Editing Carl Friedrich Gauss (section)

==== Non-Euclidean geometry ====
{{Main|Non-Euclidean geometry}}
[[File:Bendixen - Carl Friedrich Gauß, 1828.jpg|thumb|upright|Lithography by [[Siegfried Bendixen]] (1828)]]
During Gauss' lifetime, the [[Parallel postulate]] of [[Euclidean geometry]] was heavily discussed.{{sfn|Stäckel|1917|pp=19–20}} Numerous efforts were made to prove it in the frame of the Euclidean [[axiom]]s, whereas some mathematicians discussed the possibility of geometrical systems without it.{{sfn|Bühler|1981|pp=100–102}} Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s  that a non-Euclidean geometry without the parallel postulate could solve the problem.{{sfn|Klein|1979|pp=57–60}}{{sfn|Stäckel|1917|pp=19–20}} In a letter to [[Franz Taurinus]] of 1824, he presented a short comprehensible outline of what he named a "[[non-Euclidean geometry]]",<ref name=":0">{{Cite journal | last = Winger | first = R. M. | date = 1925 | title = Gauss and non-Euclidean geometry | url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-31/issue-7/Gauss-and-non-euclidean-geometry/bams/1183486559.full | journal = Bulletin of the American Mathematical Society | volume = 31 | issue = 7 | pages = 356–358 | doi = 10.1090/S0002-9904-1925-04054-9 | issn = 0002-9904 | doi-access = free }}</ref> but he strongly forbade Taurinus to make any use of it.{{sfn|Klein|1979|pp=57–60}} Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well.<ref>{{Cite book |last=Bonola |first=Roberto |url=https://archive.org/details/noneuclideangeom00bono/page/64 |title=Non-Euclidean Geometry: A Critical and Historical Study of its Development |publisher=The Open Court Publishing Company |year=1912 |pages=64–67 |language=en}}</ref><ref name=":0" /><ref>{{Cite book |last=Klein |first=Felix |url=https://archive.org/details/elementarymathem0000klei/page/176 |title=Elementary Mathematics from an Advanced Standpoint: Geometry |publisher=Dover Publications |year=1939 |pages=176–177 |language=en}}</ref>

The first publications on non-Euclidean geometry in the history of mathematics were authored by [[Nikolai Lobachevsky]] in 1829 and [[Janos Bolyai]] in 1832.{{sfn|Bühler|1981|pp=100–102}} In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion.{{sfn|Klein|1979|pp=57–60}}<ref>{{Cite journal | last1 = Jenkovszky | first1 = László | last2 = Lake | first2 = Matthew J. | last3 = Soloviev | first3 = Vladimir | date = 12 March 2023 | title = János Bolyai, Carl Friedrich Gauss, Nikolai Lobachevsky and the New Geometry: Foreword | journal = Symmetry| volume = 15 | issue = 3 | pages = 707 | doi = 10.3390/sym15030707 | arxiv = 2303.17011 | bibcode=2023Symm...15..707J | issn = 2073-8994 | doi-access = free}}</ref> Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai<ref>{{Cite web|url=https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up|title=Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai|first=Carl Friedrich Gauss|last=Farkas Bólyai|date=22 April 1899|publisher=B. G. Teubner|via=Internet Archive}}</ref> claiming that these were congruent to his own thoughts of some decades.{{sfn|Klein|1979|pp=57–60}}<ref>{{cite book | last = Krantz | first = Steven G. | author-link = Steven G. Krantz | title = An Episodic History of Mathematics: Mathematical Culture through Problem Solving|url=https://books.google.com/books?id=ulmAH-6IzNoC&pg=PA171 | access-date = 9 February 2013 | date = 2010 | publisher = [[The Mathematical Association of America]] | isbn = 978-0-88385-766-3| pages = 171f}}</ref> However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.{{sfn|Bühler|1981|pp=100–102}}

[[Wolfgang Sartorius von Waltershausen|Sartorius]] first mentioned Gauss's work on non-Euclidean geometry in 1856, but only the publication of Gauss's [[Nachlass]] in Volume VIII of the Collected Works (1900) showed Gauss's ideas on the matter, at a time when non-Euclidean geometry was still an object of some controversy.{{sfn|Klein|1979|pp=57–60}}