Editing Non-Euclidean geometry (section)

===Terminology===
It was Gauss who coined the term "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.) p. 176.</ref> He was referring to his own work, which today we call ''hyperbolic geometry'' or ''Lobachevskian geometry''. Several modern authors still use the generic term ''non-Euclidean geometry'' to mean ''hyperbolic geometry''.<ref>For example:
{{citation |last=Kulczycki |first=Stefan |year=1961 |title=Non-Euclidean Geometry |publisher=Pergamon |url=https://archive.org/details/noneuclideangeom0000stef/page/53 |url-access=limited |page=53 }}<br/ >
{{citation |last=Iwasawa |first=Kenkichi |year=1993 |title=Algebraic Functions |publisher=American Mathematical Society |page=140 |isbn=978-0-8218-4595-0 |url=https://archive.org/details/algebraicfunctio0000iwas/page/140/ |url-access=limited }}
</ref>

[[Arthur Cayley]] noted that distance between points inside a conic could be defined in terms of [[logarithm]] and the projective [[cross-ratio]] function. The method has become called the [[Cayley–Klein metric]] because [[Felix Klein]] exploited it to describe the non-Euclidean geometries in articles<ref>F. Klein, Über die sogenannte nichteuklidische Geometrie, ''Mathematische Annalen'', '''4'''(1871).</ref> in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.

Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry ''parabolic'', a term that generally fell out of use<ref>The Euclidean plane is still referred to as ''parabolic'' in the context of [[conformal geometry]]: see [[Uniformization theorem]].</ref>). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.

There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.<ref>for instance, {{harvnb|Manning|1963}} and Yaglom 1968</ref>

There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in the conventional meaning of "non-Euclidean geometry", such as more general instances of [[Riemannian geometry]].