Abstract
This Chapter deals with the early physico-mathematics of Descartes in 1618–1620, which he pursued at first in conjunction with his mentor, Isaac Beeckman, who also had conveyed to him his first inkling of corpuscular-mechanism as an approach to natural philosophy. In the physico-mathematical program, the traditional view of the mixed mathematical sciences as subordinate to natural philosophy, and devoid of explanatory power, was challenged. The mixed mathematical disciplines were intended to become more integrally linked to questions of matter and cause; in other words, to questions of a natural philosophical type. In the case of Descartes and Beeckman, this meant an unsystematized, but firmly held, corpuscular-mechanism. The Chapter deals with three case studies of Descartes’ physico-mathematics: his manuscript on hydrostatics and the hydrostatic paradox; his work with Beeckman on the nature of accelerated fall, which is treated here in a new way as an exercise in physico-mathematics; and a widely overlooked, but extremely important, geometrical and physical optical fragment on refraction of light, adapted from bits of the work of Kepler.
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- 1.
Beeckman was born in Middelburg on 10 December 1588. He was first intended for the reformed ministry and studied theology at Leiden between 1607 and 1610. There he also came in contact with Rudolph Snel, the Ramist practical mathematician and pedagogue. This connection is of potentially great significance for the interpretation of Beeckman’s career, for Snel offers a prime example of the tendency of late sixteenth century Ramism to concern itself with problems of the practice and pedagogy of the mechanical arts and applied mathematics. See Hooykaas (1981) and biographical note by C. de Waard in Mersenne (1932–88) ii. 217; Mahoney (1981); Ong (1958), 305; Vollgraff (1913). The most important work on Beeckman of the last generation without doubt is Klaas van Berkel (1983). The author informs me that the long awaited English translation of this work is presently being prepared by Maartin Ultee for the Johns Hopkins University Press. (Personal communication 15 September 2009.)
- 2.
The last extant letter from this period dates from 29/4/1619, AT. X. p. 164.
- 3.
Descartes to Beeckman, 23/4/1619, AT. X. pp. 162–3.
- 4.
With a few notable and important exceptions—Gaukroger (1995), van Berkel (1983), Shea (1991), and Garber (1992)—who all published subsequent to my (1977). Cartesian scholars have tended to minimize the import of Descartes’ friendship with Beeckman. The literature has understandably focused attention on the metaphysical and epistemological aspects of Descartes’ thought, and to the extent that it has dealt with Descartes’ natural philosophy at all, it has usually stressed the novelty of his enterprise. The lack of appreciation of the similarities between the natural philosophical enterprises of Descartes and Beeckman has perhaps been reinforced by an implicit bias toward accepting Descartes’ account of their relations. That account derives in large measure form Descartes’ correspondence concerning a dispute with Beeckman which broke out in the early 1630s when Descartes was writing Le Monde. Descartes’ complaint rested on Beeckman’s remarks to Marin Mersenne to the effect that the had been Descartes’ ‘master’ for ten years; that he had taught Descartes whatever he knew about music; and that he had invented many natural philosophical ideas and recorded them in his Journal long before Descartes decided to put similar ones into print. Additionally, as Descartes had learned, Beeckman was also on the verge of preparing his disparate natural philosophical manuscripts for publication (see below notes 11 and 12 and corresponding texts). Beeckman’s claims are undoubtedly exaggerated, and the novelty of Descartes’ natural philosophical vision, emerging in Le Monde, cannot be denied. But one should not ignore the very real, if somewhat elusive nature of Beeckman’s influence on Descartes’ career in natural philosophy, and hence one should not dismiss Beeckman’s dismay at the prospect of Descartes publishing a system of natural philosophy. Descartes’ debt to Beeckman was quite complex, not only on the basis of their early interaction, but also in the light of Descartes’ dealings with him in the late 1620s, after an absence of 10 years. Descartes was to emulate some of the natural philosophical concerns of Beeckman and part of his style of explanation; but, he also found inadequacies in Beeckman’s work and developed his own ideas partly in response to them, as we shall see in Sect. 10.3.
- 5.
Descartes alludes to the study of ‘mechanics’ and ‘geometry’ in the correspondence with Beeckman: 26/3/1619, AT, X. p. 159 l.13; and 23/4/1619, AT. X. p. 162 l.15.
- 6.
- 7.
See examples of this below, and in Beeckman’s case later in 1620s in the context of celestial mechanical speculations in Sect. 10.3.
- 8.
As is the case with all the early writings, no exact date can be assigned to the hydrostatics manuscript. Some internal evidence suggests that Descartes composed it shortly before Beeckman left Breda at the beginning of 1619: see AT x. 69 l.15 and 74 l.23 which seem to imply that Beeckman and Descartes had recently discussed these problems in person. Adam and Tannery note that the ‘Physico-Mathematica’ were misplaced in Beeckman’s Journal, having been transcribed along with the Compendium Musicae between two entries for 20 April 1620 (AT. x. 26–7). By that time Descartes himself was in Germany and no longer in contact with Beeckman. If, as seems to be the case, the ‘Physico-Mathematica’ were composed around the same time as the Compendium of Music which was a New Year’s gift to Beeckman, then it again seems very likely that the hydrostatics manuscript dates from late 1618 or early 1619. The Compendium is not treated in this chapter on Descartes as a physico-mathematician for the simple reason that this early work of Descartes shows hardly any traits of physico-mathematics, staying almost entirely within the realm of traditional mixed mathematics. Zarlino’s views on consonance are followed, but derived as much as possible from geometrical considerations. There is a brief early passage, inserted according to a suggestion by Beeckman, dealing with the physical vibrations actually made by a string, but it does not affect the tenor of the bulk of the piece. At no point do physico-mathematical protocols of the sort we will unpack here make an appearance. On the content and tenor of the Compendium the key work is by Floris Cohen (1984). See also the discussion in Gaukroger (1995), pp. 74–80, who points out that at this early stage Descartes is oblivious to recent developments by Benedetti, Vincenzo Galileo and Beeckman himself, who all recognised difficulties with Zarlino’s arithmetical treatment of consonance and thus turned their attention to conceptualizing consonance, and its problems, as due to the coincidence of sound vibrations (variously physically explicated). Descartes keeps the Compendium within the realm of mixed mathematics, rather than opening up this potentially physico-mathematical domain.
- 9.
Of course, in his own natural philosophizing Descartes would eventually employ a very different notion of just what the principles of mechanics are which provide the causal dimension of his mechanical philosophy. In addition, unlike Beeckman, Descartes would later be drawn into serious concern about the metaphysical grounding of his natural philosophy and the epistemic status of his claims. Nevertheless, from Beeckman came the inspiration for a new species of natural philosophy, as well as a considerable portion of its content.
- 10.
It was edited by his brother Abraham after Beeckman’s death and appeared as Beeckman (1644).
- 11.
For the details of this episode, see van Berkel (2000).
- 12.
Beeckman (1939–53) ii. 99.
- 13.
See Paolo Rossi (1970), 1–62, and our historiographical observations above Sect. 2.7.
- 14.
Beeckman (1939–53) i. 25.
- 15.
Beeckman to Mersenne, 1 October 1629, Mersenne (1932–88) ii, 283, ‘nihil enim in philosophia admitto quam quod imaginationi velut sensile representatur.’ Cf. the demands that Descartes was to place on mathematics and ‘mathematical’ natural philosophy in the latter portions of the Regulae, written in the late 1620s, as well as his insistence on the ‘figurate’ representation of problems to be solved, both in mathematics and in optics and natural philosophy generally, on which in general see Sepper (2000) and which we will see illustrated in the early physico-mathematical work below.
- 16.
Beeckman (1939–53) ii. 77–8. Similarly, Aristotelian ‘philosophical’ arguments against the existence of the void carried less weight against his atomism than the transdiction of the ‘metaphysical’ objection that perfectly hard atoms lacking pores cannot undergo rebound (ibid. p. 100). He was obviously disturbed by his inability to conceive of a convincing macroscopic model for hard body rebound. Mechanical common sense seemed to indicate atoms do not exist.
- 17.
Prior to 1616 Beeckman had spent a few years in the trade of candle making and also followed his father’s craft of laying water conduits, especially for breweries. Many of the notes in his Journal reveal that Beeckman saw connections between practical questions raised in relation to his craft activities and the teachings he had received from the elder Snel, as well as the writings of Willebrord Snel and Simon Stevin. Beeckman, however, did not plan on remaining a practitioner of the mechanical arts, albeit a highly educated and philosophically literate one. In 1618 he took an M.D. degree at Caen. From November 1619 he was Conrector of the Latin School at Utrecht, and in December 1620 he moved to Rotterdam, where his brother was Rector of the Latin School. Beeckman gave lessons and became Conrector in 1624. He also founded a ‘collegium mechanicum’, or society for craftsmen and scholars interested in natural philosophical questions with technical import. In 1627 he became Rector of the Latin School at Dordrecht, a position he held until his death in 1637.
- 18.
Beeckman (1939–53) ii. 86.
- 19.
Ibid, pp. 86, 96; cf. Beeckman (1939–53) iii. 138, ‘Ignis minimum non est atomus sed homogeneum ex atomis compositum.’
- 20.
Beeckman (1939–53) ii. 100–1.
- 21.
Beeckman (1939–53) iii. 31 Beeckman’s theory of light provides a good example: He held light to be corporeal and to consist in the finest particles of elemental heat or fire. Because light can be reflected and refracted (to Beeckman refraction was a form of internal reflection), it cannot consist in isolated atoms; therefore, light, heat and fire had to be conceived as second order homogenous composites made up of numerous atoms and void space.
- 22.
See Gaukroger and Schuster (2002), p. 545. For example, the basic principle behind the Mechanica’s treatment of the lever (set out in a number of passages in Aristotle) ‘holds that the same force will move two bodies of different weights, but it will move the heavier body more slowly, so that the velocities of the two bodies are inversely proportional to their weights. When these weights are suspended from the ends of a lever, we have two forces acting in contrary directions, and each body moves in an arc with a force proportional to its weight times the length of the arm from which it suspended. The one with the greater product will descend in a circular arc, but if the products are equal, they will remain in equilibrium.’ In contrast, the purely statical and mathematical approach of Archimedes ‘makes statics a mathematical discipline independent of any general theory of motion, whilst that of the Mechanica makes statics simply a limiting case of a general dynamical theory of motion, a theory which is resolutely physical. In other words, the Mechanica account comes as part of a package which is driven by Aristotelian dynamics, above all by the principle of the proportionality of weight and velocity. This did not stop a number of mathematicians, such as Benedetti, Tartaglia, and Galileo, from trying to revise the package, hoping they could salvage the dynamical interpretation of the beam balance and simple machines while jettisoning the natural philosophy that lay behind it, but the pivotal role this natural philosophy had played meant that such a revision could never be successful, as we shall see below when we consider Galileo’s attempt to realize this program. The Archimedean account, by contrast, comes without any dynamical, or more broadly speaking physical, commitments: put more strongly, it comes without any physical content.’
On the influence of the Mechanica in the sixteenth and seventeenth centuries, see Duhem (1905–6), Rose and Drake (1971), and Laird (1986). The Mechanica, which is probably the work of Strato or Theophrastus, was traditionally attributed to Aristotle, an attribution which Duhem and Carteron (1923) follow. The work is Aristotelian in tenor, but has the peculiar feature that whereas Aristotelian natural philosophy confines itself to natural processes, for it is these that follow from the nature of things, the subject matter of the Mechanica, as is explained in the opening sentence of the work is ‘those phenomena that are produced by art despite nature, for the benefit of mankind.’
- 23.
Beeckman (1939–53) I. 24–5. I have employed the typescript translation by the late Michael S. Mahoney, Princeton University.
- 24.
Ibid.
- 25.
Beeckman even tried to explain the centrifugal tendency of bodies moving in circular motion in resisting media as the result of the combination of circular inertia and differential resistance of the medium on different parts of the body. Beeckman (1939–53) i. 253.
- 26.
Ibid, 25.
- 27.
See Appendix I in Mersenne (1932–88) ii. 632–44, which includes de Waard’s notes.
- 28.
Beeckman’s rules fall into two broad categories: (1) cases in which one body is actually at rest prior to collision, and (2) cases which are notionally reduced to category (1). The concept of inertia and the stipulation that only external impacts can change the state of motion of a body provide the keys to interpreting instances of the first category. The resting body is a cause of the change of speed of the impacting body and it brings about this effect by absorbing some of the quantity of motion of the moving body. Beeckman invokes an implicit principle of the directional conservation of quantity of motion to control the actual transfer of motion. In each case the two bodies are conceived to move off together after collision at a speed calculated by distributing the quantity of motion of the impinging body over the combined quantities of matter of the two bodies. For example, in the simplest case, in which one body strikes an identical body at rest, ‘…each body will be moved twice as slowly as the first body was moved…since the same impetus must sustain twice as much matter as before, they must proceed twice as slowly.’ And he adds, analogizing the situation to the mechanics of the simple machines, ‘…it is observed in all machines that a double weight raised by the same force which previously raised a single weight, ascends twice as slowly.’ (Beeckman 1939–53, i. 265–6) Instances of the second category of collision are assessed in relation to the fundamental case of collision of equal speeds in opposite directions (ibid, 266). Being perfectly hard and hence lacking the capacity to deform and rebound, the two atoms annul each other’s motion, leaving no efficacious residue to be redistributed to cause subsequent motion. This symmetrical case, which was also generalized to cases of equal and opposite quantities of motion arising from unequal bodies moving with compensating reciprocally proportional speeds, derives from a dynamical interpretation of the equilibrium conditions of the simple machines. Instances in which the quantities of motion of the bodies are not equal are handled by annulling as much motion of the larger and/or faster moving body as the smaller and/or slower body possesses (Beeckman 1939–53, i. 266.) This in effect reduces the smaller and/or slower body to rest. The outcome of the collision is then calculated by distributing the remaining unannulled motion of the larger and/or swifter body over the combined quantities of matter of the two bodies (ibid). It is obvious that Beeckman viewed this case through a two-fold reference to the simple machines; for the first extracts as much motion as can conduce to the equilibrium condition for symmetrical cases, and then he invokes the principle cited just above in this note to determine the final outcome.
- 29.
Beeckman (1939–53) iii. 133–4. Consider for example his commentary in 1629 upon a remark made by Mersenne in his Traité de l’Harmonie universelle (1627) to the effect that, ‘Vitesse ou tardivité du mouvement cause de tout ce qui se fait par bilances.’ As Beeckman’s entry shows, he fundamentally agreed with this dynamicist interpretation of the principles of the simple machines: ‘The reason for this fact can be rendered very easily by those things which I wrote a little before concerning motion. For it follows from them that a sphere twice as heavy [as another sphere], that is, having twice as much matter, but moving twice as slowly [as the other sphere], will be stopped after colliding with it, that is, both spheres will be at rest. For I specified that mass and motion compensate for one another [se reciprocari]. The same thing must also be concluded concerning the balance.’ Despite some confusions Beeckman introduced in the explication of this point, his central contention is clear enough: even macroscopic equilibrium is a consequence of the laws of motion and impact, because it can be explained through a dynamical interpretation of countervailing motions on the model of the laws of collision. He closes with a clear statement of this point: ‘One should not doubt how an account is given here of the theory of equilibrium [in isorhopicis] by means of motion. For even if there is no motion when bodies hang in equilibrium, motions would however take place immediately if an external force, a weight, etc. were to displace these weights from equilibrium. Moreover, all bodies that return to their own places as soon as they are moved from them never change their places of their own accord. Thus stones never ascend spontaneously and in the absence of an external force. Bodies which are at rest in our vicinity never spontaneously move.... The cause of equilibrium therefore can be motion, even if the bodies in equilibrium are not moved. For the cause of equilibrium is past and future motion. During the present, to be sure, the body is at rest because past and future motions occasion rest.’
- 30.
AT x. 52. ‘Physico-mathematici paucissimi’. In this regard Beeckman was to note in 1628 that his own work was deeper than that of Bacon on the one hand and Stevin on the other just for this very reason. Beeckman (1939–53) iii. 51–2, ‘Crediderim enim Verulamium [Francis Bacon] in mathesi cum physica conjugenda non satis exercitatum fuisse; Simon Stevin vero meo judico nimis addictus fuit mathematicae ac rarius physicam ei adjunxit.’
- 31.
The text, Aquae comprimentis in vase ratio reddita à D. Des Cartes which derives from Beeckman’s diary, appears at AT x., 67–74, as the first part of the Physico-Mathematica. See also the related manuscript in the Cogitationes Privatae, AT x. 228, introduced with, ‘Petijt e Stevino Isaacus Midlleburgensis quomodo aqua in funda vasis b…’.The expression ‘hydrostatics manuscript’ appears in Schuster (1977, 1980, 2005), Gaukroger (1995), and Gaukroger and Schuster (2002).
- 32.
- 33.
Ibid, i. 415. ‘We have to prove that on the bottom EF there rests a weight equal to the gravity of the water of the prism GHFE. If there rests on the bottom EF more weight than that of the water GHFE, this will have to be due to the water beside it. Let this, if it were possible, be due to the water AGED and HBCF. But this being assumed, there will also rest on the bottom DE, owing to the water GHFE, because the reason is the same, more weight than that of the water AGED; and on the bottom FC also more weight than that of the water HBCF; and consequently on the entire bottom DC there will rest more weight than that of the whole water ABCD, which (in view of ABCD being a corporeal rectangle) would be absurd. In the same way it can also be shown that on the bottom EF there does not rest less than the water GHFE. Therefore, on it there necessarily rests a weight equal to the gravity of the water of the prism GHFE.’
- 34.
Ibid, i. 417. ‘Let there again be put in the water ABCD a solid body, or several solid bodies of equal specific gravity to the water. I take this to be done in such a way that the only water left is that enclosed by IKFELM. This being so, these bodies do not weight or lighten the base EF any more than the water first did. Therefore we still say, according to the proposition, that against the bottom of EF there rests a weight equal to the gravity of the water having the same volume as the prism whose base is EF and whose height is the vertical GE, from the plane AB through the water’s upper surface MI to the base EF.’
- 35.
AT x. 68–9. This is the second of the four puzzles posed in the text, the others are: ‘(First), the vase A along with the water it contains will weigh as much as vase B with the water it contains. … Third, vase D and its water together weigh neither more nor less than C and its water together, into which embolus E has been fixed. Fourth, vase C and its water together will weigh more than B and its water. Yesterday I was deceived on this point.’ (Descartes’ latter point is not to be confused with his proof in the text that the water in vase B and vase C will weigh equally upon their respective bases—another case exemplifying the hydrostatic paradox and argued in a manner similar to the case we are treating in detail.)
- 36.
Beeckman (1939–53) iii. 51–2.
- 37.
‘In order to set out fully my reasoning concerning the questions which have been proposed, I would first have to explain a great deal concerning the foundations of my Mechanics; but, since time does not permit this, I shall try to explain the matter briefly.’ AT X p. 67.
- 38.
AT X. 68.
- 39.
AT x. 68. In the Cogitationes Privatae (AT X. 228) the inclination to motion is described as being evaluated ‘in ultimo instanti ante motum’.
- 40.
AT x. 68.
- 41.
- 42.
Milhaud (1921), 34.
- 43.
Descartes consistently fails to distinguish between ‘points’ and finite parts. But he does tend to assimilate ‘points’ to the finite spaces occupied by atoms or corpuscles. Throughout we shall assume that Descartes intended his points to be finite and did not want his ‘proofs’ to succumb to the paradoxes of the infinitesimal.
- 44.
AT x. 70.
- 45.
AT X. 70–1.
- 46.
On the aerostatics see Sect. 8.2.3.3 below. On the cosmological version of mechanistic optics in Le Monde, see Sect. 10.7 below. By ‘cosmological’ mechanistic optics, I mean the physical theory of light as a mechanical tendency to motion caused by the corpuscular-mechanical character of the sun and other stars, as well as their vortices, and governed by certain rules of dynamics.
- 47.
This mode of explanation haunts so much of Descartes’ physical thought that one could venture to suggest that it goes a long way toward accounting for the curiously tendentious and idiosyncratic character of much of his later natural philosophical discourse. On the one hand, we can say, Whiggishly, that, after all, this style of explanation really consists in a connected sequence of ad hoc manipulations. The manipulations masquerade as clarifications, while in fact they condition a progressive loss of contact with the original aims of the problem. They close in on themselves, forming a superficially tidy universe of discourse increasingly irrelevant to the problem at hand and insulated from any fruitful return to new empirical information. On the other hand, and taking Descartes’ part as it were, we shall argue in Chap. 10, concerning Le Monde that the persistence of these protocols and motifs shows the manner in which Descartes’ first system of natural philosophy still bore the traces of (and was partially constituted by) his previous engagements with ‘physico-mathematics’.
- 48.
See Gaukroger (1995) 172–81, and Sepper (1996), 157–208. Indeed, Descartes would over time drive an agenda favoring analysis over synthesis in all mathematical pursuits: his view being that a geometrical demonstration does not reveal to us how a mathematical result is generated. Algebraic proofs, by contrast, have a transparency which reveals the path by which the conclusion is produced. The problem-solving, analytical approach to physico-mathematics in the manuscript hints at this later maturing agenda. In Chap. 5 we shall examine his early work in mathematics along these lines between 1619 and about 1625 and see how his pure mathematical work developed, and intersected with his physico-mathematical natural philosophizing, by means of his unifying dreams of first, a so-called ‘universal mathematics’, and then, his universal method.
- 49.
My use of the term ‘figuring up’, here and throughout, to denote Descartes’ idiosyncratic protocols for problem preparation in physico-mathematics was suggested to me by reflecting on Sepper’s seminal work on Descartes’ early use of imagination in ‘figuring things out’ and ‘figurate solution of problems’ (Sepper 2000).
- 50.
AT x. 68.
- 51.
AT x. 70–1.
- 52.
AT x. 71.
- 53.
AT x. 71.
- 54.
AT x. 72.
- 55.
AT x. 72.
- 56.
AT x. 159 l.11-12; and 162 l.15.
- 57.
I say to a first approximation, because whilst superficially this seems to comport with the physico-mathematics of Beeckman, we shall soon see that it is, in underlying terms, much more radical—and intentionally so.
- 58.
The representation of corpuscular tendencies to motion by means of geometrical lines is a symptom of the more radical intentions of Descartes’ species of physico-mathematics and also a partial indicator of its links to his aspirations for an analytical (rather than demonstrative) approach to mathematics, including mixed mathematical disciplines, which he intends to render more ‘physical’ or organically articulated to natural philosophizing.
- 59.
Milhaud (1921), 34–7.
- 60.
Rodis-Lewis (1971) vol. 1, 30–1.
- 61.
As was the case with many master practitioners of the practical mathematical disciplines, Stevin envisions the applications of these results to more properly practical ends; that is, a key mathematical result will command a wide domain of application in a number of practical fields. Cf. Bennett (1998).
- 62.
- 63.
AT ii. 385.
- 64.
See below Sect. 4.6 and Schuster (2000).
- 65.
Beeckman’s Journal (Beeckman 1939–53) contains Beeckman’s statement of the problem, his remarks on the mathematical arguments of Descartes and his own further comments. Journal f105v-106r, cited in AT X 58–61. The Journal also contains a set of two short essays by Descartes which have been published under the title ‘Physico–mathematica’, AT X 75–78. The first essay, as we have seen, concerns the hydrostatics. The second essay contains Descartes’ version of his contribution to the discussion of accelerated fall. Finally, in the early fragments of Descartes, published in the Adam-Tannery edition under the title Cogitationes Privatae, one entry directly concerns the matters discussed with Beeckman about fall and several others on the related theme of the mathematical representation of motions. AT X 219–222.
- 66.
- 67.
See also Jullien and Charrak (2002) 19–20, 89–96, 100–104, 107–112.
- 68.
My treatment of the material on fall differs considerably from that in Schuster (1977), which had been taken up and improved by Gaukroger (1995) 80–84. The chief difference is owing to my emphasis here on a new and improved understanding of the early physico-mathematical aspirations of Descartes. This frames the entire presentation and much of its content, although many technical details remain the same.
- 69.
AT X. p. 58.
- 70.
AT X p. 58, Koyré (1978) 80. Moventur res deorsum ad centrum terrae, vacuo intermedio spatio existente, hoc pacto: Primo momento, tantum spacium conficit, quantum per terrae tractionem fieri potest. Secundo, in hoc motu perseverando superadditur motus novas tractionis, ita ut duplex spacium secundo momento peragretur. Tertio momento, duplex spacium perseverat, cui superadditur ex tractione terrae tertium, ut uno momento triplum spacii primi peragretur. The translation has been slightly modified.
- 71.
AT X p. 61 Beeckman often speaks in this kind of shorthand for actually intended corpuscular-mechanical explanations.
- 72.
AT X p. 60.
- 73.
AT X pp. 59–60 Koyré (1978) 80–81. Cum autem momenta haec sint individua, habebit spacium per quod res una hora cadit, ADE. Spatium per quod duabus horis cadit, duplicat proportionem temporis, id est ADE ad ACB, quae est duplicata: proportio AD ad AC. Sit enim momentum spatii per quod res una hora cadit alicuius magnitudinis videlicet ADEF. Duabus hours perficiet talia tria momenta, scilicet AFEGBHCD, Sed AFED constat ex ADE cum AFE; atque AFEGBHCD constat ex ACB cum AFE & EGB, id est cum duplo AFE.
Sic, si momentum sit AIRS, erit proportio spatii ad spatium, ut ADE cum klmn, ad ACB cum klmnopqt, id est etiam duplum klmn. At klmn est multo minus quam AFE. Cum igitur proportio spatii peragrati ad spatium peragratum constet ex proportione trianguli ad triangulum, adjectis utrique termino aequalibus, cumque haec aequalia adjecta semper eo minors fiant, quo momenta spatii minora sunt; sequitur haec adjecta nullius quantitatis fore, quando momentum nullius quantitatis statuitur. Tale autem momentum est spatii per quod res cadit. Restat igitur spatium per quod res cadit una hora, se habere ad spatium per quod cadit duabus horis, ut triangulum ADE ad triangulum ACB.
- 74.
AT X p. 58.
- 75.
AT X pp. 75–7, Koyré (1978) 82–83 (translation slightly modified). In proposita quaestione, ubi imaginatur singulis temporibus novam addi vim qua corpus grave tendat deorsum, dico vim illam eodem pacto augeri, quo augentur Iineae transversae de, fg, hi. & aliae infinitae transversae, quae inter illas possum imaginari. Quod ut demonstrem, assumam pro primo minimo vel puncto motus, quod causatur a prime quae imaginari potest attractiva vi terrae, quadratum alde. Pro secundo minimo motus, habebimus duplum, nempe dmgf: pergit enim ea vis quae erat in primo minimo, & alia nova accedit illi aequalis. Item in tertio minimo motus, erunt 3 vires; nempe primi, secundi & tertii minimi temporis etc. Hic autem numerus est triangularis, ut alias forte fusius explicabo, & apparet hunc figuram triangularem abc representare. Immo, inquies, sunt partes protuberantes ale, emg, goi, etc., quae extra trianguli figuram exeunt. Ergo figura triangulari illa progressio non debet explicari. Sed respondeo illas partes protuberantes oriori ex eo quod latitudinem dederimus minimis, quae indivisibilia debent imaiginare & nullis partibus constantia. Quod ita demonstratur. Dividam illud minimum ad in duo aequalia in q; iamque arsq est primum minimum motus, et gted secundum minimum motus, in quo erunt duo minima virium. Eodem pacto dividamus df, fh, etc., Tune habebimus partes protuberantes ars, ste, etc., Minores sunt parte protuberante ale, ut patet. Rursum, si pro minimo assumam minorem, ut aα, partes protuberantes erunt adhuc minores, ut aβδ, etc., Quod si denique pro illo minimo assumam verum minimum, nempe punctum, tum illae partes protuberantes nullae erunt, quia non possunt esse totem punctum, ut patet, sed tantum media pars minimi alde; atqui puncti media pars nulla est.
- 76.
AT X p. 77. Koyré (1978) p.83 …si imaginetur, verbi gratia lapis ex a ad b trahi a terra in vacuo per vim quae aequaliter ab illa semper fluat, priori remanante, motum primum in a se habere ad ultimum qui est in b, ut punctum a se habet ad lineam bc; mediam vero partem gb triplo celerius pertransiri a lapide, quam alia media pars ag, quia triplo majori vi a terra trahitur: spatium enim fgbc triplum est spatii afg, ut facile probatur; & sic proportione dicendum de caeteris partibus.
- 77.
Koyré (1939) pt 2, pp. 32–33, 37; (1978 pp.83–84). ‘C’est lorsqu’il essaie de traduire les résultats de son integration (of minima motūs) en terms d’espace que, emporté par l’élan de la representation imaginative et de sa tendence à la géométrisation à outrance, il tombe dans l’erreur’. Cf Hanson (1958) 45–46, ‘The point of the problem of free fall eludes Descartes’. We shall see, and indeed already have seen in regard to the hydrostatics manuscript, that Descartes’ views on causation within natural philosophy were marked not by a géométrisation à outrance, but if anything, by a ‘dynamicization’ à outrance’—a concern with imputing forces and tendencies to bodies at particular instants in their motions or tendencies to motion.
Koyré’s indictment, however, runs to further particulars. In his view, the specific mistake of Descartes the mathematician was to have failed to exploit Beeckman’s ‘intellectual conquest’, the principle of conservation of motion (1939 pt 2, 36; 1978, 83). To Koyré, Beeckman’s notion that the conservation of uniform motion does not require a cause or explanation was clearly in the line of development of classical mechanics. By reintroducing the metaphysical concept of an internal moving force, Descartes fell back into the ‘impetus physics’ of the fourteenth century (1939 pt 2, 36; 1978, 83). It is correct to point out the contrast between Descartes’ view of mechanics and that of Beeckman. In his later work Descartes would further develop the idea that the inertial motion of a body is caused by the continued action of an internal force of motion. In fact, much of his natural philosophy and mechanistic optics will be built around the analysis of the magnitude and components of directional magnitude of the force of motion possessed by a body at each moment of its motion. By contrast, Beeckman always seems to have entertained a ‘modern’ concept of motion, just because he did not mention impressed or internal moving forces. Nevertheless, Koyré’s view can be shown to have been doubly misguided. In the first place, as Koyré himself showed, and subsequent research confirmed, the inertial concepts of both Descartes and Newton had significant residues of notions of impetus-like internal moving forces. Beeckman may have had a modern textbook notion of inertia, but the modern view itself emerged from the tradition of mechanics in which Descartes and then Newton forged the concept with strong dynamical overtones. Therefore, it is of little conceptual or historical significance to credit the ‘progressive’ nature of Beeckman’s ideas over those of Descartes. Secondly, and more pertinently, it is erroneous to imply, as did both Koyré and Hanson, that Descartes’ so-called impetus physics was responsible for his mistakes. (Koyré 1939 pt 2, 36; 1978, 83; Hanson 1958, 48). We are about to see that Descartes’ concept of a conserved internal moving force mediates between the reiterated applications of the tractive force and the consequent motion actually produced and conserved. It is a conceptual elaboration, explicating the problem of the cause of the continued motion of the body. It need not have posed any mathematical difficulties. Beeckman’s ‘correct’ demonstration of the time-space relation can be rewritten, substituting for momenta spatii more involved phrases relating increments of impressed internal moving force to minima motūs and thence to momenta spatii. Nor is this surprising, since the impetus theorists of the Parisian School following from Oresme were the first to derive the triangular representation of ‘uniformly difform motions’ in general (Clagett 1961, 331–418). Finally, it is amusing to note that if we reflect Koyré’s views back onto our reading of the hydrostatics manuscript, we can speculate that had Descartes in fact been more of a pure geometer, and less of a physico-mathematician, he probably would have left Stevin’s findings in the rather more rigorous form in which he had found them!
- 78.
In the hydrostatics manuscript, where of course no actual translation takes place, only instantaneously exerted tendencies to motion, the instants obviously are instants of time not of space or distance.
- 79.
- 80.
Remember that Beeckman himself is very clear from the first statement of the problem that reiteration of a moving force is imparting increments of motion, expressed in the condensed locution, italicized here: ‘Secundo, in hoc motu pervererando supperadditur motus novus tractionis, ita ut duplex spacium secundo momento peragretur.’ At one level the differences of conception and expression between the two physico-mathematicians were quite small.
- 81.
AT X pp. 77, 219.
- 82.
A signification that Beeckman, of course, made quite explicit!
- 83.
In addition to Thomas Kuhn’s well known and seriously intended metaphor of expert, problem-oriented, scientific research as craftsman’s work, see the profound development of that conceit by J.R. Ravetz (1971) and the convincing articulation of the notion in very many examples of the subsequent literature on ‘sociology of scientific knowledge’, particularly in the works of Karen Knorr-Cetina (1981), Trevor Pinch (1985) and Andrew Pickering (1995). I deploy the idea of pitfalls looming at the research coal-face, and of initial recognition of them, followed, one hopes, by gradual, crafty finding one’s way around them, in the spirit of Ravetz’s deep and still useful discussion.
- 84.
Recent luminaries of ‘science studies’ have described partially overlapping dimensions of the sort of craftsman-like grappling with scientific research problems alluded to here: Knorr-Cetina (1981) discussed ‘tinkering toward success’; Latour (1987) borrowed and attractively articulated Levi-Strauss’s conceit of ‘bricolage’, whilst Pickering (1995) describes ‘the mangle of practice’. It is surely better to look to these authorities for heuristic guidance in understanding Descartes’ problem-solving styles and struggles, than to spin out fantasy tales of ‘methodological control’, or, with Koyré, tales of ingrained, congenital epistemological blockages.
- 85.
AT X p. 61 Si vero momentum minimum spatii sit alicuius quantitatis, erit arithmetica progressio. Nec poterit sciri ex uno casu, quantum singulis horis perficiat; set opus erit duobus casibus, ut inde sciamus quantitatem primi momenti.
- 86.
Ibid.
- 87.
AT X p. 219 Aliter autem proponi potest haec quaestio, ita ut semper vis attractiva terrae aequalis sit illi quae primo momento fuit: nova producitur, priori remanante. Tunc quaestio solvetur in pyramide.
- 88.
Koyré (1939) pt 2, p. 32 ‘Comment un tel accroisement de la force attractive serait-t-it possible? Descartes ne se le demande pas. En fait ce n’est pas en physicien, c’est en mathematicien pur, en pur geométre, qu’il voit le problem.’ This surely will not do, however, because there was no criterion of contemporary relevance to Beeckman and Descartes permitting a distinction between Beeckman’s ‘physics’ and Descartes ‘geometry’. Neither man had any firm basis for asserting the physical reality of any particular law of fall—as we are in the process of learning. Nor was the speculative corpuscular-mechanical explanation of one law any less plausible than that of another. Descartes’ law of increasing force could be ‘explained’ just as well as Beeckman’s law of uniform periodic impulse.
- 89.
Hanson (1958) 45–6. As we shall see, the ‘cubic relation’ that holds here—provided the force acts continuously, from instant to instant, a physical matter about which we have seen Beeckman and Descartes might have doubts— is that the distance travelled (or as Descartes would say, the sum of instantaneously exerted ‘minima motus’) will be as the cube of the time of fall. Hanson’s text reads in full: ‘(Descartes) proposes another possible case, one in which the attractive force grows from moment to moment. In the second moment of its fall a body is attracted with twice the force of the first moment, in the third moment with a triple force. In this solution the velocities increase in a cube-like way and not as squares. Descartes never asks about the physical possibility of this hypothesis of growing force. It is a case of geometry—one more mathematical possibility.’ Apparently, behind the façade of discussing natural philosophy, Descartes was really playing a mathematician’s game of altering variables and solving new mathematical puzzles. What Hanson should have said, of course, is that if Beeckman’s speculations were physics—that is physico-mathematics!—so were those of Descartes.
- 90.
Koyré forgets that in Newtonian physics the acceleration of locally falling bodies also increases if only in a small manner.
- 91.
AT X pp. 77–78. Koyré (1978) 85 Aliter vero potest haec quaestio proponi difficilius, hoc pacto. Imaginetur lapis in puncto a manere, spatium inter a & b vacuu; iamque primum, verbi gratia, hodie hora nona Deus creet in b vim attractivam lapidis et singulis postea momentis novam et novam vim creet, quae aequalis sit illi quam primo momento creavit; quae iuncta cum vi ante creata fortius lapidem trahat & fortius iterum, quia in vacuo quod semel motum est semper movetur; tandemque lapis, qui erat in a, perveniat ad b hodie hora decima. Si petatur quanto tempore primam mediam partem spatii confecerit, nempe ag, & quanto reliquam: respondeo lapidem descendisse per lineam ag tempore 1/8 horae; per spatium autem gb 7/8 horae. Tunc enim debet fieri pyramis supra basim trangularem, cuius altitudo sit ab, quae quocunque pacto dividatur una cum tota pyramide per lineas transversas aeque distantes ab horizonte. Tanto celerius lapis inferiores partes lineae ab percurret, quanto majoribus insunt totium pyramidis sectionibus.
- 92.
Descartes’ argument thus moves entirely within the confines set by the procedure of establishing an arithmetical series expressive of a force law (or causal regime) and then conceiving of a representative figure by intuitively reducing the ‘moments’ of application of the force to instants. He wrote in the Cogitationes Privatae (AT X p. 220 l.5-9) ‘Ut autem huius scientiae fundamenta jaciam, motus ubique aequalis linea representabitur, vel superficie rectangula, vel parallelogrammo, vel parallelpipedo; quod augetur ab una causa, triangulo; a duabus, pyramide, ut supra; a tribus, aliis figuris.’ This might at first glance seem reminiscent of the treatment of ‘latitude of forms’ stemming from Oresme and involving a looser kind of inquiry involving classification of types of motion mapped by reference to types of figural representations. Taking the entire exchange into account, however, it would seem that what Descartes envisions is just what we have been describing, a physico-mathematical inquiry into the modes of representation of various possible causal regimes covering natural fall. There were many possibilities, as no agreed, exact mixed mathematical law of fall had eventuated, and many causal regimes could be imagined, and geometrically represented. No closure of the physico-mathematical inquiry was reached, and it petered out in ramifying possibilities.
- 93.
AT X pp. 242–3: ‘Lux quia non nisi in materia potuit generari, ubi plus est materiae, ibi facilius generatur, caeteris paribus; ergo facilius penetrat per medium densius quam per rarius. Unde fit ut refractio fiat in hoc a perpendiculari, in alio ad perpendicularem.’
- 94.
- 95.
- 96.
Kepler held that light is an immaterial emanation propagated spherically in an instant from each point of a luminous object. Refraction, he maintained, is a surface phenomenon, occurring at the interface between media. The movement of the expanding surface of light is affected by the surface of the refracting medium, because, according to Kepler, like affects like, hence surface can only affect surface, and the surface of the refracting medium ‘partakes’ in the density of the medium. He analyzed the effect of the refracting surface upon the incident light, by decomposing its motion into components normal and parallel to the surface. The surface of a denser medium weakens the parallel component of the motion of the incident light, bending the light toward the normal; a rarer refracting medium facilitates or gives way more easily to the parallel component of the motion of the incident light, deflecting it away from the normal. (The normal component of the motion of light is also affected at the surface by the density of the refracting medium, weakening or facilitating its passage, but not contributing to the change of direction). Ad Vitellionem Paralipomena, Chap. 1 Prop. 12, 13, 14, 20, in Kepler (1938ff) vol. II, 21–3, 26–7. I have termed this Kepler’s official theory of refraction, because it is not his only articulated discussion of the causes of refraction (and their geometrical representation) offered in Ad Vitellionem, as we are now about to see.
- 97.
Descartes’ familiarity with Kepler new theory of vision and image formation has important implications for our reconstruction, in Chap. 4, of his later discovery of the law of refraction. Some time ago Dr Albrecht Heeffer, University of Gent, explored the Kepler/Descartes relation regarding these passages in the context of reconstructing Descartes’ discovery and explanation of the law of refraction. During the course of an interesting and erudite discussion, ‘The logic of disguise: Descartes’ discovery of the sine law’, Dr Heefer did not cite my work (1977) and (2000; 2005) This was apparently a seminar or working paper at the University of Gent, History of Science Institute. I had the opportunity to confer with Dr Heeffer whilst he visited the HPS Unit, University of Sydney, March 2011, during which he kindly directed me to his published version of the original text, Heeffer (2006), which does cite my (1977) extensively.
- 98.
Ad Vitellionem Paralipomena Kepler (1938ff), vol. II, 81–5.
- 99.
Kepler (1938ff) 86.
- 100.
- 101.
For example, Kepler’s official theory of refraction (Note 99 above) dealt with the parallel and normal components of the motion of the light, asserting that both are weakened at the interface, whilst attributing the refraction to the alteration in the parallel component alone. In the traditional optical literature it was also thoroughly commonplace to attend to the comportment of the normal and parallel components of the motion of light when discussing its refraction and reflection.
- 102.
Again, our interpretation should be compared to Sepper’s (2000) interesting thesis about how the young Descartes solved problems via strategies of figural representation. Here Descartes uses the routine representation of the components of rays to represent and articulate Kepler’s interesting physical hypothesis. He was, in the language we developed above to describe his physico-mathematical practices, ‘figuring up’ the problem, by imposing upon Kepler’s inviting conjecture and diagram the customary component analysis of rays.
- 103.
See above Note 97. Sabra (1967) drew attention to Descartes signalling the entailment, but incorrectly interpreted Descartes’ premise concerning ‘greater penetration in denser media’ as applying independently of angle of incidence—thus allowing Sabra to deduce the sine law of refraction from the ‘text’. We have seen that Descartes’ premise applied to the normal component of the force or motion of the incident ray. Had Descartes carried out the resulting deduction, he would have arrived a tangent law of refraction.
- 104.
This tactic curiously foreshadows a similar process of adaption, criticism and modification which Beeckman and then Descartes would adopt toward Kepler’s celestial mechanics in the late 1620s, a process that had its outcome in Descartes’ vortex theory of planetary motion, a very seriously worked out theory indeed, as we shall learn in Chap. 10 Cf. Schuster (2005).
- 105.
AT x. 242–3. ‘…omnium autem maxima refractio esset densissimum, a quo iterum exiens radius egrederetur per eundem angulum.’ In his analysis of the fragment, Sabra did not cite or discuss this remark; yet, it is of vital importance in understanding Descartes as a ‘physico-mathematical’ reader and-interpreter of Ad Vitellionem. See Sect. 4.6.
- 106.
Kepler (1938ff) II 107. ‘In medio densissimo omnes refractiones fiunt ad ipsas perpendiculares suntque aequales inclinationibus.’
- 107.
loc. cit. pp.89–90. ‘…si perpendas, quid fieri debeat, medio existente plane densissimo (seu infinitae densitatis), deprehendes ex analogia mediorum caeterorum, oportere, si quod esset, omnes omnino radios ab uno puncto in superficiem huiusmodi illapsos, refringi plenarie, hoc est, coincidere post refractionem cum ipsos perpendicularis.’
- 108.
Had Descartes assumed that the parallel component varies either directly or inversely with the density, he would have again deduced ‘tangent laws’ with slightly differing indices of refraction. There seems no way to proceed directly from the assumptions of 1620 to the sine law of refraction, unless one is prepared to introduce Newtonian complications about the variation in components as functions of the angle of incidence, a way of conceiving the problem foreign to Descartes in 1620, 1626, as well as 1637. Sabra, of course, assumed that penetration varied with density regardless of the angle of incidence, an assumption that does indeed yield the sine law, when conjoined with the assumption that the parallel component of the motion, force or penetration of the incident ray is unaffected by refraction. Sabra’s error consisted in his construal of the first premise: Descartes was envisioning that the normal component of penetration varied with density. These matters are discussed in more detail below in Chap. 4.
- 109.
And in the case of the study of the physico-mathematics of fall, Beeckman and Descartes had both seemingly spoken a surface language of attractions and forces, arguably covering corpuscular-mechanical commitments about the causes of fall.
- 110.
e.g. Beeckman (1939–53) III, 27–28.
- 111.
- 112.
For Descartes’ similar reaction to Beeckman’s celestial mechanical speculations see Schuster (2005) and Sect. 10.3 below.
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Schuster, J. (2012). ‘Recalled to Study’—Descartes, Physico-Mathematicus . In: Descartes-Agonistes. Studies in History and Philosophy of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4746-3_3
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