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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 ...
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.
1830: Non-Euclidean geometry (hyperbolic geometry) – Nikolai Ivanovich Lobachevsky (1830), János Bolyai (1832); preceded by Gauss (unpublished result) c. 1805. 1831: Electromagnetic induction was discovered by Michael Faraday in England in 1831, and independently about the same time by Joseph Henry in the U.S. [37]
[202] [200] In a letter to Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "non-Euclidean geometry", [203] but he strongly forbade Taurinus to make any use of it. [202] Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well. [204] [203 ...
Non-Euclidean geometry models proposed by Klein (left) and Poincaré (right) In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric.
It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.
Consequently, hyperbolic geometry has been called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors 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, [ 64 ] though he did not publish.
Karl Wilhelm Feuerbach (1800–1834) – Euclidean geometry; Julius Plücker (1801–1868) János Bolyai (1802–1860) – hyperbolic geometry, a non-Euclidean geometry; Christian Heinrich von Nagel (1803–1882) – Euclidean geometry; Johann Benedict Listing (1808–1882) – topology; Hermann Günther Grassmann (1809–1877) – exterior algebra