as special cases.
The more comprehensive law enables us to criticize Kepler's laws from a
higher standpoint, to see how far they are exact and how far they are
only approximations. They are, in fact, not precisely accurate, but the
reason for every discrepancy now becomes abundantly clear, and can be
worked out by the theory of gravitation.

We may treat Kepler's laws either as immediate consequences of the law
of gravitation, or as the known facts upon which that law was founded.
Historically, the latter is the more natural plan, and it is thus that
they are treated in the first three statements of the above notes; but
each proposition may be worked inversely, and we might state them
thus:--

1. The fact that the force acting on each planet is directed to the sun,
necessitates the equable description of areas.

2. The fact that the force varies as the inverse square of the distance,
necessitates motion in an ellipse, or some other conic section, with the
sun in one focus.

3. The fact that one attracting body acts on all the planets with an
inverse square law, causes the cubes of their mean distances to be
proportional to the squares of their periodic times.

Not only these but a multitude of other deductions follow rigorously
from the simple datum that every particle of matter attracts every other
particle with a force directly proportional to the mass of each and to
the inverse square of their mutual distance. Those dealt with in the
_Principia_ are summarized above, and it will be convenient to run over
them in order, with the object of giving some idea of the general
meaning of each, without attempting anything too intricate to be readily
intelligible.

[Illustration: FIG. 70.]

No. 1. Kepler's second law (equable description of areas) proves that
each planet is acted on by a force directed towards the sun as a centre
of force.

The equable description of areas about a centre of force has already
been fully, though briefly, established. (p. 175.) It is undoubtedly of
fundamental importance, and is the earliest instance of the serious
discussion of central forces, _i.e._ of forces directed always to a
fixed centre.

We may put it afresh thus:--OA has been the motion of a particle in a
unit of time; at A it receives a knock towards C, whereby in the next
unit it travels along AD instead of AB. Now the area of the triangle
CAD, swept out by the radius vector in unit time, is 1/2_bh_; _h_ being
the perpend