by János Bolyai and Nikolai Lobachevsky

Revealed to the world by János Bolyai and Nikolai Lobachevsky in the 19th century, hyperbolic geometry is built from axioms closely related to the axioms of Euclidean geometry [1].

Nikolai Lobachevsky

Rather than finding a contradiction, Nikolai Lobachevsky invented hyperbolic geometry, which is not only consistent as a mathematical system but has actually found use (those bleeping physicists again).

by János Bolyai

In the 1800’s hyperbolic geometry was discovered by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom whom sometimes is named.

Bolyai, and Lobachevski

Hyperbolic geometry, sometimes called non-Euclidean geometry, was discovered independently by Gauss, Bolyai, and Lobachevski in the 19th century as a way of finally demonstrating that the parallel postulate of plane geometry is not a logical consequence of the other postulates.

Felix Klein (1849-1925)

The terms HYPERBOLIC GEOMETRY, ELLIPTIC GEOMETRY, and PARABOLIC GEOMETRY were introduced by Felix Klein (1849-1925) in 1871 in 'Über die sogenannte Nicht-Euklidische Geometrie'

Mathematicians Janos Bolyai and Nikolai Lobachevsky

Mathematicians Janos Bolyai and Nikolai Lobachevsky would develop hyperbolic-geometry and Bernhard Riemann, elliptical geometry, both of these systems would deny the parallel postulate and later become known as Non-Euclidean geometries.

Girolamo Saccheri in Girolamo Saccheri's Euclides Vindicatus (1733)

Girolamo Saccheri in Girolamo Saccheri's Euclides Vindicatus (1733) essentially discovered Hyperbolic Geometry, by building around the hypothesis that the angles of a triangle add up less than 180°.

Eugenio Beltrami

This viewpoint was vindicated when, in 1868, Eugenio Beltrami produced a model of hyperbolic geometry.

János Bolyai and Nikolai Ivanovich Lobachevsky, after whom whom sometimes is named

In the 1800’s hyperbolic geometry was discovered by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom whom sometimes is named.

Euclid

Omar Khayyám (1050–1123), a Persian, made the first attempt at formulating a non-Euclidean postulate as an alternative to the parallel postulate, and Euclid was the first to consider the cases of elliptical geometry and hyperbolic geometry, though Euclid excluded the latter.