Foundations of geometry different sets of axioms for Euclidean geometry, any of these can replace the Euclidean parallel postulate. The following partial list indicates some of these theorems that are of historical interest. Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. In Euclid's "Elements", the first 28 propositions and Proposition I.31 avoid using the parallel postulate,
Ptolemy Philadelphus (son of Cleopatra) Ptolemy Philadelphus (son of Cleopatra) Ptolemy Philadelphus (, "Ptolemy the brother-loving", August/September 36 BC – 29 BC) was a Ptolemaic prince and was the youngest and fourth child of Greek Ptolemaic Queen Cleopatra VII of Egypt, and her third with Roman Triumvir Mark Antony. Ptolemy was of Greek and Roman heritage. He was born in Antioch, Syria (now a part of modern Turkey). Ptolemy was named after the original Ptolemy II Philadelphus (the second Pharaoh of the Ptolemaic dynasty) and Cleopatra's intention was recreating the former Ptolemaic Kingdom. In late 34 BC, at the Donations of Alexandria, Ptolemy was made