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 II of Telmessos a paternal first cousin called Antipater Epigonos and likely he had a paternal second cousin called Epigonos. Through his father, Ptolemy II was a direct descendant of Lysimachus who was one of the Diadochi of the Greek King Alexander the Great who was King of Thrace, Asia Minor and Macedonia and Ptolemy I Soter another of the Diadochi of the Greek King Alexander the Great who was the founder of the Ptolemaic dynasty of Ancient Egypt and his wife, Berenice I of Egypt. Berenice I was the great-niece of the powerful Regent Antipater, through her maternal grandfather Cassander, the brother