opposite ends f b towards each other at the point d.  And from the two lines you will have drawn the straight line c d which cuts the middle of the intersection of these two lines at the point n without any intersection.  For, you imagine these two lines as having breadth, it is evident that by this motion the first will entirely cover the other—­being equal with it—­without any intersection, in the position c d.  And this is sufficient to prove our proposition.

81.

HOW THE INNUMERABLE RAYS FROM INNUMERABLE IMAGES CAN CONVERGE TO A
POINT.

Just as all lines can meet at a point without interfering with each other—­being without breadth or thickness—­in the same way all the images of surfaces can meet there; and as each given point faces the object opposite to it and each object faces an opposite point, the converging rays of the image can pass through the point and diverge again beyond it to reproduce and re-magnify the real size of that image.  But their impressions will appear reversed—­as is shown in the first, above; where it is said that every image intersects as it enters the narrow openings made in a very thin substance.

Read the marginal text on the other side.

In proportion as the opening is smaller than the shaded body, so much less will the images transmitted through this opening intersect each other.  The sides of images which pass through openings into a dark room intersect at a point which is nearer to the opening in proportion as the opening is narrower.  To prove this let a b be an object in light and shade which sends not its shadow but the image of its darkened form through the opening d e which is as wide as this shaded body; and its sides a b, being straight lines (as has been proved) must intersect between the shaded object and the opening; but nearer to the opening in proportion as it is smaller than the object in shade.  As is shown, on your right hand and your left hand, in the two diagrams a b c n m o where, the right opening d e, being equal in width to the shaded object a b, the intersection of the sides of the said shaded object occurs half way between the opening and the shaded object at the point c.  But this cannot happen in the left hand figure, the opening o being much smaller than the shaded object n m.

It is impossible that the images of objects should be seen between the objects and the openings through which the images of these bodies are admitted; and this is plain, because where the atmosphere is illuminated these images are not formed visibly.

When the images are made double by mutually crossing each other they are invariably doubly as dark in tone.  To prove this let d e h be such a doubling which although it is only seen within the space between the bodies in b and i this will not hinder its being seen from f g or from f m; being composed of the images a b i k which run together in d e h.