The name non-Euclidean was used by Gauss to describe a system of
geometry which differs from Euclid's in its properties of
parallelism. Such a system was developed independently by Bolyai in
Hungary and Lobatschewsky in Russia, about 120 years ago. Another
system, differing more radically from Euclid's, was suggested later
by Riemann in Germany and Cayley in England. The subject was
unified in 1871 by Klein, who gave the names of parabolic,
hyperbolic, and elliptic to the respective systems of
Euclid-Bolyai-Lobatschewsky, and Riemann-Cayley. Since then, a vast
literature has accumulated. The Fifth edition adds a new chapter,
which includes a description of the two families of 'mid-lines'
between two given lines, an elementary derivation of the basic
formulae of spherical trigonometry and hyperbolic trigonometry, a
computation of the Gaussian curvature of the elliptic and
hyperbolic planes, and a proof of Schlafli's remarkable formula for
the differential of the volume of a tetrahedron.
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