right angles to the picture plane
into any number of given measurements. Let _SA_ be the given line. From
_A_ measure off on the base line the divisions required, say five of
1 foot each; from each division draw diagonals to point of distance _D_,
and where these intersect the line _AC_ the corresponding divisions will
be found. Note that as lines _AB_ and _AC_ are two sides of the same
square they are necessarily equal, and so also are the divisions on _AC_
equal to those on _AB_.

[Illustration: Fig. 53.]

The line _AB_ being the base of the picture, it is at the same time a
perspective line and a geometrical one, so that we can use it as a scale
for measuring given lengths thereon, but should there not be enough room
on it to measure the required number we draw a second line, _DC_, which
we divide in the same proportion and proceed to divide _cf_. This
geometrical figure gives, as it were, a bird's-eye view or ground-plan
of the above.

[Illustration: Fig. 54.]

XV

HOW TO PLACE SQUARES IN GIVEN POSITIONS

Draw squares of given dimensions at given distances from the base line
to the right or left of the vertical line, which passes through the
point of sight.

[Illustration: Fig. 55.]

Let _ab_ (Fig. 55) represent the base line of the picture divided into a
certain number of feet; _HD_ the horizon, _VO_ the vertical. It is
required to draw a square 3 feet wide, 2 feet to the right of the
vertical, and 1 foot from the base.

First measure from _V_, 2 feet to _e_, which gives the distance from the
vertical. Second, from _e_ measure 3 feet to _b_, which gives the width
of the square; from _e_ and _b_ draw _eS_, _bS_, to point of sight. From
either _e_ or _b_ measure 1 foot to the left, to _f_ or _f'_. Draw _fD_
to point of distance, which intersects _eS_ at _P_, and gives the
required distance from base. Draw _Pg_ and _B_ parallel to the base, and
we have the required square.

Square _A_ to the left of the vertical is 2-1/2 feet wide, 1 foot from
the vertical and 2 feet from the base, and is worked out in the same
way.

_Note._--It is necessary to know how to work to scale, especially in
architectural drawing, where it is indispensable, but in working out our
propositions and figures it is not always desirable. A given length
indicated by a line is generally sufficient for our requirements. To
work out every problem to scale is not only tedious and mechanical, but
wastes time, and also takes