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° 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.
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.
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.
GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED
As at first there may be a little difficulty in realizing the resemblance between geometrical and perspective figures, and also about certain expressions we make use of, such as horizontals, perpendiculars, parallels, &c., which look quite different in perspective, I will here make a note of them and also place side by side the two views of the same figures.
[Illustration: Fig. 51 A. The geometrical view.]
[Illustration: Fig. 51 B. The perspective view.]
[Illustration: Fig. 51 C. A geometrical square.]