504 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where the lof>;arithm of the cross-ratio is divided by 2i instead of by 2 

 as above. The choice of the divisor 2i is due to the desire to have 

 the angle real when the fixed lines are conjugate imaginary lines and 

 to have the total angle about a point ecjual to 27r as in Euclidean 

 geometry; this is not, however, in any way suggested by projective 

 geometry. In our non-Euclidean geometry, where we have taken a 

 different set of postulates for rotation, the real divisor 2 is more natural. 

 We have seen that from the point of view of the postulates of trans- 

 lation or the parallel transformation our geometry and the ordinary 

 Euclidean geometry fall into one class, while such geometries as the 

 Lobatchewskian and the Riemannian belong to another class. With 

 respect to the postulates of rotation, however, the Euclidean and most 

 of the non-Euclidean geometries which have been studied lie in one 

 class, to which our geometry does not belong. The methods of pro- 

 jective geometry are applicable to all these classes. 



If the ray r is perpendicular to the rays r' and r", the latter two being 

 in the same line but oppositely directed, it is evident that we must 

 choose arbitrarily the sign of the angle =t 5 wi between r and r' ; but 

 we shall assume that if the sign of the angle rr' has been determined 

 the sign of the angle rr" will be the same. Thus the angle r'r" is 

 zero. This means that a pair of intersecting lines determine but one 

 angle except for sign; thus any angle is identical, except for sign, with 

 its supplement. 



The angle from a line to a second line and the angle from the first 

 line to the perpendicular to the second will be called complementary. 

 The complement of a real angle is a complex angle, and vice versa. 



65. Hitherto we have chosen to avoid the use of the term distance, 

 and have used the word interval to represent a positive number 

 expressing the measure of length. If r is a line drawn from the origin, 

 the interval of r has been defined as Vx- — y^ or "^y- — .r^ according 

 as X is greater than y or y greater than x. This was done to avoid 

 altogether the use of imaginaries. We might, however, haye defined 

 distance as 



f 



V(/a'2 — dy\ 



where x is, for example, measured along a (7)-line, y along a perpendic- 

 ular (5)-line. Then every (7)-line would have a real, and every (5)- 

 line an imaginary distance. In this case it would be convenient to 

 consider the distance along any vector AB as the negative of the 

 distance along BA. The distance along any singular line is zero. 



