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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 requirement.
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.
Gauss thought about the basics of geometry since the 1790s years, but in the 1810s he realized that a non-Euclidean geometry without the parallel postulate could solve the problem. [ 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 ...
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
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.
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.
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.