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 the time (but see below) at which he himself wrote. If no other date information were available, all that could be known would be that he was later than Ptolemy (died c. 168 AD), whom he quotes, and earlier than Proclus (born ), who quotes him. The "Suda" states that Pappus was of the same age as Theon of Alexandria, who active in the reign of Emperor Theodosius I (372–395). A different date is given by a marginal note to a late 10th-century manuscript (a copy of a chronological table by the same Theon), which states, next to an entry