[35] Prop. 32.
[36] Prop. 33.
[37] Prop. 35.
The same theory which gave these Inequalities in the motion of the Moon produced by the disturbing force of the sun, gave also [410] corresponding 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;[38] and it was shown that the motions of their nodes would be slow by theory, as Flamsteed had found it to be by observation.[39] 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;[40] and he notices in the second edition,[41] that it follows from the theory of gravity, that the aphelia of Mercury, Venus, the Earth, and Mars, slightly progress.
[38] B. i. Prop. 66.
[39] B. iii. Prop. 23.
[40] B. iii. Prop. 13.
[41] Scholium to Prop. 14. B. iii.
In one celebrated instance, indeed, the deviation of the theory of the Principia from observation was wider, and more difficult to explain; 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 of the same amount as that which the sun exerts on the moon, he simply says,[42] “the apse of the moon moves about twice as fast.”
[42] B. i. Prop. 44, second edit. There is reason to believe, however, that Newton had, in his unpublished calculations, rectified this discrepancy.