WILSON' AM) LEWIS. — REL-VriVITY. 390 



But by III, Bl) is equal to DG. And writing AE = FB + BD, we 

 have 



ECX BD = CD X FB -{- CD X BD. 



Add to each side the product FB X EC. Then 



EC{BD + FB) = CDX BD-h FB(CD + EC). 

 Hence 



EC X AE — CDXBD = FB X AF. 



X on-Euclidean Rotation. 



9. The group of parallel geometries determined by Postulates 

 l°-9°, which, notwithstanding- its generality, gives rise, as we have 

 seen, to some interesting and important theorems, may be subdivided 

 by adding a set of postulates belonging to a second transformation 

 which by analogy may be called rotation. It is this set of postu- 

 lates which will differentiate our non-Euclidean geometry from the 

 Euclidean. 



The difference between our non-Euclidean rotation and the ordi- 

 nary kind is that in addition to a fixed point, two real lines through 

 the point remain unchanged. We may postulate for rotation: 



10°. Any one point and only that one remains fixed. 



This point may be called the center of rotation. 



11°. Two lines through this point remain unchanged. 



These lines may be called the fixed lines of the rotation. 



12°. Any half-line (or ray) from the center, and lying in one of 

 the angles determined by the fixed lines, may be turned into any other 

 ray in the same angle, and this uniquely determines the rotation. 



13°. The succession of two rotations about the same point is a 

 rotation. 



14°. The result of a rotation about and a translation from 

 to 0' is independent of the order in which the rotation and transla- 

 tion are carried out. 



It follows immediately from 14° that the fixed lines in a rotation 

 about any point are parallel to the fixed lines in a rotation about 

 any other point 0' . All lines in the plane may now be divided into 

 classes in such manner that neither translation nor rotation can 

 change the classification. Namely, 



(a) lines parallel to one of the fixed directions, 



(jS) lines parallel to the other of the fixed directions, 



