In fig. 83, you see how to put a square in perspective in another manner. The plan is not reversed as in former figures; PO' is the distance of the eye, which is made equal to O D, and being set up on the ground line м N, the effect is as if the horizontal line were brought to coincide with the ground line; oa being drawn parallel to A c, and a being brought up to the line o' D D, we obtain D', the vanishing point of A c. This construction gives the sides of the square; in order to get the angles, draw the perpendiculars from the corners of the square, to intersect MN; from these intersections, draw lines to the point o, and the perspective figure is complete.

This is a very serviceable method of finding vanishing points, where your object is to occupy little space with the necessary constructions. By comparing it with the figures in the preceding lessons on Perspective, the principle of the con. struction will be easily understood, and in practice it will be perhaps the most useful of all that we nave given.

VOL. III.

ground line; AB is then bisected in E, and the line EV is drawn to v, the vanishing point; this line divides the perspective square ABCD into halves. Drawing in each half the diagonals CE, BI, DE, AI, a line K o, drawn through the intersections of these diagonals, is parallel to the ground line, and is the perspective representation of a diameter of the original circle parallel to the ground line. The line I is the perspective representation of a diameter at right angles to Ko. Take AF, equal to one-fifth of AE, and join F and v; repeat this operation on the part B E. Then the intersections of the line F V with the diagonals DE and AI will give two points, and the intersections of the corresponding line next to B V, with the diagonals CE and BI will give two more points. These four points, with the four points E, I, Z, O, give the eight points, through which an ellipse being drawn, we have the correct perspective of a circle. You will be assisted in drawing the ellipse by first determining its centre; this is done by bisecting the line IE. The perspective centre of the circle and the centre of the ellipse are not the same point. The perspective centre of the circle is determined by the intersection of 1 x and x O.

In the examples of other objects drawn in perspective, which will follow in subsequent lessons, the perspective of plane figures will be more fully shown; but it is requisite, in completing this section, that we should now speak of lines which are not in the ground plane. Lines which are perpendicular to the ground plane, have no vanishing point, but are always perpendicular to that plane in their perspective representations, as shown in fig. 85.

Fig. 85.

In fig. 86,let B A be the height of a perpendicular line of which it is wished to obtain the perspective; let the point pin the plan, be the foot of this perpendicular, or the point where it intersects the ground plane then P' is the perspective of the point P, as obtained by the method already described; set up BA any where in the ground line, and draw lines from the points B and A, to a point o, taken any where in the horizontal line

54