; and then
from these divisions obtain dotted lines crossing the picture from one
side to the other which must all meet at some distant point on the
horizon. These act as guiding lines, and are sufficient to give us the
direction of any vanishing lines going to the same point. For those that
go in the opposite direction we proceed in the same way, as from _b_ on
the right to _V'_ on the left. They are here put in faintly, so as not
to interfere with the drawing. In the sketch of Toledo (Fig. 164) the
same thing is shown by double lines on each side to separate the two
sets of lines, and to make the principle more evident.

[Illustration: Fig. 164. Toledo.]

LXXXVIII

THE CIRCLE

If we inscribe a circle in a square we find that it touches that square
at four points which are in the middle of each side, as at _a b c d_. It
will also intersect the two diagonals at the four points _o_ (Fig. 165).
If, then, we put this square and its diagonals, &c., into perspective we
shall have eight guiding points through which to trace the required
circle, as shown in Fig. 166, which has the same base as Fig. 165.

[Illustration: Fig. 165.]

[Illustration: Fig. 166.]

LXXXIX

THE CIRCLE IN PERSPECTIVE A TRUE ELLIPSE

Although the circle drawn through certain points must be a freehand
drawing, which requires a little practice to make it true, it is
sufficient for ordinary purposes and on a small scale, but to be
mathematically true it must be an ellipse. We will first draw an ellipse
(Fig. 167). Let _ee_ be its long, or transverse, diameter, and _db_ its
short or conjugate diameter. Now take half of the long diameter _eE_,
and from point _d_ with _cE_ for radius mark on _ee_ the two points
_ff_, which are the foci of the ellipse. At each focus fix a pin, then
make a loop of fine string that does not stretch and of such a length
that when drawn out the double thread will reach from _f_ to _e_. Now
place this double thread round the two pins at the foci _ff'_ and
distend it with the pencil point until it forms triangle _fdf'_, then
push the pencil along and right round the two foci, which being guided
by the thread will draw the curve, which is a true ellipse, and will
pass through the eight points indicated in our first figure. This will
be a sufficient proof that the circle in perspective and the ellipse are
identical curves. We must also remember that the ellipse is an oblique
projection of a circle, or