draw _KM_, thus completing the outer parallel square.
Through _F_, where _PS_ intersects _MK_, draw _AV_ till it cuts the
horizon in _V_, its vanishing point. From _V_ draw _VB_ cutting side
_KE_ of outer square in _G_, and we have the four points _AFGB_, which
are the four angles of the square required. Join _FG_, and the figure is
complete.

Any other side of the square might be given, such as _AF_. First through
_A_ and _F_ draw _SC_, _SP_, then draw _Ao_, then through _o_ draw _CD_.
From _C_ draw base of parallel square _CE_, and at _M_ through _F_ draw
_MK_ cutting diagonal at _K_, which gives top of square. Now through _K_
draw _SE_, giving _KE_ the remaining side thereof, produce _AF_ to _V_,
from _V_ draw _VB_. Join _FG_, _GB_, and _BA_, and the square required
is complete.

The student can try the remaining two sides, and he will find they work
out in a similar way.

LXXXII

HOW TO DRAW SOLID FIGURES AT ANY ANGLE BY THE NEW METHOD

As we can draw planes by this method so can we draw solids, as shown in
these figures. The heights of the corners of the triangles are obtained
by means of the vanishing scales _AS_, _OS_, which have already been
explained.

[Illustration: Fig. 152.]

[Illustration: Fig. 153.]

In the same manner we can draw a cubic figure (Fig. 154)--a box, for
instance--at any required angle. In this case, besides the scale _AS_,
_OS_, we have made use of the vanishing lines _DV_, _BV_, to corroborate
the scale, but they can be dispensed with in these simple objects, or we
can use a scale on each side of the figure as _a'o'S_, should both
vanishing points be inaccessible. Let it be noted that in the scale
_AOS_, _AO_ is made equal to _BC_, the height of the box.

[Illustration: Fig. 154.]

By a similar process we draw these two figures, one on the square, the
other on the circle.

[Illustration: Fig. 155.]

[Illustration: Fig. 156.]

LXXXIII

POINTS IN SPACE

The chief use of these figures is to show how by means of diagonals,
horizontals, and perpendiculars almost any figure in space can be set
down. Lines at any slope and at any angle can be drawn by this
descriptive geometry.

The student can examine these figures for himself, and will understand
their working from what has gone before. Here (Fig. 157) in the
geometrical square we have a vertical plane _AabB_ standing on its base
_AB_. We wish to place a projection of this figure at a certain dist