THEORY OF PROJECTION. 4] 
gives a one-to-one correspondence between the points of the 
ranges on / and /’; and thus we see that the conic K deter- 
mines a new kind of projection of the line / on 1’, and also of 
the line 1’ on 1. 
Let T and T’ be the points of contact of the conic with 1 
_and l’ respectively. From T only one tangent can be drawn 
to K and that is the line / itself; this cuts /’ in O: hence 
T on / corresponds to O onl’. In like manner 1’ is the only 
tangent that can be drawn to K from T’: hence T’ on 1’ cor- 
responds to O onl. The tangent to K parallel to / cuts 1’ in 
I’; hence I’ corresponds to the point at infinity onl. In like 
manner J on / corresponds to the point at infinity on 1’. 
Projection of this kind is called non-perspective in order to 
distinguish it from the kind when the lines joining corres- 
ponding points meet in a point. 
We proceed to show that perspective projection is only a 
special case of non-perspective projection. If the conic K 
touches one of the lines / or l’ at O, it must also touch the 
other at O since it touches both. In this case the conic de- 
generates into a limited segment of a line having one ex- 
tremity at O. Let P be any point in the plane; the segment 
OP, Fig. 1, may therefore be considered as a conic touching 
both/ andl’. The tangents to this conic form a pencil of 
lines meeting in P; and this gives us a perspective projection 
with P as the center of perspective. 
THEOREM 26. The non-perspective projection of one range upon 
another is completely determined by a conic K touching both 
bases. Perspective projection is a special case of non-perspective 
projection in which the determining conic reduces to a line-segment 
with one extremity at the intersection of the bases of the two ranges. 
50. Cross-ratio Unaltered by Non-perspective Projection. 
We shall now prove the important fact that the cross-ratio of 
four points of a range is unaltered by a non-perspective pro- 
jection. We shall assume the well-known theorem that any 
