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XL VI. The Relation between the Projective and the Metrical 

 Scales, and its bearing on the Theory of Parallels. By 



LiUDWIK SlLBERSTEIN, Ph.D.* 



1. ET ns recall briefly the structure of the projective 



I A scale as given by v. Staudt. OUT being a given 

 segment of a straight line, let us affix to the points and U 

 of this segment the numbers, labels, or, as we will call them, 

 the Staudtian indices and 1, respectively. The fourth 

 harmonic to 0, 1, T, conjugate to 0, whose construction is 

 shewn in fig. 1, will then have the index, 2. Continuing this 



Fig. 1. 



graphical process we shall have the points with indices 3, 4,. 

 etc. With equal ease points with any fractional indices can 

 be constructed, and irrational indices are obtained by limit 

 considerations. 



We need not give here a detailed description of this pro- 

 jective scale construction f . Suffice it to say that, the points 

 0, 1, and T being chosen, the position of a point of the 

 segment OUT having any real positive index n is uniquely 

 determined. The point T itself will have the index n~oo . 

 To points following upon one another from 0, via U, to T 

 correspond indices n increasing from to go . Beyond 

 and beyond T we have negative indices, so that the supple- 

 ment of the segment represents analytically but one domain, 

 — co <n<0. At T itself, approached from the right, we 

 have n— — co . (Singular point ; discontinuity of n.) 



Assuming all these things to be familiar to the reader, let 

 us, however, state explicitly how to any point n of the 

 segment OUT the point with index —n (or the fourth 

 harmonic to 0, T, n, conjugate to n) is constructed, since this 

 will be needed in the sequel. Through co and draw any 

 two straight lines crossing in T y . Take on the segment 



* Communicated by the Author. 



t The reader unacquainted with the subject may consult, for instance, 

 Coolidge's 'Non-Euclidean Geometry/ or the Author's 'Projective 

 Vector Algebra/ London, Bell, 1919. 



