131 Dr. L. Silberstein : Further Contributions 



the second tangent from P will give the required m as it* 

 contact-point. 



Proceeding in the same manner with b, m as before- 

 with «, b, we can construct the point ?z, such that aOn = Za y 

 and so on. 



Combining both processes we can, from a given a, build 

 up any angle 



mat 



where m, n are any positive integers. 



Although the equality definition based on ab suffices for 

 comparing angles with one another, it is interesting to- 

 introduce another angle measure which we may call the 

 arc-measure of an angle. It will again suffice to consider 

 angles with vertex 0. Let a, b be a pair of points on the 

 conic k determining, let us say , the concave angle a, h = aOb. 

 Draw a chain of vector chords ap i9 p\p^-> etc., p n b. Consider 

 the sum of the tensors of all these vectors, 



I a V\ i + P1P2 + ■•- + pJ> ; =2As, say, 



and making the necessary analytical assumptions, pass to its- 

 limit 





which, if it exists, is obviously an ordinary scalar number,, 

 and call it the "length" of the arc or simply the arc ab of 

 our conic. It is the number of projective unit steps- 

 contained in ab. 



In particular let 5,. stand for the arc subtending a right 

 angle {any right angle), i. e. corresponding to the case in 

 which ab = 0. What the numerical value of s r is will be 

 seen presently. 



Let us now define the pure number 



e=s ab 



as the arc-measure of the angle a, b or aOb. 



It is not difficult to see that two angles which are equali 

 according to the previous definition (13) have also equaliarc- 

 measures. In fact, let 



lm = ab = ,r. 

 Then, taking m as the .i'-axis of orthogonal coordinates x, n± 



