# Concept and Inference: Descartes and Beeckman on the Fall of Bodies

Chapter

## Abstract

The discovery of the law of free fall is usually considered to be a milestone in the development of modem physics and a major step in superseding medieval ways of thought. The subject of the law is the relation between the space traversed by a falling body and the time elapsed. The law states that under certain conditions the spaces traversed measured from rest are proportional to the squares of the times elapsed.

## Keywords

Classical Mechanic Geometrical Representation Classical Physic Free Fall Uniform Motion
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## Reference

- 4.This was already tacitly used by Archimedes in the proof of Proposition 1 of “On Spirals”; see Archimedes 1953, p. 155.Google Scholar
- 5.Thus Moerbeke translates the (absurd) conclusion in
*Physics*VI.1 deduced by Aristotle from the premises of his adversaries*as: “chrw(133)*erit utique motus non ex motibus, sed ex momentis” (see Galluzzi 1979, p. 121).Google Scholar - 7.For a detailed history of the concept momentum see Galluzzi 1979, pp. 3–149.Google Scholar
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- 13.This twofold meaning of the configuration of qualities is the basis of the very different interpretations of Duhem and Maier. Whereas Duhem saw in this technique a rudimentary form of analytical geometry, Maier insisted on the “symbolic” character of the configuration (see Maier 1949, p. 125, note, and 1952, pp. 290–291, 306307, 312, and 328). Clagett (1968a, pp. 15 and 25f) discusses both interpretations.Google Scholar
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*De Configurationibus*III.vii; Oresme 1968a, 409f. On the application of the “Merton Rule” in the*De Configurationibus*,see Maier, 1952, pp. 334f and 347f, and Clagett’s introduction to Oresme 1968a, p. 46.Google Scholar - 20.
*De configurationibus*I.viii, Oresme 1968a, p. 185. See Clagett’s introduction to Oresme 1968a, pp. 19 and 82. See also Lewis 1980.Google Scholar - 21.Jacobus is presumed to be the author of
*De latitudinibus formarum*Text and translation by Clagett in Oresme 1968a, pp. 89f.Google Scholar - 22.The significance of the impetus theory as a theory of causality was stressed in various places by Maier. See, for example, her “Die Impetustheorie,” in Maier 1951, pp. 113–314, and “Das Problem der Gravitation,” in Maier 1952, pp. 143–254. Wolff (1978) has also emphasized this idea showing that proponents of the impetus theory also employed the same notion of causality as mediated by the transference of an entity from the agent to the object acted upon in other contexts as well, e.g., in the theological theory of the sacraments or in the economic theory of value production.Google Scholar
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*Quaestiones super libris quattuor de caelo et mundo*The Latin text was edited by E.A. Moody (Buridan 1942); this passage is also quoted in Maier’s “Das Problem der Gravitation” (Maier 1952, pp. 201–202). The translation is taken from Clagett 1959, pp. 560–561. Clagett observes that from Buridan’s wording it is not clear whether the velocity is proportional to time or to distance of fall, and he even doubts whether the stated dependency of speed on fall should mean proportionality at all. Clagett rather suggests that Buridan “made no clear distinction between the mathematical difference involved in saying that the velocity increases directly as the distance of fall and saying that it increases directly as the time of fall” (Clagett 1959, p. 563, cf. p. 552).Google Scholar - 24.Ex velocitatione motus per quam acquiritur quaedam habilitas vel impetus et quaedam fortificatio accidentalis ad velocius movendum.“ Quoted from Oresme’s Latin commentary to
*De caelo*in Maier 1951, p. 244.Google Scholar - 27.Explaining the oscillation to and fro at the center of the Earth Oresme says: “Et la cause est pour l’impetuosité ou embruissement que elle [la pierre] aquiert par la cressance de l’isneleté de son mouvement jouste ce que fu dit plus a plain ou. xiiie chapitre”(Oresme 1968b, p. 572). See Maier 1951, pp. 252f and 1952, pp. 203–206.Google Scholar
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*Journal*,*J*IV, 49–52, J I, 260–265, 360361; and in Descartes’*Oeuvres*,AT X, 75–78, 58–61, and 219–220. All three are given in translation in Chapter 5 below; see documents 5.1.1, 5.1.2, and 5.1.3.Google Scholar - 29.See the note by Cornelis de Waard in J IV, 49; compare also the“Advertissement” by Charles Adam in AT X, 26f.Google Scholar
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*passim*Beeckman’s principle is not restricted to rectilinear motion:*“Id*,*quod semel movetur*,*in vacuo semper movetur*,sive secundum lineam rectam seu circularemchrw(133)” (J I, 353).Google Scholar - 33.Descartes says, “those [forces] of the first, second, and third time minima”; this is overlooked by Koyré (1978, p. 84) and Hanson (1961, p. 48); Shea (1978, pp. 141f) even sums up the argument incorrectly by omitting the part of the argument cited above. Only Schuster (1977, pp. 72–84) discusses Descartes’ denotation of the minima as “minima temporis.” Accordingly, he severely criticizes the standard interpretation which can be traced back to Koyré’s mistake (pp. 84–88).Google Scholar
- 38.AT I, 71–73. See document 5.1.5 and the note there on the source of the figure.Google Scholar
- 39.This part of the letter has been a particular source of confusion for historians of science who interpret the documents anachronistically from the viewpoint of classical physics and thus overlook the fact that the values Descartes gives are not consistent with the figure but rather are calculated using the factor
^{4}/3; see section 41 The areas in Descartes’ figure for the first four spaces traversed are as 1 to 3 to 5 to 7. According to Descartes’ argument the corresponding times are as 1 to^{1}/3 to^{1}/5 to^{1}/7. The total of these times is_{176/105.}The internally correct result is therefore that the times for distances as 1 and 4 are as 1 to_{176/105}or as 105 to 176. This result, too, is of course incompatible with the correct law of fall.Google Scholar - 42.AT I, 222. See section 1.5.4.1 for a discussion of the consequences of a functional interpretation of Descartes’ figure.Google Scholar
- 43.Letter to Mersenne, August 18, 1634; AT I, 304–305.Google Scholar
- 44.AT XI, 629. The Excerpta Anatomica comprise a number of manuscripts by Descartes, later in the possession of Clerselier, many of them anatomical in content; they were transcribed by Leibniz in Paris in the 1670’s. This particular manuscript is dated around 1635 by Gabbey (1985), whose translation we have adopted. See document 5. 1. 9.Google Scholar
- 57.March 11, 1640; AT III, 36–38. Further explanations were added in a letter of June 11, 1640; AT III, 79. See documents 5.1.11 and 5.1. 12.Google Scholar
- 62.The concept of force in classical physics is not something that can survive its effect. Newton’s axioms already determine a conceptual structure which is incompatible with anything like an idea of “conservation of forces” as it is inherent in the medieval concept of impetus.Google Scholar
- 63.In contemporary analytical geometry it is customary to represent the independent variable by the horizontal dimension. Therefore, figures like Descartes’ geometrical representations of free fall are normally drawn rotated 90 degrees counterclockwise.Google Scholar
- 65.This seems to be a strong indication that, despite the widespread use of geometrical figures to represent motions, Descartes and his contemporaries probably had no direct access to genuine sources on Oresme and the Merton tradition.Google Scholar
- 75.This has been overlooked up to now by historians of science. One of the editors of Descartes’ works, Paul Tannery, in a note (AT I, 75) on the series of figures given by Descartes, explained them correctly as representing spaces traversed which are a function of powers of times elapsed with an exponent tog
^{2}/tog 4 — log 3 instead of 2. However, he erroneously considered this incorrect law of fall to be implied by Descartes’ interpretation of the vertical side of his triangular geometrical figure as representing the spaces traversed. This mistake was reproduced by de Waard in his edition of the correspondence of Mersenne (vol. 2, p. 320). He even explicitly formulated the contradictory statement that in Descartes’ figure the areas of 1, 3, 5, etc. represent the average velocities and that this is the same as to say that the spaces traversed are proportional to the powers of the times elapsed with an exponent of log^{2}/(log 4 —tog 3) instead of 2. If they had really calculated the resulting values they would have recognized the contradiction between both statements immediately. It seems that historians of science who have adopted the interpretation never questioned this conclusion or calculated the corresponding values. Thus the error can still be found in recent publications. See, for instance, Shea (1978, p. 145), who, closely following a footnote of Koyré (1978, p. 87 fn. 99), stressed the incompatibility of Descartes values with the correct law of fall but overlooked the incompatibility of these values with Descartes’ own proof. Marshall (1979) correctly reconstructed Descartes’ incorrect generalization on p. 127, and correctly reconstructed the implications of Descartes’ proof on p. 128, but did not notice the contradiction between the two.Google Scholar - 76.A method of testing theoretical assumptions by calculating numerical values is not an anachronistic suggestion: it was in fact actually applied by Galileo. See Renn 1990.Google Scholar
- 80.For an impressive account of the difficulties involved in the notion of action at a distance, see the 68th letter of Euler’s
*Letters to a German Princess*For a general account see Aiton 1972.Google Scholar

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