János Bolyai and Nikolai Ivanovich Lobachevsky

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

by Nikolai Lobachevsky

Hyperbolic geometry was also independently discovered by Nikolai Lobachevsky, who is quoted to write There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world -- and indeed, non-euclidean geometry has found many applications, for example in physics, and more recently in art (M.C. Escher), game design, data visualization and social network analysis; and even HyperRogue itself can be applied as a powerful engine to work with applied hyperbolic geometry (see RogueViz).

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'

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.

by János Bolyai

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

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.

by Janos Bolyai and Nicholay Lobatchevsky

Hyperbolic Geometry is a Non-Euclidean Geometry discovered by Janos Bolyai and Nicholay Lobatchevsky in the first half of the 19th century.

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.

by János Bolyai, N. I. Lobachevsky

Hyperbolic geometry is the non-Euclidean geometry discovered by János Bolyai, N. I. Lobachevsky, and K. F. Gauss about 200 years ago.

Bolyai

The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered hyperbolic geometry independently and published in 1832.

János Bolyai, Nikolai Ivanovich Lobachevsky, Carl Friedrich Gauss and Franz Taurinus

In the 19th century, hyperbolic geometry was explored extensively by János Bolyai, Nikolai Ivanovich Lobachevsky, Carl Friedrich Gauss and Franz Taurinus.

Beltrami and the German mathematician Felix Klein

In 1869–71 Beltrami and the German mathematician Felix Klein developed the first complete model of hyperbolic geometry (and first called the geometry “hyperbolic”).

by Lobachevsky

The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered hyperbolic geometry independently and published in 1832.