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The Principia’s second law (as Newton understood it) from Galileo to Laplace

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The same force, by acting with the same direction and in the same time on the same body whether at rest or carried on with any motion whatever, will in the meaning of this Law achieve an identical translation.... The truth of this principle, I may add, is amply enough confirmed from experience....

Isaac Newton [on the meaning of Law II, early 1690s]

Abstract

Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the major treatises on the foundations of mechanics in the eighteenth century—by Varignon, Hermann, Euler, Maclaurin, d’Alembert, Euler (again), Lagrange, and Laplace—do not record, cite, discuss, or even mention the Principia’s statement of the second law. Nevertheless, the present study shows that all of these scientists do in fact assume the principle that the Principia’s second law asserts as a fundamental axiom in their mechanics. (For what that second law asserts, we rely on Newton’s own testimony.) Some, like Varignon and Hermann, assume the axiom implicitly, apparently unaware that any assumption is being made, while others, like Maclaurin and Euler, assume the axiom explicitly, apparently unaware that the assertion assumed is the second law as Newton himself understood it. But in every case these scientists employ the principle asserted by the Principia’s second law fundamentally, unaware that they should be citing Neutonus, Prin., Phil. Nat. Math., Lex II.

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Notes

  1. In (35, VI, 540–543) Newton’s figure is redrawn. I thank George Smith for providing me with a copy of Newton’s original hand-drawn figure.

  2. D.T. Whiteside has preserved these manuscripts, accompanied by insightful commentary, in Volume VI of (35, 538–609). For more on Newton’s planned radical revisions, see Brackenridge (1990) and Pourciau (1992).

  3. In this definition and elsewhere in the Principia, Cohen and Whitman render both “celeritas” and “velocitas” as “velocity.” I am reluctant to disagree with any aspect of their fine translation, but it seems to me that “speed” would have been a less ambiguous choice for the modern reader, since “velocity” as used by physicists and mathematicians today has a direction, while “celeritas” and “velocitas,” both meaning “swiftness, fleetness, speed,” (Andrews 1863) do not.

  4. The “law” that results from misinterpreting the resting deflection \(\overrightarrow{PG}\) as the environmental resting deflection \(\overrightarrow{PG}^{E}\) would apply only in a “velocity-independent” force field, a force field which, for a given body in a given place, produces the same impressed force, no matter the (vector) velocity of the body. Such an impotent “law” would not apply to the centripetal forces in the first six sections of Book I (for it is only in Section VII that Newton begins to assume that his centripetal forces are “velocity-independent”) nor to any of the resistances forces of Book II. See Pourciau (2011).

  5. It is in fact Proposition VI, Book I—and, one might argue, its ancestors, Proposition VI of the first edition Principia, which surfaces as Corollary I of the new Proposition VI in the second and third editions, and Theorem 3 in “De Motu corporum in gyrum” (Newton 1989, 3)—which asserts a Newtonian version of the classical \(\mathbf {f}=m\mathbf {a}\).

  6. Published in three volumes (1739, 1740, and 1742), the so-called “Jesuit edition”—presenting the Latin of the 1726 Principia with extensive commentaries by the Minim (not Jesuit) friars François Jacquier and Thomas Le Seur and the professor of mathematics Jean Louis Calandrini—was very influential in the eighteenth century and lent further authority to this misreading of the Principia’s second law, especially to the restricted direction of the force. See (Newton 1739–1742, Notes 25–31) as well as Guicciardini (2015).

  7. For more on the reaction of eighteenth-century scientists to this [mis]interpretation of the second law, read Hankins (1967) and Harman (1988). Both Hankins and Harman take the meaning of Law II to be the Common Misinterpretation of the Second Law, given above.

  8. Euler was not the first to present the scalar version, \(f=ma\), of the law in differential form that we call Newton’s Second Law today. For motion along a line, Varignon in 1700 wrote \(y=dv/dt\), where y is the force and the mass has been taken to be one (Varignon 1700, 23), and Hermann in 1716 argued that \(G=MdV:dT\), where G stands for the “variable urging” (Hermann 1716, 56, §131).

  9. Note the difference between the two kinds of “velocity independence” involved here: the “velocity independence” of gravity, where a given body at a given place will be acted on by the same impressed force, whether the body is in motion or at rest, versus the Compound Second Law, where a given impressed force acting on a given body will generate the same deflection, whether the body is in motion or at rest.

  10. We do not exaggerate: we really do mean centuries of misinterpretations. As recently as 2016, for example, in side-by-side chapters of The Cambridge Companion to Newton, Cohen (2016) and the author Pourciau (2016) put forward mutually inconsistent interpretations of the Principia’s Law II. These two interpretations cannot both be correct. It follows that one of us, perhaps both of us, must have been misreading Law II. As of 2016, then, the statement of Law II in the Principia was still being misinterpreted. This amounts to \(2016-1687=329\) years or 3.29 centuries of misinterpretations.

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Acknowledgements

For help with translations from the French, Ed Sandifer and Rob Bradley; for translations from the Latin, feminist scholar Mary Ann Rossi; for helpful conversations on Newtonian mechanics, Michael Spivak and Scott Corry; for (temporarily) pulling Hermann’s Phoronomia out of a box addressed to the Huntington Library, Jessica Murphy at the (now departed) Burndy Library; and for editorial comments, intelligent and insightful, George Smith and an anonymous reader.

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Dedicated to Curtis Wilson.

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Pourciau, B. The Principia’s second law (as Newton understood it) from Galileo to Laplace. Arch. Hist. Exact Sci. 74, 183–242 (2020). https://doi.org/10.1007/s00407-019-00242-y

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