Editing Hyperbolic geometry (section)

===19th-century developments===

In the 19th century, hyperbolic geometry was explored extensively by [[Nikolai Lobachevsky]], [[János Bolyai]], [[Carl Friedrich Gauss]] and [[Franz Taurinus]]. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.<ref>{{Cite book|author=Bonola, R.|title=Non-Euclidean geometry: A critical and historical study of its development|year=1912|location=Chicago|publisher=Open Court|url=https://archive.org/details/noneuclideangeom00bono}}</ref><ref>{{cite book|author-link1=Marvin Greenberg|last1=Greenberg|first1=Marvin Jay|title=Euclidean and non-Euclidean geometries: development and history|url=https://archive.org/details/euclideannoneucl00gree_304|url-access=limited|date=2003|publisher=Freeman|location=New York|isbn=0716724464|page=[https://archive.org/details/euclideannoneucl00gree_304/page/n194 177]|edition=3rd|quote=Out of nothing I have created a strange new universe. JÁNOS BOLYAI}}</ref>

Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "[[non-Euclidean geometry]]"<ref>Felix Klein, ''Elementary Mathematics from an Advanced Standpoint: Geometry'', Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176</ref> causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

In 1868, [[Eugenio Beltrami]] provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent [[if and only if]] Euclidean geometry was.

The term "hyperbolic geometry" was introduced by [[Felix Klein]] in 1871.<ref>F. Klein. "Über die sogenannte Nicht-Euklidische Geometrie". ''Math. Ann.'' 4, 573–625 (also in ''Gesammelte Mathematische Abhandlungen'' 1, 244–350).</ref> Klein followed an initiative of [[Arthur Cayley]] to use the transformations of [[projective geometry]] to produce [[isometries]]. The idea used a [[conic section]] or [[quadric]] to define a region, and used [[cross ratio]] to define a [[metric (mathematics)|metric]]. The projective transformations that leave the conic section or quadric [[invariant (mathematics)#Invariant set|stable]] are the isometries. "Klein showed that if the [[Cayley absolute]] is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."<ref>Rosenfeld, B.A. (1988) ''A History of Non-Euclidean Geometry'', page 236, Springer-Verlag {{ISBN|0-387-96458-4}}</ref>