44 + MATHEMATICS. 
the reflection of light from other bodies about or beyond this. The body 
will therefore become again somewhat lighter, after its darkest shade. Simi- 
lar conditions must be presented in respect to the illumination of all other 
surfaces, and it will not be difficult to determine the tone of light of each 
surface, knowing the direction of the incident rays. It is evident that by 
taking this direction at other angles, any other illumination may be con- 
structed. It must not be forgotten, however, that the angles of incidence 
upon the two planes of projection must always be complements of each 
other, so that light, incident at an angle of 40° upon the plane of elevation, 
will strike the horizontal plane at an angle of 50°, &c. 
Shadows, as already mentioned, are produced when one part of a body 
projects beyond another, or one body is interposed between the source of 
light and another body. These shadows, like the bodies themselves, may be 
constructed when the measurements of all the parts are known, 7. e. the body 
itself, with all its accompaniments, may be constructed in plan and eleva- 
tion. The rules for the reception of light must always be the same as in 
the case of shades, and the same angle of incidence of the light must be 
employed for both the shades and the shadows. 
When,a shadow is to be determined, it is, first of all, necessary to deter- 
mine the lines which cast the shadow ; and after these have been found, to 
seek the projections of these lines of shadow. After this, those parts of the 
entire surface from which light is intercepted may be readily determined. 
a. Shades and Shadows upon Plane Surfaces and Curved Surfaces of 
Elevation. 
Let us suppose figs. 36-44, pil. 4, to represent the vertical, and beneath 
this the horizontal projection of a plane wall, to whose anterior face six 
bodies, of different forms, are attached, and covered above with partly cir- 
cular, partly rectilineal plates: let now the problem be, to determine the 
shadows cast by the plates upon the solids, and by both plates and solids 
upon the wall. 
Fig. 36 is a feur-sided prism, A, covered with a somewhat projecting 
plate, B, likewise four-sided. To find the various shadows, it becomes ne- 
cessary first to find the line of shadow. The direction of the rays of light 
is here, and in the following cases, assumed at 45°, for the horizontal and 
vertical projection. Draw lines in the horizontal projection, in the direction 
of the rays, to the bodies A’ and B’; then the first rays passing by the body 
will go through the points c and d. These are the projections of the two 
right edges of the bodies A and B, in vertical projection; the latter will 
therefore be two lines of shadow. If, moreover, rays be drawn at an angle 
of 45° in the vertical projection, one will pass by at a’ and others at c’, d, 
d°: the line a’d? will consequently cast a shadow. As c’ is the projection 
of the upper right hand edge of the prism, d’ that of the lower right hand 
edge of the plate, and d’ of the upper right hand edge of the same, the four 
lines just mentioned will cast shadows. Of all these edges, the line @’d’ 
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