LIGHT 581 opaque body all round. These lines form a cone. The points of contact form a line on the opaque body separating the illuminated from the non-illuminated portion of its surface. Similarly, when these lines aro produced to meet the screen, their points of intersection with it form a line which separates the illuminated from the non-illuminated parts of the screen. Ceo- This line is called the boundary of the geometrical metrical shadow. A common but beautiful instance of it is seen shadow. w } ien a ver y small gas-jet is burning in a ground-glass shade, near the wall of a room. In this case the cone, above mentioned, is usually a right cone with its axis vertical. Thus the boundary of the geometric shadow is a portion of a circle on the roof, but a portion of an hyperbola on the vertical wall. If the roof be not horizontal, we may obtain in this way any form of conic section. Interesting and useful hints in projection may be obtained by observ ing the shadows of bodies of various forms cast in this way by rays which virtually diverge from one point : e.y., how to place a plane quadrilateral of given form so that its geometric shadow may be a square ; how to place an elliptic disk, with a small hole in it, so that the shadow may be circular with a bright spot at its centre, &c. When there are more luminous points than one, we have only to draw separately the geometrical shadows due to each of the sources, and then superpose them. A new con sideration now comes in. There will be, in general, portions of all the separate geometrical shadows which overlap one another in some particular regions of the screen. In such regions we still have full shadow ; but around them there will be other regions, some illuminated by one of the sources alone, some by two, &c., until finally we come to the parts of the screen which are illuminated directly by all the sources. There will evidently be still a definite boundary of the parts wholly unillutninated, i.e., the true shadow or umbra, and also a definite boundary of the parts wholly illuminated. The region between these boundaries i.e., the partially illumined portion is called the penumbra. Fig. 1 shows these things very well. It represents the shadow of a circular disk cast by four equal luminous Pomim- 1 .ra. points arranged as the corners of a square, the disk being large enough to admit of a free overlapping of the separate shadows. The amount of want of illumination in each portion of the penumbra i.s roughly indicated by the shading. The separate shadows are circular, if the disk is parallel to the screen. If we suppose the number of sources to increase indefinitely, so as finally to give the appearance of a luminous surface as the source of light, it Sharp ness of shadows. is obvious that the degrees of darkness at different portions of the penumbra will also increase indefinitely; i.e., there will be a gradual increase of bright ness in the penumbra from total darkness at the edge next the geo- | metrical shadow to full illumination " at the outer edge. It is most in structive to contrast with the above figure that now given (fig. 2), in which the size of the disk is con siderably diminished everything else being unchanged. Here there is no true shadow only four equally bright portions of the penumbra, each illuminated by three of the sources. Thus we see at once why the shadows cast by the sun or moon are in general so much less sharp than those cast by the electric light (when it is not surrounded by a semi- opaque screen). For, practically, at moderate distances from the electric arc, it appears as a mere luminous point. But, if we place a body at a distance of a foot or two only from the arc, the shadow cast will have as much of penumbra as if the sun had been the source. The breadth of the penumbra when the source and screen are nearly equidistant from the opaque body is equal to the diameter of the luminous source. Simple as is the question from the point of view we have adopted, it may to some persons appear simpler to imagine themselves placed (as spectators) on the screen in different parts of the shadow or penumbra, and to consider what portions of the luminous source they would then be in a position to see. This is what happens to us when we observe an eclipse Eclipses of the sun. When the eclipse is total, there is a real geometrical shadow, very small compared with the penumbra (for the apparent diameters of the sun and moon are nearly equal, but their distances are as 370 : 1) ; when the eclipse is annular, the shadow is all penumbra. In a lunar eclipse, on the other hand, the earth is the shadow- casting body, and the moon is the screen, and we observe things according to our first point of view. Suppose, next, that the body which casts the shadow is a large one, such as a wall, with a hole in it. If we were to plug the hole, the whole screen would be in geometrical Light )assin S ture. Fig. 3. shadow. Hence the illumination of the screen by the light passing through the hole is precisely what would be cut off by a disk which fits the hole Fig. 3. which is the
