410 HISTORY OF PHYSICAL ASTRONOMY. 



ponding Inequalities in the motions of the Satellites of other planets, 

 arising from the same cause ; and likewise pointed out the necessary 

 existence of irregularities in the motions of the Planets arising from 

 their mutual attraction. Newton gave propositions by which the 

 Irregularities of the motion of Jupiter's moons might be deduced from 

 those of our own ; 14 and it was shown that the motions of their nodes 

 would be slow by theory, as Flamsteed had found it to be by observa- 

 tion. 15 But Newton did not attempt to calculate the effect of the 

 mutual action of the planets, though he observes, that in the case of 

 Jupiter and Saturn this effect is too considerable to be neglected ; 16 

 and he notices in the second edition, 17 that it follows from the theory 

 of gravity, that the aphelia of Mercury, Venus, the Earth, and Mars, 

 slightly progress. 



In one celebrated instance, indeed, the deviation of the theory of 

 the Principia from observation was wider, and more difficult to ex- 

 plaiu ; and as this deviation for a time resisted the analysis of Euler 

 and Clairaut, as it had resisted the synthesis of Newton, it at one 

 period staggered the faith of mathematicians in the exactness of the 

 law of the inverse square of the distance. I speak of the Motion of 

 the Moon's Apogee, a problem which has already been referred to ; 

 and in which Newton's method, and all the methods which could be 

 devised for some time afterwards, gave only half the observed motion; 

 a circumstance which arose, as was discovered by Clairaut in 1750, 

 from the insufficiency of the method of approximation. Newton does 

 not attempt to conceal this discrepancy. After calculating what the 

 motion of apse would be, upon the assumption of a disturbing force ot 

 the same amount as that which the sun exerts on the moon, he simply 

 says, 18 " the apse of the moon moves about twice as fast." 



The difficulty of doing what Newton did in this branch of the sub- 

 ject, and the powers it must have required, may be judged of from 

 what has already been stated ; — that no one, with his methods, has 

 yet been able to add any thing to his labors : few have undertaken to 

 illustrate what he has written, and no great number have understood 

 it throughout. The extreme complication of the forces, and of the 

 conditions under which they act, makes the subject by far the most 

 thorny walk of mathematics. It is necessary to resolve the action 



14 B. i. Prop. 66. is B. iii. Prop. 23. ™ B. iii. Prop. 13. 



17 Scholium to Prop. 14. B. iii. 



18 B. i. Prop. 44, second edit. There is reason to believe, however, that Newton 

 had, in his unpublished calculations, rectified this discrepancy. 



