412 D'ALEMBERT. 



extraordinary genius displayed in it. He did not com- 

 municate the whole, or even the more essential portion 

 of his investigation, but he afterwards gave it in a paper 

 to the Berlin Academy in 1747, and in another to the 

 Petersburgh Academy in 1750, the first of these con- 

 taining our earliest view of the variation of arbitrary 

 constants in differential equations, and the development 

 of the radical which expresses the relative disturbance 

 between two planets in a series of sines and co-sines 

 of angles multiples of the elongation, a series so artistry 

 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 pro- 

 blem some time before Euler. In 1743, he gave a 

 Memoir on the Moon's Orbit, according to the New- 

 tonian theory of gravitation, and it appears in the 

 volume for that year ; but this paper must be admitted 

 to have been a somewhat slight performance for so con- 

 summate a geometrician. It rather evaded the diffi- 

 culties of the problem than surmounted by enounter- 

 ing 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 distance of that body from each of 

 the two others, and the almost elliptical path of the 

 one whose orbit was sought, leaving its deviation from 

 that path alone to be sought after. Accordingly, the 

 paper of 1743 did not satisfy its illustrious author, 

 who, in 1747, produced another worthy of the sub- 

 ject and of himself. This was read 15th November, 



* 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 times greater 

 than that of the moou. 



