48 MATHEMATICS. 
b. Shades and Shadows upon hollow, straight, and curved Surfaces. 
Fig. 42 exhibits a four-cornered niche. closed above. The point @ is 
the horizontal projection of the shadow-casting line, and must itself be the 
shadow-casting point. Passing a ray, aa’, through this point, it will deter- 
mine the situation of the point of shadow upon the back wall, whose 
projection in elevation may be determined by the lines a’a’, and a‘a? at a’. 
As, however, a is the projection of the entire shadow-casting edge, the limit 
of shadow for this edge must lie in the perpendicular a’a*; a parallel, 
therefore, to the corner of the niche, drawn through a‘, will determine the 
shadow of the cover. 
Fig. 43 represents a niche, forming the half of a hexagon, and covered 
rectilineally above. The ray of light, aa’, determines the extremity of the 
shadow in horizontal projection. The intersection of the perpendicular a’a’, 
with the ray through a’, determines its position, a’, in elevation. The part 
of the perpendicular below this point, a’, will be the line of shadow cast by 
the vertical edge of the niche. As the right side of the niche is oblique to 
the surface of representation, the shadow must run obliquely from ad’; now, 
as the cover coincides with the edge of the niche in c, the shadow must run 
in this direction ; d’c will therefore be the line of shadow on this oblique 
side. | 
Fig. 44 exhibits the half of a hollow cylinder, open above, and the problem 
is, to find’ the shadow cast by the edge of the cylinder upon its inner 
surface. Its boundary in vertical projection is obtained, in the first place, 
by passing a ray through the point a, the horizontal projection of the edge 
of the cylinder. The vertical projection of the point a’, where it meets the 
inner surface of the cylinder, is obtained at a’, by the intersections of the 
lines a’a@® and a’a*. A part of the upper edge of the cylinder also casts 
a shadow. This begins in the point c’, which is the vertical projection of 
the point c', where a ray is tangent to the surface of the cylinder. The 
point c’ is obtained by drawing a line from c to the surface of the cylinder, 
at an angle of 45°. The shadow runs from c’ to a’ in the curve. This 
curve is found by obtaining individual points, as before explained. 
Pl. 4, fig. 45, exhibits the shadow of a straight cover to a semi-cylindrical 
niche. The straight part of the shadow is obtained as in the preceding 
case; the figure itself shows the method of finding the shadow of the cover- 
ing. Thus for instance, the shadow of the point ) is obtained by means of 
the lines 0b’, b’, and 6°, and the shadow ends in the point c’, where the ante- 
rior edge, a’c’, of the covering, meets the wall of the cylinder. 
Fig. 46 explains the method of finding the shadow cast upon its inner 
surface by the edge of aniche, dome-shaped above. This shadow has a very 
peculiar outline, and can only be determined for the dome by a very exact 
projection of the rays, and the accompanying subsidiary lines. The limit 
of the straight part of the shadow is easily found to lie at a, according to 
fig. 45; the extremity of the compound shadow must necessarily lie at e, 
where a ray of light would be tangent to the dome. To obtain the curve 
between a and e the following method is to be employed, which is quite 
48 
