[51] Gautier, Prob. de Trois Corps, p. 155.
[52] In the first edition of this History, I had ascribed to Lagrange the invention of the Method of Variation of Elements in the theory of Perturbations. But justice to Euler requires that we should assign this distinction to him; at least, next to Newton, whose mode of representing the paths of bodies by means of a Revolving Orbit, in the Ninth Section of the Principia, may be considered as an anticipation of the method of variation of elements. In the fifth volume of the Mécanique Céleste, livre xv. p. 305, is an abstract of Euler’s paper of 1749; where Laplace adds, “C’est le premier essai de la méthode de la variation des constantes arbitraires.” And in page 310 is an abstract of the paper of 1756: and speaking of the method, Laplace says, “It consists in regarding the elements of the elliptical motion as variable in virtue of the perturbing forces. Those elements are, 1, the axis major; 2, the epoch of the body being at the apse; 3, the eccentricity; 4, the movement of the apse; 5, the inclination; 6, the longitude of the node;” and he then proceeds to show how Euler did this. It is possible that Lagrange knew nothing of Euler’s paper. See Méc. Cél. vol. v. p. 312. But Euler’s conception and treatment of the method are complete, so that he must be looked upon as the author of it.
[53] Gautier, p. 104.
[54] Ib. p. 184.
[55] Ib. p. 196.
8. Mécanique Céleste, &c.—Laplace also resumed the consideration of the secular changes; and, finally, undertook his vast work, the Mécanique Céleste, which he intended to contain a complete view of the existing state of this splendid department of science. We may see, in the exultation which the author obviously feels at the thought of erecting this monument of his age, the enthusiasm which had been excited by the splendid course of mathematical successes of which I have given a sketch. The two first volumes of this great work appeared in 1799. The third and fourth volumes were published in 1802 and 1805 respectively. Since its publication, little has been added to the solution of the great problems of which it treats. In 1808, Laplace presented to the French Bureau des Longitudes, a Supplement to the Mécanique Céleste; the object of which was to improve still further [372] the mode of obtaining the secular variations of the elements. Poisson and Lagrange proved the invariability of the major axes of the orbits, as far as the second order of the perturbing forces. Various other authors have since labored at this subject. Burckhardt, in 1808, extended the perturbing function as far as the sixth order of the eccentricities. Gauss, Hansen, and Bessel, Ivory, MM. Lubbock, Plana, Pontécoulant, and Airy, have, at different periods up to the present time, either extended or illustrated some particular part of the theory, or applied it to special cases; as in the instance of Professor Airy’s calculation of an inequality of Venus and the earth, of which the period is 240 years. The approximation of the Moon’s motions has been pushed to an almost incredible extent by M. Damoiseau, and, finally, Plana has once more attempted to present, in a single work (three thick quarto volumes), all that has hitherto been executed with regard to the theory of the Moon.
I give only the leading points of the progress of analytical dynamics. Hence I have not spoken in detail of the theory of the Satellites of Jupiter, a subject on which Lagrange gained a prize for a Memoir, in 1766, and in which Laplace discovered some most curious properties in 1784. Still less have I referred to the purely speculative question of Tautochronous Curves in a resisting medium, though it was a subject of the labors of Bernoulli, Euler, Fontaine, D’Alembert, Lagrange, and Laplace. The reader will rightly suppose that many other curious investigations are passed over in utter silence.
[2d Ed.] [Although the analytical calculations of the great mathematicians of the last century had determined, in a demonstrative manner, a vast series of inequalities to which the motions of the sun, moon, and planets were subject in virtue of their mutual attraction, there were still unsatisfactory points in the solutions thus given of the great mechanical problems suggested by the System of the Universe. One of these points was the want of any evident mechanical significance in the successive members of these series. Lindenau relates that Lagrange, near the end of his life, expressed his sorrow that the methods of approximation employed in Physical Astronomy rested on arbitrary processes, and not on any insight into the results of mechanical action. But something was subsequently done to remove the ground of this complaint. In 1818, Gauss pointed out that secular equations may be conceived to result from the disturbing body being distributed along its orbit so as to form a ring, and thus made the result conceivable more distinctly than as a mere result of calculation. And it appears [373] to me that Professor Airy’s treatise entitled Gravitation, published at Cambridge in 1834, is of great value in supplying similar modes of conception with regard to the mechanical origin of many of the principal inequalities of the solar system.
Bessel in 1824, and Hansen in 1828, published works which are considered as belonging, along with those of Gauss, to a new era in physical astronomy.[56] Gauss’s Theoria Motuum Corporum Celestium, which had Lalande’s medal assigned to it by the French Institute, had already (1810) resolved all problems concerning the determination of the place of a planet or comet in its orbit in function of the elements. The value of Hansen’s labors respecting the Perturbations of the Planets was recognized by the Astronomical Society of London, which awarded to them its gold medal.
[56] Abhand. der Akad. d. Wissensch. zu Berlin. 1824; and Disquisitiones circa Theoriam Perturbationum. See Jahn. Gesch. der Astron. p. 84.