424 



Dr. L. Silberstein on the Relation between 



It will be well to restate this result : — The knowledge of 

 the mid-point Afoi a segment AB enables us to draw through 

 any point Y the parallel to AB (fig. 4), and vice versa, the 

 knowledge of a parallel to AB enables us to find the point M 

 (by means o£ a straight-edge alone). 



Fi-. 4. 



We can express this also by saying that with every 

 Euclidean segment is associated a certain characteristic point, 

 its mid-point, the unique image * of both the (coalescent) 

 points I — + oo and I = — x> of the straight line AB. 



We shall see that in the space of Lobatchevsky there is 

 on every straight segment a pair of distinct points which 

 take over the role of the mid-point. The latter is thus a 

 double point, the coalescence of the pair. 



4. Lobatchevsky an segment. — Let the curvature of the 

 contemplated hyperbolic space be K= — l/R' 2 , and let us 

 take R, a real length, as unit length. Then, our previous 

 OUT being a straight segment in this space, we have to 

 put in equation (2) i= v' — 1 instead of R. Thus the relation 

 between the metrical and the Staudtian scales becomes 



Cot 1= - [Cot h + (n- 1) Cot Z w ], 



(6) 



where Cot stands for the hyperbolic cotangent. 



Without dwelling any further upon the general relation 

 (6), let us specialize it by taking for the Staudtian n~l the 

 metrical mid-point M of the segment. Thus_, if 2X be the 

 total length of the segment, with R as unit, let us put lx = \ 

 l m =2\. Then 



CotZ=i[Cot\H-(n-l)Cot2X], 



n 



for any I. Solve for n and, aiming at the Lobatchevskyan 



* Or fourth harmonic of A, B, and the point at infinity, conjugate to 

 the latter. 



