DESCRIPTIVE GEOMETRY. 45 
alone casts a shadow upon A; all the rest, and even a part of a’d’, cast 
shadows upon the wall only. 
As the direction of every line is determined by several points lying in it, 
to determine the boundary of shadow in both projections, we need two ih 
for a straight line, and a greater number for curves. 
The siete of the line a’d* falls upon the body A, and it becomes neces- 
sary to obtain the point 0°, from which the shadow must run parallel to the 
shadow-casting edge, the two surfaces of the body and the plate being 
parallel. in any case, the point which casts its shadow upon 0° must lie in 
the line a’d’, whose horizontal projection is ad, and this, in a direction of 
45°. If, therefore, in the horizontal projection, a line be drawn from the left 
corner of A’ to ad, at an angle of 45°, it will determine 6b as a shadow-cast- 
ing point. This point is then transferred to the vertical projection, by 
means of the perpendicular bb’, and b’ is then exhibited as the shadow-casting 
point in the latter. Drawing a line from 0’, at an angle of 45°, its intersec- 
tion with the left edge, b*, of the prism determines the shadow of the point 
b’, and the direction of the line of shadow. As the line a’d’ determines the 
boundary of the surface which prevents the incidence of light upon the body, 
A, that part of A above the line passing through 0”, lies in the shadowed 
portion. 
With respect to the shadows cast by the prism and plate upon the wall, 
the edge a casts a shadow, which necessarily begins in the point where this 
edge touches the wall. Drawing a line from a’, at an angle of 45°, this line, 
a'a’, will be the line of shadow. The side of the plate behind the line d’d' 
will likewise cast a shadow, whose direction is determined by the two fines 
drawn at an angle of 45°, in the vertical projection. The length of this 
shadow is obtained by considering that the point d, the horizontal projection 
of the line d’d*, and consequently the edge which is projected through it, 
casts its shadow to d’. The edge a’d’ also casts a shadow upon the wall, 
which must run parallel to the edge, the edge and the wall being themselves 
parallel. The point d* is the point of shadow for d’; if then, through da‘, a 
parallel to a’d? be drawn, this will be the line of shadow of the plate on the 
wall behind it. Finally, the side of the prism lying behind c casts also a 
shadow upon the wall, whose limits will be the shadow of the edge which 
is projected through c. If then the tangential ray cc’ be drawn, and the 
point c’ be projected on the vertical plane, the perpendicular through this 
point will determine the line of shadow, whose upper point still remains to 
be determined. The point c’ in the vertical projection answers to c in the 
horizontal; then, if we draw a line, at an angle of 45°, through c’, it will in- 
tersect the above mentioned perpendicular in e, and this point, e, will be 
the shadow of c, or, what is the same, of c?, and will limit the shadow of the 
edge of the prism. 
Pl. 4, fig. 37, exhibits the half of a hexagonal prism, covered by a four- 
sided plate, under the same conditions as in the preceding case. The shade 
of the body is found according to the principles already laid down. The 
surface receives the strongest light to the left, the rays here falling perpen- 
dicularly ; the light, however, fades somewhat towards the extreme left. 
45 
