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 Philadelphus in the Library of Alexandria Ptolemy II Philadelphus in the Library of Alexandria Ptolemy Philadelphus in the Library of Alexandria is an 1813 oil on canvas painting by Vincenzo Camuccini. It is now in the National Museum of Capodimonte in Naples. The work was commissioned by Napoleon I along with "Charlemagne Ordering Italian Scholars to Found the University of Paris", both for the central hall of Rome's Palazzo del Quirinale. It shows Ptolemy II Philadelphus in red robes among several philosophers in the Library of Alexandria. The artist had recently returned to Italy from Paris and decided to accept the commission. "Ptolemy" was then moved