SEQUEL TO THE GENERALIZATION. 371 



mines the secular changes of each of the elements or determining quan- 

 tities of the orbit. In 1773, Laplace also attacked this subject of 

 secular changes, and obtained expressions for them. On this occasion, 

 he proved the celebrated proposition that, "the mean motions of the 

 planets are invariable :" that is, that there is, in the revolutions of the 

 system, no progressive change which is not finally stopped and re- 

 versed ; no increase, which is not, after some period, changed into de- 

 crease ; no retardation which is not at last succeeded by acceleration ; 

 although, in some cases, millions of years may elapse before the system 

 reaches the turning-point. Thomas Simpson noticed the same conse- 

 quence of the laws of universal attraction. In 1774 and 1776, La- 

 grange 6 still labored at the secular equations; extending his researches 

 to the nodes and inclinations ; and showed that the invariability of the 

 mean motions of the planets, which Laplace had proved, neglecting 

 the fourth powers of the eccentricities and inclinations of the orbits, 7 

 was true, however far the approximation was carried, so long as the 

 squares of the disturbing masses were neglected. He afterwards im- 

 proved his methods ; 8 and, in 1783, he endeavored to extend the calcu- 

 lation of the changes of the elements to the periodical equations, as 

 well as the secular. 



8. Mecanique Celeste, <kc. — Laplace also resumed the consideration 

 of the secular changes; and, finally, undertook his vast work, the 

 Mecanique Celeste, 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 ex- 

 cited 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 

 Mecanique Celeste; the object of which was to improve still further 



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 Mic. Cel. 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. 

 6 Gautier, p. 104. ■ lb. p. 184. « lb p. 196. 



