The outline of a sphere, projected by the beholder against any plane
surface behind it, is a circle only when a perpendicular, let fall on
the plane from the eye, passes through the centre of the sphere. In
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.
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