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.

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).

Felix Klein

The term "hyperbolic geometry" was introduced by Felix Klein in 1871.

by János Bolyai

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

Eugenio Beltrami

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

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.

Lobachevsky in 1829/1830, while 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 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.

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°.

Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai

Nikolai Ivanovich Lobachevsky, around 1830, and János Bolyai, in 1832, published treatises on what is now called 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.

William Thurston

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem.

the parallel postulate , called hyperbolic geometry

Around 1830 though, the Hungarian János 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.

Gregoire de Saint-Vincent

The geometric mean is an ancient concept, but hyperbolic angle was developed in this configuration by Gregoire de Saint-Vincent.

by Gauss,

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 Lobachevsky

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

by Einstein

Hyperbolic geometry, once purely theoretical, was later adopted by Einstein to model Einstein's general theory of relativity—a landmark of scientific discovery.

János Bolyai

The mathematicians János Bolyai occupies an interesting place in the history of mathematics for the development of hyperbolic geometry.

Men such as Karl Gauss, Nikolai Lobachevsky, and Janós Bolyai

Men such as Karl Gauss, Nikolai Lobachevsky, and Janós Bolyai began to question the Euclid parallel-lines axiom and discovered hyperbolic geometry, the first non-Euclidean geometry.