Here, for instance, are two aspects of the same thing: the geometrical square A, which is facing us, and the perspective square B, which we suppose to lie flat on the table, or rather on the perspective plane. Line A·C· is the perspective of line AC. On the geometrical square we can make what measurements we please with the compasses, but on the perspective square B· the only line we can actually measure is the base line. In both figures this base line is the same length. Suppose we want to find the

perspective of point P (Fig. 146), we make use of the diagonal CA. From P in the geometrical square draw PO to meet the diagonal in O; through O draw perpendicular fe; transfer length fB, so found, to the base of the perspective square; from f draw fS to point of sight; where it cuts the diagonal in O, draw horizontal OP·, which gives us the point required. In the same way we can find the perspective of any number of points on any side of the square.

Fig. 146.
[ LXXIX]
Perspective of a Point Placed in any Position within the Square

Let the point P be the one we wish to put into perspective. We have but to repeat the process of the previous problem, making use of our measurements on the base, the diagonals, &c.

Fig. 147.

Indeed these figures are so plain and evident that further description of them is hardly necessary, so I will here give two drawings of triangles which explain themselves. To put a triangle into perspective we have but to find three points, such as fEP, Fig. 148 A, and then transfer these points to the perspective square 148 B, as there shown, and form the perspective triangle; but these figures explain themselves. Any other triangle or rectilineal

figure can be worked out in the same way, which is not only the simplest method, but it carries its mathematical proof with it.

Fig. 148 A. Fig. 148 B.