The Nub of the Lunar Problem: From Euler to G. W. Hill

Euler J. A. , “Réflexions sur la Variation de la Lune”, Histoire de l'Académie Royale des Sciences et Belles-Lettres, xxii (1766), 334–53. I have capitalized the first letters of “Variation” and “Lune” to indicate that they are used here as proper nouns.

I translate from Euler's French. In view of both the technical expertise of the paper and its bold declarative style, I am strongly inclined to regard the senior Euler, rather than his less distinguished son, as the author.

In Book I of the Principia, Prop. 66, Corollaries 2–5, and in Book III, Props. 26–29.

The two papers can be found in The collected mathematical works of George William Hill (4 vols, Washington, DC, 1905–7), i, 243–70 and 284–335. Their titles and places of original publication are given in the next two references.

Hill G. W. , “Researches in the lunar theory”, American journal of mathematics, i (1878), 5–26, 129–47, 245–60.

Hill G. W. , “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon”, separately published, Cambridge, MA, 1877; reprinted in Acta mathematica, viii (1886), 1886–36.

Newton , Principia, Book I, Prop. 4, Corollary 2.

See Van Helden Albert , “Measuring solar parallax: The Venus transits of 1761 and 1769 and their nineteenth-century sequels”, in Taton René Wilson Curtis (eds), Planetary astronomy from the Renaissance to the rise of astrophysics, Part B: The eighteenth and nineteenth centuries (Cambridge, 1995), 153–68.

The result can be obtained from Newton's Principia, Prop. 4 of Book I.

On Euler's views, see Wilson Curtis , “Euler on action-at-a-distance and fundamental equations of continuum mechanics”, in Harman P. M. Shapiro Alan E. (eds), The investigation of difficult things: Essays on Newton and the history of the exact sciences in honour of D. T. Whiteside (Cambridge, 1992), 399–420. For Clairaut, see Waff Craig B. , “Clairaut and the motion of the lunar apse: The inverse-square law undergoes a test”, Planetary astronomy from the Renaissance to the rise of astrophysics, Part B (ref. 8), 35–46.

See his second lunar theory: Theoria motus lunae exhibens omnes eius inequalitates (Berlin, 1753), in Leonardi Euleri, Opera omnia, ser. 2, xxiii, 64–336.

Euler Leonhard , Recherches sur la question des inégalités du movement de Saturne et de Jupiter, sujet proposé pour le prix de l'année 1748, par l'Académie Royale des Sciences de Paris , in Leonardi Euleri, Opera omnia, ser. 2, xxv, 45–117.

On 23 November 1785 Laplace announced to the Paris Academy that the anomalies in the mean motions of Jupiter and Saturn could be accounted for on the assumption of universal gravitation. See Wilson Curtis , “The Great Inequality of Jupiter and Saturn from Kepler to Laplace”, Archive for history of exact sciences, xxxiii (1985), 15–290.

Leonhard Euler, [E398] = “Nouvelle méthode de determiner les dérangemens dans le movement des corps célestes, causés par leur action mutuelle”, Mémoires de l'Académie des Sciences de Berlin , xix (1763), 1770, 141–79. To be published in Leonardi Euleri, Opera omnia, ser. 2, xxvi.

Leonhard Euler, [E401] = “Nouvelle manière de comparer les observations de la Lune avec la théorie”, Mémoires de l'Académie des Sciences de Berlin , xix (1763), 1770, 221–34. Opera Omnia, ser. 2, xxiv, 90–100.

The term ‘inequality’ has been used at least since the seventeenth century to refer to sinusoidal terms that have to be added to the mean motion of a planet or satellite in order to obtain the true motion.

See Prop. 28 of Book III of the Principia, where Newton derives the ratio 70:69 of the two axes of the Variation Curve.

In the notation used by Hill and Brown, m = n′/(n — N′), where n′ is the mean motion of the Sun and (n — N′) the Moon's mean synodic motion. Here n is the Moon's mean sidereal motion, not to be confused with Euler's n = 1/m.

Hill , “Researches in the lunar theory”, Collected mathematical works of G. W. Hill (ref. 4), i, 286.

Ibid.

On Hill's undergraduate studies at Rutgers under Theodore Strong, and their emphasis on Euler, see Hogan Edward R. , “Theodore Strong and ante-bellum American mathematics”, Historia mathematica, viii (1981), 439–55.

Hill G. W. , “Demonstration of the differential equations employed by Delaunay in the lunar theory”, The analyst, iii (1876), 161–227.

Hill , “Researches in the lunar theory”, Collected mathematical works of G. W. Hill (ref. 4), i, 285.

Hill G. W. , “Researches in the lunar theory”, Collected mathematical works of G. W. Hill (ref. 4), i, 323–4.

Hill G. W. , “On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon”, Collected mathematical works of G. W. Hill (ref. 4), i, 243–70; separately published, Cambridge, MA, 1877, and reprinted in Acta mathematica, viii (1886), 1–36.

Jacobi C. G. J. , “Sur le mouvement d'un point et sur un cas particulier du problème des trois corps”, Comptes rendus de l'Académie de Paris, iii (1836), 59–61; reprinted in C. G. J. Jacobi's Gesammelte Werke, iv, ed. by Weierstrass K. (Berlin, 1886), 35–38. According to Jacobi in his Vorlesungen über Dynamik (ed. by Clebsch A. ) in Gesammelte Werke, Supplementband (Berlin, 1884), 10, Euler regarded the vis viva integral as valid only about a fixed centre of attraction; Jacobi got his Jacobian integral by applying it to a moving centre.

A presentation of the derivation less cryptic and more reader-friendly than Hill's is given by Brouwer D. Clemence G. M. , Methods of celestial mechanics (New York and London, 1961), 336–66. In the 1930s, E. W. Brown showed that Hill's results could be got by a much simpler route; see Brouwer Clemence , op. cit., 370–3, “Brown's method of differential correction”.

See, for instance, Fine Henry B. , A college algebra (Boston, 1901), chap. 31, “Determinants and elimination”.

Another possible source is Briot C. Bouquet T. , Théorie des fonctions doublement périodiques (Paris, 1859).

See “The President's Address”, Monthly notices of the Royal Astronomical Society , lxvii (1907), 310. Cf. Poincaré Henri , Les methods de la mécanique céleste (repr. New York, 1957), i, chap. 4.

See Fotheringham J. K. , “The longitude of the Moon from 1627 to 1918”, Monthly notices of the Royal Astronomical Society , lxxx (1920), 289–307, and Jeffreys Harold , “Chief cause of the lunar secular acceleration”, ibid., 309–17.

Brown E. W. , “The evidence for changes in the rate of rotation of the Earth …”, Transactions of the Astronomical Observatory of Yale University, no. 3 (1926), 205–35 + 3 plates.

Jones H. Spencer , “The rotation of the Earth, and the secular accelerations of the Sun, Moon and planets”, Monthly notices of the Royal Astronomical Society, xcix (1939), 541–58.

See Lambeck Kurt , The Earth's variable rotation: Geophysical causes and consequences (Cambridge, 1980).