Newton realised that, while planets and satellites are chiefly controlled by the central body about which they revolve, the new law must involve irregularities, due to their mutual action—such, in fact, as Horrocks had indicated. He determined to put this to a test in the case of the moon, and to calculate the sun’s effect, from its mass compared with that of the earth, and from its distance. He proved that the average effect upon the plane of the orbit would be to cause the line in which it cuts the plane of the ecliptic (i.e., the line of nodes) to revolve in the ecliptic once in about nineteen years. This had been a known fact from the earliest ages. He also concluded that the line of apses would revolve in the plane of the lunar orbit also in about nineteen years; but the observed period is only ten years. For a long time this was the one weak point in the Newtonian theory. It was not till 1747 that Clairaut reconciled this with the theory, and showed why Newton’s calculation was not exact. Newton proceeded to explain the other inequalities recognised by Tycho Brahe and older observers, and to calculate their maximum amounts as indicated by his theory. He further discovered from his calculations two new inequalities, one of the apogee, the other of the nodes, and assigned the maximum value. Grant has shown the values of some of these as given by observation in the tables of Meyer and more modern tables, and has compared them with the values assigned by Newton from his theory; and the comparison is very remarkable. Newton. Modern Tables. ° ’ " ° ’ " Mean monthly motion of Apses 1.31.28 3.4.0 Mean annual motion of nodes 19.18.1,23 19.21.22,50 Mean value of “variation” 36.10 35.47 Annual equation 11.51 11.14 Inequality of mean motion of apogee 19.43 22.17 Inequality of mean motion of nodes 9.24 9.0 The only serious discrepancy is the first, which has been already mentioned. Considering that some of these perturbations had never been discovered, that the cause of none of them had ever been known, and that he exhibited his results, if he did not also make the discoveries, by the synthetic methods of geometry, it is simply marvellous that he reached to such a degree of accuracy. He invented the infinitesimal calculus which is more suited for such calculations, but had he expressed his results in that language he would have been unintelligible to many.