CALCULUS OP PARTIAL DIFFERENCES. 49 



paper read at the Academy, 17th February, 1734, we find a 

 passage that certainly looks towards that calculus, and shows 

 that he used a new algorithm as requisite for conducting his 

 operation: " J'ai ete oblige," he says, " de faire varier les 

 memes lignes en deux manieres differentes. II a fallu de- 

 signer leurs variations differemment." " J'ai marque les unes 

 commes les geometres Anglais par des fluxions (points) ; 

 les autres par des differences (d x) a notre maniere ; de sorte 

 qu'ici d x ne sera pas la meme chose que x, d arque x" (p. 18). 

 " II pent y avoir," he afterwards adds, " des problemes qui 

 dependroient de cette methode fluxio-differentielle." 



Nothing that has now been said can, in any manner, detract 

 from the renown justly acquired by D'Alembert and Lagrange 

 as the first who fully expounded the two great additions to the 

 Differential Calculus first applied them systematically to the 

 investigation of physical as well as mathematical questions, 

 and therefore may truly be said to have first taught the use 

 of them as instruments of research to geometricians.* 



In the year 1746 the Academy of France proposed, as the 

 subject of its annual prize essay for 1748, the disturbances 

 produced by Jupiter and Saturn mutually on each other's 

 orbits. Euler's Memoir gained the prize ; and it contains the 

 solution of the famous Problem of the Three Bodies namely, 

 to find the path which one of those bodies describes round 

 another when all three attract each other with forces varying 

 inversely as the squares of their distances, their velocities 

 and masses being given, and their directions in the tangents 

 of their orbits.t This, which applies to the case of the 

 Moon, would be resolved were we in possession of the 

 solution for the case of Jupiter and Saturn, which, instead of 



* There was nothing in the observation of Fontaine that can be tertned 

 an anticipation of Lagrange, though D'Alembert, unknown to himself, 

 had certainly been anticipated by Euler. 



t The problem of the Three Bodies, properly speaking, is more general ; 

 but, in common parlance, it is confined to the particular case of gravita- 

 tion, and indeed of the sun, earth, and moon, as three bodies attracting 

 each other by the law of gravitation, and one of which is incomparably 

 larger than the other two. 



E 



