400 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(7) lines which He in one of the pairs of vertical angles determined 

 by the fixed directions, 



(6) lines which lie in the other pair of vertical angles determined 

 by the fixed directions. 



The lines of fixed direction, namely, the (a) -lines and (jS) -lines, 

 will be called singular lines. 



A system of measurement may be set up for angles between rays ^^ 

 which issue from a point into one of the angles determined by the 

 fixed lines through the point. For a succession of rotations may be 

 used (in the same manner as the succession of translations was used 

 to establish the measure of interval along a line). Thus if a line 

 a is carried into a line a' and at the same time the line a' is carried 

 into the line a", the angles between a and a' and between a' and a" 

 are congruent and the measures of the angles are said to be equal. 

 Now as the rotation may be repeated any number of times without 

 reaching the fLxed line, it is possible to find an angle aa^"'^ which shall 

 be 71 times the angle aa'. AVe shall assume the postulate, analogous 

 to the Archimedean: 



15°. If a sufficient number of equal angles be laid off about a 

 point from any initial ray, any ray of that class may be surpassed. 



It thus appears that the angles between any given line and other 

 lines of the same class may be placed into one-to-one correspondence 

 with all positive and negative real numbers, just as the intervals 

 from a point on a line may be thus correlated. ^^ This constitutes a 

 very great difference between our geometry and the Euclidean. 



It is impossible to show from the preceding statements that any 

 given figure maintains a constant area during rotation. ^^ We shall 

 therefore lay down the additional postulate: 



10 The relations of order of all lines of a given class, (7) or (5), are the same 

 as those of points on a line, as in 4°. 



11 The angle between two singular lines (a) and {0) can obviously not be 

 measured. Such an angle, and also the angle between any line and a line of 

 fixed direction, must be regarded as infinite. 



12 This matter may readily be discussed analytically. As axes of reference 

 choose the fixed lines, and let u, v denote coordinates. As rotation is a linear 

 transformation, the point P {u, v) and the transformed point P' {u', v') are 

 connected by the equations 



u' = au + bv + c, v' = du + ev + /. 



As the hues u = and v — are fixed, these equations reduce to u' = au, 

 v' = ev; and as rotation depends on only one parameter, we may write 

 e = 4>{a). The succession of two rotations is then expressed by 



{ u' = au ( u" = bii' ^ u" — abu 



\v'= (f>ia)v, \ v" = <t>ib)v', I v" = 4>{a)(t>{b)v, 



