P_ to the point of distance _D_ and this line _PD_ will be at an angle
of 45 deg, or at the same angle as the diagonal of a square. See
definitions.

[Illustration: Fig. 45.]

XI

THE SQUARE

Draw a square in parallel perspective on a given length on the base
line. Let _ab_ be the given length. From its two extremities _a_ and _b_
draw _aS_ and _bS_ to the point of sight _S_. These two lines will be at
right angles to the base (see Fig. 43). From _a_ draw diagonal _aD_ to
point of distance _D_; this line will be 45 deg to base. At point _c_,
where it cuts _bS_, draw _dc_ parallel to _ab_ and _abcd_ is the square
required.

[Illustration: Fig. 46.]

We have here proceeded in much the same way as in drawing a geometrical
square (Fig. 47), by drawing two lines _AE_ and _BC_ at right angles to
a given line, _AB_, and from _A_, drawing the diagonal _AC_ at 45 deg
till it cuts _BC_ at _C_, and then through _C_ drawing _EC_ parallel
to _AB_. Let it be remarked that because the two perspective lines
(Fig. 48) _AS_ and _BS_ are at right angles to the base, they must
consequently be parallel to each other, and therefore are perspectively
equidistant, so that all lines parallel to _AB_ and lying between them,
such as _ad_, _cf_, &c., must be equal.

[Illustration: Fig. 47.]

So likewise all diagonals drawn to the point of distance, which are
contained between these parallels, such as _Ad_, _af_, &c., must be
equal. For all straight lines which meet at any point on the horizon are
perspectively parallel to each other, just as two geometrical parallels
crossing two others at any angle, as at Fig. 49. Note also (Fig. 48)
that all squares formed between the two vanishing lines _AS_, _BS_, and
by the aid of these diagonals, are also equal, and further, that any
number of squares such as are shown in this figure (Fig. 50), formed in
the same way and having equal bases, are also equal; and the nine
squares contained in the square _abcd_ being equal, they divide each
side of the larger square into three equal parts.

[Illustration: Fig. 48.]

[Illustration: Fig. 49.]

From this we learn how we can measure any number of given lengths,
either equal or unequal, on a vanishing or retreating line which is at
right angles to the base; and also how we can measure any width or
number of widths on a line such as _dc_, that is, parallel to the base
of the picture, however remote it may be from that base.

[Illustratio