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
Associated With
Includes
Source Eugenio Beltrami (MacTutor)
Source MacTutor biography of Felix Klein, St Andrews
Source Ferdinand Karl Schweikart (Wikipedia)
Source Franz Taurinus (Wikipedia)
Source Georg Simon Klügel (Wikipedia)
Source Wikipedia: Giovanni Girolamo Saccheri
Source Giuseppe Battaglini (MacTutor)
Source Harold Scott MacDonald Coxeter (MacTutor)
Source János Bolyai (Wikipedia)
Source Johann Heinrich Lambert (MacTutor)
Source Marcel Grossmann (Wikipedia)
Source Nikolai Lobachevsky (MacTutor)
Source Wilhelm Killing (Wikipedia)
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.
In Group: Differential and Non-Euclidean Geometry, lead paragraph
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.
View the Source Johann Heinrich Lambert (MacTutor)
Includes: Johann Heinrich Lambert, Biography, parallel postulateQuote, Includes: Johann Heinrich Lambert, Biography, parallel postulate
he managed to deduce a large number of non-euclidean results
Nikolai Lobachevsky (MacTutor)
Includes: Nikolai Lobachevsky, SummaryQuote, Includes: Nikolai Lobachevsky, Summary
Nikolai Lobachevsky published his work on non-Euclidean geometry, the first account of the subject to appear in print.
Harold Scott MacDonald Coxeter (MacTutor)
Includes: Harold Scott MacDonald Coxeter, Summary, non-euclidean geometryQuote, Includes: Harold Scott MacDonald Coxeter, Summary, non-euclidean geometry
contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics
Eugenio Beltrami (MacTutor)
Includes: Eugenio Beltrami, Summary, non-Euclidean geometryQuote, Includes: Eugenio Beltrami, Summary, non-Euclidean geometry
gave a concrete realisation of the non-euclidean geometry
Giuseppe Battaglini (MacTutor)
Includes: Giuseppe Battaglini, Biography, Giornale di matematicheQuote, Includes: Giuseppe Battaglini, Biography, Giornale di matematiche
Battaglini edited the journal, aimed at university students, which became the main outlet for papers in non-Euclidean geometry in Italy.
János Bolyai (Wikipedia)
Includes: János Bolyai, LeadQuote, Includes: János Bolyai, Lead
who developed absolute geometry
Marcel Grossmann (Wikipedia)
Includes: Marcel Grossmann, Career, researchQuote, Includes: Marcel Grossmann, Career, research
He continued to do research on non-Euclidean geometry
Franz Taurinus (Wikipedia)
Includes: Franz Taurinus, LeadQuote, Includes: Franz Taurinus, Lead
known for his work on non-Euclidean geometry
Georg Simon Klügel (Wikipedia)
Includes: Georg Simon Klügel, Article textQuote, Includes: Georg Simon Klügel, Article text
a study of 30 attempted proofs of the parallel postulate
Ferdinand Karl Schweikart (Wikipedia)
Includes: Ferdinand Karl Schweikart, Lead, field of workQuote, Includes: Ferdinand Karl Schweikart, Lead, field of work
an astral geometry before the discovery of non-Euclidean geometry
Wikipedia: Giovanni Girolamo Saccheri
Includes: Giovanni Girolamo Saccheri, LeadQuote, Includes: Giovanni Girolamo Saccheri, Lead
He is considered the forerunner of non-Euclidean geometry. == Biography == The son of a lawyer, Saccheri was born in Sanremo, Genoa (now Italy) on September 5, 1667.
Wilhelm Killing (Wikipedia)
Wikimedia FoundationIncludes: Wilhelm Killing, Lead paragraphQuote, Includes: Wilhelm Killing, Lead paragraph
Wilhelm Karl Joseph Killing (10 May 1847-11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
MacTutor biography of Felix Klein, St Andrews
Includes: Felix Klein, BiographyQuote, Includes: Felix Klein, Biography
best known for his work in non-euclidean geometry
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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