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Galileo’s quanti: understanding infinitesimal magnitudes

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Abstract

In On Local Motion in the Two New Sciences, Galileo distinguishes between ‘time’ and ‘quanto time’ to justify why a variation in speed has the same properties as an interval of time. In this essay, I trace the occurrences of the word quanto to define its role and specific meaning. The analysis shows that quanto is essential to Galileo’s mathematical study of infinitesimal quantities and that it is technically defined. In the light of this interpretation of the word quanto, Evangelista Torricelli’s theory of indivisibles can be regarded as a natural development of Galileo’s insights about infinitesimal magnitudes, transformed into a geometrical method for calculating the area of unlimited plane figures.

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Notes

  1. See Giusti (1986) and Giusti(1993), Palmieri (2001).

  2. See Drake (1974a, 1979), Hill (1986), Naylor (1990), Palmieri (2006), Wisan (1974).

  3. First edition 1612, now available on line at the url: http://vocabolario.sns.it/html/index.html.

  4. The original: “che ha quantità”.

  5. Drake (1974a, p. xxxvi).

  6. I shall quote Galileo’s original texts from Galilei (1890–1909), by giving the volume in Roman and the page in Arabic numbers.

  7. The English translation from which I shall quote is Drake (1974a).

  8. Drake (1974a, p. 190), Galilei (1890–1909, VIII, p. 190).

  9. Drake (1974a, p. 154). The original Latin reads “a quiete recedens, temporibus aequalibus aequalia celeritatis momenta sibi superaddit” (Galilei 1890–1909, VIII, p. 198).

  10. Galileo’s original text is “gradum seu momentum velocitatis” (Galilei 1890–1909, VIII, p. 198, l. 10). “Momentum of swiftness” must not be interpreted as “mechanical moment”, which points to the static effect of the heaviness of a body, a technical word which belongs to pre-modern studies on machines. Mechanical moment varies according to the distance of the body from the centre of rotation that is the length of a lever arm as in the principle of the lever. As far as the evolution of the meaning of “momentum of swiftness” is concerned, see Galluzzi (1979), in particular p. 364, footnote 2.

  11. Drake (1974a, p. 157), Galilei (1890–1909, VIII, p. 200).

  12. Drake (1974a, p. 157), Galilei (1890–1909, VIII, pp. 200–201).

  13. Drake (1974a, p. 157), Galilei (1890–1909, VIII, p. 201).

  14. Drake (1974a, p. 22), Galilei (1890–1909, VIII, p. 61).

  15. Drake (1974a, pp. 27–28), Galilei (1890–1909, VIII, p. 67).

  16. Drake (1974a, p. 28), Galilei (1890–1909, VIII, p. 68).

  17. The problem of the wheel (see Fig. 1) is the paradox of two concentric circles rotating around their common centre (\(A\)) and on a surface (BF). In the first case, two points (\(C\) and \(B\)) on each one of them draw two circumferences with different length around the common centre \(A\). In the second case, the same points draw two identical straight lines (CE and BF). It was solved firstly by Jean Jacques d’Ortous de Mairan.

    Fig. 1
    figure 1

    The diagram exemplifying the problem of the wheel (Drake 1974a, p. 29; Galilei 1890–1909, VIII, p. 68)

    Full size image

  18. Drake (1974a, pp. 32–33), Galilei (1890–1909, VIII, p. 71).

  19. Drake (1974a, pp. 31–32), Galilei (1890–1909, VIII, p. 70).

  20. Drake (1974a, p. 32), Galilei (1890–1909, VIII, p. 71).

  21. Galileo is considering a circumference consisting of a countable infinity of points. Therefore, its length is the sum of a countable infinity of magnitudes.

  22. The modern technical term infinitesimal is used here to describe a magnitude that, as Galileo understands it, is extremely small, as small as can possibly be, but not zero in size.

  23. Galileo seems to have established a sort of relation between points and numbers as if he wanted to measure points in some ways. In Euclidean geometry, a point has no dimension, neither length, width, nor depth, so we can identify the dimension of a point as zero.

  24. Drake (1974a, p. 33), Galilei (1890–1909, VIII, p. 71).

  25. Drake (1974a, p. 33), Galilei (1890–1909, VIII, pp. 71–72).

  26. Drake (1974a, p. 33), Galilei (1890–1909, VIII, p. 72).

  27. (Galluzzi (2001, p. 74).

  28. “We are concerned, in the problem of ARISTOTLE’S Wheel, with two matters, (1) the problem of motion, particularly the composition of motions, and (2) the point-to-point correspondence of paths of different lengths. These matters are, to be sure, only apparently independent, for both are, at least from one view-point, ultimately bound up with the problems of continuity, infinity, and the number system” (Drabkin 1950, p. 169).

  29. Drake (1974a, p. 42), Galilei (1890–1909, VIII, p. 80).

  30. Drake (1974a, pp. 38–40), Galilei (1890–1909, VIII, p. 77).

  31. Drake (1974a, p. 43), Galilei (1890–1909, VIII, p. 81).

  32. Drake (1974a, p. 40), Galilei (1890–1909, VIII, p. 78).

  33. Drake (1974a, pp. 44–45), Galilei (1890–1909, VIII, p. 82).

  34. Drake (1974a, p. 44), Galilei (1890–1909, VIII, p. 81).

  35. Drake (1974a, p. 53), Galilei (1890–1909, VIII, pp. 91–92).

  36. Drake (1974a, p. 53), Galilei (1890–1909, VIII, p. 91).

  37. Drake (1974a, p. 53–54), Galilei (1890–1909, VIII, p. 92).

  38. For studies of Torricelli, see Bortolotti (1925, 1928, 1939), Jacoli (1875), Belloni (1987), De Gandt (1987, 1991a, b, 1992, 1995), Giusti (1989), Festa (1992), Loria (1897).

  39. I shall quote Torricelli’s original texts from Torricelli (1919–1944), by giving the volume in Roman and the page in Arabic numbers. Here, Torricelli (1919–1944, I, part 2, pp. 1–43 and pp. 20–23).

  40. Torricelli (1919–1944, I, part 2, pp. 20–27). There exists another folder, which has an identical title and in which there are notes very similar (Torricelli 1919–1944, I, part 2, pp. 46–48). There are also examples involving curves.

  41. Torricelli (1919–1944, I, part 2, pp. 415–432).

  42. Torricelli (1919–1944, I, part 2, pp. 417–426).

  43. Cavalieri’s scholars have proved that the notion of a whole of lines, called aggregate, is neither to be thought of as a set of lines in the modern sense nor as a figure compounded of line-like indivisibles. See Giusti (1980), Andersen (1985), De Gandt (1991b).

  44. Torricelli (1919–1944, I, part 2, p. 320). Translation of Curtis Wilson in De Gandt (1995, p. 188). I shall quote English translations of Torricelli’s texts by referring to De Gandt (1995). When not specified, the translation is mine.

  45. Torricelli (1919–1944, I, Part 2, pp. 320–321).

  46. Drake (1974a, p. 48), Galilei (1890–1909, VIII, p. 86).

  47. Drake (1974a, p. 60), Galilei (1890–1909, VIII, p. 99).

  48. Drake (1974a, p. 56), Galilei (1890–1909, VIII, p. 95).

  49. Numerous scholars have studied Galileo’s physical atomism. Here is a selection of the major works: Baldini (1977), Redondi (1985), Murdoch (2001), Shea (2001), Galluzzi (2001), Biener (2004).

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    Tiziana Bascelli

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Bascelli, T. Galileo’s quanti: understanding infinitesimal magnitudes. Arch. Hist. Exact Sci. 68, 121–136 (2014). https://doi.org/10.1007/s00407-013-0124-2

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