ance
and at a given angle in space. First of all we transfer it to the side
of the cube, where it is seen in perspective, whilst at its side is
another perspective square lying flat, on which we have to stand our
figure. By means of the diagonal of this flat square, horizontals from
figure on side of cube, and lines drawn from point of sight (as already
explained), we obtain the direction of base line _AB_, and also by means
of lines _aa'_ and _bb'_ we obtain the two points in space _a'b'_. Join
_Aa'_, _a'b'_ and _Bb'_, and we have the projection required, and which
may be said to possess the third dimension.

[Illustration: Fig. 157.]

In this other case (Fig. 158) we have a wedge-shaped figure standing on
a triangle placed on the ground, as in the previous figure, its three
corners being the same height. In the vertical geometrical square we
have a ground-plan of the figure, from which we draw lines to diagonal
and to base, and notify by numerals 1, 3, 2, 1, 3; these we transfer to
base of the horizontal perspective square, and then construct shaded
triangle 1, 2, 3, and raise to the height required as shown at
1', 2', 3'. Although we may not want to make use of these special
figures, they show us how we could work out almost any form or object
suspended in space.

[Illustration: Fig. 158.]

LXXXIV

THE SQUARE AND DIAGONAL APPLIED TO CUBES AND SOLIDS DRAWN THEREIN

[Illustration: Fig. 159.]

As we have made use of the square and diagonal to draw figures at
various angles so can we make use of cubes either in parallel or angular
perspective to draw other solid figures within them, as shown in these
drawings, for this is simply an amplification of that method. Indeed we
might invent many more such things. But subjects for perspective
treatment will constantly present themselves to the artist or
draughtsman in the course of his experience, and while I endeavour to
show him how to grapple with any new difficulty or subject that may
arise, it is impossible to set down all of them in this book.

[Illustration: Fig. 160.]

LXXXV

TO DRAW AN OBLIQUE SQUARE IN ANOTHER OBLIQUE SQUARE
WITHOUT USING VANISHING POINTS

It is not often that both vanishing points are inaccessible, still it is
well to know how to proceed when this is the case. We first draw the
square _ABCD_ inside the parallel square, as in previous figures. To
draw the smaller square _K_ we simply draw a smaller parallel square