DESCRIPTIVE GEOMETRY. 43 
parabola, in its development, appears divided into two symmetrical parts, 
owing to its falling on the line in which the convex surface of the cone is 
supposed to be divided. By these various constructions, we obtain 
the symmetrical figure, H*E*F’B*C*A*D*E*H’G’, which forms that part 
of the convex surface of the cone bounded by the three conic sections. 
B. PROJECTION OF SHADOWS. 
By projection of shadows is to be understood the method of representing 
bodies as they appear to an observer under illumination from. a certain 
direction. It is evident that both the direction and the nature of the illumi- 
nation (whether from a point or a surface) must greatly influence this mode 
of representing objects. If the illumination be supposed to proceed from a 
single point, it involves a department of the subject which will not be treated 
of in this place, as it more properly belongs to another part of our work. 
We here treat only of that description of shadows produced by an infinitely 
creat luminous surface, considered as the source of light. In the former 
case, the rays of light form a cone, and diverge the more the nearer the 
source of light (the apex of the cone) lies to the object illuminated. 
In the latter case, and the one to be now treated of, the rays are all parallel: 
to each other. In what follows, we suppose the plane of illumination to be so 
situated with respect to the surface of representation; as that all the rays 
come in the direction of the diagonal of a cube, 1. e. incident at an angle of 
45° on both the horizontal and vertical plane. The rays of light are 
supposed to come over the left shoulder, and to fall upon the paper and the 
object to be represented. 
The general head of shadows embraces two subdivisions, viz. shadows 
proper and shades. The shade of a body is that part of its own surface 
from which light has been intercepted by some other part of the body itself. 
The shadow of a body is that part of indefinite space from which light is 
excluded by the body. The shadow on a body is that portion of its surface 
from which light has been intercepted by some other body, placed between 
it and the source of light. 
With respect to the shades of bodies, it is evident that the rays a light. 
can exert their greatest powers of illumination only when they fall at aon 
angles upon a surface, and that the illumination will be less, the more oblique 
the angle of incidence. The deepest shade must be produced where the 
rays are only tangent to the body, as they there no longer illuminate the 
body. Taking for illustration a half cylinder, as exhibited in horizontal pro- 
jection in pl. 4, fig. 38, the line 0b’ will be perpendicular to the surface, 
and the illumination of the cylinder will consequently be greatest in this 
part. This is the point of highest ight. The ray of light dd’ will only be 
tangent to the circumference of the cylinder: here then will be the darkest 
shade. Between b’ and d the rays of light will fall more and more obliquely, ' 
and the illumination become less and less. The same must bé the case from 
b’ towards a. Beyond d the body would be. entirely dark, were it not for- 
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