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Uniform Acceleration, Space, and Time

Published online by Cambridge University Press:  05 January 2009

Stillman Drake
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Extract

The most reliable source for a reconstruction of Galileo's progress toward a science of motion is the series of undated fragmentary notes on that subject preserved in Codex A of the Galilean manuscripts at Florence. A gathering of such fragments was published by Favaro in the National Edition of Galileo's works, following the Discorsi. The more sophisticated fragments are clearly associated with the composition of that work, and show a definite and consistent understanding of acceleration. Eliminating those, it will be found that the earlier notes fall into recognizable groups. First, there are some that refer to “moment of gravity”, or to the impetus of a body along a line of descent, and are associated with the discussion of inclined planes in De motu. Second, some refer to descent along arcs and chords of circles, associated with Galileo's letter of 29 November 1602 to Guido Ubaldo. These first two groups of notes do not explicitly refer to accelerated motion, and should not be assumed to do so implicity, where such an assumption can be avoided.

Type
Research Article
Information
The British Journal for the History of Science , Volume 5 , Issue 1 , June 1970 , pp. 21 - 43
Copyright
Copyright © British Society for the History of Science 1970

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References

1 For example, the first diagram on p. 380, Opera, viiiGoogle Scholar, suggests that accelerated fall along the diameter is under consideration. But reference to p. 378 shows that the proposition was probably derived without any thought of acceleration. The note accompanying it is a memorandum for the order of proofs in the projected treatise of 1609, discussed below, in which static, dynamic and kinematic considerations were integrated.

2 Opere, viii, 383.Google Scholar It is possible that the document is not a copy, but an attempt by Guiducci to resurrect space-proportionality, like Sagredo in the Discorsi. But Guiducci copied many other Galilean notes, preserved like this one with Galileo's own papers, so it is probably authentic. The copies were probably made after 1616, when Galileo had to drop astronomy and return to physics.

3 Ibid., 388. Cum assumptum sit almost certainly refers to the opening word, Assumo, of the document copied by Guiducci.

4 Ibid., 389.

5 Ibid., 380. For the two substituted propositions, see pp. 376–377.

6 Opere, x, 248–249.Google Scholar Valerio's reply is confusing because he ignored the phenomenon of acceleration in one of the propositions but not in the other. Valerio also acknowledges the receipt of a theorem, which was probably the Latin version of the 1604 fragment which Valerio called “most elegant, and worthy of you”.

7 Opere, viii, 380.Google Scholar

8 Ibid., 388.

9 Opere, xi, 85.Google Scholar

10 Opere, viii, 384–385.Google Scholar

11 Ibid., 378.

12 Ibid., 375; cf. vii, 48–50 and Dialogue, tr. Drake, S. (Berkeley, 1967), 24–26.Google Scholar The same paradox had been noted by Oresme; see Clagett, M., Nicole Oresme and the Medieval Geometry of Qualities and Motions (Madison, 1968), p. 279.Google Scholar

13 Ibid., 381.

14 Ibid., 380.

15 Ibid., 387.

16 Opere, i, 297Google Scholar; cf. Galileo on Motion and on Mechanics, tr. Drabkin, I. E. and Drake, S. (Madison, 1960), p. 64.Google Scholar

17 “The 1604 Fragment on Falling Bodies, Galileo Gleanings XVIII”, The British Journal for the History of Science, iv (1968–1969), 340–358.Google Scholar

18 Koyré, A., Newtonian Studies (Cambridge, Mass., 1965), p. 36.CrossRefGoogle Scholar

19 Koyré, A., Études Galiléennes (Paris, 1939), ii, p. 98Google Scholar, n. 2.

20 Two New Sciences, tr. Crew, H. and Salvio, A. De (New York, 1914 and later eds.), p. 168.Google Scholar

21 Unterredungen und mathematische Demonstrationen, tr. von Oettingen, A. (Leipzig, 1891 and later eds.), vol. 2, p. 16.Google Scholar

22 See, for example, Koyré, A., Études, ii, pp. 95–99Google Scholar; Cohen, I. B., “Galileo's Rejection of the Possibility of Velocity Changing Uniformly with Respect to Distance”, Isis, 47 (1956), pp. 231–235CrossRefGoogle Scholar; Hall, A. R., “Galileo's Fallacy”, Isis, 49 (1958), pp. 342–346CrossRefGoogle Scholar; Discorsi, ed. Carugo, A. and Geymonat, L. (Torino, 1958), pp. 776–778.Google Scholar

23 Opere, viii, p. 203.Google Scholar

24 Tenneur, J. A., De motu naturaliter accelerato tractatus … (Paris, 1649), p. 8.Google Scholar

25 That is, every doubled space is associated with a doubled velocity.

26 Tenneur, , op. cit., pp. 10–11.Google Scholar The diagram eliminates some additional lines used by Cazrae, but not referred to in Tenneur's present argument.

27 Two New Sciences, pp. 31 ff.

28 See note 1, above.

29 Les Nouvelles Pensées de Galilée (Paris, 1639), p. 184.Google Scholar

30 Ibid., p. 188.

31 Cf. Two New Sciences, p. 180 n.Google Scholar

32 Cf. Dialogue, tr. Drake, S. (Berkeley, 1967), pp. 227–230.Google Scholar

33 Raccolta di opuscoli scientifici, e letterarj (Ferrara, 1779), p. 64.Google Scholar

34 Mach, E., Science of Mechanics (La Salle, 1942), p. 309.Google Scholar

35 Cohen, , op. cit., p. 235.Google Scholar

36 Swineshead, , The Book of Calculations, tr. Clagett, Marshall, The Science of Mechanics in the Middle Ages (Madison, 1959), p. 291.Google Scholar

37 Cf. Wolfson, H. A., Crescas' Critique of Aristotle (Cambridge, Mass., 1929), pp. 189–191Google Scholar; 346–347. For this reference I am indebted to Rabbi Nachum L. Rubinovitch, whose paper “A 14th-Century Insight on Affinity” will soon appear in Isis.