NEWTON S PRINCIPIA. 93 



Mars, and collects from thence that of the Earth, Venus, 

 and Mercury, by the law which regulates the motion of 

 the apsides, namely, the sesquiplicate proportion to the 

 distances. By this he makes the motion of the Earth s 

 aphelion 17 40&quot; in a century, or 10&quot; 36&quot; yearly, being 

 not a second and a half different from what it is now 

 understood to be. 



The calculation of the motion of the moon s apsides, 

 however, which he deduced from these propositions, differed 

 widely from the truth. He made it, as we have seen, 

 amount to little more than a degree and a half each revo 

 lution*, or about one-half of the truth ; and for the dis 

 crepancy between the theory and the phenomena he seems 

 to have failed in accounting. Others, in the earlier part 

 of the eighteenth century, having applied to the subject 

 a different investigation, but founded upon his principles, 

 obtained a different result, but erring by excess ; for they 

 made the motion 3 27 each revolution, or nearly 45 in 

 the year instead of 39. About the year 1745 the three 

 great mathematicians of that age, Clairaut, Euler, and 

 D Alembert, investigated the subject ; and applying the 

 whole resources of analysis to its discussion as a case of 

 the problem of three bodies, obtained general solutions of 

 great beauty. However, they still found the theory differ 

 with the fact nearly as much as Newton himself had done; 

 and Clairaut was even driven by this to devise a new law 

 for the purpose of explaining the apparent discrepancy. He 



supposed the centripetal force to be not as -j- 2 but as 



-TJ + -74* In a very short time, however, he candidly 



gave up this theory, and announced the important fact 

 that he had found the whole error to arise from his hav- 



* 1 31 28&quot;. 



