ance-point does not come inside the
picture, so we take a fourth of the base and a fourth of the distance
and draw a line from 1/4 base to 1/4 distance. We shall find that it
passes precisely through the same point _o_ as the other lines _aD_, &c.
We are thus able to find the required point _o_ without going outside
the picture.

Of course we could in the same way take an 8th or even a 16th distance,
but the great use of this reduced distance, in addition to the above,
is that it enables us to measure any depth into the picture with the
greatest ease.

It will be seen in the next figure that without having to extend the
base, as is usually done, we can multiply that base to any amount by
making use of these reduced distances on the horizontal line. This is
quite a new method of proceeding, and it will be seen is mathematically
correct.

XXXIV

HOW TO DRAW A LONG PASSAGE OR CLOISTER BY MEANS OF THE REDUCED DISTANCE

[Illustration: Fig. 86.]

In Fig. 86 we have divided the base of the first square into four equal
parts, which may represent so many feet, so that A4 and _Bd_ being the
retreating sides of the square each represents 4 feet. But we found
point 1/4 D by drawing 3D from 1/4 base to 1/4 distance, and by
proceeding in the same way from each division, _A_, 1, 2, 3, we mark off
on _SB_ four spaces each equal to 4 feet, in all 16 feet, so that by
taking the whole base and the 1/4 distance we find point _O_, which is
distant four times the length of the base _AB_. We can multiply this
distance to any amount by drawing other diagonals to 8th distance, &c.
The same rule applies to this corridor (Fig. 87 and Fig. 88).

[Illustration: Fig. 87.]

[Illustration: Fig. 88.]

XXXV

HOW TO FORM A VANISHING SCALE THAT SHALL GIVE THE HEIGHT, DEPTH,
AND DISTANCE OF ANY OBJECT IN THE PICTURE

If we make our scale to vanish to the point of sight, as in Fig. 89, we
can make _SB_, the lower line thereof, a measuring line for distances.
Let us first of all divide the base _AB_ into eight parts, each part
representing 5 feet. From each division draw lines to 8th distance; by
their intersections with _SB_ we obtain measurements of 40, 80, 120,
160, &c., feet. Now divide the side of the picture _BE_ in the same
manner as the base, which gives us the height of 40 feet. From the side
_BE_ draw lines 5S, 15S, &c., to point of sight, and from each
division on the base line also draw lines 5S, 10S, 15S, &c.