On the principle that all bodies attract each other with forces directly as their masses, and inversely as the squares of their distances, Newton showed that all the movements of the celestial bodies may be accounted for, and that Kepler’s laws might all have been predicted—­ the elliptic motions—­the described areas the relation of the times and distances.  As we have seen, Newton’s contemporaries had perceived how circular motions could be explained; that was a special case, but Newton furnished the solution of the general problem, containing all special cases of motion in circles, ellipses, parabolas, hyperbolas—­that is, in all the conic sections.

The Alexandrian mathematicians had shown that the direction of movement of falling bodies is toward the centre of the earth.  Newton proved that this must necessarily be the case, the general effect of the attraction of all the particles of a sphere being the same as if they were all concentrated in its centre.  To this central force, thus determining the fall of bodies, the designation of gravity was given.  Up to this time, no one, except Kepler, had considered how far its influence reached.  It seemed to Newton possible that it might extend as far as the moon, and be the force that deflects her from a rectilinear path, and makes her revolve in her orbit round the earth.  It was easy to compute, on the principle of the law of inverse squares, whether the earth’s attraction was sufficient to produce the observed effect.  Employing the measures of the size of the earth accessible at the time, Newton found that the moon’s deflection was only thirteen feet in a minute; whereas, if his hypothesis of gravitation were true, it should be fifteen feet.  But in 1669 Picard, as we have seen, executed the measurement of a degree more carefully than had previously been done; this changed the estimate of the magnitude of the earth, and, therefore, of the distance of the moon; and, Newton’s attention having been directed to it by some discussions that took place at the Royal Society in 1679, he obtained Picard’s results, went home, took out his old papers, and resumed his calculations.  As they drew to a close, he became so much agitated that he was obliged to desire a friend to finish them.  The expected coincidence was established.  It was proved that the moon is retained in her orbit and made to revolve round the earth by the force of terrestrial gravity.  The genii of Kepler had given place to the vortices of Descartes, and these in their turn to the central force of Newton.

In like manner the earth, and each of the planets, are made to move in an elliptic orbit round the sun by his attractive force, and perturbations arise by reason of the disturbing action of the planetary masses on one another.  Knowing the masses and the distances, these disturbances may be computed.  Later astronomers have even succeeded with the inverse problem, that is, knowing the perturbations or disturbances, to find the place and the mass of the disturbing body.  Thus, from the deviations of Uranus from his theoretical position, the discovery of Neptune was accomplished.