THEORY OF PROJECTION. 39 
called perspective ranges, or are said to be in perspective 
position. 
THEOREM 24. Two ranges in perspective position have a one- 
to-one correspondence; the lines joining corresponding points meet 
in a point, the center of the perspective projection; the point of in- 
tersection of the two lines is a self-corresponding point on the two 
ranges. 
48. Invariance of Cross-ratios. The cross-ratio of the 
four points A, B, C, D is defined by the function 
= _ ACAD 
b= ABCD) ap 
The cross-ratio of the four corresponding points A’, B’, C’, D’is 
a / InN A'C’ : A'D! } 
k' = (A'B'C'D') = 3a : gp: We wish to show that these 
two cross-ratios are equal. 
The triangles APJ and A’PI’, Fig. 1, are similar; also the 
triangles CPJ and C’ PI’ are similar. 
eee ia leer and sO — el Ol 
: lets lel pers Jedi 
Bre Ae AIT and JC = cr: 
Subtracting, we get 
AC=JC—JA= =" (AV —-Cl) = 2 - AC. 
Al. CP A’. Cl 
In like manner we get 
JP. PY 
BOS ay ee, 
EA URS E iuoetal ty 
A ig) CD 
JP. PI LT 
BaD erga BL « 
Dividing, we get 
AC . AD A'C! | A'D! 
BOiat IBD BIC MD Ie ey) 
Hence the two cross-ratios (ABCD) and (A’B’C’D’) are 
equal. 
