IMAGES AND SHADOWS. 



671 



body is completely screened from the rays. The cone thus constructed 

 is called the shadow-cone, and its intersection with any surface behind 

 the opaque body defines the shadow cast upon that surface. In the 

 case which we have been supposing — that of a luminous point — the 

 shadow-cone and the shadow itself will be sharply defined." 



Actual sources of light are not, however, mere points ; they have 

 finite and variable dimensions, and this complicates the effect in the 

 geometry of shadows. These effects are well illustrated by the dia- 

 gram (Fig. 7). Let the central sphere represent a source of light 

 placed between two opaque spheres, one larger and the other smaller 

 than itself. We shall then have two kinds of shadow, the total 

 shadow or umbra, and the partial shadow or penumbra. If the 

 opaque sphere, seen at the right in the diagram, be smaller than the 

 luminous body, the umbra terminates at a finite distance depending 

 upon the relative magnitudes and the proximity of the two bodies. 

 If the opaque body is larger than the source of illumination, as illus- 

 trated in the left portion of the diagram, the cone of the umbra is 

 projected to an infinite distance. The penumbral cones represented 



Fig. 7. 



Umbra and Penumbra. 



by a lighter shading, it is seen, are wider than the cones of total 

 shadow, and include them. It will be noticed that all points lying 

 within the penumbral cones are excluded from the view of some por- 

 tion of the luminous body, and are thus partially shaded by the 

 opaque bodies. Points that are very near the outer boundaries are 

 very slightly darkened, if near the total shadow they are almost com- 

 pletely shaded. The penumbra is, therefore, not a uniform shadow, 

 but gradually fades into darkness from its outer limit to the total ob- 

 scuration of the umbra. It follows from this that the shadow of an 

 opaque body falling upon a screen will not have a sharply-defined 

 edge, but will show a transition from total shadow to the complete 

 absence of shadow. To have the shadow clean and sharp at its edge, 



