Within the framework of Euclid’s other four postulates (and a few that he omitted), there were also possible elliptic and hyperbolic geometries. In plane elliptic geometry there are no parallels to a given line through a given point; it may be viewed as the geometry of a spherical surface on which antipodal points have been identified and all lines are great circles. This was not viewed as revolutionary. More exciting was plane hyperbolic geometry, developed independently by the Hungarian mathematician János Bolyai (1802–60) and the Russian mathematician Nikolay Lobachevsky (1792–1856), in which there is more than one parallel to a given line through a given point. This geometry is more difficult to visualize, but a helpful model presents the hyperbolic plane as the interior of a circle, in which straight lines take the form of arcs of circles perpendicular to the circumference.
Another way to distinguish the three geometries is to look at the sum of the angles of a triangle. It is 180° in Euclidean geometry, as first reputedly discovered by Thales of Miletus (fl. 6th century bc), whereas it is more than 180° in elliptic and less than 180° in hyperbolic geometry. See Figure 2![Figure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]](http://media-2.web.britannica.com/eb-media/52/2352-003-90DB3EDD.gif)
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Zenos-paradox-illustrated-by-Achilles-racing-a-tortoiseFigure 1: Zeno’s paradox, illustrated by Achilles racing a tortoise.[Credits : Encyclopædia Britannica, Inc.]
Contrasting-triangles-in-Euclidean-elliptic-and-hyperbolic-spacesFigure 2: Contrasting triangles in Euclidean, elliptic, and hyperbolic spaces.[Credits : Encyclopædia Britannica, Inc.]
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