46 MATHEMATICS. 
The anterior surface receives the light more obliquely, and thus appears 
somewhat darker and uniformly illuminated, the surface being parallel to the 
plane of projection. The light passes entirely by the right surface, which 
consequently appears entirely dark, brightening a little, however, towards 
the wall, where it receives a certain amount of reflected light. The 
shadows on the wall are constructed as in fig. 36 ; those cast by the plate on 
the body, are obtained as follows: that for the anterior surface is obtained, 
as in jig. 36, by the lines bb’, b’b*, as this surface is parallel to the surface 
of representation, and to the anterior surface of the incumbent plate. The 
shadow cast by the corner a of the plate, will be determined by a ray of 
light passed through the point. It will fall upon the left lateral surface at a’, 
which will therefore be the projection of this shadow. To obtain this point 
in vertical projection, draw the perpendicular a’a’*, and intersect this by a 
ray through a, a’ will be the point desired. If this be connected with the 
extremity of the shadow or the anterior surface, the broken line through 
a*b’, gives the shadow of the anterior face of the plate. Since @ is the pro- 
jection of the horizontal lateral edge of the plate, it follows that every ray 
through that edge must be parallel to aa’; the direction of the shadow, 
therefore, from a’ towards the left, will be regulated by that of the ray. 
Fig. 38 represents a half cylinder, covered by a four-cornered plate. The 
shade of the body is obtained by drawing a line from the left-hand side to 
the centre, at an angle of 45°. Where this line cuts the convex surface of 
the cylinder, the light will be greatest, the rays falling here in a plane 
perpendicular to the surface. A second ray, tangent to the surface, deter- 
mines the line of deepest shade, up to which the light decreases more and 
more, and beyond which reflected light comes into play. The shadows of 
the body and the plate on the wall are obtained as before. The shadow 
cast by the plate on the body must be a curve, the body itself being curved. 
To obtain this shadow, find first the shadow of the point 6, the corner of the 
plate, which is done by means of the lines bd’, b°b°. As it is a curve that 
we are seeking, it will be necessary to obtain a number of points in it, so 
_as to determine its direction.. The points casting shadows lie, however, in 
the line be. .One of the points of shadow is given by the line cc’, c’c*, and 
as many more can be obtained as is necessary for the required degree 
of accuracy. The shadow naturally ceases where the ray, dd’, is tan- 
gent to the cylinder: the shadow then passes into the shade of the body. 
To find the direction of the shadow from 0° to the left, seek first the shadow 
of the point a. This falls at a’, and the vertical projection of this point 
must lie on the line a*. But b? is the projection of the left side of the plate ; 
the point @ lies, therefore, behind 0’, and if a ray be passed through 6? 
(actually through d’), it will determine the point a’. This lies in the line 
b°b®, and all the other points of the shadows cast by the lateral edge will fall 
in the direction 070’. ) 
Pl. 4, fig. 39, represents the half of a hexagonal prism, covered by a 
semicircular plate. The shade of the body has been already constructed in 
fig. 37, as well as the shadow of the body. It now remains to determine the 
shadow of the plate on the wall. As the plate is circular, its shadow must 
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