WILSON AND LEWIS. — RELATIVITY. 503 



The cross-ratio formed by the four Hues, .r, /•, a, /3 is 



_ sin Z (^, r) sin Z U, a) 

 sin Z (r, a) sin Z (/3, .r)' 



where the angles are measured in EucHdean fashion. Hence 



^ ^ sin|^ +^ ^ 1 _^ t^^g ^ 1^ Janh^ ^ ^,^ 

 V A 1 — tan^ 1 — tanhc^ 



Or 



sinf;;-^ 



<l> = UogX. 



Hence the non-EucUdean angle is measured by one-half the log- 

 arithm of the cross-ratio of four rays. Although the Euclidean 

 point of view has been adopted for simplicity, the final result, depend- 

 ing as it does only on the cross-ratio, is projective ; it is therefore 

 independent of the particular assumptions that the rays a and /9 are 

 perpendicular and that the initial line bisects the angle between them. 



Consider next a ray /' such that in the Euclidean sense 



Z (a, r') = Z (r, a). 



(In the non-Euclidean sense r and / are perpendicular). In forming 

 the cross-ratio it is evident that X' = — X. Hence for the non-Eucli- 

 dean angle 0' between x and r' 



4>' = HogX' = Uog(— X) = + Hogi— 1). 

 Hence 



The angle (p' — 4>, that is, the angle between two lines perpendicular in 

 the non-Euclidean sense is therefore ± ^iri. This result also is projec- 

 tive and independent of our special assumptions. It is only natural 

 that the angle between two lines in different classes should appear as a 

 complex number, owing to the fact that it is impossible to rotate 

 one line into the other. 



In setting up a projective measure of angle by means of cross-ratios, 

 it is customary among mathematicians to define the angle as 



<t> = log X, 



2 V— 1 



