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
Library of Alexandria the "Letter of Aristeas" is very late and it contains information that is now known to be inaccurate. Other sources claim that the Library was instead created under the reign of Ptolemy I's son Ptolemy II Philadelphus (283–246 BC). Modern scholars agree that, while it is possible that Ptolemy I may have laid the groundwork for the Library, it probably did not come into being as a physical institution until the reign of Ptolemy II. By that time, Demetrius of Phalerum had fallen out of favor with the Ptolemaic court and could not, therefore, have had any role in establishing