Editing Non-Euclidean geometry (section)

===Development of non-Euclidean geometry===
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.
Circa 1813, [[Carl Friedrich Gauss]] and independently around 1818, the German professor of law [[Ferdinand Karl Schweikart]]<ref>In a letter of December 1818, Ferdinand Karl Schweikart (1780–1859) sketched a few insights into non-Euclidean geometry.  The letter was forwarded to Gauss in 1819 by Gauss's former student Gerling.  In his reply to Gerling, Gauss praised Schweikart and mentioned his own, earlier research into non-Euclidean geometry.  See:
* Carl Friedrich Gauss, ''Werke'' (Leipzig, Germany:  B. G. Teubner, 1900), vol. 8, [http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN236010751&DMDID=DMDLOG_0058&LOGID=LOG_0058&PHYSID=PHYS_0187 pp. 180–182.]
* English translations of Schweikart's letter and Gauss's reply to Gerling appear in:[http://www.math.uwaterloo.ca/~snburris/htdocs/noneucl.pdf Course notes:  "Gauss and non-Euclidean geometry", University of Waterloo, Ontario, Canada]; see especially pages 10 and 11.
* Letters by Schweikart and the writings of his nephew [[Franz Adolph Taurinus]], who also was interested in non-Euclidean geometry and who in 1825 published a brief book on the parallel axiom, appear in:  Paul Stäckel and Friedrich Engel, ''Die theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung der nichteuklidischen Geometrie'' (The theory of parallel lines from Euclid to Gauss, an archive of non-Euclidean geometry), (Leipzig, Germany:  B. G. Teubner, 1895), [http://quod.lib.umich.edu/u/umhistmath/abq9565.0001.001/254?rgn=full+text;view=pdf pages 243 ff.]</ref> had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew [[Franz Taurinus]] did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.<ref>{{citation|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>

Then, in 1829–1830 the [[Russia]]n mathematician [[Nikolai Ivanovich Lobachevsky]] and in 1832 the [[Hungary|Hungarian]] mathematician [[János Bolyai]] separately and independently published treatises on hyperbolic geometry.  Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. [[Carl Friedrich Gauss|Gauss]] mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,<ref>In the letter to Wolfgang (Farkas) Bolyai of March 6, 1832 Gauss claims to have worked on the problem for thirty or thirty-five years {{Harvard citation|Faber|1983|loc=p. 162}}. In his 1824 letter to Taurinus {{Harvard citation|Faber|1983|loc=p. 158}} he claimed that he had been working on the problem for over 30 years and provided enough detail to show that he actually had worked out the details. According to {{harvtxt|Faber|1983|loc=p. 156}} it wasn't until around 1813 that Gauss had come to accept the existence of a new geometry.</ref> though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter&nbsp;''k''. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.

[[Bernhard Riemann]], in a famous lecture in 1854, founded the field of [[Riemannian geometry]], discussing in particular the ideas now called [[manifold]]s, [[Riemannian metric]], and [[curvature]].
He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in [[Euclidean space]]. The simplest of these is called [[elliptic geometry]] and it is considered a non-Euclidean geometry due to its lack of parallel lines.<ref>However, other axioms besides the parallel postulate must be changed to make this a feasible geometry.</ref>

By formulating the geometry in terms of a curvature [[tensor]], Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature.