When we call the equation **f** = *m***a** “Newton’s second law,” how much historical truth lies behind us? Many textbooks on introductory physics and classical mechanics claim that the *Principia*’s second law becomes **f** = *m***a**, once Newton’s vocabulary has been translated into more familiar terms. But there is nothing in the *Principia*’s second law about acceleration and nothing about a rate of change. If the *Principia*’s second law does not assert **f** = *m***a**, what does it assert, and is there some other axiom or some proposition in the *Principia* that does assert **f** = *m***a**? Is there any historical truth behind us when we call **f** = *m***a** “Newton’s second law”? This article answers these questions.

## REFERENCES

1.

Isaac Newton,

*Philosophiae Naturalis Principia Mathematica*or*Mathematical Principles of Natural Philosophy*[*Natural Philosophy*meaning*Physics*], first published in 1687, with a second edition in 1713, and a third, just a year before Newton died, in 1726. Devoted to “rational mechanics”—“the science,” as Newton puts it, “expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever”—the*Principia*consists of two preliminary sections (*Definitions*and*Axioms, or the Laws of Motion*) followed by three books: Books I and II (both titled*The Motion of Bodies*, but with Book II concentrating on*resisted*motion) together present mathematical principles, while Book III (*The System of the World*) applies these principles to planets and moons of our solar system. All quotations from the*Principia*come from the recent English translation: I. Newton,*The Principia, Mathematical Principles of Natural Philosophy*, translated into English from the Latin of the third (1726) edition by I. Bernard Cohen and Anne Whitman, assisted by Julia Budenz, and preceded by “A guide to Newton’s*Principia*” by I. Bernard Cohen (University of California Press, Berkeley,1999

).2.

Reference 1, p.

416

.3.

According to this standard interpretation, the “change in motion” is the change Δmv = mΔv in linear momentum, where Δv stands for a generated change in speed, and the second law applies directly only to an instantaneous impulse, which produces an instantaneous change in either speed or direction. Generally, the meaning of the “motive force” remains a mystery in this reading. Under this interpretation, the second law would apply to a (continuous) force only indirectly, through a complicated limit argument involving impulses in series, and to an oblique force only after finding the component parallel to the direction of motion. See, for instance, I. B. Cohen, “Newton’s Concept of Force and Mass, With Notes on the Laws of Motion

,” in The Cambridge Companion to Newton

, edited by I. B.

Cohen

and G. E.

Smith

(Cambridge U.P.

, Cambridge

, 2002

), pp. 65

–67

;M.

Blay

, “Force, Continuity, and the Mathematization of Motion at the End of the Seventeenth Century

,” in Isaac Newton’s Natural Philosophy

, edited by J. Z.

Buchwald

and I. B.

Cohen

(The MIT Press

, Cambridge, MA

, 2001

), pp. 226

–227

; andH.

Erlichson

, “Motive Force and Centripetal Force in Newton’s Mechanics

,” Am. J. Phys.

59

, 842

–849

(1991

), p. 844.4.

See, for example, these excellent mechanics texts:

J. M.

Knudsen

and P. G.

Hjorth

, Elements of Newtonian Mechanics, Including Nonlinear Dynamics

, revised and enlarged 3rd ed. (Springer-Verlag

, Berlin

, 2000

), p. 28

;J. B.

Marion

and S. T.

Thornton

, Classical Dynamics of Particles and Systems

, 4th ed. (Saunders

, Fort Worth

, 1995

), p. 49

;5.

For more detail on the interpretation of the

*Principia*’s second law presented here, see B. Pourciau, “Newton’s interpretation of Newton’s second law,”Arch. Hist. Exact Sci.

**60**, 157–207 (2006

).6.

“At rest” in Newton’s sense of being at rest relative to “that immovable space in which the bodies truly move.” Newton was well aware of the weaknesses in his own theory—that it depends on “immovable space … [which] makes no impression on the senses,” (Ref. 1, p. 414) for example, or that it provides no explanation for the proportionality of inertial and gravitational mass.

7.

Newton applies the Principia’s second law only to forces (such as centripetal forces) that would move a body at rest along a line

.8.

Today we would detect any deviation from uniform straight line motion or rest with a nonzero (vector) acceleration. Working instead with the moving and resting deflections, Newton could accommodate deviations generated by instantaneous impulses, when the acceleration, being infinite, does not exist. (For an instantaneous impulse, the arc segment

*PQ*would be a line segment.) Such impulses play a central role early in the*Principia*, where they are used in series to approximate the motion of a body under the influence of a (continuous) centripetal force in the argument for Proposition I.9.

To clarify a possible confusion: A charged particle at rest at a point

*P*in a magnetic field will remain at rest, while the same particle projected through*P*with a nonzero velocity (not parallel to the magnetic field) will be deflected from its uniform straight line motion. This difference does not violate the principle that the resting and moving deflections must be the same (for a given force on a given body). Although the origin or mechanism (the magnetic field) remains the same for the particle in both cases, the force (or “thrust”) experienced by the particle is not the same. If we were to imagine applying the same “thrust” experienced by the moving particle to the resting particle, then the resting deflection would equal the moving deflection, according to this natural sequel to the first law. That this natural sequel is relativistically false does not worry us, because we are working within an earlier paradigm: the mechanics of the*Principia*.10.

The Correspondence of Isaac Newton, 1688–1694

, edited by H.

Turnbull

(Cambridge U.P.

, Cambridge

, 1961

), Vol. 3

, pp. 155

–156

.11.

*Christiaan Huygens’ The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks*, translated and with notes by R. Blackwell, based on the original 1673 edition (Iowa State U.P., Ames,

1986

), p. 33.12.

Reference 1, p.

382

.13.

The Mathematical Papers of Isaac Newton

, edited and with extensive commentaries by D. T.

Whiteside

(Cambridge U.P.

, Cambridge

, 1974

), Vol. 6

, pp. 538

–609

.14.

See Ref.13, pp. 539–543, or

I. B.

Cohen

, “Newton’s Second Law and the Concept of Force in the Principia

,” in The Annus Mirabilis of Sir Isaac Newton 1666–1966

, edited by R.

Palter

(The MIT Press

, Cambridge, MA

, 1970

), pp. 143

–185

.15.

Nowhere else in his work on mechanics, published or unpublished, does Newton use the phrase “in the meaning of this law” in a passage that describes the second law of motion. Indeed, nowhere else does he describe how he interprets the second law. The passage we have quoted is the only known record, in his own words, of Newton’s understanding of his second law.

16.

Newton often writes that two line segments are “parallel and equal,” or that a quantity “takes place along” a specified line. Such directed line segments are all over the

*Principia*. The anachronistic overhead arrow we have used captures this notion perfectly, with no distortion of Newton’s intended meaning.17.

In Newton’s mechanics, a given force and its observable sign are conceptually distinct, the observable sign being the deflection (or the acceleration): intuitively, a thrust is seen as distinct from the observed effect of that thrust. In any development of classical mechanics where this distinction disappears, the Compound Second Law becomes a vacuous tautology. Within such developments, a meaningful analogue may be phrased in terms of reference frames and Galilean relativity: Two observers, moving with constant relative (vector) velocity, record the same deflection….

18.

Reference 1, p.

407

.19.

Because the mathematics of the

*Principia,*for the most part, is a geometrical version of limits and calculus, Newton preferred to work with proportions rather than equations. But we lose nothing and we gain a more modern presentation treating these proportions as equalities. Hence, we write the “motive force*is*…,” rather than the “motive force is proportional to….”20.

reprinted in

A.

Koyré

, Metaphysics and Measurement

(Chapman and Hall

, London

, 1968

), pp. 89

–117

.Also see

J. G.

Yoder

, Unrolling Time: Christiaan Huygens and the Mathematization of Nature

(Cambridge U.P.

, Cambridge

, 1988

), Chaps. II–IV.21.

For example, we can gauge how fast 1 – cos

*h*approaches 0 as*h*→ 0^{+}by comparing it to the quantities*h*,*h*^{2}, and*h*^{3}, which approach 0 increasingly fast: (1 – cos*h*)/*h*→ 0, $(1-cosh)/h2\u219212$, and (1 – cos*h*)/*h*^{3}→ +*∞*. These limits tell us that 1 – cos*h*approaches 0 faster than*h*, at the same rate as*h*^{2}, and slower than*h*^{3}. Newton would write that 1 – cos*h*“will be as”*h*^{2}(as*h*→ 0). We have the good approximation $1-cosh\u224812h2$ for*h*sufficiently small; so we can also think of the Newtonian code “will be as” as signaling an “ultimate proportion”: 1 – cos*h*is nearly proportional to*h*^{2}for sufficiently small*h*. It does not matter that the limit of the ratio is 1/2, only that the limit is nonzero and finite.22.

Reference 1, p.

655

.23.

Reference 1, pp.

437

–438

.24.

A (finite) force, by definition, generates a finite acceleration: thus $2(LQ\u2192/h2)=2(LQ\u2192/h)/h$ approaches the finite limit

**a**(*t*_{0}). When a ratio has a finite limit and its denominator [*h*] approaches zero, then the numerator [$LQ\u2192/h$] must also approach zero. The same is true for $PG\u2192/h$. Thus, as we claimed earlier, the motive force $M(PG\u2192/h)$ and the change in motion $M(LQ\u2192/h)$ are relative, not absolute, measures of the force.25.

J.

Lagrange

, Analytical Mechanics

, translated and edited by Auguste Boissonnade

and Victor N.

Vagliente

(Kluwer Academic Publishers

, Boston

, 1997

), p. 171

.26.

The major treatises on mechanics in the 18th century—by Varignon, Hermann, Euler, MacLaurin, d’Alembert, Euler (again), Lagrange, and Laplace—all acknowledge the enormous influence of Newton’s

*Principia*, yet none of these works states or cites the second law as it appears in the*Principia*. Given that Newton evidently regarded the*Principia*’s second law as a fundamental axiom of mechanics, this omission is odd. But the explanation is easy. Confused by the ambiguous account of the second law in the*Principia*, these scientists misinterpreted the second law in ways that made it appear irrelevant to their developments of mechanics, as a law applying to impulses only, for example. Nevertheless, all of them did apply the property asserted by the*Principia*’s second law (as Newton understood it, in the form of the Compound Second Law), using it as a fundamental assumption of their own mechanics, without realizing that they should have cited the second law in the*Principia*. See, for example, J. Hermann,*Phoronomia, sive de viribus et motibus corporum solidorum et fluidorum, libri duo*(R. & G. Wetstenios, Amsterdam,1716

), pp. 68–69; C. MacLaurin, *A Treatise of Fluxions in Two Books*(T. W. and T. Ruddimans, Edinburgh,1742

), Vol. **1**, p. 347; and L. Euler,*Theoria motus corporum solidorum seu rigidorum: ex primus nostrae cognitionis principiis stabilita et ad omnes motus, qui in huis modi corpora cadere possunt, accommodata*, Leonhardi Euleri Opera Omnia, Series II: Opera Mechanica et Astronomica Vol.**3**, edited by C. Blanc, originally published by Orell Füssli in 1765 (Birkhäuser, Berlin,1948

), pp. 68–69. For more along these lines, read B. Pourciau, “Newton’s second law (as Newton understood it) from Galileo to Laplace,” unpublished.27.

Reference 1, p.

454

.28.

For details on this and on Proposition VI more generally, read B. Pourciau, “

Force, deflection, and time: Proposition VI of Newton’s Principia

,” Hist. Math. **34**, 140–172 (2007

).29.

See Ref. 28, p. 160. Our $PG\u2192$ and $LQ\u2192$ notation in Proposition VI hides a technical point: if these two deflections were generated by exactly the same force (in direction and magnitude) during the brief time

*h*, then we would have $LQ\u2192=PG\u2192$ by the Compound Second Law and the ratio would be*M*even for*h*> 0 and not just in the limit. But in Proposition VI, the force which generates $LQ\u2192$ is only approximately the same as the force that generates $PG\u2192$. Nevertheless, we still obtain*M*in the limit as*h*→ 0.30.

Reference 1, p.

467

.31.

We have concluded that Proposition VI is the

*Principia*’s version of**f**=*m***a**. Although alike underneath, Proposition VI and**f**=*m***a**are not alike on the surface; hence the need for the modifier “version of.” A natural question arises: Who was the first to write the equation**f**=*m***a**in something like its present form? The answer, like so many answers in the history of mathematics and physics, is Leonhard Euler. His “Discovery of a new principle in mechanics,” published in 1750, recorded the equation in Cartesian coordinates: 2*Mddx*=*Pdt*^{2}, 2*Mddy*=*Qdt*^{2}, 2*Mddz*=*Rdt*(L. Euler, “Découverte d’un nouveau principe de mécanique

,” Mémoires de l’Académie des Sciences de Berlin **6**, 185–217 (1750

). Also see *Leonhardi Euleri Opera Omnia*, Serie II, Vol.**5**, pp.81

–108

.) Euler could legitimately regard this principle as new, even though its ancestor had appeared 63 years earlier as Newton’s Proposition VI, because, for the first time, it was written in a fixed orthogonal coordinate system and applied, not just to point-masses, but to extended bodies. For further details, read G. Maltese, “The Ancients’ Inferno: The Slow and Tortuous Development of ‘Newtonian’ Principles of Motion in the Eighteenth Century,” in *Essays on the History of Mechanics: In Memory of Clifford Ambrose Truesdell and Edoardo Benvenuto*, edited by Antonio Becchi (Birkhäuser Verlag, Basel,2003

), pp. 199–221. Also see C. Truesdell, The Rational Mechanics of Flexible or Elastic Bodies 1638–1788, Introduction to Leonhardi Euleri Opera Omnia, Series II, Vol. **10**and**11**(Orell Füssli, Zürich,1960

), p. 251.© 2011 American Association of Physics Teachers.

2011

American Association of Physics Teachers

AAPT members receive access to the

*American Journal of Physics*and*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.