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
Pappus of Alexandria Pappus of Alexandria Pappus of Alexandria (; ; ) was one of the last great Greek mathematicians of Antiquity, known for his "Synagoge" (Συναγωγή) or "Collection" (), and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than, (from his own writings) that he had a son named Hermodorus, and was a teacher in Alexandria. "Collection", his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics, including geometry, recreational mathematics, doubling the cube, polygons and polyhedra. Pappus was active in the 4th