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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