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 Reign of Cleopatra The reign of Cleopatra VII of the Ptolemaic Kingdom of Egypt began with the death of her father, the ruling pharaoh Ptolemy XII Auletes, by March 51 BC. It ended with her death on 10 or 12 August 30 BC. Following the reign of Cleopatra, the country of Egypt was transformed into a province of the Roman Empire and the Hellenistic period came to an end. During her reign she ruled Egypt and other territories as an absolute monarch, in the tradition of the Ptolemaic dynasty's founder Ptolemy I Soter (r. 305–283 BC) as well as Alexander