its stands out as
one of the most conspicuous events in the history of astronomy. It may,
in fact, be doubted whether any other discovery in the whole range of
science has led to results of such far-reaching interest.

We must here adventure for a while into the field of science known as
geometry, and study therein the nature of that curve which the
discovery of Kepler has raised to such unparalleled importance. The
subject, no doubt, is a difficult one, and to pursue it with any detail
would involve us in many abstruse calculations which would be out of
place in this volume; but a general sketch of the subject is
indispensable, and we must attempt to render it such justice as may be
compatible with our limits.

The curve which represents with perfect fidelity the movements of a
planet in its revolution around the sun belongs to that well-known group
of curves which mathematicians describe as the conic sections. The
particular form of conic section which denotes the orbit of a planet is
known by the name of the _ellipse_: it is spoken of somewhat less
accurately as an oval. The ellipse is a curve which can be readily
constructed. There is no simpler method of doing so than that which is
familiar to draughtsmen, and which we shall here briefly describe.

We represent on the next page (Fig. 37) two pins passing through a sheet
of paper. A loop of twine passes over the two pins in the manner here
indicated, and is stretched by the point of a pencil. With a little care
the pencil can be guided so as to keep the string stretched, and its
point will then describe a curve completely round the pins, returning to
the point from which it started. We thus produce that celebrated
geometrical figure which is called an ellipse.

It will be instructive to draw a number of ellipses, varying in each
case the circumstances under which they are formed. If, for instance,
the pins remain placed as before, while the length of the loop is
increased, so that the pencil is farther away from the pins, then it
will be observed that the ellipse has lost some of its elongation, and
approaches more closely to a circle. On the other hand, if the length of
the cord in the loop be lessened, while the pins remain as before, the
ellipse will be found more oval, or, as a mathematician would say, its
_eccentricity_ is increased. It is also useful to study the changes
which the form of the ellipse undergoes when one of the pins is altered,
while the len