Fortunately, however, the law of
elliptic motion established by Kepler has suggested the means of
defining the identity of a comet with absolute precision.

After Newton had made his discovery of the law of gravitation, and
succeeded in demonstrating that the elliptic paths of the planets around
the sun were necessary consequences of that law, he was naturally
tempted to apply the same reasoning to explain the movements of comets.
Here, again, he met with marvellous success, and illustrated his theory
by completely explaining the movements of the remarkable body which was
visible from December, 1680, to March, 1681.

[Illustration: Fig. 69.--The Parabolic Path of a Comet.]

There is a certain beautiful curve known to geometricians by the name of
the parabola. Its form is shown in the adjoining figure; it is a curved
line which bends in towards and around a certain point known as the
focus. This would not be the occasion for any allusion to the
geometrical properties of this curve; they should be sought in works on
mathematics. It will here be only necessary to point to the connection
which exists between the parabola and the ellipse. In a former chapter
we have explained the construction of the latter curve, and we have
shown how it possesses two foci. Let us suppose that a series of
ellipses are drawn, each of which has a greater distance between its
foci than the preceding one. Imagine the process carried on until at
length the distance between the foci became enormously great in
comparison with the distance from each focus to the curve, then each end
of this long ellipse will practically have the same form as a parabola.
We may thus look on the latter curve represented in Fig. 69 as being one
end of an ellipse of which the other end is at an indefinitely great
distance. In 1681 Doerfel, a clergyman of Saxony, proved that the great
comet then recently observed moved in a parabola, in the focus of which
the sun was situated. Newton showed that the law of gravitation would
permit a body to move in an ellipse of this very extreme type no less
than in one of the more ordinary proportions. An object revolving in a
parabolic orbit about the sun at the focus moves in gradually towards
the sun, sweeps around the great luminary, and then begins to retreat.
There is a necessary distinction between parabolic and elliptic motion.
In the latter case the body, after its retreat to a certain distance,
will turn round and again dra