econd case all
the horizontals _e_, _f_, _g_, _h_, drawn between _A'O_ and _B'O_ also
represent 5 feet each. So that by the aid of this scale we can give the
exact perspective height and width of any object in the picture, however
far it may be from the base line, for of course we can increase or
diminish our measurements at _AB_ and _A'B'_ to whatever length we
require.

As it may not be quite evident at first that the points _O_ may be taken
at random, the following figure will prove it.

XXVI

THE VANISHING SCALE CAN BE DRAWN TO ANY POINT ON THE HORIZON

From _AB_ (Fig. 77) draw _AO_, _BO_, thus forming the scale, raise
vertical _C_. Now form a second scale from _AB_ by drawing _AO' BO'_,
and therein raise vertical _D_ at an equal distance from the base.
First, then, vertical _C_ equals _AB_, and secondly vertical _D_ equals
_AB_, therefore _C_ equals _D_, so that either of these scales will
measure a given height at a given distance.

[Illustration: Fig. 77.]

(See axioms of geometry.)

[Illustration: Fig. 79. Schoolgirls.]

[Illustration: Fig. 80. Cavaliers.]

XXVII

APPLICATION OF VANISHING SCALES TO DRAWING FIGURES

In this figure we have marked off on a level plain three or four points
_a_, _b_, _c_, _d_, to indicate the places where we wish to stand our
figures. _AB_ represents their average height, so we have made our scale
_AO_, _BO_, accordingly. From each point marked we draw a line parallel
to the base till it reaches the scale. From the point where it touches
the line _AO_, raise perpendicular as _a_, which gives the height
required at that distance, and must be referred back to the figure
itself.

[Illustration: Fig. 78.]

XXVIII

HOW TO DETERMINE THE HEIGHTS OF FIGURES ON A LEVEL PLANE

_First Case._

This is but a repetition of the previous figure, excepting that we have
substituted these schoolgirls for the vertical lines. If we wish to make
some taller than the others, and some shorter, we can easily do so, as
must be evident (see Fig. 79).

Note that in this first case the scale is below the horizon, so that we
see over the heads of the figures, those nearest to us being the lowest
down. That is to say, we are looking on this scene from a slightly
raised platform.

_Second Case._

To draw figures at different distances when their heads are above the
horizon, or as they would appear to a person sitting on a low seat. The
height of the head