l be exactly half the area of the
larger one; for its side will be equal to half the diagonal of the
larger square, as can be seen by studying the following figures. In Fig.
170, for instance, the side of small square _K_ is half the diagonal of
large square _o_.

[Illustration: Fig. 170.]

[Illustration: Fig. 171.]

In Fig. 171, _CB_ represents half of diagonal _EB_ of the outer square
in which the circle is inscribed. By taking a fourth of the base _mB_
and drawing perpendicular _mh_ we cut _CB_ at _h_ in two equal parts,
_Ch_, _hB_. It will be seen that _hB_ is equal to _mn_, one-quarter of
the diagonal, so if we measure _mn_ on each side of _D_ we get _ff'_
equal to _CB_, or half the diagonal. By drawing _ff_, _f'f_ passing
through the diagonals we get the four points _o o o o_ through which to
draw the smaller square. Without referring to geometry we can see at a
glance by Fig. 172, where we have simply turned the square _o o o o_ on
its centre so that its angles touch the sides of the outer square, that
it is exactly half of square _ABEF_, since each quarter of it, such as
EoCo, is bisected by its diagonal _oo_.

[Illustration: Fig. 172.]

[Illustration: Fig. 173.]

XCII

HOW TO DRAW A CIRCLE IN ANGULAR PERSPECTIVE

Let _ABCD_ be the oblique square. Produce _VA_ till it cuts the base
line at _G_.

[Illustration: Fig. 174.]

Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure it
on each side of _E_, and so obtain _Ef_ and _Ef'_, and proceed to draw
_fV_, _EV_, _f'V_ and the diagonals, whose intersections with these
lines will give us the eight points through which to draw the circle. In
fact the process is the same as in parallel perspective, only instead of
making our divisions on the actual base _AD_ of the square, we make them
on _GD_, the base line.

To obtain the central line _hh_ passing through _O_, we can make use of
diagonals of the half squares; that is, if the other vanishing point is
inaccessible, as in this case.

XCIII

HOW TO DRAW A CIRCLE IN PERSPECTIVE MORE CORRECTLY,
BY USING SIXTEEN GUIDING POINTS

First draw square _ABCD_. From _O_, the middle of the base, draw
semicircle _AKB_, and divide it into eight equal parts. From each
division raise perpendiculars to the base, such as _2 O_, _3 O_, _5 O_,
&c., and from divisions _O_, _O_, _O_ draw lines to point of sight,
and where these lines cut the diagonals _AC_, _DB_, draw horizontals
p