Abstract
This paper is inspired by a specific conception of the history of science. This latter, like any historical description, must in the first place study facts: establishing when and how a particular result is obtained, discerning heritage and genuine creation and thus preparing an accurate assessment of the successive contributions which gave rise to the major scientific theories — such is clearly its first task. But there also exists another way of looking at the history of science: more concerned with internal logic than with chronological details, it aims above all at bringing out which, at the level of conceptual initiatives, opened the way to the new doctrines, allowed them to take form, and thereby to acquire this systematicity and deep consistency without which no intellectual construction can really endure. So construed — and without neglecting the facts — it mainly focuses on the changes — be they abrupt or progressive — thanks to which an ever more precise representation of reality is built. In other words one will find, in the following pages, neither discussions of unpublished documents, nor chronological conjectures, nor even a comparative study. We begin by an observation: the creation by Galileo, within a few decades, of a mathematical theory dealing with the natural motion of heavy bodies. This theory does not appear in a conceptual void and in no way is free from technical constraints. Galileo finds himself facing a severely weakened but not totally ruined conception which he must initially combat and, later, replace piece by piece. At the same time, he uses a very specific type of mathematics — basically Euclidean geometry — guided (or held back) by requirements which were rendered obsolete by subsequent development in mathematics. Keeping these different aspects in mind, I should like to restore, in their internal logic and originality, some of the initiatives which made Galileo the creator of an entirely new discourse about nature.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
Notes
On Motion, in On Motion and On Mechanics, translated respectively I. E. Drabkin and Stillman Drake, University of Wisconsin Press, 1960; for a systematic analysis of the position elaborated by Galileo, one may consult our book The Natural Philosophy of Galileo (abridged NPhG), The MIT Press, 1974, Chapter 3.
A negative value naturally means upward motion.
Two New Sciences, translated by Stillman Drake, The University of Wisconsin Press, 1974, p. 77.
The weaknesses of the aristotelian doctrine are shown in On Motion; cf. NPhG pp. 120 sq.
This critic is taken up again in the Two New Sciences, pp. 70–71.
Galileo never clearly distinguished sliding and rolling.
In accordance with the expression used in the Dialogue Concerning The Two Chief World Systems, translated by Stillman Drake, University of California Press, 1967, p. 147.
On Mechanics, p. 171.
Such is the formulation given in the Dialogue, p. 149. The Fourth Day of the Two New Sciences, as one knows, will replace this spherical plane by a horizontal plane. A letter of Benedetto Castelli allows us to assert that Galileo had reached this conclusion at least as early as 1607; cf. Opere di Galileo Galilei, a cura di A. Favaro, Firenze, 1890-1909, Vol. X, p. 170.
Two New Sciences, p. 149.
Ibid., p. 148.
Galileo devotes several lines of commentary to this term in a Note. In a word, the term “any” is necessary to avoid considering as uniform motions in which distances still equal would be covered in times of a certain determined duration, but yet in which “the spaces completed in smaller parts of those same times, themselves equal, are not equal”.
Ibid., p. 148.
Ibid.
Ibid., pp. 149-50.
Ibid., p. 150.
It is Descartes who first, by asserting the principle of a correspondence between the number and the continuous magnitude, will lift this interdict and will open the way to a mathematical genesis of the physical concepts; cf. Pierre Boutroux, Principes del’analyse mathématique, tome I, Paris, 1914, pp. 121 sq.
Two New Sciences, p. 148–49.
Ibid., p. 149.
As we have alreday noted, Theorem I is practically taken from Archimedes, On Spirals, I. If Galileo makes the resort to definition 5 of Book V of the Elements more explicit, he brings nothing substantially new.
According to the Elements, VI, proposition 23; by definition, A/B is a ratio composed of two ratios if one has A/B = A/C × C/B. Theorem IV, notably, will play a decisive role for the theory of accelerated motion.
Or inversely, by the excess of gravity of the medium on that of the bodies, in the case of upward motion.
I am reconstructing, by abridging it, the long analysis developed by Galileo in the First Day of the Two New Sciences, pp. 71 sq.
These numbers are from Galileo, ibid., pp. 79-80.
Ibid., p. 75. In the still unsure vocabulary of Galileo, ‘resistance’ simply signifies ‘density’.
Ibid., p.86.
“… the internal heaviness [interna gravita] of different moveables has no part at all in diversifying their speeds …, ” ibid., p. 90.
Two New Sciences, III, p. 154.
Ibid., pp. 155-61.
Ibid., p. 170-71. If \({\rm{SX}} = \sqrt {ST \times SV}\), then Tst/Tsv = ST/SX or Tsv/Tst = SV/SX.
Theorem II is indeed never used directly … except for establishing this corollary.
Ibid., p. 175.
Ibid., pp. 166-67.
Ibid., pp. 165-66.
Moreover, we come across it several times, for instance farther in the Third Day, proposition 23, Scolie, or already in the Dialogue, Second Day, pp. 227–28. Galileo discusses its difficulties — but also its plausibility — at length in the First Day of the Two New Sciences.
Besides, he is not at all its inventor: it dates back to the 14th century when, with its help, the philosophers from Oxford and Paris could revive in a remarkable way the traditional analysis of motion, and even got close on certain points to the threshold of great discoveries; cf. NPhG Chapter 2, pp. 62–83. One must obviously not conclude that Galileo would content himself with perfecting the works of the 14th centruy, ibid., chapter 6, pp. 292-98.
If É can be taken as the equivalent of an integration, it is of course by a way which does not resemble at all what we call like that nowadays, and does not anticipate it either. The same procedure reappears in the Scolie of proposition 23: same assimilation of aggregates of lines — via the method of indivisibles — to determined geometrical figures, same postulate of the proportionality of the space covered to these aggregates geometrically specified.
Two New Sciences, pp. 175–76.
Ibid., p. 162. The Author in question is that of the treatise that the interlocutors are supposed to read, i.e., Galileo himself.
Ibid., pp. 163-64.
If the impeto acquired by a moving body after various descents, but of constant height, did not always take it back to this same height, nothing would prevent us to imagine devices which, by deflecting the moving body from an itinerary to another, would allow to make it go back up to a greater height; cf. NPhG, pp. 365–66.
NPhG, pp. 366–67.
Ibid., pp. 345-59.
Let’s not forget that this principle does not intervene at all in the geometrical constructions of the Third Day.
The principle of conservation holds on a horizontal surface, and not spherical anymore; as for the principle of composition, it is stated in answer to an objection of one of the interlocutors, after Theorem I which uses it.
For the reconstruction of this method, cf. NPhG, Appendix 6.
Which Galileo recalls, for instance, on the threshold of his definition of the naturally accelerated motion, Two New Sciences, p. 153.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 D. Reidel Publishing Company, Dordrecht, Holland
About this chapter
Cite this chapter
Clavelin, M. (1983). Conceptual and Technical Aspects of the Galilean Geometrization of the Motion of Heavy Bodies. In: Shea, W.R. (eds) Nature Mathematized. The University of Western Ontario Series in Philosophy of Science, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6957-5_2
Download citation
DOI: https://doi.org/10.1007/978-94-009-6957-5_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-6959-9
Online ISBN: 978-94-009-6957-5
eBook Packages: Springer Book Archive