These superb discoveries were, however, but the starting point from which Newton entered on a series of researches, which disclosed many of the profoundest secrets in the scheme of celestial mechanics.  His natural insight showed that not only large masses like the sun and the earth, and the moon, attract each other, but that every particle in the universe must attract every other particle with a force which varies inversely as the square of the distance between them.  If, for example, the two particles were placed twice as far apart, then the intensity of the force which sought to bring them together would be reduced to one-fourth.  If two particles, originally ten miles asunder, attracted each other with a certain force, then, when the distance was reduced to one mile, the intensity of the attraction between the two particles would be increased one-hundred-fold.  This fertile principle extends throughout the whole of nature.  In some cases, however, the calculation of its effect upon the actual problems of nature would be hardly possible, were it not for another discovery which Newton’s genius enabled him to accomplish.  In the case of two globes like the earth and the moon, we must remember that we are dealing not with particles, but with two mighty masses of matter, each composed of innumerable myriads of particles.  Every particle in the earth does attract every particle in the moon with a force which varies inversely as the square of their distance.  The calculation of such attractions is rendered feasible by the following principle.  Assuming that the earth consists of materials symmetrically arranged in shells of varying densities, we may then, in calculating its attraction, regard the whole mass of the globe as concentrated at its centre.  Similarly we may regard the moon as concentrated at the centre of its mass.  In this way the earth and the moon can both be regarded as particles in point of size, each particle having, however, the entire mass of the corresponding globe.  The attraction of one particle for another is a much more simple matter to investigate than the attraction of the myriad different points of the earth upon the myriad different points of the moon.

Many great discoveries now crowded in upon Newton.  He first of all gave the explanation of the tides that ebb and flow around our shores.  Even in the earliest times the tides had been shown to be related to the moon.  It was noticed that the tides were specially high during full moon or during new moon, and this circumstance obviously pointed to the existence of some connection between the moon and these movements of the water, though as to what that connection was no one had any accurate conception until Newton announced the law of gravitation.  Newton then made it plain that the rise and fall of the water was simply a consequence of the attractive power which the moon exerted upon the oceans lying upon our globe.  He showed also that to a certain extent the sun produces tides, and he was able to explain how it was that when the sun and the moon both conspire, the joint result was to produce especially high tides, which we call “spring tides”; whereas if the solar tide was low, while the lunar tide was high, then we had the phenomenon of “neap” tides.