50 DYNAMICAL PRINCIPLE. 



revolving round each other, revolve round the third body. 

 Euler's investigation did not appear quite satisfactory ; and 

 in 1750 the same subject was announced for 1752, when he 

 again carried oif the prize by a paper exhausting the subject, 

 and affording such an approximation to the solution as the 

 utmost resources of the integral calculus can give. But 

 while we admit, because its illustrious author himself ad- 

 mitted, the justice of the Academy's views respecting his first 

 solution, we must never forget the extraordinary genius dis- 

 played in it. He did not communicate the whole, or even 

 the more essential portion of his investigation ; but he after- 

 wards gave it in a paper to the Berlin Academy in 1747, and 

 in another to the Petersburg!! 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 disturbance between 

 two planets in a series of sines and cosines 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 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 admitted to have been a some- 

 what slight performance for so consummate a geometrician. 

 It rather evaded the difficulties of the problem than sur- 

 mounted 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 times greater 

 than that of the moon. 



