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.
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.