160. x z, x w, being drawn, will represent the lines in which the plane of the shadow intersects those of the parallel faces of the solid (94). If wo suppose planes parallel to that of the picture to pass through the various points of the cube, as a, these will intersect the two original planes In lines, as a a, a a', parallel to Y z, z w ; and a line, a a', through the point of the cube, parallel to the auxiliary vanishing line, will meet a a' in the point a',* which will be the oblique pro jection of the point a on the plane of the shadow. Therefore, by drawing lines through the points a, b, c, d, parallel to Y z, to cut a z in a, $, . . a, then lines parallel to w z, through a, /I, ... 3, will cut lines parallel to w c', drawn through a, b, c, d, in the oblique projections of those points on the plane of the shadow, and by referring e, f, g . .. to w x, in the same way, we obtain the oblique projections of the other angles of the cube.
161. Since the sides of the cube a b, cd, ef, &c., are parallels, their oblique projections will be parallels (59), consequently the images of these parallels a' 6', c' d', e f, &c., will have a common vanishing point P' in the vanishing line of the plane in which the oblique projections lie ; for the same reason, a'd', c', f &c., will have a common vanish ing point in v X. Now it is obvious that the vanishing points P' e are, by an extension of the principle, the oblique projections on the plane of the shadow of the vanishing points P, of the original sides of the cube ; consequently the former may be determined from the last-named vanishing points by simply drawing lines through them parallel to w c to cut Y x in 162. If k had been given as the image of the point in which a line through the luminary perpendicular to the plane Y z met that plane, the image of the luminary would be determined by making k`, drawn to the auxiliary vanishing point q, the image of the given perpendicular height of the luminary above the original plane. A line through * parallel to w e will meet c'k produced in 1, the oblique projection of the luminary on the plane Its oblique projection 1' ou the plane of the shadow may be either determined as those of a, b, c, d, &c , were, or by drawing a line, as al, at pleasure, to cut the vanishing line x P in some vanishing point; this vanishing point may be transferred to Y by a parallel to c w; than a line drawn through a, the oblique projection of a, to this transferred vanishing point, will cut 1 pro duces! in l', the oblique projection of the luminary on the plane of the shadow.
163. Draw lines through l' and through the oblique projection a, 6', e, d', e', intersecting each such line by the luminous ray a, 6, c, &c., in the shadows a, , e", &e., of the angles of the cube, and these points being joined, the figure thus produced will be the image of the shadow of the cube on the plane as proposed.
164. The oblique projecting lines a a, &c., were assumed parallel to the picture and its auxiliary plane, simply for facility of construction, or else the points a, b, c, e, f, &c., might, as well as the luminary, have been projected on the plane of the shadow by lines in any direction, provided these lines were parallels according to the above principle (158).
165. Draw lines through and P, rand Q, the vanishing poiuts of the aides of the cube ; then the as ceb" , d", c" , of the shadows of the parallel lines of the original solid will meet in a point in the correspouding line drawn through and the vanishing point of those originals. For the planes passing through and the parallels ab, cd, ef, &c., must intersect in a common line, passing through and parallel to those originals : this common intersection will there fore have the same vanishing point, as those originals ; the line r, represents that common intersection. Now the shadows a' ' h" , c" d', ef' are the intersections of the before-mentioned planes by another, namely, by the plane of the shadow, and these intersections, a" b", must meet each other in a point in r, the common intersection of the planes passing through and the original lines. The same reasoning applies to the other shadows of the corresponding parallel sides of the cube.