D ALEMBERT. 423 



be said to have first taught the use of them as instru- 

 ments 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, f This, which applies to the case of the Moon, 

 would be resolved were we in possession of the solu- 

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

 of revolving round each other, revolve round the third 

 body. Euler's investigation did not appear quite satis- 

 factory; and, in 1750 the same subject was announced 

 for 1752, when he again carried off the prize by a paper 

 exhausting the subject, and affording such an approxi- 

 mation to the solution as the utmost resources of the 

 integral calculus can give. But while we admit, because 

 its illustrious author himself admitted, the justice of the 

 Academy's views respecting his first solution, we must 



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

 termed 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 gravitation, 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. 



