DESCRIPTIVE GEOMETRY. 49 
analogous to those already mentioned, and peculiar only in the construction 
of the perpendiculars, by means of which the vertical projection of those 
points is found, whose horizontal projection has already been ascertained. 
These perpendiculars, which in all the other constructions have appeared 
as straight lines, here present themselves, in the space of the dome, as curves. 
To determine this curvature, horizontal sections are passed through the 
vertical projection, at 1, 2, 3, &c., projected as straight lines here, but as 
semicircles in the horizontal view. Suppose in the ground-plan a ray to be 
passed from any point in the anterior edge of the dome, against the inner 
wall ; this will intersect the projections of ail the above mentioned horizon- 
tal projections. . It will then be easy to determine, from the preceding 
observations, the vertical projections of these intersection lines, which will 
be straight to the commencement of the dome, and will then curve accord- 
ing to the curvature of the dome. If these lines be intersected by a line, 
at an angle of 45°, from the vertical projection of the point from which the 
ray has been drawn in the horizontal projection, we shall obtain one point 
in the curve of shadow; the other points necessary fully to determine this 
curve, may be obtained in the same manner. 
Fig. 47 teaches the method of finding the shadow cast upon the inner 
wall of a dome-shaped, closed, half-round niche, where a part of the dome 
is cut away above, and the niche continued in a semi-cylinder. This is a 
combination of the cases treated in figs. 44 and 46, so that it would merely 
be necessary to construct two shadows, one after another, but for the fact 
that a part of the dome (that lying between b’ and c’), has been cut out. 
Here it is not the contour of the dome that casts the shadow, but the 
boundary of the section. In this manner a part of the shadow seen in fig. 
46 is cut away, as shown in fig. 47. 
It is necessary, before concluding these remarks on shades and shadows, 
to advert to the cases where the body is not attached to a wall, but stands 
at some distance from it. Although these cases present now no difficulty 
whatever, it may be advisable to give an example, as in fig. 48. Let a six- 
sided prism stand in front of a wall, as shown by the horizontal projection 
of fig. 48. The problem is, to ascertain the shadow cast by the pyramid, 
in horizontal projection upon the floor, and in vertical projection upon the 
wall behind. First, to find the point where the apex g casts its shadow. 
‘Draw rays from g and g’, in horizontal and vertical projection, and com- 
bining them in the usual way, find their intersection at g’, the vertical plane. 
The line g’g* is the projection of the shadow of the axis of the prism, and 
g’ is a point in this shadow. We obtain the shadow-casting point of the 
axis by drawing a line to that axis, from g’, at an angle of 45°. If a hori- 
zontal plane, d’b", be passed through g", this plane will be projected in plan 
as a small hexagon. Pass tangent rays to this hexagon ; they will determine 
the points d’ and b’ as the projections of the shadow-casting points of the 
pyramid, in horizontal projection. If then from d and b the lines dd’ and 
bb’ be drawn, these will determine the outline of the tapering shadow of 
the pyramid. The lines d’g* and b’g* will determine the outline of the 
ICONOGRAPHIC ENCYCLOPADIA.—VOL. I. 4 49 
