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

  • Peter Damerow
  • Gideon Freudenthal
  • Peter McLaughlin
  • Jürgen Renn
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)


The discovery of the law of free fall is usually considered to be a milestone in the development of modern 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.


Classical Mechanic Geometrical Representation Free Fall Uniform Motion Quadratic Relation 
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  1. 1.
    See Koyré 1978, p. 79 (originally published in French, 1939). Koyré’s interpretation of the documents is essentially shared by Clagett (1959), Hanson (1961), and Shea (1978). In the following we shall not criticize these interpretations on each occasion of disagreement, since in our view they in principle impute an anachronistic framework to the texts interpreted. The differences to our interpretation follow from their reading the geometrical figures used as rudimentary representations of velocity (in the sense of classical physics) depending on time respectively space. They thus reduce the difficulty in elaborating the concepts involved to deciding between the alternative of velocity being proportional to the time elapsed or to the space traversed in fall and then deriving the space-time function without modern integral calculus. Closest to our approach is Schuster’s (1977) interpretation that stresses the application of medieval concepts (Part 1, pp. 72–93). However, limited to the 1618/19 documents, his interpretation does not comprehensively account for the entire deductive structure of the conceptual system involved. Dijksterhuis (1961) also stresses the importance of the relation between the medieval concepts and the concepts of pre-classical mechanics. However, he deals only with Beeckman’s proof of the law of falling bodies (pp. 329–333) because he assumes that Beeckman and not Descartes is the original author of this proof. This opinion seems to us to be incompatible with the sources.Google Scholar
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    John Buridan, 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
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    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
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    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. 28The documents are printed in Beeckman’s 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
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    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
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    AT I, 69–75 and 82–104; here p. 91. A later letter (Oct./Nov. 1631; AT I, 226–232) reiterates the position. See documents 5.1.5, 5.1.6, and 5.1. 7.Google Scholar
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    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
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    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
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    October 11, 1638; AT II, 380–402. See 3.1, 3.8, and document 5.3. 1.Google Scholar
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    See Descartes’ letter to Huygens, February 18, 1643 (AT III, 805–814; document 5.4.1) and the Epilogue (Chapter 4) below.Google Scholar
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    Euler 1912, p. 21 (Chapter 1, Proposition 4).Google Scholar
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    See Galileo 1967, p. 222. See section 1.3.5.Google Scholar
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    See Schuster (1977, p. 67) on Beeckman, J III, 133f.Google Scholar
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    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

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Peter Damerow
    • 1
  • Gideon Freudenthal
    • 2
  • Peter McLaughlin
    • 3
  • Jürgen Renn
    • 1
  1. 1.Max Planck Institute for the History of ScienceBerlinGermany
  2. 2.The Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityTel AvivIsrael
  3. 3.Philosophisches SeminarUniversität HeidelbergHeidelbergGermany

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