re _g_ is the intensity of terrestrial
gravity, a thing easily measured; being, indeed, numerically equal to
twice the distance a stone drops in the first second of free fall. [See
table p. 205.] Hence, expressing that the weight of a body is due to
gravity, and remembering that the centre of the earth's attraction is
distant from us by one earth's radius (R), we can write

_Vm_E
_mg_ = ------,
R^2

or

_V_E = gR^2 = 95,522 cubic miles-per-second per second.

But we already know _v_E, in terms of the moon's motion, as

4[pi]^2r^3
-----------
T^2

approximately, [more accurately, see preceding note, this quantity is
_V_(E + M)]; hence we can easily see if the two determinations of this
quantity agree.[20]

All these deductions are fundamental, and may be considered as the
foundation of the _Principia_. It was these that flashed upon Newton
during that moment of excitement when he learned the real size of the
earth, and discovered his speculations to be true.

The next are elaborations and amplifications of the theory, such as in
ordinary times are left for subsequent generations of theorists to
discover and work out.

Newton did not work out these remoter consequences of his theory
completely by any means: the astronomical and mathematical world has
been working them out ever since; but he carried the theory a great way,
and here it is that his marvellous power is most conspicuous.

It is his treatment of No. 7, the perturbations of the moon, that
perhaps most especially has struck all future mathematicians with
amazement. No. 7, No. 14, No. 15, these are the most inspired of the
whole.

No. 7. The moon is attracted not only by the earth, but by the sun also;
hence its orbit is perturbed, and Newton calculated out the chief of
these perturbations.

Now running through the perturbations (p. 203) in order:--The first is
in parenthesis, because it is mere excentricity. It is not a true
perturbation at all, and more properly belongs to Kepler.

(_a_) The first true perturbation is what Ptolemy called "the evection,"
the principal part of which is a periodic change in the ellipticity or
excentricity of the moon's orbit, owing to the pull of the sun. It is a
complicated matter, and Newton only partially solved it. I shall not
attempt to give an account of it.

(_b_) The next, "the variation," is a much simpler affair. It is caused
by the fact that as the moon revolves round