Newton had shewn (chapter IX., § 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 when required, as in the cases of precession and of other motions of a planet or satellite about its centre, and of the corresponding action of a non-spherical planet on its satellites; to this group of problems belongs also that of the tides and other cases of the motion of parts of a body of any form relative to the rest. Again, the solar system happens to be so constituted that each body’s motion can be treated as determined primarily by one other body only. A planet, for example, moves nearly as if no other body but the sun existed, and the moon’s motion relative to the earth is roughly the same as if the other bodies of the solar system were non-existent. The problem of the motion of two mutually gravitating spheres was completely solved by Newton, and was shewn to lead to Kepler’s first two laws. Hence each body of the solar system could be regarded as moving nearly in an ellipse round some one body, but as slightly disturbed by the action of others. Moreover, by a general mathematical principle applicable in problems of motion, the effect of a number of small disturbing causes acting conjointly is nearly the same as that which results from adding together their separate effects. Hence each body could, without great error, be regarded as disturbed by one body at a time; the several disturbing effects could then be added together, and a fresh calculation could be made to further diminish the error. The kernel of Newton’s problem is thus seen to be a special case of the so-called =problem of three bodies=, viz.:— _Given at any time the positions and motions of three mutually gravitating bodies, to determine their positions and motions at any other time._ Even this apparently simple problem in its general form entirely transcends the powers, not only of the mathematical methods of the early 18th century, but also of those that have been devised since. Certain special cases have been solved, so that it has been shewn to be possible to suppose three bodies initially moving in such a way that their future motion can be completely determined. But these cases do not occur in nature.