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
Ptolemy IV Philopator Ptolemy IV Philopator Ptolemy IV Philopator (, "Ptolemaĩos Philopátōr" "Ptolemy, lover of his Father"; 245/4–204 BC), son of Ptolemy III and Berenice II, was the fourth Pharaoh of Ptolemaic Egypt from 221 to 204 BC. The decline of the Ptolemaic dynasty began under the reign of Ptolemy IV. Among the children of Ptolemy IV Philopator and his sister-wife Arsinoe III of Egypt was Ptolemy V Epiphanes, who married Cleopatra I Syra, daughter of Antiochus III the Great and Laodice III. Ptolemy IV's reign was inaugurated by the murder of his mother, and he was always under the dominion of favourites,