In order to prove that Euclid's Fifth Postulate was right, Saccheri used the reductio ad absurdum method; he considered the Parallel Postulate was false, thus being allowed the existence of a quadrilateral with right (a "normal" quadrilateral), acute or obtuse summit angles -- the Saccheri Quadrilateral. He named these last two hypotheses acute and obtuse angle hypotheses, AAH and OAH, respectively. He then proceeded to refute them, by finding contradictions in them, thus proving the Parallel Postulate right.

Regarding the OAH, i have found multiple versions about the method used by Saccheri. One of them (which can be found here) explains that Saccheri used Euclid's Proposition I.16 (how, though?), admitting, as the latter did, the infinitude of the straight line. Since the OAH contradicts this axiom (which is similar to Euclid's Second Postulate), for the curving line would eventually meet itself at some point, this hypothesis would be refuted. Very well. Makes sense to me. The hypothesis was false because it would lead to the inexistence of infinite lines, something that had been (wrongly, as we can now understand through elliptical geometry) admitted by Euclid as an axiom since the beginning.

This is where things get strange, because i have read someplace else that Saccheri did assume the infinitude of the straight line, as stated before, only this time it's said that he showed that the OAH implied the fifth postulate itself! And this is where I lose track of everything because every paper I read doesn't actually explain this step, they just assume it. This is one of the things I would like to know: what did Saccheri do in order to try to refute the OAH, how did he show that the OAH implied the fifth postulate.

If he used Euclid's Proposition I.16 and concluded that the OAH was inconsistent with it (i.e. with the infinitude of the straight line and also with the angle measures of the triangle), very well, I understand it all. Now, if he somehow showed that the OAH implies the fifth postulate, i ask: how did he do that?

On the other hand, he also tried to refute the AAH; for this, he brought up concepts about elements at infinity (check the last link) and i'm not quite sure how this would work as a refutation of this hypothesis. I know that not even Saccheri himself was too convinced about his work here; but what exactly was his idea? How does it work?

I'll be waiting for your replies. Thank you.

  • $\begingroup$ You can read Saccheri's work by yourself here: archive.org/download/girolamosacceri00saccuoft/… $\endgroup$ – Aretino Aug 24 '17 at 15:52
  • $\begingroup$ @Aretino thank you so much! I spent so much time trying to find an english version of Saccheri's papers and couldn't find a single complete one... Had i known about that website and i would have spared some of you from reading this post...sorry guys $\endgroup$ – onthecomplexplane Aug 24 '17 at 16:04

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