528] 
409 
528. 
ON THE NON-EUCLIDIAN GEOMETRY. 
[From the Mathematische Annalen, vol. v. (1872), pp. 630—634.] 
The theory of the Non-Euclidian Geometry as developed in Dr Klein’s paper 
“ Ueber die Nicht-Euklidische Geometrie ” may be illustrated by showing how in such 
a system we actually measure a distance and an angle and by establishing the 
trigonometry of such a system. I confine myself to the “ hyperbolic ” case of plane 
geometry; viz. the absolute is here a real conic, which for simplicity I take to be a 
circle; and I attend to the points within the circle. 
I use the simple letters a, A,.. to denote (linear or angular) distances measured 
in the ordinary manner; and the same letters, with a superscript stroke, a, A,.. to 
denote the same distances measured according to the theory. The radius of the 
absolute is for convenience taken to be = 1; the distance of any point from the centre 
can therefore be represented as the sine of an angle. 
C. VIII. 
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