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 II of Telmessos Telmessos, as he ruled from 206 BC to until at least 181 BC. According to surviving inscriptions at Telmessos, Ptolemy II didn’t seem to have a royal title nor it is clear his relationship with the Pharaohs in Alexandria. It seems likely that his family had relative autonomy from Ptolemaic control increased. As Ptolemaic power declined rapidly and dramatically outside of Egypt after the death of Ptolemy III Euergetes in 222 BC, probably Ptolemy II’s father with his family had the motive and opportunity for divorcing themselves from Ptolemaic suzerainty. At an unknown date during his father’s reign, Lysimachus with