e is from Serlio's _Architecture_ (1663), showing what
excellent proportion can be obtained by the square and diagonals. The
width of the door is one-third of the base of square, the height
two-thirds. As a further illustration we have drawn the same figure in
perspective.

[Illustration: Fig. 140.]

[Illustration: Fig. 141.]

LXXVI

HOW TO MEASURE DEPTHS BY DIAGONALS

If we take any length on the base of a square, say from _A_ to _g_, and
from _g_ raise a perpendicular till it cuts the diagonal _AB_ in _O_,
then from _O_ draw horizontal _Og'_, we form a square AgOg', and thus
measure on one side of the square the distance or depth _Ag'_. So can we
measure any other length, such as _fg_, in like manner.

[Illustration: Fig. 142.]

[Illustration: Fig. 143.]

To do this in perspective we pursue precisely the same method, as shown
in this figure (143).

To measure a length _Ag_ on the side of square _AC_, we draw a line from
_g_ to the point of sight _S_, and where it crosses diagonal _AB_ at _O_
we draw horizontal _Og_, and thus find the required depth _Ag_ in the
picture.

LXXVII

HOW TO MEASURE DISTANCES BY THE SQUARE AND DIAGONAL

It may sometimes be convenient to have a ready method by which to
measure the width and length of objects standing against the wall of a
gallery, without referring to distance-points, &c.

[Illustration: Fig. 144.]

In Fig. 144 the floor is divided into two large squares with their
diagonals. Suppose we wish to draw a fireplace or a piece of furniture
_K_, we measure its base _ef_ on _AB_, as far from _B_ as we wish it to
be in the picture; draw _eo_ and _fo_ to point of sight, and proceed as
in the previous figure by drawing parallels from _Oo_, &c.

Let it be observed that the great advantage of this method is, that we
can use it to measure such distant objects as _XY_ just as easily as
those near to us.

There is, however, a still further advantage arising from it, and that
is that it introduces us to a new and simpler method of perspective, to
which I have already referred, and it will, I hope, be found of infinite
use to the artist.

_Note._--As we have founded many of these figures on a given square in
angular perspective, it is as well to have a ready and certain means of
drawing that square without the elaborate setting out of a geometrical
plan, as in the first method, or the more cumbersome and extended system
of the second method. I