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The Third Law in Newton's Mechanics

Published online by Cambridge University Press:  05 January 2009

Roderick W. Home
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Extract

Most modern analysts of Newton's laws of motion, whether they have approached the subject from a historical or from a philosophical viewpoint, have tended to concentrate on the status of the first two laws; the third law has largely been overlooked, or else it has been dismissed as somehow less interesting. My purpose in this paper is to reverse this approach—I intend to investigate some of the historical aspects of the third law, particularly the empirical background to Newton's statement of it, and in so doing, I intend to skirt most of the questions which have been raised concerning the status of the other two laws. In concentrating on the historical aspects of the third law, I shall also by-pass Mach's controversial re-interpretation of its role in mechanics, for while Mach saw the law as the basis for an operational definition of “mass”, it is quite clear that Newton did not so regard it. On the contrary, Newton seems to have regarded all three of his laws as straightforward statements of fact about the world, so that a knowledge of the factual background to the laws is a fundamental pre-requisite to an understanding of Newton's thought.

Type
Research Article
Information
The British Journal for the History of Science , Volume 4 , Issue 1 , June 1968 , pp. 39 - 51
Copyright
Copyright © British Society for the History of Science 1968

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References

1 Newton, Isaac, Mathematical Principles of Natural Philosophy (trans. Motte, Andrew, revised by Florian Cajori, Berkeley, 1962), i, 13Google Scholar. In the original Latin, the law reads: “Actioni contrariam semper & aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.”

2 Ibid., 17, 19.

3 Ibid., 164 ff.

4 Ibid., 22–25.

5 Ibid., 26–28.

6 A hint to this effect was given some years ago by Lenzen, V. F. in his article, “Newton's Third Law of Motion”, Isis, xxvii (1937), 258–60CrossRefGoogle Scholar, but he did not take the matter any further.

7 Newton, Isaac, Opticks, or a Treatise of the Reflections, Refractions, Inflections & Colours of Light (New York, 1952, based on 4th edn., London, 1730), 344.Google Scholar

8 Ibid., 339.

9 It is perfectly true, of course, that in terms of Newton's particulate theory of light, the action and reaction between a body and light may be reduced ultimately to accelerative forces between particles of the body and particles of light. The first passage just quoted shows, however, that the reduction to such forces is not necessary in order to assess the action and reaction. The action is to be assessed directly in terms of the light emitted, the reaction is to be assessed in terms of the heat generated. Hence my point stands. It is interesting to note that in the passage leading up to his own formulation of the action-reaction law (composed in 1695), Leibniz left the meaning of “action” and “reaction” equally vague, even though his discussion was confined to impact phenomena, one field to which Newton had applied his third law in very precise terms. Leibniz's discussion was as follows:

“the passion of every body is spontaneous, or arises from internal force, although upon external occasion. I understand here, however, passion proper, which arises from percussion … For since the percussion is the same, to whatever at length true motion corresponds, it follows that the result of the percussion is distributed equally between both, and thus both act equally in the encounter, and thus half the result arises from the action of the one, the other half from the action of the other; and since half, also, of the result or passion is in one, half in the other, it is sufficient that we derive the passion which is in one from the action also which is in itself, and we need no influence of the one upon the other, although by the action of one an occasion is furnished the other for producing a change in itself. Certainly, while A and B meet, the resistance of the bodies, united with their elasticity, causes them to be compressed because of the percussion, and the compression is equal in each … Then the balls A and B, restoring themselves by the force of their own violent … elasticity, mutually repel each other by turns, and spread out, as it were, in an arc, and, with a force equal on both sides, each is driven back by the other, and so, not by the force of the other, but by its own force, it recedes from that one … From what has been said, it is understood that the action of bodies is never without reaction, and both are equal to each other, and directly contrary.”

(Leibniz, Gottfreid Wilhelm, New Essays concerning Human Understanding (trans. Langley, Alfred Gideon, New York, 1896, 688–689)Google Scholar. How are we to measure the “compression”, or the “force of [a body's] own violent elasticity”? We are not told.

10 Principia, 21.Google Scholar

11 Dugas, René, Mechanics in the Seventeenth Century (trans. Jacquot, Freda, Neuchâtel, , Switzerland, 1958), 350.Google Scholar

12 Gilbert, William, On the Magnet (trans. Thompson, Silvanus P., ed. Price, Derek J., New York, 1958), 67.Google Scholar

13 Boyle, Robert, Experiments and Notes about the Mechanical Origine or Production of Electricity (facsimile, ed., New York, 1945), 17–20.Google Scholar

14 Principia, 25–26.Google Scholar

15 The argument just described, and also the statement, “This law takes place also in attractions, as will be proved in the next Scholium,” were added to the second edition of the Principia as a result of Newton's correspondence with Cotes. It has already been argued elsewhere (Koyré, Alexandre, “Études Newtoniennes III: Attraction, Newton and Cotes”, Archives internationales d'histoire des sciences, xiv (1961), 225–236)Google Scholar that the only way we can explain Newton's reasoning is “to admit that for Newton ‘attraction’, all the pseudo-positivistic and agnostic talk notwithstanding, was a real force (though not a mechanical and perhaps not even a ‘physical’ one) by which bodies really acted upon each other”.

16 Le Meccaniche (ca. 1600). Trans. Stillman Drake, and published in Galilei, Galileo, On Motion and On Mechanics (Madison, Wisconsin, 1960).Google Scholar

17 Ibid., 180.

18 Particularly in his Discourse on Bodies in Water (Urbana, Illinois, 1960)Google Scholar, and in the socalled “Sixth Day” intended for his Two New Sciences (published in French translation in Moscovici, S., “Remarques sur le dialogue de Galilée ‘De la force de la percussion’”, Revue d'histoire des sciences, xvi (1963), 97–137)Google Scholar. See, for example, pp. 6–8 of the former work, or p. 125 of Moscovici's article.

19 E.g. in a letter to Mersenne dated 12 September 1638, and in another, also to Mersenne, dated 2 February 1643. Both are published in Adam, Charles and Tannery, Paul (eds.), Oeuvres de Descartes (Paris, 1897–1913)Google Scholar. The first is in vol. ii, 352–362, the second is in vol. iii, 611–615.

20 Principia, 26.Google Scholar

21 Ibid. 2.

22 Ibid. 13.

23 Ibid., 27.

24 Ibid., 28 (my italics).

25 It should be added that a formal difference remained between the case of simple machines and the others. Whenever the product involved a vis inertiae, the velocity which formed the other term in the product was obtained as the vector difference of two other velocities—in other words, it was always a change in velocity. In the case of the machines, the velocities used in forming the products were always genuine ones (though virtual), and not vector differences.

26 Descartes, , Oeuvres (op. cit. (19)), ix, 89–93.Google Scholar

27 Oeuvres complètes de Christiaan Huygens (La Haye, 1888–1950), xvi, 180.Google Scholar

28 Wallis, John, “A Summary Account… of the General Laws of Motion …”, Phil. Trans. Roy. Soc., iii (1668), 864–866.CrossRefGoogle Scholar

29 Cambridge University Library, Add. MS. 4004, f. 13.

30 A diagram accompanying the text makes it clear that pressures “towards w” and “towards v” are in fact acting in opposite directions.

31 Wren initiated the experimental study of impacts before the Royal Society some time during the 1660's, while Mariotte's work first appeared in 1673 under the title Traité de la percussion ou du choc des corps (reprinted in Oeuvres de M. Mariotte, de l'Académie Royale des Sciences (La Haye, 1740), 1–116)Google Scholar. There has been some dispute concerning the extent to which Mariotte's work was original. Huygens, for instance, stated flatly that

“Mariotte has taken everything from me, as those of the Académie des Sciences, M. du Hamel, M. Gallois, and the records, can testify: the machine, the experiment on the elasticity of glass spheres, the experiment on one or several balls pushed together against a row of similar balls, the theorems I had published. He should have mentioned my name. I told him so one day, and he did not know what to answer…” (Quoted by Dugas, , op. cit. (11), 289).Google Scholar Again, Tait has argued (P. G. Tait, “Note on a Singular Passage in the Principia”, Proc. Roy. Soc. Edinburgh, 19 January 1885; reprinted in Tait, 's Scientific Papers (Cambridge, 1900), ii, 110–114)Google Scholar that Newton's reference to Mariotte was extremely sarcastic, though the sarcasm has been lost in Motte's translation. Tait's supposition is that Newton knew that most of the ideas claimed as his own by Mariotte had been put forward earlier by Wren. Such criticism is hardly fair to Mariotte, for while many of the ideas he espoused were undoubtedly borrowed without acknowledgement, others were clearly his own, and certainly the synthesis he achieved was entirely his own.

32 Principia, 24.Google Scholar

33 Ibid.