Essay Review: The Mathematical Principles Underlying the Principia Revisited: The Key to Newton's Dynamics: The Kepler Problem and the Principia,Newton's Principia: The Central Argument

Russell J. L. , “Kepler's law of planetary motion: 1609–66”, The British journal for the history of science , ii (1964), 1–24. In particular, this article points out that Kepler's area law was not generally accepted during Newton's time.

In De motu, as translated from the Latin by Mary Ann Rossi (B, 71).

The first 17 propositions of Book 1 discuss the motion of a body about a fixed centre of force, and therefore the concept of mass does not play any role. Newton deals here only with the accelerative force, Definition 7, which is measured by (or is equal to) the acceleration imposed.

For example, both Whiteside D. T. Aiton Eric have claimed that there were subtleties that Newton had not considered when taking the continuum limit in Prop. 1, Book 1 (see Newton's mathematical papers , vi, ed. by D. T. Whiteside (Cambridge, 1974), 35–37, fn. 19, and Eric Aiton, “Polygons and parabolas: Some problems concerning the dynamics of planetary motion”, Centaurus , xxxi (1989), 207–21). Actually Newton justified this limit correctly by referring to Cor. 4 of Lemma 3. Hence, when one is taking this limit by augmenting the number of triangles while decreasing their widths indefinitely, the vertices of the polygonal path are assumed to lie on a fixed curve. However, this important condition is generally overlooked. While not stated explicitly, in Cors. 2 and 3 to Prop. 1, added in the second edition of the Principia, Newton referred to the sides of the polygon as chords of arcs. However, Densmore claims that “Newton has not offered an argument that the limiting case in the proposition is a unique curve, with a unique orbit…. Remember that our polygon of finite tangential motions does not circumscribe or in any other way follow some ‘ghost curve’” (D, 99). Actually, without such a curve Newton would have had to specify how to scale the impulses (proportional to the square of the time interval) when changing the size of the triangles, which he did not do in Prop. 1, and then he would not have been able to resort to any of his lemmas for the existence of a continuum limit.

The weakness of the extended Cor. 1 to Prop. 13 is that Newton claimed without proof that “… it is impossible to describe two mutually tangent orbits with the same centripetal force and the same velocity”. However, such a proof is contained in Prop. 41, Book 1. Densmore offers the explanation “that Newton suggests that the body has no means of deciding to go off in some exotic curve that happens to have not only the same curvature but also the same force law and initial velocity as a body in a conic section at that point”, adding a metaphysical remark: “(Perhaps the body would need some additional guidance to do so, as we saw for example would be required in Proposition 7, a situation unexceptionable mathematically, but disturbing to physical intuition.) This argument has some persuasive power.” The reader can judge the validity and usefulness of such explanations for himself, but the reference to Prop. 7, Book 1 is based on an error as has been shown in our discussion of this proposition. For further discussions of Cor. 1 and the disputes in the literature see B, 219–20 and Nauenberg M. , “Newton's Principia and inverse square orbits”, The college mathematics journal , xxv (1994), 212–22.

See, for example, Nauenberg M. , “Hooke, orbital motion and Newton's Principia”, American journal of physics , lxii (1994), 331–50, p. 336.

In Densmore's as well as in Brackenridge's book Newton's description of the geometrical construction for this Lemma, translated literally from the Latin, leads to some confusion. Donahue W. H. translates this construction as “… let AG, BG be erected perpendicular to the latter subtense AB and to the tangent AD, meeting at G …” (D, 80), while Mary Ann Rossi translates it as “… perpendicular to the latter subtense AB and to the tangent AD, erect AG and BG, meeting at G …” (B, 241). However, according to the diagram accompanying Lemma 11, AG is perpendicular to AD while BG is perpendicular to AB. The often maligned Motte-Cajori translation states correctly “Draw BG perpendicular to the subtense AB, and AG perpendicular to the tangent AD, meeting in G …”, which is not what Newton wrote, but clarifies what he meant.

Newton's critical assumption is revealed in his remark that “on account of the common distance SP from the centre S therefore equal centripetal forces [my italics], MN [deviation for a circle] and QR [deviation for the ellipse] are equal [for a given time interval]” (B, 121). Note that, if the “centripetal force is reciprocal as L x SP²” as Newton concludes in Prop. 11, Book 1, then this force would depend on L as well as on the distance SP, and in this case MN and QR would be different.

Likewise , in Prop. 14, Book 1, Newton assumed that “… the very small line QR, given the time, is as the generating centripetal force, that is (by hypothesis) reciprocally as SP²…” (B, 262). It is interesting that in the formulation of this proposition Newton referred to “several bodies [which] revolve about a common centre” rather than to a single body revolving in different orbits, thus preparing the ground for the application of Prop. 15, Book 1 (Kepler's third law) to celestial mechanics.

In the Cor. to Prop. 14, Book 1, Newton concluded that “the total area of the ellipse, and proportional to it, the rectangle [generated] by the axis [a and b], is in the ratio compounded of the latus rectum and the whole ratio of the periodic time” (B, 262), i.e., ab ∝√[LT], where T is the periodic time.

In the proof of Prop. 15, Newton invoked”… the corollary of the sixth theorem (Cor. Prop. 14)” (B, 163); see ref. 10.

Hence δt ∝ (QT x SP)/h, which is valid for any general central force and has the correct dimension of time.

Herivel J. , The background to Newton's Principia (Oxford, 1965), 291, fn. 32.

See ref. 4 above.

In Props. 59 and 60, Book 1, Newton obtained the correction to Kepler's third law due to the finiteness of the mass M of the Sun by invoking the third Law of Motion to solve the two-body problem. In this case the relative acceleration between the planet and the Sun depends also on the mass m of the planet and increases by a factor (1 + m/M). Therefore if Kepler's third law is written in the form a³ = KT², where a is the major axis of the ellipse, T is the period and K is a constant when the centre of force (the Sun) is fixed, then for the two-body case K becomes K (1 + m/M). This predicts a dependence on the masses of the different planets which turns out to be small because m/M « 1.

We now call this principle the equivalence of inertial mass (determined by Newton's third law of motion), and gravitational mass (determined by the ratio of weight to gravitational acceleration). Newton's theory of the pendulum appears in Prop. 24, Book 2, and is discussed in Densmore's book. However, Corollaries 2–5 are left out. In Cor. 5, Newton announces his theoretical result in complete form, “that the quantity of matter [mass] in the pendulous body is directly as the weight and the square of the time [period of oscillation] and inversely as the length of the pendulum”. To verify this relation he carried out careful experiments with “gold, silver, lead, glass, sand, common salt, wood, water and wheat”, Prop. 6, Book 3, by hanging equal weights of these materials in boxes suspended with equal lengths (eleven feet) from the centre of oscillation. Newton found that “they went to and fro together for a very long time with equal oscillations …”, and concluded that “… in bodies of the same weight, a difference no greater than the thousand part [1/1000] of the whole matter could have been plainly perceived in these experiments” (D, 330) The reason for this arrangement is that it insured “similar… air resistance”, which Newton's theory did not take into account, and limits the total number of oscillations before the pendular motions are damped. For example, a relative inertial to gravitational mass difference of 1/1000 would lead to a difference of 1/8 of one oscillation in 250 oscillations of two pendula with different material.

Newton attributed the slow rotation of the lunar apogee (about 3.3° per revolution) to the effect of the solar gravitational perturbation, and he devoted considerable efforts to evaluate this perturbation. His simplest approximate calculation (Prop. 45, Cor.2) gave the correct order of magnitude and sign of the rotation, but it was smaller than observation by a factor of 2.

In his book, Newton's Principia for the common reader (Oxford, 1995) 364–8, Presents Chandrasekhar S. a calculation on the “variation of the major axis” of the satellite's orbit due to the effect of the solar perturbation that he attributes to Newton. However, this variation is an unobservable effect unrelated to Newton's calculation. Moreover, Chandrasekhar claims that Newton's result is given by the relation.

Δr/R = (4/9)(d/e — 1) = 1/2250.

This relation is manifestly unphysical as it is independent of the masses of Jupiter and the Sun. In his recent review of Chandrasekhar's book in this journal ( JHA , xxvii (1996, 353–62), George Smith considers Chandrasekhar's calculation a reasonable candidate for Newton's computation, but remarks “that doubts must remain … because the numbers differ non-trivially”.

Densmore claims to obtain Newton's result by applying the relation f_M/f_J = r_J²/r_M², where f_J is the accelerative force of Jupiter towards the Sun, f_M is the corresponding force on a moon of Jupiter, and r_J and r_M are the radial distances of Jupiter and the moon from the Sun (D, 341). However, this relation is based on the assumption that the gravitational accelerative force of the Sun is independent of the mass of Jupiter and its moon. Therefore it can not be applied to derive Newton's result, which tests the assumption that the Sun's gravitational accelerative forces f_J and f_M do depend on the mass of Jupiter and its moons. Newton assumes that f_M/f_J deviates from unity by only one-thousandth of the whole of gravity in the case that Jupiter and its moon are located at “equal distances from the Sun”, i.e., r_J = r_M. In this case the relation used by Densmore gives f_M/f_J = 1. Instead Densmore sets f_M/f_J = 1001/1000 and obtains r_J/r_M = 2001/2000, concluding that this will give “the difference in distance … about 1/2000 as Newton says” (D, 342).

See ref. 15.