Compare with Slowik, Cartesian Spacetime, 114: ‘In fact, bearing in mind that Cartesian motion does not take place at the level of instants, it becomes difficult to draw a sharp distinction between instantaneous tendencies towards motion, as measured by [the quantity of motion], and infinitesimal displacement, as incorporated in the [General Statical Principle]’.
75 Garber, Descartes' Metaphysical Physics, 273.
76 Garber, Descartes' Metaphysical Physics, 273.