184. The motion of the moon presents special difficulties, but Newton, who was evidently much interested in the problems of lunar theory, succeeded in overcoming them much more completely than the corresponding ones connected with the planets.
The moon’s motion round the earth is primarily due to the attraction of the earth; the perturbations due to the other planets are insignificant; but the sun, which though at a very great distance has an enormously greater mass than the earth, produces a very sensible disturbing effect on the moon’s motion. Certain irregularities, as we have seen (chapter II., [§§ 40], 48; chapter V., [§ 111]), had already been discovered by observation. Newton was able to shew that the disturbing action of the sun would necessarily produce perturbations of the same general character as those thus recognised, and in the case of the motion of the moon’s nodes and of her apogee he was able to get a very fairly accurate numerical result;[107] and he also discovered a number of other irregularities, for the most part very small, which had not hitherto been noticed. He indicated also the existence of certain irregularities in the motions of Jupiter’s and Saturn’s moons analogous to those which occur in the case of our moon.
185. One group of results of an entirely novel character resulted from Newton’s theory of gravitation. It became for the first time possible to estimate the masses of some of the celestial bodies, by comparing the attractions exerted by them on other bodies with that exerted by the earth.
The case of Jupiter may be given as an illustration. The time of revolution of Jupiter’s outermost satellite is known to be about 16 days 16 hours, and its distance from Jupiter was estimated by Newton (not very correctly) at about four times the distance of the moon from the earth. A calculation exactly like that of § 172 or § 173 shews that the acceleration of the satellite due to Jupiter’s attraction is about ten times as great as the acceleration of the moon towards the earth, and that therefore, the distance being four times as great, Jupiter attracts a body with a force 10 × 4 × 4 times as great as that with which the earth attracts a body at the same distance; consequently Jupiter’s mass is 160 times that of the earth. This process of reasoning applies also to Saturn, and in a very similar way a comparison of the motion of a planet, Venus for example, round the sun with the motion of the moon round the earth gives a relation between the masses of the sun and earth. In this way Newton found the mass of the sun to be 1067, 3021, and 169282 times greater than that of Jupiter, Saturn, and the earth, respectively. The corresponding figures now accepted are not far from 1047, 3530, 324439. The large error in the last number is due to the use of an erroneous value of the distance of the sun—then not at all accurately known—upon which depend the other distances in the solar system, except those connected with the earth-moon system. As it was necessary for the employment of this method to be able to observe the motion of some other body attracted by the planet in question, it could not be applied to the other three planets (Mars, Venus, and Mercury), of which no satellites were known.
186. From the equality of action and reaction it follows that, since the sun attracts the planets, they also attract the sun, and the sun consequently is in motion, though—owing to the comparative smallness of the planets—only to a very small extent. It follows that Kepler’s Third Law is not strictly accurate, deviations from it becoming sensible in the case of the large planets Jupiter and Saturn (cf. chapter VII., [§ 144]). It was, however, proved by Newton that in any system of bodies, such as the solar system, moving about in any way under the influence of their mutual attractions, there is a particular point, called the centre of gravity, which can always be treated as at rest; the sun moves relatively to this point, but so little that the distance between the centre of the sun and the centre of gravity can never be much more than the diameter of the sun.
It is perhaps rather curious that this result was not seized upon by some of the supporters of the Church in the condemnation of Galilei, now rather more than half a century old; for if it was far from supporting the view that the earth is at the centre of the world, it at any rate negatived that part of the doctrine of Coppernicus and Galilei which asserted the sun to be at rest in the centre of the world. Probably no one who was capable of understanding Newton’s book was a serious supporter of any anti-Coppernican system, though some still professed themselves obedient to the papal decrees on the subject.[108]
187. The variation of the time of oscillation of a pendulum in different parts of the earth, discovered by Richer in 1672 (chapter VIII., [§ 161]), indicated that the earth was probably not a sphere. Newton pointed out that this departure from the spherical form was a consequence of the mutual gravitation of the particles making up the earth and of the earth’s rotation. He supposed a canal of water to pass from the pole to the centre of the earth, and then from the centre to a point on the equator (B O a A in fig. 72), and then found the condition that these two columns of water O B, O A, each being attracted towards the centre of the earth, should balance. This method involved certain assumptions as to the inside of the earth, of which little can be said to be known even now, and consequently, though Newton’s general result, that the earth is flattened at the poles and bulges out at the equator, was right, the actual numerical expression which he found was not very accurate. If, in the figure, the dotted line is a circle the radius of which is equal to the distance of the pole B from the centre of the earth O, then the actual surface of the earth extends at the equator beyond this circle as far as A, where, according to Newton, a A is about 1∕230 of O B or O A, and according to modern estimates, based on actual measurement of the earth as well as upon theory (chapter X., [§ 221]), it is about 1∕293 of O A. Both Newton’s fraction and the modern one are so small that the resulting flattening cannot be made sensible in a figure; in fig. 72 the length a A is made, for the sake of distinctness, nearly 30 times as great as it should be.
Fig. 72.—The spheroidal form of the earth.