424 D'ALEMBERT. 



never forget the extraordinary genius displayed in it. 

 He did not communicate the whole, or even the more 

 essential portion of his investigation, but he afterwards 

 gave it in a paper to the Berlin Academy in 1740, and 

 in another to the Petersburgh Academy in 1750, the first 

 of these containing our earliest view of the variation of 

 arbitrary constants in differential equations, and the 

 development of the radical which expresses the relative 

 distance between two planets in a series of sines and 

 co-sines of angles multiples of the elongation, a series 

 so artistly framed that every three consecutive terms 

 are related together in such a manner as to give the 

 whole series from a determination of the first two 

 terms. Clairaut appears to have turned his attention 

 to the same problem some time before Euler. In 1743, 

 he gave a Memoir on the Moon's Orbit, according to 

 the Newtonian theory of gravitation, and it appears 

 in the volume for that year; but this paper must be ad- 

 mitted to have been a somewhat slight performance for 

 so consummate a geometrician. It rather evaded the diffi- 

 culties of the problem than surmounted by encountering 

 them; for he assumed the orbit of the moon to differ 

 imperceptibly from a circle ; and his differential equation 

 could not have been integrated without this supposition. 

 Now, the only assumptions which had been conceived 

 permissible were the incomparably greater mass of one 

 body than those of the two others,* the nearly equal 



* In truth, the mass of the sun being 355,000 times that of the 

 earth, and that of the earth being between sixty-eight and sixty-nine 

 times that of the moon, the mass of the sun is twenty-five millions 

 of time* greater than that of the 



