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which AB, BC, are the front edges) are in sight, and that therefore only the two openings figured 1, 2; 3, 4 are seen. Now, as the sides of these openings are one-third the length of the sides of the cube, and as the four edges of these openings figured 1, 2, 3, 4 will be seen of their actual lengths, to show them in the position they occupy on the cube, set off in points y z from the lines ac, a'c' (Fig. 117) the front and back faces of the cube the distance (a 1 or b 3) that the edges of the square opening abed are from the edges of the cube. Through these points draw short lines in the upper and lower half of the figure parallel to ac a'c", and from points 1, 2, 3, 4 (Fig. 116), and parallel to aa or cc (Fig. 117) draw lines cutting those drawn through y z in 1' 1", 2' 2", 3' 3" 4' 4". The outer edges of the two openings seen from x will thus have been obtained. Two of their inner edges, one at e and another at f, are also seen, the first at e being that of the top side of the upper opening, and that at f the corresponding edge of the lower opening. Projecting these points over on Fig. 117, and drawing in the lines they represent, will give their elevation. To complete the view of 68 FIRST PRINCIPLES OF the cube required, we have yet to show the position of the two open- ings in its front and back faces, represented in Fig. 1 1 6 by the square abed. This is found by drawing in in dotted lines in Fig. 117, the projection of the two corners a and c of that square ; the other corners b and d being directly behind the edge of the cube at B, and coinciding with it, will not be seen in the required elevation. Figs. 118 and 119 are sectional side elevations of the same original object, the first being that of the cube cut by a plane passing through ts diagonal AC, and the second by a similar plane at SP parallel to