Non-Euclidean geometry comprises the geometries built on axioms that replace Euclid's parallel postulate, describing spaces where the familiar rules of flat, Euclidean space no longer hold. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhether Euclid's fifth postulate, that through a point not on a line exactly one parallel to that line can be drawn, is an independent axiom or a theorem provable from the other four, a question that troubled geometers for more than a thousand years before it was settled by exhibiting consistent geometries in which the postulate fails. 1 Key DebateWho first discovered that a consistent geometry could deny Euclid's parallel postulate. Carl Friedrich Gauss privately told Bolyai's father that he had developed such a geometry years earlier without publishing it, a claim that left Janos Bolyai and Nikolai Lobachevsky, who published independently, to share credit for a discovery Gauss never publicly claimed himself. 1 Classification
Pure or Applied Connections
Sources
1. Non-Euclidean Geometry (Wikipedia)
WikipediaLead section
Non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.
Background
For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four.
Development of non-Euclidean geometry
Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish.
View the Source Dissenting Readings (1 dissenting reading)
Key Debate
Gauss's defenders hold that Gauss deserves priority for discovering non-Euclidean geometry, pointing to his own account, given to Bolyai's father in 1832, that he had already worked out the consistency of a geometry denying the parallel postulate years earlier. On this reading Gauss withheld publication specifically to avoid controversy with readers committed to Kant's view that Euclidean space was a necessary truth of reason, not from any doubt in the result. Mainstream historians of mathematics reject the priority claim as unverifiable and beside the point: a result kept private cannot be checked, built on, or credited as a discovery in the ordinary sense, so credit belongs to Bolyai and Lobachevsky, who each risked publishing a complete, checkable system.
A dissenting reading, from Carl Friedrich Gauss and his later biographersNon-Euclidean Geometry (Wikipedia), Wikipedia
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