THEORY OF PROJECTION. 43 
(A,B,C,D,). Evidently R, and R, have a one-to-one corre- 
spondence and (A,B,C,D,) =(A,B,C,D,). Hence R, and R, 
are projective. 
Two projective ranges may be situated on the same line. 
Thus, for example, we might project perspectively a range 
A, B,C, Don | from two different points, Pand P,, and cut the 
two pencils thus formed by a second line /,. The ranges 
formed on 1, by the section of the two pencils are each pro- 
jective with the given range and hence projective with each 
other and situated on the same line. This case is of frequent 
occurrence. 
Two ranges of points are projective when and only when 
they satisfy these two conditions: first, they have a one-to- 
one correspondence ; second, the cross-ratios of any four cor- 
responding points are equal. 
52. Number of Points Determining Projectivity. We now 
proceed to the question of the number of points on each line 
which it is necessary to know in order to determine the pro- 
jection of one line upon the other. We shall assume the 
theorem that a conic is completely determined by any five 
independent conditions; in particular, that it is determined 
by any five tangents. The conic K which determines the pro- 
jection of | upon l’ must touch both / andl’; this gives two 
conditions for K. If now we select any three points on /, as 
A, B, C, and any three points on I’, as A’, B’, C’, to be respect- 
ively their corresponding points, the conic is completely 
determined; for the conic K must touch the lines AA’, BB’, 
CC’, 1, ’. When K is once found all pairs of corresponding 
points on the two lines / and l/ are determined by the tan- 
gents to K. Hence three points on / and their corresponding 
points on /’ are necessary and sufficient to determine a pro- 
jection of the non-perspective kind. 
In the case of perspective projection the projection is com- 
pletely determined as soon as the center of the projecting 
pencil isknown. This is determined by choosing two pairs of 
corresponding points on / and 1’ and drawing the lines joining 
