Newton's Apsidal Precession Theorem and Eccentric Orbits

Cajori F. , Sir Isaac Newton's Mathematical Principles of Natural Philosophy and his System of the World (Berkeley, 1947), 407. For an account of Newton's treatment of the problem of the lunar precession and of Clairaut's eventual solution of it, see Waff Craig B. , “Clairaut and the motion of the lunar apse: The inverse square law undergoes a test”, chap. 16 in The general history of astronomy, vol. 2B, ed. by Taton R. and Wilson C. (Cambridge, 1995), 35–46.

Cajori , op. cit., 546.

Hall A. , “A suggestion in the theory of Mercury”, The astronomical journal, xiv (1894), 49–51.

Le Verrier U. J. J. , “Théorie du mouvement de Mercure”, Annates de l'Observatoire Impériale de Paris, v (1859), 1–196, esp. pp. 98–106; Newcomb Simon , “Discussion and results of observations on transits of Mercury from 1677 to 1881”, Astronomical papers prepared for the use of the American Ephemeris and Nautical Almanac, i (1882), 473.

Bertrand J. , “Théorème relatif au mouvement d'un point attiré vers un centre fixe”, Comptes rendus des séances de l'Académie des Sciences, Séance du lundi 20 Octobre 1873, xxvii/10 (1873), 849–53. Bertrand shows that if m is the number by which π must be multiplied to give the angle between upper and lower apse, then the force law is given by If we put the exponent of r equal to N, we obtain the formula used by Hall.

See Danby J. M. A. , Fundamentals of celestial mechanics, 2nd edn (Richmond, Virginia, 1988), 62, equation (4.3.2).

For the development that follows we are indebted to Professor Michael Nauenberg of UC Santa Cruz.

Clemence G. M. , “The relativity effect in planetary motions”, Reviews of modern physics, xix (1947), 361–4.

Brown E. W. , “On the degree of accuracy in the new lunar theory”, Monthly notices of the Royal Astronomical Society, lxiv (1903), 524–34, esp. p. 532; Roseveare N. T. , Mercury's perihelion from Le Verrier to Einstein (Oxford, 1982), 67.

Einstein A. , “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie”, Königlich Preussische Akademie der Wissenschaften [Berlin]: Sitzungsberichte, 1915, 831–9; Roseveare , op. cit. (ref. 9); Earman J. and Janssen M. , “Einstein's explanation of the motion of Mercury's perihelion”, in The attraction of gravitation: New studies in the history of general relativity , ed. by Earman J. Janssen M. and Norton J. D. (Einstein studies, v; Boston, 1993), 129–72.

Whittaker E. T. , A treatise on the analytical dynamics of particles and rigid bodies, 4th edn (Cambridge, 1961), 81–82.

In a private communication.

The idea of examining the functions θ(N) for fixed values of the eccentricity, and their first derivatives ∂θ/∂N in the neighbourhood of N = 2, was suggested to us by Professor George Smith of Tufts University. He also derived for us the formulas (11) and (12) given in the text. So far as we are aware, these have not appeared previously in the literature.

We initially undertook to determine the derivatives ∂θ/∂N by applying Lagrangian interpolation to the data-points given in our Tables I-IV, along with the value θ = 180° for N = 2. Later Professor George Smith calculated the same derivatives from the finite differences of the same data points. The results of the two processes were in close agreement with each other; the discrepancies from the correct values as calculated below are less than 1%, and are no doubt attributable to the paucity of data-points. We conclude that the curves θ(e) are smooth.

For recent discussions of the Runge-Lenz vector and its history, see Goldstein Herbert , “Prehistory of the ‘Runge-Lenz’ vector”, American journal of physics, xliii (1975), 737–8, and “More on the prehistory of the Laplace or Runge-Lenz vector”, ibid., xliv (1976), 1123–24; Caplan S. Fuerstenberg H. Hayes C. Kane D. , and Raboy S. , “The Kepler orbit from initial conditions via the Lenz vector”, ibid., xlv (1977), 1089–90; Price Michael P. and Rush William F. , “Nonrelativistic contribution to Mercury's perihelion precession”, ibid., xlvii (1979), 531–4; Garavaglia T. , “The Runge-Lenz vector and Einstein perihelion precession”, ibid, lv (1987), 164–5.

Whiteside D. T. (ed.), The mathematical papers of Isaac Newton (London and New York, 1967–80), iii, 186–7.

On the flaw in the proof, see Erlichson H. , “Newton's solution of the equiangular spiral problem and a new solution using only the equiangular property”, Historia mathematica, xix (1992), 402–13; and Wilson Curtis , “Newton on the equiangular spiral: An addendum to Erlichson's account”, ibid., xxi (1994), 196–203.

A strong case for the hypothesis just stated is made by Nauenberg Michael in “Newton's early computational method for dynamics”, Archive for history of exact sciences, xlvi (1994), 221–52.

The correspondence of Isaac Newton , ii, ed. by Turnbull H. W. (Cambridge, 1960), 308. On Newton's calculation in this letter, see Nauenberg , op. cit. (ref. 18).

Turnbull (ed.), op. cit. (ref. 19), 309.

In Propositions 11–13 of Book I of the Principia Newton proved that conic-section orbits, with centre of force at a focus, imply an inverse-square field of force; and in Corollary I to Proposition 13, in the first edition, he asserted but did not demonstrate the converse. In the second edition he sketched a possible demonstration; the steps of the sketched proof have been filled in by Pourciau Bruce , “On Newton's proof that inverse-square orbits must be conics”, Annals of science, xlviii (1991), 159–72, and “Newton's solution of the one-body problem”, Archive for history of exact sciences, xliv (1992), 125–46. Arnol'd V. I. in Huygens and Barrow, Newton and Hooke (Basel, 1990), 30–33, points out that, because Newton's solutions in Proposition 17 depend smoothly on the initial conditions, their uniqueness is unproblematic; and the converse then follows.

Whiteside (ed.), op. cit. (ref. 16), vi, 151–3.

See Chandrasekhar S. , Newton's Principia for the common reader (Oxford, 1995), 114–18, for a resumé of what Newton accomplishes in establishing duality between laws of centripetal force.

This consequence was pointed out to us by Professor Michael Nauenberg of the University of California, Santa Cruz.

Cajori edition (ref. 1), 141.

For both ideas and references, we are indebted in our concluding section to an unpublished essay, “Newton and hidden symmetry”, by Pesic Peter D. of St John's College in Santa Fe, New Mexico.