f smaller squares. Nor need the figure be
necessarily a square; it is just as easy to make it an oblong, as _ABEF_
(Fig. 136); for although we begin with a square we can extend it in any
direction we please, as here shown.

[Illustration: Fig. 135.]

[Illustration: Fig. 136.]

LXXIII

OF PARALLELS AND DIAGONALS

[Illustration: Fig. 137 A.]

[Illustration: Fig. 137 B.]

[Illustration: Fig. 137 C.]

To find the centre of a square or other rectangular figure we have but
to draw its two diagonals, and their intersection will give us the
centre of the figure (see 137 A). We do the same with perspective
figures, as at B. In Fig. C is shown how a diagonal, drawn from one
angle of a square _B_ through the centre _O_ of the opposite side of the
square, will enable us to find a second square lying between the same
parallels, then a third, a fourth, and so on. At figure _K_ lying on the
ground, I have divided the farther side of the square _mn_ into 1/4,
1/3, 1/2. If I draw a diagonal from _G_ (at the base) through the half
of this line I cut off on _FS_ the lengths or sides of two squares;
if through the quarter I cut off the length of four squares on the
vanishing line _FS_, and so on. In Fig. 137 D is shown how easily any
number of objects at any equal distances apart, such as posts, trees,
columns, &c., can be drawn by means of diagonals between parallels,
guided by a central line _GS_.

[Illustration: Fig. 137 D.]

LXXIV

THE SQUARE, THE OBLONG, AND THEIR DIAGONALS

[Illustration: Fig. 138.]

[Illustration: Fig. 139.]

Having found the centre of a square or oblong, such as Figs. 138 and
139, if we draw a third line through that centre at a given angle and
then at each of its extremities draw perpendiculars _AB_, _DC_, we
divide that square or oblong into three parts, the two outer portions
being equal to each other, and the centre one either larger or smaller
as desired; as, for instance, in the triumphal arch we make the centre
portion larger than the two outer sides. When certain architectural
details and spaces are to be put into perspective, a scale such as that
in Fig. 123 will be found of great convenience; but if only a ready
division of the principal proportions is required, then these diagonals
will be found of the greatest use.

LXXV

SHOWING THE USE OF THE SQUARE AND DIAGONALS IN DRAWING DOORWAYS,
WINDOWS, AND OTHER ARCHITECTURAL FEATURES

This exampl