Note 52 — Chapter VI · Sequel to the Generalization of the Principles of Mechanics.—Period of Mathematical Deduction.—Analytical Mechanics.

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.