Branches of Mathematics
Non-Euclidean Geometry
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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.
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 Cross-Tradition Connections
Sources
1. Non-Euclidean Geometry (Wikipedia)
WikipediaLead sectionQuote, Lead section
Non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.
1. Non-Euclidean Geometry (Wikipedia)
WikipediaBackgroundQuote, 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.
1. Non-Euclidean Geometry (Wikipedia)
WikipediaDevelopment of non-Euclidean geometryQuote, 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.
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