42 THEORY OF PROJECTION. 
four tangents to a conic is cut by any fifth tangent in four 
points whose cross-ratio is constant.* Let AA’, BB’, CC’, 
PP’, Fig. 2, be any four tangents to the conic K. These four 
tangents are cut by the tangents / and 1’ in the four points 
A, B,C, P, and A’, B’, C’, P’, respectively, whose cross-ratios are 
equal by the above theorem. Thus the cross-ratio of any four 
points is unaltered by a non-perspective projection. 
THEOREM 27. If two ranges of points on intersecting lines are 
related by a non-perspective projection, the cross-ratio of any four 
points of one range is equal to that of the four corresponding points 
on the other range. 
51. Projective Ranges. We have thus far defined two dif- 
ferent methods of projecting one range of points into another, 
viz.: perspective and non-perspective projection. We observe 
that the properties of these two kinds of projection are very 
nearly the same. They both set up a one-to-one correspond- 
ence between the points of the two ranges, which is exception- 
less when we assume a single point at infinity in each range. 
They both leave the cross-ratio of four points invariant. 
They differ only in the fact that one (perspective projection ) 
gives a self-corresponding point in the two ranges, while the 
other does not. In fact, as we have shown, perspective pro- 
jection is only a special case of non-perspective projection. 
Two ranges of points on intersecting lines are called project- 
ive ranges, or said to be projectively related, or projective to 
one another when one of them is derived from the other 
either by a perspective or a non-perspective projection. 
Two ranges which are each projective with a third range are 
projective with one another. To show this let us suppose that 
a range FR on lis projective with R, on /, and also with R, on 
l,. Since R and R, are projective ranges they have a one-to- 
one correspondence and the cross-ratios of any four correspond- 
ing points are equal. Thus (ABCD) =(A,B,C,D,). Also R 
and FR, have a one-to-one correspondence and (ABCD) = 
*Salmon’s Conic Sections, p. 252. 
