Hyperbolic Geometry - History

History

A number of geometers made attempts to prove the parallel postulate by assuming its negation and trying to derive a contradiction, including Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám, Nasir al-Din al-Tusi, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, and Legendre. Their attempts failed, but their efforts gave birth to hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.

In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions and computed the area of a hyperbolic triangle.

In the nineteenth century, hyperbolic geometry was extensively explored by János Bolyai and Nikolai Ivanovich Lobachevsky, after whom it sometimes is named. Lobachevsky published in 1830, while Bolyai independently discovered it and published in 1832. Carl Friedrich Gauss also studied hyperbolic geometry, describing in a 1824 letter to Taurinus that he had constructed it, but did not publish his work. In 1868, Eugenio Beltrami provided models of it, and used this to prove that hyperbolic geometry was consistent if Euclidean geometry was.

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

For more history, see article on non-Euclidean geometry, and the references Coxeter and Milnor.