Essay Review: Chandrasekhar's Principia: Newton's Principia for the Common Reader

Unfortunately, Chandrasekhar has used the Motte-Cajori translation. A new translation by Cohen Bernard I. and Whitman Anne , with the assistance of Julia Budenz, will be published by University of California Press in 1997.

From the “Prologue”.

Chandrasekhar's theory of white dwarfs was put forward in 1935 while he was a Fellow of Trinity College, Cambridge, two years after he received his Ph.D. degree from Cambridge University. He died suddenly of a heart attack in August 1995. For details of his career, see Wali Kameshwar C. , Chandra: A biography of S. Chandrasekhar (Chicago and London, 1991).

P. 93.

Chandrasekhar's approach to the curvature-based demonstrations follows the lead of Arnol'd's V. I. Huygens and Barrow, Newton and Hooke (Basel, 1990); for a more detailed treatment, see Brackenridge's Bruce recent The key to Newton's dynamics: The Kepler problem and the Principia (Berkeley, 1995).

Kriloff A. N. , “On Sir Isaac Newton's method of determining the parabolic orbit of a comet”, Monthly notices of the Royal Astronomical Society, lxxxv (1925), 640–56, and “On a theorem of Sir Isaac Newton”, ibid., lxxxiv (1924), 392–5.

Fraser Duncan C. , “Newton and interpolation”, in Isaac Newton, 1642–1727, ed. by Greenstreet W. J. (London, 1927), 45–54.

Proudman J. , “Newton's work on the theory of the tides”, ibid., 87–95.

Tisserand F. , Traité de mécanique céleste (Paris, 1894), especially vol. iii, chap. III, “Théorie de la lune de Newton”, pp. 27–45.

Pollard Harry , Mathematical introduction to celestial mechanics (Englewood Cliffs, 1966).

Scarborough James B. , The gyroscope: Theory and applications (New York, 1958).

See Chandrasekhar S. , Ellipsoidal figures of equilibrium (New Haven, 1968).

Chandrasekhar contends that the potential-theory results on spheres were more important to Newton than is generally suggested. He argues that Newton was deterred from pursuing his thoughts about inverse-square gravity any further in the late 1660s not by the numerical discrepancy in the “Moon test”, but by doubts about whether gravity acts as if all the mass were concentrated at the centres of heavenly bodies (p. 6).

Pp. 179f.

The correspondence of Isaac Newton , iii, ed. by Turnbull H. W. (Cambridge, 1961), 348–54 (especially note 1); and Whiteside D. T. , The mathematical papers of Isaac Newton, vi (Cambridge, 1974), 354, n. 214 and 435–8. The memorandum was also incorporated into Gregory's (unpublished) commentary on the Principia (p. 44).

Pp. 364–8. Gregory (op. cit., 181f) has offered a less elaborate, more obvious candidate for Newton's calculation, using the fact that 1 — √(1001/1000) = 1/2000, very nearly, to supplement what Newton says in the text.

P. 273.

Newton's dot notation, his most felicitous, postdates the Principia. For an example of how nonperspicuous Newton's earlier notations tended to be, see Hall Rupert A. and Hall Boas Marie , Unpublished scientific papers of Isaac Newton (Cambridge, 1962), 15–64.

Chapters 13 and 14, pp. 219–68, cover the results given in the corollaries to Proposition 66 of Book 1, and chap. 22, pp. 419–54, covers the quantitative results given in Book 3.

P. 206.

Newton adopts the Horrocksian model not just in Corollary 9 of Proposition 66 and the Scholium following Proposition 35 in Book 3, but also in his “Theory of the Moon's motion” that was included in Gregory's David Elements of physical and geometrical astronomy of 1702; see Isaac Newton's theory of the Moon's motion , ed. by Cohen Bernard I. (Folkestone, 1975). Horrocks's model is described in Wilson's Curtis A. “Predictive astronomy in the century after Kepler”, in The general history of astronomy, ii: Planetary astronomy from the Renaissance to the rise of astrophysics, Part A: Tycho Brahe to Newton, ed. by Taton R. and Wilson C. (Cambridge, 1989), 161–206, pp. 194–201; and Newton's use of it, in Wilson's “The Newtonian achievement in astronomy”, ibid., 233–74, pp. 265f. I am indebted to Dr Wilson for assistance on this matter.

See Chandrasekhar's equation (9), p. 251, where the term with the 31-day period has a coefficient of 9, and the other two terms have coefficients of 1 and 2.

Newton succeeds in reaching the second-order in the case of the lunar variation, but his approach is not one of varying the elements; rather, it is ad hoc, and, ingenious as it is, it yields good numerical results only because Newton's guess of an ellipse for the variational orbit holds true so long as the number of lunations per year is not small. (Chandrasekhar fails to mention this contingency in Newton's treatment of the variation; see pp. 423–30, as well as pp. 242–7.)

Although Whiteside's comments are generally not on the Principia itself, but on manuscripts that were immediate forerunners of it, the differences between them tend to be minor, and when they are not he usually comments on the versions in the Principia.

Hooke raised this question in one of the letters that started Newton on the path to the Principia: “This Curve truly Calculated will shew the error of those many lame shifts made use of by astronomers to approach the true motions of the planets with their tables” (The correspondence of Isaac Newton, ii, ed. by Turnbull H. W. (Cambridge, 1960), 309.)

Chandrasekhar is scarcely the first to inject conservation of energy into the discussion of Section 8. Moreover, as anachronisms go, this is not the most extreme example in his book. In Lemma 3 of Book 3 Newton, in the manner of Descartes, sums the motion — i.e. the mass times the scalar velocity — of all the particles forming the rotating Earth. Chandrasekhar's comment: “It has, to the writer's knowledge, never been recognized that Newton defined the notion of circulation some two hundred years before Lord Kelvin to whom it is commonly credited” (p. 471).

Galileo's postulate reads, “The speeds acquired by one and the same body moving down planes of different inclinations are equal when the heights of these planes are equal”. In later editions of Two new sciences Galileo offered a demonstration of this assumption. Torricelli devised the principle named after him in order to obtain a more compelling demonstration of it, and Huygens , calling attention to the need for a proof, extended this principle in demonstrating it in his Horologium oscillatorium. All these efforts, however, were restricted to uniform gravity.

Chandrasekhar summarizes the achievement of Newton's theory as follows: “One may in truth say that there is hardly anything in any modern textbook on celestial mechanics (e.g., Danby J. M. A. , Fundamentals of celestial mechanics (Willmann-Bell, Inc. Richmond VA., U.S.A., 1989) Chapter 11 §119) [sic] and the entire Chapter 12 on ‘The Motion of the Moon’ that one cannot find in the propositions that we have enumerated, and indeed with deeper understanding” (p. 419).

The law is given as T = 2πa^3/2 on p. 104, inviting the mistaken inference that T²/a³ is a universal constant, and not an indicator of the mass of the central body.

P. 592. This is a mistranslation of minutorum tertiorum from the first edition, where Newton says that the time of a single arc of an 8-inch pendulum is 28 ¾ thirds. See Isaac Newton's Philosophiae naturalis principia mathematica, ed. by Koyré Alexandre and Cohen Bernard I. (Cambridge, 1972), i, 530.

P. 422.

P. 358.

Torge Wolfgang , Geodesy (Berlin, 1991), 56f. The value 1/289 is the ratio of the centrifugal force at the equator to gravity. Chandrasekhar himself gave 1/294 as the value of the ellipticity in his Ellipsoidal figures of equilibrium, 3.

P. 475.

P. 113.

See Cohen Bernard I. , An introduction to Newton's ‘Principia’ (Cambridge, 1978), chap. 7, and Whiteside , op. cit. (ref. 15), 538, n. 1 and 548f, n. 25. Also see Brackenridge , op. cit. (ref. 5), chaps. 8 and 9.

P. 475. Chandrasekhar is also careful to point out the fundamental error in Newton's treatment of the tides (pp. 401ff). He fails to note other significant errors, however. He skips over Newton's mistaken qualitative account of the interaction of Jupiter and Saturn in Proposition 13 of Book 3, a place where the limitations of Newton's geometric approach stand out especially clearly. He also bypasses Newton's error in balancing forces instead of torques in Propositions 51 and 52 of Book 2 concerning the vortices set up by rotating cylinders and spheres — propositions that prepare the way for the central argument for universal gravity by raising objections to Descartes's vortex theory. This error, which Chandrasekhar could hardly have missed given his own work on Couette flow (Hydrodynamic and hydromagnetic stability (Oxford, 1961)), raises questions about how well Newton understood rotational motion.

P. 475.

See Wilson Curtis , “The precession of the equinoxes from Newton to d'Alembert and Euler”, in The general history of astronomy, ii: Planetary astronomy from the Renaissance to the rise of astrophysics, Part B: The eighteenth and nineteenth centuries, ed. by Taton R. and Wilson C. (Cambridge, 1995), 47–54.

Westfall Richard S. , “Newton and the fudge factor”, Science, clxxix, issue of 23 February 1973, 751–8, especially, pp. 755–8; and Never at rest: A biography of Isaac Newton (Cambridge, 1980), 736–9. Both of these offer grounds for rebutting the several complaints Chandrasekhar lodges against Cotes.