Non-Euclidean geometry that are both perpendicular to a third line: Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work "Elements" was written. In the "Elements", Euclid began with a limited number of assumptions (23 definitions, five common notions, and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of the postulates is often
Spherical geometry geometries exist; see elliptic geometry. The earliest mathematical work of antiquity to come down to our time is "On the rotating sphere" (Peri kinoumenes sphairas) by Autolycus of Pitane, who lived at the end of the fourth century BC. Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote the Sphaerics, a book on the geometry of the sphere, and Menelaus of Alexandria, who wrote a book on spherical trigonometry called "Sphaerica" and developed Menelaus' theorem. "The book of unknown arcs of a sphere" written by the Islamic mathematician Al-Jayyani