_A_ is
the same length as line _B_ its horizontal trace is longer than that of
the other; that the projection of a curve (_C_) in this upright position
is a straight line, that of a horizontal line (_D_) is equal to it, and
the projection of a perpendicular or vertical (_E_) is a point only.
The projections of lines or points can likewise be shown on a vertical
plane, but in that case we draw lines parallel to the horizontal plane,
and by this means we can get the position of a point in space; and by
the assistance of perspective, as will be shown farther on, we can carry
out the most difficult propositions of descriptive geometry and of the
geometry of planes and solids.

[Illustration: Fig. 18.]

The position of a point in space is given by its projection on a
vertical and a horizontal plane--

[Illustration: Fig. 19.]

Thus _e'_ is the projection of _E_ on the vertical plane _K_, and
_e''_ is the projection of _E_ on the horizontal plane; _fe''_ is the
horizontal trace of the plane _fE_, and _e'f_ is the trace of the same
plane on the vertical plane _K_.

VI

SCIENTIFIC DEFINITION OF PERSPECTIVE

The projections of the extremities of a right line which passes through
a vertical plane being given, one on either side of it, to find the
intersection of that line with the vertical plane. _AE_ (Fig. 20) is the
right line. The projection of its extremity _A_ on the vertical plane is
_a'_, the projection of _E_, the other extremity, is _e'_. _AS_ is the
horizontal trace of _AE_, and _a'e'_ is its trace on the vertical plane.
At point _f_, where the horizontal trace intersects the base _Bc_ of the
vertical plane, raise perpendicular _fP_ till it cuts _a'e'_ at point
_P_, which is the point required. For it is at the same time on the
given line _AE_ and the vertical plane _K_.

[Illustration: Fig. 20.]

This figure is similar to the previous one, except that the extremity
_A_ of the given line is raised from the ground, but the same
demonstration applies to it.

[Illustration: Fig. 21.]

And now let us suppose the vertical plane _K_ to be a sheet of glass,
and the given line _AE_ to be the visual ray passing from the eye to the
object _A_ on the other side of the glass. Then if _E_ is the eye of the
spectator, its projection on the picture is _S_, the point of sight.

If I draw a dotted line from _E_ to little _a_, this represents another
visual ray, and _o_, the point where it passes through t