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Non-Euclidean geometry

Euclid's *Elements* contained five postulates that form the basis for Euclidean geometry. One of these, the parallel postulate has been the subject of debate among mathematicians for many centuries. It states that on any plane on which there is a straight line *L*_{1} and a point *P* not on *L*_{1}, there is only one straight line*L*_{2} on the plane that passes through the point *P* and is parallel to the straight line *L*_{1}. Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.^{[12]} Around 1830 though, the HungarianJános Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinitenumber of parallel lines pass through the point *P*. Consequently the sum of angles in a triangle is less than 180^{o} and the ratio of a circle's circumference to its diameter is greater than pi. In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in which no parallel lines pass through *P*. In this geometry, triangles have more than 180^{o} and circles have a ratio of circumference-to-diameter that is less than pi.
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