other positions the projection of the sphere becomes an ellipse, or one
of its varieties, the parabola and hyperbola. The parabola is the
boundary of the projection of a sphere upon a plane, when the eye is
just as far from the plane as the outer edge of the sphere is, and the
hyperbola is a similar curve formed by bringing the eye still nearer to
the plane.

By these metamorphoses the circle loses much of its monotony, without
losing much of its simplicity. The law of the projection of a sphere
upon a plane is simple, in whatever position the plane may be. And if
we seek a law for the ellipse, or either of the conic sections, which
shall confine our attention to the plane, the laws remain simple. There
are for these curves two centres, which come together for the circle,
and recede to an infinite distance for the parabola; and the simple law
of their formation is, that the curve everywhere makes equal angles
with the lines drawn to these two centres. According to the fundamental
canon, a conic section should be a beautiful curve; and the proof that
it is so is to be found in the attention which these curves have always
drawn upon themselves from artists and from mathematicians. Plato,
equally great in mathematics and in metaphysics, is said to have been
the first to investigate the properties of the ellipse. For about a
century and a half, to the time of Apollonius, the beauty of this
curve, and of its variations, the parabola and hyperbola, so fascinated
the minds of Plato's followers, that Apollonius found theorems and
problems relating to these figures sufficient to fill eight books with
condensed truths concerning them. The study of the conic sections has
been a part of polite learning from his day downward. All men confess
their beauty, which so entrances those of mathematical genius as
entirely to absorb them. For eighteen centuries the finest spirits of
our race drew some of their best means of intellectual discipline from