stration: Fig. 106.]

Its sides are no longer drawn to the point of sight as in Fig. 7, nor
its diagonal to the point of distance, but to some other points on the
horizon, although the same rule holds good as regards their parallelism;
as for instance, in the case of _bc_ and _ad_, which, if produced, would
meet at _V_, a point on the horizon called a vanishing point. In this
figure only one vanishing point is seen, which is to the right of the
point of sight _S_, whilst the other is some distance to the left, and
outside the picture. If the cube is correctly drawn, it will be found
that the lines _ae_, _bg_, &c., if produced, will meet on the horizon at
this other vanishing point. This far-away vanishing point is one of the
inconveniences of oblique or angular perspective, and therefore it will
be a considerable gain to the draughtsman if we can dispense with it.
This can be easily done, as in the above figure, and here our geometry
will come to our assistance, as I shall show presently.

L

HOW TO PUT A GIVEN POINT INTO PERSPECTIVE

Let us place the given point _P_ on a geometrical plane, to show how far
it is from the base line, and indeed in the exact position we wish it to
be in the picture. The geometrical plane is supposed to face us, to hang
down, as it were, from the base line _AB_, like the side of a table, the
top of which represents the perspective plane. It is to that perspective
plane that we now have to transfer the point _P_.

[Illustration: Fig. 107.]

From _P_ raise perpendicular _Pm_ till it touches the base line at _m_.
With centre _m_ and radius _mP_ describe arc _Pn_ so that _mn_ is now
the same length as _mP_. As point _P_ is opposite point _m_, so must it
be in the perspective, therefore we draw a line at right angles to the
base, that is to the point of sight, and somewhere on this line will be
found the required point _P'_. We now have to find how far from _m_ must
that point be. It must be the length of _mn_, which is the same as _mP_.
We therefore from _n_ draw _nD_ to the point of distance, which being at
an angle of 45 deg, or half a right angle, makes _mP_' the perspective
length of _mn_ by its intersection with _mS_, and thus gives us the
point _P'_, which is the perspective of the original point.

LI

A PERSPECTIVE POINT BEING GIVEN, FIND ITS POSITION
ON THE GEOMETRICAL PLANE

To do this we simply reverse the foregoing problem. Thus let _P_ be the
given persp