## Geometry and Trigonometry## Other Geometries |

## Who developed the two alternatives to Euclidean geometry? |

Hyperbolic geometry was first announced by Russian mathematician Nikolai Ivanovich Lobachevski (1792-1856; also seen as Lobatchevsky) in 1826. He challenged Euclid’s fifth postulate that one and only one line parallel to a given line can be drawn through a fixed point external to the line. Instead, he developed a self-consistent system of geometry in which the “flawed” postulate was replaced by one allowing more than one parallel through the fixed point.

This idea was already developed independently by Hungarian János (or Johann) Bolyai (1802–1860) in 1823 (after several attempts to prove the Euclidean parallel postulate, he developed his system by assuming that a geometry could be constructed without the parallel postulate) and German mathematician, physicist, and astronomer Karl Friedrich Gauss (1777-1855; also seen as Johann Carl [or Karl] Friedrich Gauss) in 1816. But, as often happens in science and mathematics, the person who publishes an idea first gets most of the credit; this time, Lobachevski was the first to publish.