Also Newton saw that his law of gravitation would afford an explanation of the rise and fall of the tides. The water on the side of the earth toward the moon, being nearer to the moon, would be more strongly attracted toward it, and therefore raised in a tide. And the water on the farther side of the earth away from the moon, being at a greater distance than the earth itself, the moon would attract the earth more strongly than this mass of water, tending therefore to draw the earth away from the water, and so raising at the same time a high tide on the side of the earth away from the moon. As the earth turns round on its axis, therefore, two tidal waves continually follow each other at intervals of about twelve hours.
The sun, too, joins its gravitating force with that of the moon, raising tides nearly half as high as those which the moon produces, because the sun's vaster mass makes up in large part for its much greater distance. At first and third quarters of the moon, the sun acts against the moon, and the difference of their tide-producing forces gives us "neap tides"; while at new moon and full, sun and moon act together, and produce the maximum effect known as "spring tides."
Newton passed on to explain, by the action of gravitation also, the precession of the equinoxes, a phenomenon of the sky discovered by Hipparchus, who pretty well ascertained its amount, although no reason for it had ever been assigned. The plane of the earth's equator extended to the celestial sphere marks out the celestial equator, and the two opposite points where it intersects the plane of the ecliptic, or the earth's path round the sun, are called the equinoctial points, or simply the equinoxes. And precession of the equinoxes is the motion of these points westward or backward, about 50 seconds each year, so that a complete revolution round the ecliptic would take place in about 26,000 years.
Newton saw clearly how to explain this: it is simply due to the attraction of the sun's gravitation upon the protuberant bulge around the earth's equator, acting in conjunction with the earth's rotation on its axis, the effect being very similar to that often seen in a spinning top, or in a gyroscope. The moon moving near the ecliptic produces a precessional effect, as also do the planets to a very slight degree; and the observed value of precession is the same as that calculated from gravitation, to a high degree of precision.
Newton died in 1727, too early to have witnessed that complete and triumphant verification of his law which ultimately has accounted for practically every inequality in the planetary motions caused by their mutual attractions. The problems involved are far beyond the complexity of those which the mathematical astronomer has to deal with, and the mathematicians of France deserve the highest credit for improving the processes of their science so that obstacles which appeared insuperable were one after another overcome.
Newton's method of dealing with these problems was mainly geometric, and the insufficiency of this method was apparent. Only when the French mathematicians began to apply the higher methods of algebra was progress toward the ultimate goal assured. D'Alembert and Clairaut for a time were foremost in these researches, but their places were soon taken by Lagrange, who wrote the "Mécanique Analytique," and Laplace, whose "Mécanique Céleste" is the most celebrated work of all. In large part these works are the basis of the researches of subsequent mathematical astronomers who, strictly speaking, cannot as yet be said to have arrived at a complete and rigorous solution of all the problems which the mutual attractions of all the bodies of the solar system have originated.
It may well be that even the mathematics of the present day are incompetent to this purpose. When the brilliant genius of Sir William Hamilton invented quaternion analysis and showed the marvelous facility with which it solved the intricate problems of physics, there was the expectation that its application to the higher problems of mathematical astronomy might effect still greater advances; but nothing in that direction has so far eventuated. Some astronomers look for the invention of new functions with numerical tables bearing perhaps somewhat the relation to present tables of logarithms, sines, tangents, and so on, that these tables do to the simple multiplication table of Pythagoras.