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158 FIRST PRINCIPLES OF points d and d' t they being the top and bottom ends of that line ; produce the axis a a of the pyramid, and on it draw the half-square d x d' as shown. Now the assumed section plane, represented by the line 3 b a in No. 1, cuts through the base of the pyramid at the distance d 3 from its vertical diagonal. To find the line on the half -base so made, through d in No. 1 draw a line indefinitely, parallel to the IL ; with d as centre and radius d a, draw an arc, cutting the line through d pro- duced, in the point x, and from the same centre, with a radius equal to d3, describe an arc cutting xd in 3'. As the line xd, No. 1, is a plan of the half -square d x d', No. 2, swung round on its diagonal d, d' as a hinge, until it is parallel to the IL, on this half-square or front half- base of the pyramid can now be found the exact length of the base of the isosceles triangle produced by the cutting plane, and from it the section itself, and the lines of intersection of the sides of the prism and pyramid. For the triangular section, draw a projector into the VP through point 3', in the line xd, No. 1, and it will cut the lines xd and xd' in No. 2, in points 3, 3' ; the distance between which is the length of the base of the triangular section. For its sides, cut by projectors drawn through points 3, 3', parallel to the IL, the lines d a, a d', in points 4, 4', and through these and the apex a draw faint lines as shown in No. 2 ; then the points 5, 5', where these faint lines or edges of the tri- angular section cut the vertical edge b b' of the prism, are two of the points sought. Join 1' in the line a a' to these points by straight lines, and they will be the lines of penetration on the left front face of the