Comments by "EebstertheGreat" (@EebstertheGreat) on "How One Line in the Oldest Math Text Hinted at Hidden Universes" video.

If Euclid had not included his fifth postulate, his theory would have been too weak to prove most of the theorems he wanted to prove, and it never would have been an influential book in the first place. Then Archimedes or someone else would have come along and improved it by adding a similar postulate. There's really no way around it, and it wasn't a "choice" per se.
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@ansfridaeyowulfsdottir8095 WTF are you talking about? It's not only not "obvious" that this is true, it is in fact not true in some geometries. Which was the whole point of the video. Did you watch it?
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I think Proclus and his contemporaries were "right" in a certain sense. They all attempted to prove Euclid's V not from his other postulates but by introducing a new postulate that they considered conceptually simpler. When Proclus said that Euclid's V should be a theorem, he meant that it assumed more than was necessary. Proclus instead used the postulate that two rays from the same point get arbitrarily far apart, and from this he produced a valid proof of Euclid's V. Much later, John Playfair used an even simpler postulate, that given a line and a point not on it, there is only one line through the point parallel to the given line. I think this is much better, and it is what is usually taught today. From this postulate, you can again prove Euclid's V.
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@bowserpouncer64 Bro, it's literally impossible to prove. The equiconsistency of absolute and Euclidean geometry shows this beyond a shadow of a doubt.
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@bowserpouncer64 It's only true in Euclidean geometry, not in absolute geometry. So without the fifth postulate (or something equivalent), you can't prove it. Of course you can prove it if you assume it is true.
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