Page:A short history of astronomy(1898).djvu/354

288 observation to be nearly spherical in form, and the rest were generally supposed to be so also.

Newton had shewn, with a considerable degree of probability, that these bodies attracted one another according to the law of gravitation; and there was no reason to suppose that they exerted any other important influence on one another's motions.

The problem which presented itself, and which may conveniently be called Newton's problem, was therefore:—

Given these 18 bodies, and their positions and motions at any time, to deduce from their mutual gravitation by a process of mathematical calculation their positions and motions at any other time; and to shew that these agree with those actually observed.

Such a calculation would necessarily involve, among other quantities, the masses of the several bodies; it was evidently legitimate to assume these at will in such a way as to make the results of calculation agree with those of observation. If this were done successfully the masses would thereby be determined. In the same way the commonly accepted estimates of the dimensions of the solar system and of the shapes of its members might be modified in any way not actually inconsistent with direct observation.

The general problem thus formulated can fortunately be reduced to somewhat simpler ones.

Newton had shewn (chapter, § 182) that an ordinary sphere attracted other bodies and was attracted by them, as if its mass were concentrated at its centre; and that the effects of deviation from a spherical form became very small at a considerable distance from the body. Hence, except in special cases, the bodies of the solar system could be treated as spheres, which could again be regarded as concentrated at their respective centres. It will be convenient for the sake of brevity to assume for the future that all "bodies" referred to are of this sort, unless the contrary is stated or implied. The effects of deviations from spherical form could then be treated separately