A sphere composed of uniform material, or of materials arranged in concentric strata, can be shown to attract external bodies as if its mass were concentrated at its centre. A hollow sphere, similarly composed, does the same, but on internal bodies it exerts no force at all.
Hence, at all distances above the surface of the earth, gravity decreases in inverse proportion as the square of the distance from the centre of the earth increases; but, if you descend a mine, gravity decreases in this case also as you leave the surface, though not at the same rate as when you went up. For as you penetrate the crust you get inside a concentric shell, which is thus powerless to act upon you, and the earth you are now outside is a smaller one. At what rate the force decreases depends on the distribution of density; if the density were uniform all through, the law of variation would be the direct distance, otherwise it would be more complicated. Anyhow, the intensity of gravity is a maximum at the surface of the earth, and decreases as you travel from the surface either up or down.
No. 14. The earth's equatorial protuberance, being acted on by the attraction of the sun and moon, must disturb its axis of rotation in a calculated manner; and thus is produced the precession of the equinoxes.
Here we come to a truly awful piece of reasoning. A sphere attracts as if its mass were concentrated at its centre (No. 12), but a spheroid does not. The earth is a spheroid, and hence it pulls and is pulled by the moon with a slightly uncentric attraction. In other words, the line of pull does not pass through its precise centre. Now when we have a spinning body, say a top, overloaded on one side so that gravity acts on it unsymmetrically, what happens? The axis of rotation begins to rotate cone-wise, at a pace which depends on the rate of spin, and on the shape and mass of the top, as well as on the amount and leverage of the overloading.
Newton calculated out the rapidity of this conical motion of the axis of the earth, produced by the slightly unsymmetrical pull of the moon, and found that it would complete a revolution in 26,000 years—precisely what was wanted to explain the precession of the equinoxes. In fact he had discovered the physical cause of that precession.
Observe that there were three stages in this discovery of precession:—
First, the observation by Hipparchus, that the nodes, or intersections of the earth's orbit (the sun's apparent orbit) with the plane of the equator, were not stationary, but slowly moved.
Second, the description of this motion by Copernicus, by the statement that it was due to a conical motion of the earth's axis of rotation about its centre as a fixed point.
Third, the explanation of this motion by Newton as due to the pull of the moon on the equatorial protuberance of the earth.
The explanation could not have been previously suspected, for the shape of the earth, on which the whole theory depends, was entirely unknown till Newton calculated it.