to which the velocity of the planet varies. The adjoining figure (Fig.
39) shows a planetary orbit, with, of course, the sun at the focus S. We
have taken two portions, A B and C D, round the ellipse, and joined
their extremities to the focus. Kepler's second law may be stated in
these words:--

"_Every planet moves round the sun with such a velocity at every
point, that a straight line drawn from it to the sun passes over
equal areas in equal times._"

[Illustration: Fig. 39.--Equal Areas in Equal Times.]

For example, if the two shaded portions, A B S and D C S, are equal in
area, then the times occupied by the planet in travelling over the
portions of the ellipse, A B and C D, are equal. If the one area be
greater than the other, then the times required are in the proportion of
the areas.

This law being admitted, the reason of the increase in the planet's
velocity when it approaches the sun is at once apparent. To accomplish a
definite area when near the sun, a larger arc is obviously necessary
than at other parts of the path. At the opposite extremity, a small arc
suffices for a large area, and the velocity is accordingly less.

These two laws completely prescribe the motion of a planet round the
sun. The first defines the path which the planet pursues; the second
describes how the velocity of the body varies at different points along
its path. But Kepler added to these a third law, which enables us to
compare the movements of two different planets revolving round the same
sun. Before stating this law, it is necessary to explain exactly what is
meant by the _mean_ distance of a planet. In its elliptic path the
distance from the sun to the planet is constantly changing; but it is
nevertheless easy to attach a distinct meaning to that distance which is
an average of all the distances. This average is called the mean
distance. The simplest way of finding the mean distance is to add the
greatest of these quantities to the least, and take half the sum. We
have already defined the periodic time of the planet; it is the number
of days which the planet requires for the completion of a journey round
its path. Kepler's third law establishes a relation between the mean
distances and the periodic times of the various planets. That relation
is stated in the following words:--

"_The squares of the periodic times are proportional to the cubes
of the mean distances._"

Kepler knew that the diffe