We consider three versions of Euclid's parallel postulate. The two most important are Euclid's own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair ...

A postulate is supposed to be self-evident, but this business about parallel lines struck a nerve with mathematicians. It didn’t seem as intuitively convincing — Euclid didn’t even invoke the Fifth ...

Later Euclid proved this, but only by using his fifth axiom. The parallel postulate was essential to Euclid in many of his proofs. But he could have chosen another axiom.