I'm reading Greenberg's "Euclidean and Non-Euclidean Geometries" and a few other sources on the history of the Parallel postulate. Now most of the failed proofs of Euclid's fifth axiom seem to be presented as proofs of Proclius' axiom, which we now of course know is equivalent to Euclid's if we assume the other axioms. My question is:
Who first realized that Euclid's fifth postulate i.e.
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
and the Parallel postulate stated as
If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other.
are equivalent? In particular, did Proclius think he had a "better" axiom, that implies the Euclid one (without realizing they are equivalent), or did he realize they are equivalent and he actually thought he was straight up proving Euclid's axiom?