How did Saccheri prove Euclid's Fifth Postulate?

[diogo_sg]

Hello people. I've been reading some papers online (if any links are needed, please let me know) regarding the foundation of the non-euclidean geometries, but i just can't figure out one or two details about Saccheri's contribution to said matter. In his attempts to prove the Parallel Postulate using the reductio ad absurdum method, according to which he designed the Saccheri Quadrilateral, he disposed of two of the three hypotheses: the acute angle and obtuse angle ones. I understand the basis of the Quadrilateral and "how it works" and why the right angle hypothesis is equivalent to the Parallel Postulate. My question is about how Saccheri proved the other two hypotheses wrong. All the papers i find on the subject don't go too deep into it and provide almost no mathematical proof, only focusing on the "theoretical part" of the proofs. Plus, the only edition of Saccheri's original paper that's available online is in latin and my latin skills are, well, inexistent. So if anyone is willing to share their knowledge or some obscure paper on the matter, please be my guest; i'd be extremely grateful for it, because this has been troubling my mind for several weeks as of now. Thank you all.

P. S. I'd like to point out that Saccheri never actually proved the Parallel Postulate, although the title of this post may make it seem like he did.

 

[fresh_42]

According to Jean Dieudonné,

My question is about how Saccheri proved the other two hypotheses wrong.

this question cannot be answered as

When he finally came to the conclusion that the two hypotheses were to be rejected, it was not because he had encountered the expected formal contradictions, but because he came to conclusions which he did not consider acceptable.

If your German is better than your Latin, you might be interested in
https://archive.org/details/dietheoriederpar00stuoft

 

[diogo_sg]

According to Jean Dieudonné,

this question cannot be answered asIf your German is better than your Latin, you might be interested in
https://archive.org/details/dietheoriederpar00stuoft

But Saccheri did try to prove the 'strange' hypotheses wrong, didn't he? Even though he didn't succeed in his attempts, i would like to know what exactly did he do (that is, the propositions he came up with, which i cannot find anywhere online) to dismiss the two odd hypotheses. If i do recall, he tried to refute the obtuse angle hypothesis using Euclid's Propositions I.16 and I.18, these two being based on the notion that straight lines are infinite; with this, Saccheri was able to prove said hypothesis "wrong", but the way he worked things out is an unsolved mystery to me. The same goes with the acute angle hypothesis and the "elements at infinity" thing.

Regarding the german, I'm not a speaker either, but i do appreciate the link; i'll try to find an english version of that paper.

 

[fresh_42]

Dieudonné writes, that Saccheri basically developed parts of a non Euclidean geometry without recognizing it. The main reference of these early achievements based on the work of the Arabic scientist Nasir ad-din at Tusi (1201-1274) of the 13th century, so maybe it could help to search for this. Apparently Saccheri, Lambert and Legendre knew this work. The subject of the parallel postulate even dates back to the year ≈1000 and Alhazen (see also the links there): On the eliminations of doubts in the book of Euclid on the elements.

Unfortunately Dieudonné only quotes Engel and Stäckel and simply writes "Saccheri deduced long and extensively conclusions from the hypothesis both angles were acute and then both angles were obtuse ..." with the original Latin paper as source - same as Wikipedia. But some sources (Latin, English, German) are mentioned on the German Wikipedia page: https://de.wikipedia.org/wiki/Saccheri-Viereck
Perhaps you can go on from there.

 

[diogo_sg]

Dieudonné writes, that Saccheri basically developed parts of a non Euclidean geometry without recognizing it. The main reference of these early achievements based on the work of the Arabic scientist Nasir ad-din at Tusi (1201-1274) of the 13th century, so maybe it could help to search for this. Apparently Saccheri, Lambert and Legendre knew this work. The subject of the parallel postulate even dates back to the year ≈1000 and Alhazen (see also the links there): On the eliminations of doubts in the book of Euclid on the elements.

Unfortunately Dieudonné only quotes Engel and Stäckel and simply writes "Saccheri deduced long and extensively conclusions from the hypothesis both angles were acute and then both angles were obtuse ..." with the original Latin paper as source - same as Wikipedia. But some sources (Latin, English, German) are mentioned on the German Wikipedia page: https://de.wikipedia.org/wiki/Saccheri-Viereck
Perhaps you can go on from there.

Thanks a lot for the links. I'll be sure to check them as soon as possible

 

[diogo_sg]

So, i checked the links and realized that perhaps i didn't formulate my question properly. So i'll try to be more specific this time.
Saccheri admitted that the Parallel Postulate was false, thus being possible the existence of a quadrilateral with acute or obtuse summit angles. He named these two hypotheses acute and obtuse angle hypotheses, AAH and OAH, respectively. He then proceeded to refute them.
Regarding the OAH, i have found multiple versions about the method used by Saccheri. One of them (which can be found here https://books.google.pt/books?id=2w...q=saccheri obtuse angle contradiction&f=false) explains that Saccheri used Euclid's Proposition I.16, 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 (https://books.google.pt/books?id=-U...AA#v=onepage&q=acute angle hypothesis&f=false) 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.
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 hope i explained it better that i had before. I'll be waiting for new replies. Thank you

 

[diogo_sg]

I posted this question on a maths forum, hoping i might get more specific answers. Now i don't know whether i should delete this one or not.
Either way, i'll leave the link to my new post: https://math.stackexchange.com/ques...-saccheri-try-to-prove-the-parallel-postulate
Thanks