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?

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    $\begingroup$ This question might be better-suited to the History of Science and Mathematics StackExchange. $\endgroup$ – Blue Jun 20 '19 at 13:02
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    $\begingroup$ See Proclus on the Parallel Postulate for an extract from Proclus concerning the parallel postulate. $\endgroup$ – Mauro ALLEGRANZA Jun 20 '19 at 13:03
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    $\begingroup$ On what ground are you asserting that "Proclus himself thought that his own "axiom" was also provable" ? $\endgroup$ – Mauro ALLEGRANZA Jun 20 '19 at 13:29
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    $\begingroup$ @MauroALLEGRANZA On page 150 of Greenberg's book he gives said proof, from the beginning of the sentence "Proclus attempted to prove the parallel postulate as follows[...]". He then gives Proclus' supposed attempt to prove his axiom (not the equivalent Euclid's statement). $\endgroup$ – TheMountainThatCodes Jun 20 '19 at 13:33
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    $\begingroup$ @MauroALLEGRANZA Thanks for the discussion, if you wish to make a short answer summarizing these comments, I would be happy to accept it. $\endgroup$ – TheMountainThatCodes Jun 20 '19 at 14:01

See Proclus' Commentary, Prop XXIX : Proclus' axiom is "proved" from a principle used by Aristotle [De Caelo, I] :

"if from one point two straight lines forming an angle be produced indefinitely, the distance between the said straight lines produced indefinitely will exceed any finite magnitude."

Then Proclus proves Euclid's Fifth postulate from his [Proclus] own axiom.

I think that all ancient attempts (at least up to Saccheri) to prove Euclid Fifth postlate was grounded on the belief that it is (obviously true but) not "evident"; thus, it must be proved on more evident principles.

Only after Saccheri, Lambert and modern non-euclidean geometry the independence of Euclid Fifth postulate was understood.

Thus, prior to this understanding, the idea of its equivalence with other principle was , if not meaningless, quite "uninteresting".

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