760 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



on f, or not. The constructions for equal angles are shown in Figures 

 12 and 13. _ _ 



The equal angles are ab and cd. In Figure 12, 



h-k = l 



m. 



24. In the discussion of distance we found besides the fixed line f 

 (locus of exceptional points in our algebra) a fixed point F', that need 



FiGTTRE 13. 



not be the fixed point of our algebra. Again in the discussion of angle 

 we have a fixed point F (locus of exceptional lines) and a fixed line f , 

 which need not be the fixed line occurring in the algebra. If we trans- 

 form the plane by a coUineation leaving F', and f fixed, the ratios of dis- 

 tances will not be changed. If then there are to exist fixed relations 

 between distances and angles in a figure (between sides and angles of 

 a triangle, for example) the ratios of angles must be at the same time 

 unchanged. We therefore choose the same fixed elements for distance 

 and angle. These must then be the exceptional elements F and f in 

 our algebra. 



Distance as here discussed has a definite algebraic sign and hence 

 along every line is assigned a definite positive direction. In Euclidean 

 geometry there is no definite direction along a line because by a rotation 

 preserving distance one direction can be turned into the other. That 

 it is not possible, by our construction for equal distances, to rotate one 



