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
Reign of Cleopatra could be classified as Greeks if they abandoned their original cultures, received a Greek education, labeled their gods and goddesses with Greek names, and embraced the Greek lifestyle. Native Egyptians had been largely excluded from serving in the military by the reign of Ptolemy II, replaced by Greek and Jewish landholders called cleruchs. By the reign of Ptolemy IV in the late 3rd century BC they were reintroduced as phalangite soldiers. Large migrations of Greeks to Egypt ceased by the 2nd century BC, so that the Greek minority in Ptolemaic Egypt remained demographically small. Although Egyptian priests were often wealthy