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 and introduced relief efforts such as releasing grain from the royal granary. Both Ptolemy XII and Cleopatra VII found it necessary to debase Ptolemaic coinage due to financial troubles. No gold coins are known from Cleopatra's reign, while use of bronze coins was revived (absent since the reign of Ptolemy IX) and silver currency was debased roughly 40% by the end of her reign. Coins struck under Cleopatra's reign came from a wide geographical expanse, including sites in Egypt like Alexandria, but also the island of Cyprus, Antioch, Damascus and Chalcis ad Belum in Syria, Tripolis in Phoenicia, Askalon in