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
Piri Reis map Alexandria, Ptolemy I, who ruled Alexandria in the 4th century BC, and Claudius Ptolemy, the Greek geographer and cartographer who lived in Alexandria during the 2nd century AD. Gregory McIntosh states "Arab writers often confused Claudius Ptolemy, the geographer of the second century C.E., with Ptolemy I, one of Alexander's generals... Piri Reis has undoubtedly made the same error, resulting in his believing the charts and maps were from the time of Ptolemy I instead of Claudius Ptolemy." The map was discovered serendipitously on 9 October 1929, through the philological work of the German theologian Gustav Adolf Deissmann (1866–1937). He