4. 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.

Another and smaller motion of a somewhat similar kind has been worked
out since: it is due to the unsymmetrical attraction of the other
planets for this same equatorial protuberance. It shows itself as a
periodic change in the obliquity of the ecliptic, or so-called recession
of the apses, rather than as a motion of the nodes.[21]

No. 15. The waters of the ocean are attracted towards the sun and moon
on one side, and whirled a little farther away than the solid earth on
the other side: hence Newton explained all the