nts of distance _DD"_. These lines will
intersect each other, and so form the squares of the pavement; to ensure
correctness, lines should also be drawn from these points 1, 2, 3, to
the point of sight _S_, and also horizontals parallel to the base, as
_ab_.

[Illustration: Fig. 71.]

XXIII

THE CUBE AT AN ANGLE OF 45 DEG.

Having drawn the square at an angle of 45 deg, as shown in the previous
figure, we find the length of one of its sides, _dh_, by drawing a line,
_SK_, through _h_, one of its extremities, till it cuts the base line at
_K_. Then, with the other extremity _d_ for centre and _dK_ for radius,
describe a quarter of a circle _Km_; the chord thereof _mK_ will be the
geometrical length of _dh_. At _d_ raise vertical _dC_ equal to _mK_,
which gives us the height of the cube, then raise verticals at _a_, _h_,
&c., their height being found by drawing _CD_ and _CD"_ to the two
points of distance, and so completing the figure.

[Illustration: Fig. 72.]

XXIV

PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45 DEG.

[Illustration: Fig. 73.]

[Illustration: Fig. 74.]

The square at 45 deg will be found of great use in drawing pavements,
roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out one
square it can be divided into four or more equal squares, and any figure
or tile drawn therein. Begin by making a geometrical or ground plan of
the required design, as at Figs. 73 and 74, where we have bricks placed
at right angles to each other in rows, a common arrangement in brick
floors, or tiles of an octagonal form as at Fig. 75.

[Illustration: Fig. 75.]

XXV

THE PERSPECTIVE VANISHING SCALE

The vanishing scale, which we shall find of infinite use in our
perspective, is founded on the facts explained in Rule 10. We there find
that all horizontals in the same plane, which are drawn to the same
point on the horizon, are perspectively parallel to each other, so that
if we measure a certain height or width on the picture plane, and then
from each extremity draw lines to any convenient point on the horizon,
then all the perpendiculars drawn between these lines will be
perspectively equal, however much they may appear to vary in length.

[Illustration: Fig. 76.]

Let us suppose that in this figure (76) _AB_ and _A'B'_ each represent
5 feet. Then in the first case all the verticals, as _e_, _f_, _g_, _h_,
drawn between _AO_ and _BO_ represent 5 feet, and in the s