132 Dr. L. Silbersteiii : Further Contributions 



definition, the equality of all angles (such as a and |3 in fig. 7)' 

 whose sides are coterminal. Such was the only case of: angle 

 comparison treated in P.V.A., Appendix A, where it was 

 shown to be sufficient to prove that the non-metrical sum o£ 



Fig. 7. 



angles in every triangle (not traversed by the T-line nor 

 having it as one of its sides) is equal to a straight angle or, 

 as we now can say, to two rigid angles. We may return to- 

 this triangle property later on. Meanwhile let us proceed 

 with the application of the definition (13) of .ingle equality 

 to cases in which 1, m are vectoria ly different from a, b. 



Without loss to generality we can assume all these four 

 vectors to be co-initial in 0, so that a, b, Z, m are on the 

 conic k. Let us put the question in the form of the problem : 

 Given a, b, the sides of the angle a, and given 1, one side of 

 the angle ft, to construct m, so that 



i. e. so that lm = ab. 



From b draw the normal to Oa meeting it in .A 7 ; through 

 2s imagine drawn the conic homologous to k : let this cut 

 01 in A 1 '. . Through A"' .draw the normal to 1 cutting k in 

 two points ; either of these will be the end-point of the 

 required m, giving /3=a. It is, for our purpose, by no 

 means necessary to execute this construction in detail.. 

 (And a more speedy construction will be given in Section 9.) 

 It is enough to contemplate it in order to see that, what- 

 ever 1, an angle equal to a can be constructed on either side 

 of 1, in a perfectly determined manner. 



In particular, if I coincides with b, when the problem 

 consists in the doubling of a, the actual construction becomes 

 a very simple process. For, involuntarily, we have already 



