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
Ptolemaic cult of Alexander the Great who had succeeded to temporarily evict her son from Alexandria. As women could not occupy a supreme priesthood in the Greek world, she had to content herself with placing one of her supporters in the post, as a public sign of her new dominance. After Artemidorus, however, the name of Ptolemy IX was subsequently added in the papyrus, which means that he managed to return to Alexandria in the same year. In 107 BC, Cleopatra III managed to expel Ptolemy IX for good from Alexandria, and raised her second son, Ptolemy X, to the throne as her co-ruler and priest