104 HISTORY OF MECHANICS. 



of the sun and earth, the relative times of the solar 

 and lunar revolutions, the eccentricities and incli- 

 nations 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 continue 

 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 labour and the 

 complexity 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 rapidly does it 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 favoured 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 suc- 

 cess, in consequence of the absence of the favourable 

 circumstances just referred to, and, accordingly, 

 Clairaut, after obtaining the six equations to which 



