NOTES TO BOOK VI. 129 



of representing the paths of bodies by means of a Revolving 

 Orbit, in the Ninth Section of the Principia^ may be con- 

 sidered as an anticipation of the method of variation of 

 elements. In the fifth volume of the Mecanique Celeste, 

 livre xv. p. 305, is an abstract of Euler's paper of 1749 ; 

 where Laplace adds, " (Test le premier essai de la methode 

 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 excentricity ; 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 Mec. Cl. 

 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. 



(E.) p. 109. 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 approxi- 

 mation employed in physical astronomy rested on arbitrary 

 processes, and not on any insight into the results of mecha- 

 VOL. II. K 



