t if the earth is supposed to move, the earth and moon together will
be revolved about their common centre of gravity. And the moon (by Prop,
lx.) will in the same periodic time, 27 days 7 hr. 43 min., with the
same circum-terrestrial force diminished in the duplicate proportion of
the distance, describe an orbit whose semi-diameter is to the
semi-diameter of the former orbit, that is, to the sixty semi-diameters
of the earth, as the sum of both the bodies of the earth and moon to the
first of two mean proportionals between this sum and the body of the
earth; that is, if we suppose the moon (on account of its mean apparent
diameter 31-1/2 min.) to be about 1/42 of the earth, as 43 to (42 +
42^2)^1/3 or as about 128 to 127. And, therefore, the semi-diameter of
the orbit--that is, the distance of the centres of the moon and
earth--will in this case be 60-1/2 semi-diameters of the earth, almost
the same with that assigned by Copernicus; and, therefore, the duplicate
proportion of the decrement of the force holds good in this distance.
(The action of the sun is here disregarded as inconsiderable.)

This proportion of the decrement of the forces is confirmed from the
eccentricity of the planets, and the very slow motion of their apsides;
for in no other proportion, it has been established, could the
circum-solar planets once in every revolution descend to their least,
and once ascend to their greatest distance from the sun, and the places
of those distances remain immovable. A small error from the duplicate
proportion would produce a motion of the apsides considerable in every
revolution, but in many enormous.

_The Tides_

While the planets are thus revolved in orbits about remote centres, in
the meantime they make their several rotations about their proper axes:
the sun in 26 days, Jupiter in 9 hr. 56 min., Mars in 24-2/3 hr., Venus
in 23 hr., and in like manner is the moon revolved about its axis in 27
days 7 hr. 43 min.; so that this diurnal motion is equal to the mean
motion of the moon in its orbit; upon which account the same face of the
moon always respects the centre about which this mean motion is
performed--that is, the exterior focus of the moon's orbit nearly.

By reason of the diurnal revolutions of the planets the matter which
they contain endeavours to recede from the axis of this motion; and
hence the fluid parts, rising higher towards the equator than about the
poles, would lay the solid parts about th