Q R, Q R will be the vanishing-line of the plane of shade, and let it cut v L', the vanishing-line of the horizon, in r ; and L' u, the vanishing-line of the vertical planes, Br c L and D E /%1 N, of the walls, in r ; v u, the vanishing-line of the gables, in ; s a', the vanishing-line of the main roof, in z; and T the vanishing-line of the penthouse, in z' : all which does but prepare for drawing the shadow of the pole, w x, upon the horizontal plane and upon the building. Now pro duce A B to meet w x in x, then x will he the point where the pole rests upon the ground, or horizontal plane: draw x Y, cutting n E in a; draw u a, cutting n NI ill b ; draw b cut ting m o in c; draw c z', cutting r o in d ; draw a d, cutting L at a; and draw L z, cutting the ridge, r II, at f; then x a b c d L f will be the whole shadow of the pole.

For, since the shadow first begins at the foot of the pole, or line, in the plane of the horizon, and since the intersec tion of the vanishing-line of a plane on which the shadow is to be thrown, with the intersection of the vanishing-line of the plane of shade, gives the vanishing-line of the shadow upon that plane ; v becomes the intersection of the vanish ing-line of the plane of shade with the vanishing-line of the horizon ; therefore y is the vanishing-point of the shadow of the line w x upon the plane of the horizon. The next plane on which the shadow is thrown is DENM; now a u is the vanishing-line of the plane D E N a, and u is the point where the vanishing-line of the plane of shade cuts a' u ; therefore u is the vanishing point of the shadow upon the plane D E N. The next plane on which the shadow is projected is the plane c D nr o: now v R is the vanishing-line of the plane c D a o, and it intersects the vanishing-line of the plane of shade in R; therefore R is the vanishing-point of the shadow upon the plane c D nr O. The next surface on which the shadow is pro jected is the plane, a N P o, of the roof of the penthouse: now z' is the intersection of the vanishing-line of the plane of shade with the vanishing-line T a' of the plane a N P o; therefore z' is the vanishing-point of the shadow on the plane N r o. The next surface on which the shadow is projected is the plane, B F G L, of the wall ; but u has already been shown to be the vanishing-point of the shadow. The plane of the roof is the last surface on which the shadow is pro jected : now s L' is its vanishing-line, and it meets the vanish ing-line of the plane of shade in z, therefore z is the vanish ing-point of the shadow upon the roof.

ln carrying the shadow of a line across several planes, it will not be surprising if some little inaccuracy takes place from the obliquity of intersections; it might be a great chance, whether, when the part of the shadow, d L, which fulls upon the plane B F 0 L, is drawn from the vanishing point, u, through the point d, it will meet the pole at L, as it ought to do. To remedy this, begin with the shadow, L d,

and proceed in the reverse order, until it meets the line w x at x, which it must in principle, and will not be liable to vary much in practice.

The points which direct the shadows upon the several planes, might also be found by the methods shown in the article PROJECTION.

The following observations will be useful in the practice of shadows.

When a straight line, that throws a shadow, is parallel to the picture, it is then represented parallel to the original. In this case it has no vanishing-point ; or, in other words, the vanishing-point of the line may be said to be at an infinite distance : and, therefore, instead of the vanishing-point of the line being joined to the vanishing-point of the sun's rays, draw a straight line from the vanishing-point of those rays parallel to the projection of the line which throws the sha dow, and it will be the vanishing-line of the plane of shade ; therefore the intersection of the vanishing-line of the plane of shade with the vanishing-line of the plane on which the shadow is to be thrown, will give the vanishing-point of the shadow on that plane, after the same analogy as lines which are inclined to the picture. This ease is similar to that of the sun's rays being parallel to the picture : for here, also, the vanishing-point of the rays is at an infinite distance ; but as the plane of shade will still have a vanishing-line, this line will be found by drawing a straight line through the vanishing-point of the line that throws the shadow parallel to the sun's rays, as shown in a former example.

Of shadows projecled from a given point ; as by the light of a candle or lamp.

It is evident, if the representation of the luminous point be given, with its scat upon any plane together with the representation of any point in space, and its representation upon t hat plane, the shadow of the point will be found by drawing a straight line from the luminous point through the point in space, and by drawing another straight line from the seat of the luminous point through the scat of the point in space ; and the intersection of the two lines thus drawn will represent the shadow of the point upon the plane. But when the relation of several planes represented in a picture, the representation of the light with its seat, and the repre sentation of a point in space with its seat, are given, to pro ject the shadow of the point on the other planes, other con siderations become necessary.

Figure 2.—For this purpose, let Apcn be the inside of a room, consisting internally of the vertical planes A E