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The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines. [12] 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 ...
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 k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the ...
A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry").
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric ...
Gauss called it "non-Euclidean geometry" [13] causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry.
There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.
This results from a construction of a non-Euclidean geometry inside Euclidean geometry, whose inconsistency would imply the inconsistency of Euclidean geometry. A well known paradox is Russell's paradox , which shows that the phrase "the set of all sets that do not contain themselves" is self-contradictory.
In non-Euclidean geometry, the concept of a straight line is replaced by the more general concept of a geodesic, a curve which is locally straight with respect to the metric (definition of distance) on a Riemannian manifold, a surface (or higher-dimensional space) which may itself be curved.