GEOMETRIC CONSTRUCTION. 75 
for both sets of lines are cut by / in the same set of four 
points. 
THEOREM 7. Perspective projection of one plane upon another 
establishes a one-to-one correspondence between the points and also 
between the lines of the two planes. The line of intersection of the 
two planes is a self-corresponding line and every point on it a self- 
corresponding point; the cross-ratio of any four collinear points or 
concurrent lines in one plane is equal to the cross-ratio of their four 
corresponding points or lines in the other plane. 
92. Non-perspective Projection. In Chapter I, $6, we 
found two methods of projecting a range of points on a line 
into a range on another line, viz.: perspective projection and 
non-perspective projection. Two ranges rendered projective 
to one another by either of these methods were found to have 
precisely the same properties, differing only in the fact that 
two perspective ranges have a self-corresponding point, while 
two non-perspective ranges do not. Projectivity was found 
to be one and the same property in each case, the difference 
being only a result of position. Perspective projection was 
shown to be only a special case of non-perspective projection. 
We have thus far in the present section defined perspective 
projection of one plane upon another and have investigated 
the properties of two fields of points and lines connected by a 
perspective projection. We wish now to consider something 
analogous to the non-perspective projection of Chapter I, arti- 
cle 49. Let us take two fields of points a and ’ related by a 
perspective projection and, while the correspondence remains 
unaltered, shift one of the planes, say x, into a new position so 
that 2 and x’ will intersect in a new line l’. The lines joining 
corresponding points will no longer meet in a point P and 
there will be no self-corresponding points of the two fields. 
We wish to find a method of constructing the point A’ in 7’ 
corresponding to a given point A in x, and the line q@’ in 7’ 
corresponding to a given line a in ~. Such a method, if 
found, might well be called a non-perspective projection of x 
ona’. Judging by analogy we should expect to find perspec- 
