the Projective and the Metrical Scales. 



427 



Thus if we wish to characterize palpably a particular 

 Lobatchevskyan space, we have only to draw a segment and 

 mark upon it the " gap " or its extremities P and P' . If we 

 do the same thing for a pencil of equal segments from M, 

 we have, within the circle of radius \, a concentric circle of 

 radius rj •; and similarly within a sphere of radius A a con- 

 centric sphere of radius 77 determined by (12). This little 

 sphere is an image of the inaccessible locus of " points at 

 infinity," which is thus seen also to be a real quadric "at 

 infinity/'' It is certainly agreeable to have its natural image 

 or correlate near at hand. 



If R becomes infinite, or better, if X/P becomes smaller 

 and smaller, the gap becomes small of the second order, 

 until the two points coalesce into the Euclidean characteristic 

 point, the ordinary mid-point of the segment. 



5. Verification of formula (12). — Since some readers may 

 find the above method based on Staudt's scale and especially 

 our deduction of (2), not wholly convincing, it has seemed 

 well to ' deduce formula (1 2) directly by the aid of Lobat- 

 chevskyan trigonometry or the equivalent analytic geometry,, 

 and thus to verify it *. 



This can be done most easily in Weierstrass coordinates. 



Fig. 7. 



In fact, let the mid-point M of the given segment AB be 

 the origin, MB and the perpendicular MY (fig. 7) the axes. 

 Then, if i\ r be any point of the plane and a, b, r its shortest 



* Another way of deducing (2) and (12), based on Cayley's definition 

 of distance by the logarithm of a cross-ratio, may he left to the care of 

 the reader. 



