DESCRIPTIVE GEOMETRY. 47 
be acurve. The method of finding this curve is easily determinable from 
the preceding considerations. Take a certain number of points in the hori- 
zontal projection, obtain their shadows in this projection, and transfer them 
to the vertical projection. ‘Thus, the point c will cast its shadow to c’, and 
this must lie on the perpendicular line, c’c’, of the vertical projection ; 
transfer the point c to c’, and from this point draw the ray c’c* ; we thus obtain 
the projection of the point of shadow ; and after a sufficient number of points 
has been obtained, an indication of the curve of shadow produced by the 
lower edge of the plate. As the upper edge must cast a similar shadow, 
this is to be obtained in the same way. The point d* determines the extreme 
point of this curve, which is closed by a perpendicular representing the 
shadow of the vertical edge or line of shade of the plate. The shadow of 
the plate upon the prism must also be curved, as it falls from a curved upon 
a plane surface. First of all, it is necessary to find the point of shadow 
upon the edge. For this purpose, draw the line aa’ in horizontal projection ; 
we shall thus obtain the shadow-casting point, @’, in the vertical projection, 
and @ as the point of shadow. According to the method described in 
jig. 38, the points b°,c*, &c., are then obtained, and consequently the 
curve of shadow upon that side of the prism which lies in the light. 
Fig. 40 exhibits the half of a truncated cone, covered by a four-sided 
plate. The construction of the shade of the body, and the shadow of the cone 
and plate upon the wall, differs very little from what has been described in 
fig. 38; it is different, however, with respect to the shadow on the cone.. This 
shadow is analogous to that of the cylinder; as, however, the surface of the 
cone is not perpendicular, but deviates every moment from the perpendicular, 
its shadow must fall somewhat differently. To obtain this shadow, suppose 
several horizontal planes to be passed through the vertical projection of the 
cone, appearing in it as straight lines, and in the horizontal projection as 
semicircles; they are indicated. by the figures 1, 2, 3, &c. Suppose the 
rays necessary for producing the shadow to be drawn in the horizontal 
projection, they will intersect the cone, and as the sections are parallel to 
the axis of the cone, these sections will appear in the vertical projection as 
hyperbolas, or at least parts of such, and may be constructed according to 
fig. 30. These hyperbolas serve instead of the perpendiculars employed in 
jig. 38, and by means of them, and of the projections of rays as bb’, b°b*, the 
curve of the shadow may be very readily determined. 
Fig. 41 represents a half cylinder, covered by a semicircular plate. 
Here one plane casts a shadow upon another parallel to it; the shadow will 
therefore be parallel to the shadow-casting line, and to determine this 
shadow nothing more is necessary than to pass a ray through the corner 
where the shadow begins. From the point where this ray intersects the 
edge of the cylinder, draw a line parallel to the plate: this will be the line 
of shadow. | 
Innumerable cases might be adduced, but the general principles involved 
in all are nearly such as have been explained and illustrated in the preceding 
instances. 
AT 
