LAPLACE. 141 



verance, and success. Tbe profound and long-continued researches of 

 the illustrious geometer established with complete evidence that the 

 planetary ellipses are perpetually variable ; that the extremities of their 

 major axes make the tour of the heavens; that, independently of an 

 oscillatory motion, the planes of their orbits experience a displacement 

 in virtue of which their intersections with the plane of the terrestrial 

 orbit are each year directed toward difierent stars. In the midst of 

 this apparent chaos there is one element which remains constant, or is 

 merely subject to small periodic changes, namely,themajor axis of each 

 orbit, and consequently the time of revolution of each i)lanet. This is 

 the element which ought to have chiefly varied according to the learned 

 speculations of Newton and Euler. 



The principle of universal gravitation suffices for preserving the 

 stability of the solar system. It maintains the forms and inclinations 

 of the orbits in a mean condition which is subject to slight oscillations ; 

 variety does not entail disorder ; the universe offers the example of 

 harmonious relations, of a state of perfection which Newton himself 

 doubted. This depends on circumstances which calculation disclosed 

 to Laplace, and which, upon a superhcial view of the subject, would 

 not seem to be capable of exercising so great an influence. Instead of 

 planets revolving all in the same direction, in slightly eccentric orbits, 

 and in planes inclined at small angles toward each other, substitute 

 different conditions, and the stability of the universe will again be put 

 in jeopardy, and, according to all probability, there will result a fright- 

 ful chaos.* 



* The researches on the secular variations of the eccentricities and inclinations of the 

 planetary orbits depend upon the solution of an algebraic equation eqiial in dejjree to the 

 number of planets whose mutual action is considered, and the co-efficients of which involve 

 the values of the masses of those bodies. It may be shown that if the roots of this equa- 

 tion be equal or itnagiuary, the corresponding element, whether the eccentricity or the in- 

 clination, will increase indefinitely with the time in the case of each planet ; but that if the 

 roots, on the other hand, be real and unequal, the value of the element will oscillate in 

 every instance within fixed limits. Laplace proved by a general analysis that the roots of 

 the equation are real and unequal ; whence it followed that neither the eccentricity nor the 

 inclination will vary in any case to an indefinite extent. But it still remained uncertain 

 whether the limits of oscillation were not in any instance so far apart that the variation of 

 the element (whether the eccentricity or the inclination) might lead to a complete destruc- 

 tion of the existing physical condition of the planet. Laplace, indeed, attempted to prove, 

 by means of two well-known theorems relative to the eccentricities and inclinations of the 

 planetary orbits, that if those elements were once small they would always remain so, pro- 

 vided the planets all revolved around the sun in one common direction, and their masses 

 were inconsiderable. It is to these theorems that M. Arago manifestly alludes in the text. 

 Le Verrier and others have, however, remarked that they are inadequate to assure the per- 

 manence of the existing physical condition of several of the planets. In order to arrive at 

 a definite conclusion on this subject, it is indispensable to have recourse to the actual solu- 

 tion of the algebraic question above referred to. This was the course adopted by the illus- 

 trious Lagrange in his researches on the secular variations of the planetary orbits. (Mem. 

 Acad. Berlin, 1783-'84.) Having investigated the values of the masses of the planets, he 

 then determined, by an approximate solution, the values of the several roots of the algebraic 

 equation upon which the variations of the eccentricities and inclinations cf the orbits de- 



