40) THEORY OF PROJECTION. 
THEOREM 25. When two ranges of points are related by a per- 
spective projection the cross-ratio of any four points of one range is 
equal to that of their four corresponding points in the other range. 
49. Non-perspective Projection. We now proceed to con- 
sider a more general method of projecting one range into 
another, which method will be shown to contain the perspec- 
tive method as a special case. 
Fig. 2. 
Take as before two lines / and 1’, Fig. 2, intersecting in O ; 
draw any conic, for simplicity an ellipse, touching both / and 
l’, We shall assume as known the fundamental property of 
a conic that from any point outside the conic two and only 
two tangents can be drawn to the conic. Let P be any point 
onl; from P draw the two tangents to the conic K. One of 
them is the line / and the other intersects the line /’ in some 
point as P’. We call P and P’ corresponding points of the 
two ranges on/ andl’. We readily see that from every point 
on / one and only one tangent other than / can be drawn to 
the conic K; and in like manner from any point on /’ one and 
only one other tangent can be drawn to K. This construction 
