gth of the loop remains unchanged. If the two pins be
brought nearer together the eccentricity will decrease, and the ellipse
will approximate more closely to the shape of a circle. If the pins be
separated more widely the eccentricity of the ellipse will be increased.
That the circle is an extreme form of ellipse will be evident, if we
suppose the two pins to draw in so close together that they become
coincident; the point will then simply trace out a circle as the pencil
moves round the figure.

[Illustration: Fig. 37.--Drawing an Ellipse.]

The points marked by the pins obviously possess very remarkable
relations with respect to the curve. Each one is called a _focus_, and
an ellipse can only have one pair of foci. In other words, there is but
a single pair of positions possible for the two pins, when an ellipse of
specified size, shape, and position is to be constructed.

The ellipse differs principally from a circle in the circumstance that
it possesses variety of form. We can have large and small ellipses just
as we can have large and small circles, but we can also have ellipses of
greater or less eccentricity. If the ellipse has not the perfect
simplicity of the circle it has, at least, the charm of variety which
the circle has not. The oval curve has also the beauty derived from an
outline of perfect grace and an association with ennobling conceptions.

The ancient geometricians had studied the ellipse: they had noticed its
foci; they were acquainted with its geometrical relations; and thus
Kepler was familiar with the ellipse at the time when he undertook his
celebrated researches on the movements of the planets. He had found, as
we have already indicated, that the movements of the planets could not
be reconciled with circular orbits. What shape of orbit should next be
tried? The ellipse was ready to hand, its properties were known, and the
comparison could be made; memorable, indeed, was the consequence of this
comparison. Kepler found that the movement of the planets could be
explained, by supposing that the path in which each one revolved was an
ellipse. This in itself was a discovery of the most commanding
importance. On the one hand it reduced to order the movements of the
great globes which circulate round the sun; while on the other, it took
that beautiful class of curves which had exercised the geometrical
talents of the ancients, and assigned to them the dignity of defining
the highways of the unive