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 X Alexander I throne, and replacing the former sarcophagus with one made of glass. The citizens of Alexandria were so outraged at this act that they exiled him again almost immediately and attempted to wipe out all record of him. He would later be killed on his way to Cyprus. When Ptolemy X died, Ptolemy IX regained the throne. When Ptolemy IX died in 81 BC, Ptolemy X's wife Berenice III took over the throne for six months. Ptolemy X Alexander I Ptolemy X Alexander I (, "Ptolemaĩos Aléxandros") was King of Egypt from 110 BC to 109 BC and 107 BC till