Comets were known to have come on many occasions into the region of space occupied by the solar system, and there were reasons to believe that one of them at least (chapter X., § 200) was a regular visitor; they were, however, scarcely regarded as belonging to the solar system, and their action (if any) on its members was ignored, a neglect which subsequent investigation has completely justified. Many thousands of fixed stars had also been observed, and their places on the celestial sphere determined; they were known to be at very great though unknown distances from the solar system, and their influence on it was regarded as insensible. The motions of the 18 members of the solar system were tolerably well known; their actual distances from one another had been roughly estimated, while the _proportions_ between most of the distances were known with considerable accuracy. Apart from the entirely anomalous ring of Saturn, which may for the present be left out of consideration, most of the bodies of the system were known from 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.[130] 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.