SEQUEL TO THE GENERALIZATION. 369 



and moon, their apogees, the moon's nodes, and other quantities ; and 

 by the variety of combinations of which these admit, the terms become 

 very numerous and complex. The magnitude of the terms depends 

 also upon various circumstances ; as the relative force of the sun and 

 earth, the relative times of the solar and lunar revolutions, the eccen- 

 tricities and inclinations of the two orbits. These are combined so as 

 to give terms of different orders of magnitudes ; and it depends upon 

 the skill and perseverance of the mathematician how far he will con- 

 tinue this series of terms. For there is no limit to their number : and 

 though the methods of which we have spoken do theoretically enable 

 us to calculate as many terms as we please, the labor and the complex- 

 ity of the operations are so serious that common calculators are stopped 

 by them. None but very great mathematicians have been able to walk 

 safely any considerable distance into this avenue, so rapidiy does : * 

 darken as we proceed. And even the possibility of doing what has 

 been done, depends upon what we may call accidental circumstances ; 

 the smallness of the inclinations and eccentricities of the system, and 

 the like. " If nature had not favored us in this way," Lagrange used to 

 say " there would have been an end of the geometers in this problem." 

 The expected return of the comet of 1682 in 1759, gave a new interest 

 to the problem, and Clairaut proceeded to calculate the case which was 

 thus suggested. When this was treated by the methods which had suc- 

 ceeded for the moon, it offered no prospect of success, in consequence 

 of the absence of the favorable circumstances just referred to, and, 

 accordingly, Clairaut, after obtaining the six equations to which he re- 

 duces the solution, 5 adds, " Integrate them who can" (Integre main- 

 tenant qui pourra). New methods of approximation were devised for 

 this case. 



The problem of three bodies was not prosecuted in consequence ot 

 its analytical beauty, or its intrinsic attraction ; but its great difficul- 

 ties were thus resolutely combated from necessity ; because in no other 

 way could the theory of universal gravitation be known to be true or 

 made to be useful. The construction of Tables of the Moon, an 

 object which offered a large pecuniary reward, as well as mathemat- 

 ical glory, to the successful adventurer, was the main purpose of these 

 labors. 



The Theory of the Planets, presented the Problem of Three Bodies 

 in a new form, and involved in peculiar difficulties ; for the approxima- 



Journal des Sevang, Aug. 1753. 

 VOL. I. 24 



