Newton’s Third Law and Universal Gravity

  • I. Bernard Cohen
Part of the Archives Internationales D’Histoire des Idées / International Archives of the History of Ideas book series (ARCH, volume 123)


Newton’s great Principia is doubly tripartite. It is composed of three “books” and has three major goals: to set forth new foundations and methods of rational mechanics; to disclose a new natural philosophy; and to develop a new system of the world based on gravitational celestial dynamics. Attention shall be focussed here on the crucial step that enabled Newton to develop and to use the concept of universal gravity and to state its quantitative law. A plausible explanation shall be offered for Newton’s daring in proposing a universal force that could extend over hundreds of millions of miles of empty space, a kind of force that in its basic properties went squarely against the principles of the “received” mechanical philosophy.1


Elliptical Orbit Centripetal Force Inertial Motion Planetary Motion Mathematical Construct 
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  1. 1.
    The mechanical philosophy, associated primarily with the name of Descartes, held that all phenomena of nature were to be accounted for in terms of matter and motion. See Marie Boas’s classic memoir, “The Establishment of the Mechanical Philosophy,” Osiris, 1952, 10: 412–541, reprinted under the title The Mechanical Philosophy (New York: Arno Press, 1981). Also R.S. Westfall: Force in Newton’s Physics (London: Macdonald; New York: American Elsevier, 1971), especially pp. 377–378.CrossRefGoogle Scholar
  2. 2.
    For elliptical orbits Kepler’s “average distance” from the sun is the semi-axis major of the ellipse. Newton learned this law on reading Thomas Streete’s Astronomia Carolina (London, 1661; rev. ed., London, 1710), where (p. 61) it is stated that the mean distances of the planets from the sun are “in sesquialter proportion to the Periods of their Revolution in Time.” We may observe that this third law was the only one of Kepler’s three laws of planetary motion for which Newton gave credit to Kepler in the Principia (2d ed. 1713, 3d ed. 1726, Book 3, Phenomena 1, 2, and 4; in the 1st ed., 1687, these were Hypotheses 5 and 7); see my “Kepler’s Century: Prelude to Newton’s,” Vistas in Astronomy, 1975, 18: 3–40, and my “Hypotheses in Newton’s Philosophy,” Physis, 1966, 8: 163–184.Google Scholar
  3. 3.
    On Newton’s early discovery of this rule or law in the 1660s, see note 6 infra. In the post-Newtonian inertial physics, the post-Principia era, any curved motion — such as motion along a circle at a constant or uniform speed — requires a continually acting accelerating force. The reason is that there is a continual change in the tangential direction of motion.Google Scholar
  4. One must be careful not to assume that in the 1660s, when Newton discovered the law of force for uniform circular motion, he had a simple and clear idea of “centrifugal” force. (This was decades before he had invented and named the concept of “centripetal” force). In the 1660s Newton didn’t even have the relatively simple notion of force that was to characterize his later dynamics. At this time he used the expression “force by which a body… indeavours from the center,” an echo of the Cartesian concept of “conatus.” On the state of Newton’s thinking about “force” in the 1660s and the development of his ideas, see Westfall’s Force (cited in ref. 1 supra), ch. 7.CrossRefGoogle Scholar
  5. 5.
    Since the circumference of a circle of radius r is 2πr and the time to traverse a complete radius is, by definition, the period of revolution T, it follows that the speed \( v = \frac{{2\pi r}}{T} \) whence \( {v^2} = \frac{{4{\pi^2}{r^2}}}{{{T^2}}} \) or, in the language of proportion, ν2 is proportional to \( \frac{{{r^2}}}{{{T^2}}} \). It requires very little mathematical insight to recognize that the final term in the proportion (math) needs only to be slightly modified in order to become \( \frac{{{r^3}}}{{{T^2}}} \), which is a constant for planets in the solar system. It is assumed that orbits are circular. Hence, since \( \frac{{{r^3}}}{{{T^2}}} \) a constant, it follows that ν2 is proportional to (math) is proportional to (\( \frac{1}{r} \))(\( \frac{{{r^3}}}{{{T^2}}} \)). But if the “force” is proportional to \( \frac{1}{r} \) and ν2 is proportional to \( \frac{{{v^2}}}{r} \), then the “force” is proportional to \( \frac{1}{{{r^2}}} \). See, further, my Newtonian Revolution (Cambridge: Cambridge University Press, 1980), pp. 229–230; also pp. 238–241. In his early writings on dynamics (1660s and early 1670s), Newton did not use the expression “centrifugal force” or “vis centrifuga,” but rather wrote of an “endeavour” to recede from the center, an echo of a Descartes’s “conatus recedendi a centro.”Google Scholar
  6. 6.
    Christiaan Huygens: Horologium Oscillatorium (Paris, 1673). Newton had discovered the \( \frac{{{v^2}}}{r} \) law independently, but had not published his result. See John W. Herivel: “Newton’s Discovery of the Law of Centrifugal Force,” Isis, 1960, 51: 546–553; also Herivel’s The Background to Newton’s Principia (Oxford: Clarendon Press, 1965). Huygens had apparently discovered the law of “conatus” (\( \frac{{{v^2}}}{r} \)) before Newton and had put together the results of his investigations in a manuscript entitled De Vis Centrifuga in 1659. This work was first published posthumously in 1703. In the Horologium Oscillatorium (1673), Huygens stated his results but did not give any proofs.Google Scholar
  7. 7.
    There is, however, an enormous conceptual gulf between Huygens’s formulation and the Newtonian (or post-Newtonian) general expression \( F = \frac{{m{v^2}}}{r} \). On this difference see H.J.M. Bos in Dictionary of Scientific Biography, vol. 6 (New York: Charles Scribner’s Sons, 1972), p. 607a. On Newton’s transformation of Huygens’s “vis centrifuga” into the Newtonian “vis centripeta” and Newton’s introduction of the latter term in honor of Huygens (the only one of his contemporaries of whom Newton used the adjective “summus”), see A. Koyre`@ and I.B. Cohen: “Newton and the Leibniz-Clarke Correspondence,” Archives Internationales d’Histoire des Sciences, 1962, 75: 63–126, esp. p. 122; see, further, my Introduction to Newton’s ‘Principia’ (Cambridge: Harvard University Press; Cambridge: Cambridge University Press, 1971; rev. ed., 1978), p. 53n, pp. 296–297 (suppl. 1, §6).Google Scholar
  8. 9.
    There is no satisfactory full-length study of Hooke. On Hooke’s dynamics, see Richard S. Westfall: “Hooke and the Law of Universal Gravitation,” The British Journal for the History of Science, 1967, 3: 254–261; also Louise Diehl Patterson: “Hooke’s Gravitation Theory and its Influence on Newton,” Isis, 1949, 40: 327–341; 1950, 41: 32–45.CrossRefGoogle Scholar
  9. 10.
    Published in Phil Trans, no. 80, 19 Feb. 1671/72, pp. 3075–3087. This publication and the printed documents of the ensuing controversy are reprinted in f acsimile in I.B. Cohen and Robert E. Schofield (eds.): Isaac Newton’s Papers and Letters on Natural Philosophy (Cambridge: Harvard University Press, 1958); revised edition, 1978). See, also, H.W. Turnbull (ed.): The Correspondence of Isaac Newton (Cambridge: published for the Royal Society at the University Press, 1959, 1960), vols. 1–2.Google Scholar
  10. 11.
    The Newton-Hooke correspondence is available in vol. 2 of Newton’s Correspondence. See Alexandre Koyre`@: “An Unpublished Letter of Robert Hooke to Isaac Newton,” Isis, 1952, 43: 312–327; reprinted in Koyre`@’s Newtonian Studies (Cambridge: Harvard University Press, 1965). On Newton’s diagram and its successive misrepresentation, see J.A. Lohne: “The Increasing Corruption of Newton’s Diagrams,” History of Science, 1967, 6: 69–89.Google Scholar
  11. 12.
    The occasion was the delivery to the Royal Society of the first book of the Principia. Halley reported to Newton (22 May 1686; Correspondence, vol. 2, p. 433) that Hooke “seems to expect you should make some mention of him in the preface.” See my Introduction to Newton’s ‘Principia’ (Cambridge: Harvard University Press; Cambridge: Cambridge University Press, 1971, 1978), pp. 52–54, 114–115, 132–135.Google Scholar
  12. A first-rate account of this episode is given in R.S. Westfall: Never at Rest: A Biography of Isaac Newton (Cambridge, London, New York: Cambridge University Press, 1980), pp. 381–388. Westfall carefully analyzes all of Hooke’s arguments and concludes (p. 387) that Hooke “compunded the confusion by combining his fallacious law of force with Kepler’s fallacious law of velocities.”Google Scholar
  13. 14.
    The importance of Hooke’s teaching Newton how to analyze curved or orbital motions is stressed by Westfall in his Force (cited in n. 1 supra), pp. 426–427.Google Scholar
  14. 15.
    Augustus de Morgan once wrote — Essays on the Life and Work of Newton (Chicago, London: The Open Court Publishing Company, 1914), p. 39 — that a “discovery of Newton was of a two-fold character — he made it, and then others had to find out that he had made it.”Google Scholar
  15. 16.
    For details see my Introduction (cited in n. 7 supra), pp. 47 sqq.).Google Scholar
  16. 17.
    See my Introduction, pp. 54 sqq. This document is to be found in the Register of the Royal Society, vol. 6, pp. 218 sqq. It has been published by Stephen Peter Rigaud: Historical Essay on the First Publication of Sir Isaac Newton’s Principia (Oxford: at the University Press, 1838), Appendix No. 1.Google Scholar
  17. 18.
    Chiefly at issue here is whether a certain tract was written before or after the writing of the Principia. This tract was alleged by John W. Herivel to contain “The Originals of the Two Propositions Discovered by Newton in December 1679,” Archives Internationales d’Histoire des Sciences, 1961, 14: 23–33, an identification challenged by A. Rupert Hall and Marie Boas Hall: “The Date of ‘On Motion in Ellipses’,” Archives, 1963, 16: 23–28. Herivel’s identification was revived by R.S. Westfall: “A Note on Newton’s Demonstration of Motion in Ellipses,” Archives, 1969, 22: 51–60; see, further, Westfall’s Force (cited in n. 1 supra), pp. 429–31. Westfall’s arguments were refuted by D.T. Whiteside in a review of Westfall’s article in Zentralblatt fr Mathematik und ihre Grenzegebiete, 1970, 194: 2–3; see, also, Whiteside’s edition of The Mathematical Papers of Isaac Newton, vol. 6 (Cambridge: at the University Press, 1974), pp. 14; 553–554. Westfall’s summary and rebuttal appears in his Never at Rest (cited in no. 14 supra), pp. 387–388 (n. 145). This question of dating is not of primary significance in relation to the topic of the present paper, since — as shall be seen in what follows — Newton’s great intellectual leap forward occurred in revising the tract De Motu in late 1684 or early 1685, not in the first composition of that work. It is therefore an independent question whether the first draft of De Motu represents the state of Newton’s thinking in 1679/1680 or a later state, as of 1684.Google Scholar
  18. 19.
    See Newton’s own statements printed in Suppl. 1 to my Introduction (cited in n. 12 supra). There is no single curve that answers the question of what the path must be under the action of an inverse-square force: the result can be an ellipse, a parabola, a hyperbola, or a straight line.Google Scholar
  19. 20.
    Robert Weinstock: “Dismantling a Centuries-Old Myth: Newton’s Principia and Inverse-Square Orbits,” American Journal of Physics, 1982, 50: 610–617; for a convenient summary of this dispute, see my The Birth of New Physics (second edition, New York: W.W. Norton, 1985), Supplement 13; also my Newtonian Revolution (cited in n. 5 supra), pp. 248–249.CrossRefGoogle Scholar
  20. 25.
    In the succeeding Props. 12 and 13, Newton shows that the inverse-square law holds also for a hyperbola and for a parabola. A corollary to Prop. 13 (in the first edition, 1687) states the converse: that if there is a centrally directed inverse-square force, the orbit will be one of the conic sections. In reply to direct criticism Newton (in the second edition, 1713) sketched out the stages of his proof. See D.T. Whiteside’s commentary in vol. 6 of his edition of Mathematical Papers (cited in n. 18 supra), pp. 146–149 (n. 124).Google Scholar
  21. 26.
    See n. 7 supra.Google Scholar
  22. 27.
    There are four known major versions of the tract De Motu. As mentioned above (n. 17) S.P. Rigaud first published a version in 1838. A second publication occurred in W.W. Rouse Ball’s An Essay on Newton’s “Principia” (London, New York: Macmillan and Co., 1893 — reprinted, with an introduction by I.B. Cohen, New York: Johnson Reprint Corporation, 1972), pp. 30–56. An edition, with an English translation and commentary, was published by A. Rupert Hall and Marie Boas Hall in their collection, Unpublished Scientific Papers of Isaac Newton (Cambridge: at the University Press, 1962), pp. 243-Google Scholar
  23. 292, with notes and commentary. The tract, in its several versions, also appears in John H. Herivel: The Background to Newton’s Principia (Oxford: at the Clarendon Press, 1965), part 2, sect. 9. A final scholarly version and translation, with many valuable historical and analytical notes, appeared in vol. 6 of D.T. Whiteside’s edition of Newton’s Mathematical Papers (cited in n. 18 supra), pp. 30–91. On the relation of the tract De Motu to the Principia, see my Introduction (cited in n. 14 supra), pp. 47–81, esp. 54–62.Google Scholar
  24. 28.
    On these emendations see D.T. Whiteside’s comments in Mathematical Papers, vol. 6, p. 78 (n. 10) and Curtis Wilson: “From Kepler’s Laws, so-called, to Universal Gravitation: Empirical Factors,” Archive for History of Exact Sciences, 1970, 6: 89–170.CrossRefGoogle Scholar
  25. 29.
    The first version consisted of two “books,” corresponding roughly to Book 1 of the final version and a different form of what became Book 3. The text of that early Book 1 has been published by D.T. Whiteside in vol. 6 of Mathematical Papers. The early version of what became Book 3 of the Principia was published soon after Newton’s death in both English and Latin versions under the title A Treatise of the System of the World (London, 1728; revised, 1731). See my introduction to the reprint of this work, published in London by Dawsons of Pall Mall in 1969. On these texts see my Introduction (cited in n. 14 supra), chapter 4 and Supplements 4, 6.Google Scholar
  26. Newton explains how he has used the third law on pp. 37–39 of the second edition (London, 1731). He is aware that “it may be objected, that according to this philosophy all bodies should mutually attract one another, contrary to the evidence of experiments in terrestrial bodies.” Newton replies that “the experiments in terrestrial bodies come to no account.” He shows that by considering actual magnitudes, it is easy to show that one could not observe the mutual gravitational action of one terrestrial body on another, nor the gravitational action of a terrestrial body on the earth.Google Scholar
  27. 30.
    These two rules occur in the introduction to Book 3 of the Principia, which is entitled “Liber Tertius: De Systemate Mundi.” In the second edition (1713) there were three such rules; a fourth was added in the third edition (1726). In the first edition (1687), the “Regulae” and what later were designated as “Phaenomena” were included under an all-embracing rubric of “Hypotheses.” For details, see my article on “Hypotheses in Newton’s Philosophy,” Physis, 1966, 8: 163–184.Google Scholar
  28. 32.
    On this topic see my historical introduction to Isaac Newton’s ‘Theory of the Moon’s Motion’ (1702) (London: Dawson, 1975) and the unpublished doctoral dissertation (1975: The Johns Hopkins University) by Craig Waff on Universal Gravitation and the Motion of the Moon’s Apogee: The Establishment and Reception of Newton’s Inverse-Square Law, 1687–1749. Also D.T. Whiteside: “Newton’s Lunar Theory: From High Hope to Disenchantment,” Vistas in Astronomy, 1976, 19: 317–328.CrossRefGoogle Scholar
  29. 38.
    The Origin of Forms & Qualities, from The Works of the Honourable Robert Boyle (London, 1772), vol. 3, p. 14.Google Scholar
  30. 39.
    On these various attempts to account for gravity see my “The Principia, Universal Gravitation, and the ‘Newtonian Style/ in Relation to the Newtonian Revolution in Science,” pp. 21–108 of Zev Bechler (ed.): Contemporary Newtonian Research (Dordrecht, Boston: D. Reidel Publishing Company, 1982) and the important studies by Henry Guerlac, collected in his Essays and Papers in the History of Modern Science (Baltimore: The Johns Hopkins Press, 1977). R.S. Westfall has made an extensive study of Newton’s thinking on this topic and concludes — in his Force (cited in n. 1 supra), p. 377 — that in about 1679 Newton’s “philosophy of nature” underwent a redirection of a magnitude that “can scarcely be overstated”: “What he proposed was an addition to the ontology of nature. Where the orthodox mechanical philosophy of seventeenth-century science insisted that physical reality consists solely of material particles in motion, characterised by size, shape, and solidity alone, Newton now added forces of attraction and repulsion, considered as properties of such particles, to the catalogue of nature’s ontology”Google Scholar
  31. 41.
    There are many explanations that have been given of Newton’s slogan, “Hypotheses non fingo,” which seems to have raised more questions than it solved. The General Scholium does not express simple allegiance to the mechanical philosophy, something Newton had long since abandoned in its oversimplified form (see, e.g., Westfall’s Never at Rest, cited in n. 13 supra, pp. 748–749).Google Scholar
  32. In an article on “Occult Qualities and the Experimental Philosophy: Active Principles in the Pre-Newtonian Matter Theory,” History of Science, 1986, 24: 335–381, John Henry argues that the English tradition of the mechanical philosophy differed from its Cartesian counterpart with respect to the activity of matter, so that Newton’s belief in a variety of “occult qualities” of a non-scholastic kind is more in keeping with the views of his English contemporaries and immediate predecessors than has hitherto been believed.Google Scholar
  33. 42.
    Newton to Bentley, 25 Feb. 1692/93, Correspondence, vol. 3. “Tis inconceivable that inanimate brute matter should (without the mediation of something else which is not material) operate upon & affect other matter without mutual contact.” Additionally: “That gravity should be innate inherent & essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it.”Google Scholar
  34. 43.
    I have developed this notion of a Newtonian style in the study referred to in note 39 supra and the book referred to in n. 5 supra. My point of view is supplementary to that held by R.S. Westfall, for which see note 39 supra.Google Scholar
  35. 45.
    Introduction to Sect. 11, Book 1.Google Scholar
  36. 46.
    See my “Newton’s Theory vs. Kepler’s Theory and Galileo’s Theory: An Example of a Difference between a Philosophical and a Historical Analysis of Science,” pp. 299–338 of Yehuda Elkana (ed.): The Interaction Between Science and Philosophy (Atlantic Highlands, N.J.: Humanities Press, 1974).Google Scholar
  37. 47.
    That is, he admitted, in a letter to Richard Bentley (25 Feb. 1692/3), he had a repugnance to a force which would act at-a-distance through empty space, “so that one body may act upon another at a distance through a vacuum without the mediation of anything else. But whether the “agent” causing gravity “be material or immaterial,” Newton “left to the consideration of my readers.” See Correspondence, vol. 3, p. 254.Google Scholar
  38. 48.
    Christiaan Huygens: Discours de la Pesanteur (1690), Oeuvres, vol. 21, pp. 472, 474. For a discussion of this topic and a translation of the relevant extracts see Alexandre Koyré: Newtonian Studies (Cambridge: Harvard University Press, 1965), pp. 121 sqq. and my Newtonian Revolution (cited in n. 5 supra), pp. 79–83.Google Scholar
  39. 49.
    Newton was evidently willing to accept the existence of small-range forces of attraction and repulsion (see n. 39 supra) and, to that extent, had indeed made a fundamental revision of the orthodox mechanical philosophy, whose full implications he abhorred, as Westfall had pointed out in both his Force (see n. 1 supra) and his biography of Newton (see n. 13 supra). But Newton did share the general abhorrence of those who adhered to the orthodox mechanical philosophy when it came to forces that could extend over vast distances, as would have to be the case for universal gravity since it affects the motion of comets way out beyond the visible solar system. Hence (see n.47 supra) he believed that gravity must act through the mediation of something else — material or immaterial. As John Henry has indicated (see n. 41 supra), Newton was hardly the first person to hold that there could be causes of natural phenomena that remain secret or unknown to us. But Newton did more; he insisted (in the General Scholium, introduced as a conclusion in the second edition of the Principia) that “it is enough” to be able to predict and to retrodict the observed phenomena, even though the active cause or force is unexplainable or is unacceptable according to the canons of a received philosophy. Since gravity “really exists,” as we all know, and since Newton’s gravity-like force in his final mathematical construct did produce the same effects that we observe in the world of experience, Newton held that it was more important to get on with the business of developing celestial mechanics than arguing to no end about whether a universal gravitating force does (or can) exist. His originality is discerned in his conversion of this conviction into a forcefully expressed public declaration of policy. But we must not forget that, despite Newton’s great interest in electricity and in aethereal medium in relation to gravitational action, he was known to have said that universal gravity is caused directly by God.Google Scholar

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • I. Bernard Cohen
    • 1
  1. 1.Harvard UniversityUSA

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