y are but as we see them in space, whereas geometry
represents figures not as we see them but as they are. When we have a
front view of a figure such as a square, its perspective and geometrical
appearance is the same, and we see it as it really is, that is, with all
its sides equal and all its angles right angles, the perspective only
varying in size according to the distance we are from it; but if we
place that square flat on the table and look at it sideways or at an
angle, then we become conscious of certain changes in its form--the side
farthest from us appears shorter than that near to us, and all the
angles are different. Thus A (Fig. 2) is a geometrical square and B is
the same square seen in perspective.

[Illustration: Fig. 2.]

[Illustration: Fig. 3.]

The science of perspective gives the dimensions of objects seen in space
as they appear to the eye of the spectator, just as a perfect tracing of
those objects on a sheet of glass placed vertically between him and them
would do; indeed its very name is derived from _perspicere_, to see
through. But as no tracing done by hand could possibly be mathematically
correct, the mathematician teaches us how by certain points and
measurements we may yet give a perfect image of them. These images are
called projections, but the artist calls them pictures. In this sketch
_K_ is the vertical transparent plane or picture, _O_ is a cube placed
on one side of it. The young student is the spectator on the other side
of it, the dotted lines drawn from the corners of the cube to the eye of
the spectator are the visual rays, and the points on the transparent
picture plane where these visual rays pass through it indicate the
perspective position of those points on the picture. To find these
points is the main object or duty of linear perspective.

Perspective up to a certain point is a pure science, not depending upon
the accidents of vision, but upon the exact laws of reasoning. Nor is it
to be considered as only pertaining to the craft of the painter and
draughtsman. It has an intimate connexion with our mental perceptions
and with the ideas that are impressed upon the brain by the appearance
of all that surrounds us. If we saw everything as depicted by plane
geometry, that is, as a map, we should have no difference of view, no
variety of ideas, and we should live in a world of unbearable monotony;
but as we see everything in perspective, which is infinite in its
variety