(see above).

_4[pi]^2mr/T^2

Equate the two together, and at once we get

_r^3/T^2 = V/4[pi]^2M;_

or, in words, the cube of the distance divided by the square of the
periodic time for every planet or satellite of the system under
consideration, will be constant and proportional to the mass of the
central body.

This is Kepler's third law, with a notable addition. It is stated above
for circular motion only, so as to avoid geometrical difficulties, but
even so it is very instructive. The reason of the proportion between
_r^3_ and _T^2_ is at once manifest; and as soon as the constant _V_
became known, _the mass of the central body_, the sun in the case of a
planet, the earth in the case of the moon, Jupiter in the case of his
satellites, was at once determined.

Newton's reasoning at this time might, however, be better displayed
perhaps by altering the order of the steps a little, as thus:--

The centrifugal force of a body is proportional to _r^3/T^2_, but by
Kepler's third law _r^3/T^2_ is constant for all the planets,
reckoning _r_ from the sun. Hence the centripetal force needed to hold
in all the planets will be a single force emanating from the sun and
varying inversely with the square of the distance from that body.

Such a force is at once necessary and sufficient. Such a force would
explain the motion of the planets.

But then all this proceeds on a wrong assumption--that the planetary
motion is circular. Will it hold for elliptic orbits? Will an inverse
square law of force keep a body moving in an elliptic orbit about the
sun in one focus? This is a far more difficult question. Newton solved
it, but I do not believe that even he could have solved it, except that
he had at his disposal two mathematical engines of great power--the
Cartesian method of treating geometry, and his own method of Fluxions.
One can explain the elliptic motion now mathematically, but hardly
otherwise; and I must be content to state that the double fact is
true--viz., that an inverse square law will move the body in an ellipse
or other conic section with the sun in one focus, and that if a body so
moves it _must_ be acted on by an inverse square law.

[Illustration: FIG. 59.]

This then is the meaning of the first and third laws of Kepler. What
about the second? What is the meaning of the equable description of
areas? Well, that rigorously proves that a planet is acted on by a force
directed to the centre about which