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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 ...
Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century.
[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 ...
Brahmagupta (597–668) – Euclidean geometry, cyclic quadrilaterals; Vergilius of Salzburg (c.700–784) – Irish bishop of Aghaboe, Ossory and later Salzburg, Austria; antipodes, and astronomy; Al-Abbās ibn Said al-Jawharī (c. 800–c. 860) Thabit ibn Qurra (826–901) – analytic geometry, non-Euclidean geometry, conic sections
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.
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.
According to Paul Stäckel and Friedrich Engel, [2] as well as Zacharias, [5] Taurinus must be given credit as a founder of non-Euclidean trigonometry (together with Gauss), but his contributions cannot be considered as being on the same level as those of the main founders of non-Euclidean geometry, Nikolai Lobachevsky and János Bolyai.
He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat ...