the Projective and the Metrical Scales. 423 



This is the required relation between the scale of Staudtians 

 (n) and that of metrical "divisions' (/) setup upon a straight 

 line in any space of constant curvature. For real finite R 

 the formula is ready for the Riemannian straight line ; the 

 Euclidean case corresponds to R = co : and the Lobatchev- 

 skyan or hyperbolic one to R 2 <0. 



Leaving out the first case, of no interest for our purpose, 

 let us consider the consequences of (2) for a Euclidean, and 

 then for a hyperbolic segment. The latter will offer some 

 interesting points connected with Lobatchevsky's parallels. 



3. Euclidean segment. — Putting i? = co, that is, making 

 the ratios of /, Z 1} l^ to R tend to zero and dividing by R, 

 we have at once 



l = nl x : |l+^L(r,-l)j, .... (3) 



and, conversely, 



-l-'tf « 



These equations enable us to write down the Staudtian 

 index corresponding to any metrical scale-division, and vice 

 versa. If /i, l m are rational numbers, to every rational I 

 corresponds, by (4), a rational index ?/, and the corresponding- 

 point can therefore be constructed by a finite number of 

 straight-edge operations. 



Of particular interest is the value of n corresponding to 

 /=cco (" point at infinity "). This is by (4) 



» = l-jrS (5) 



and the same index corresponds to Z= — go. We assume 

 that l m >Zj, so that the point corresponding to 



h 



lies within the segment OUT. Constructing its negative as- 

 shown in fig. 2, we find, through any point I 7 , the parallel 

 to the base line. 



In particular, if Z w =2Zi, so that n = l is the metrical 

 mid-point of OT, we have for the parallel (i. e. for /= +co) 



n= — 1, 



whence the known construction of the Euclidean parallel to 

 a bisected segment, as a sub-case of the construction repre- 

 sented by (5). 



