then
from these divisions draw lines to the measuring point, these lines
will intersect the vanishing line _AbV_ in the lengths and proportions
required. To find a measuring point for the lines that go to the other
vanishing point, we proceed in the same way. Of course great accuracy
is necessary.

Note that the dotted lines 1,1, 2,2, &c., are parallel in the
perspective, as in the geometrical figure. In the former the lines are
drawn to the same point _m_ on the horizon.

LVIII

HOW TO DIVIDE ANY GIVEN STRAIGHT LINE INTO EQUAL OR PROPORTIONATE PARTS

[Illustration: Fig. 117.]

Let _AB_ (Fig. 117) be the given straight line that we wish to divide
into five equal parts. Draw _AC_ at any convenient angle, and measure
off five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From
5C draw line to 5B. Now from each division on _AC_ draw lines 4,4, 3,3,
&c., parallel to 5,5. Then _AB_ will be divided into the required number
of equal parts.

LIX

HOW TO DIVIDE A DIAGONAL VANISHING LINE INTO ANY NUMBER
OF EQUAL OR PROPORTIONAL PARTS

In a previous figure (Fig. 116) we have shown how to find a measuring
point when the exact measure of a vanishing line is required, but if it
suffices merely to divide a line into a given number of equal parts,
then the following simple method can be adopted.

We wish to divide _ab_ into five equal parts. From _a_, measure off on
the ground line the five equal spaces required. From 5, the point to
which these measures extend (as they are taken at random), draw a line
through _b_ till it cuts the horizon at _O_. Then proceed to draw lines
from each division on the base to point _O_, and they will intersect and
divide _ab_ into the required number of equal parts.

[Illustration: Fig. 118.]

[Illustration: Fig. 119.]

The same method applies to a given line to be divided into various
proportions, as shown in this lower figure.

[Illustration: Fig. 120.]

[Illustration: Fig. 121.]

LX

FURTHER USE OF THE MEASURING POINT O

One square in oblique or angular perspective being given, draw any
number of other squares equal to it by means of this point _O_ and the
diagonals.

Let _ABCD_ (Fig. 120) be the given square; produce its sides _AB_, _DC_
till they meet at point _V_. From _D_ measure off on base any number of
equal spaces of any convenient length, as 1, 2, 3, &c.; from 1, through
corner of square _C_, draw a line to meet the horizo