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In Euclid's elements, some of the theorems (e.g. SAA congruence) can be proven using the parallel postulate, much easier than without it. But it seems that Euclid has intentionally avoided using it, when possible.

  1. Am I right?

  2. What is the reason behind this choice?

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    $\begingroup$ Possible duplicate of Why were geometers dissatisfied with the parallel postulate? Presumably because he disliked postulating it and hoped someone would prove it, hence established as much as possible without it in preparation. $\endgroup$ – Conifold Oct 23 '17 at 23:03
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    $\begingroup$ See the very similar post why-did-the-ancients-hate-the-parallel-postulate. $\endgroup$ – Mauro ALLEGRANZA Oct 24 '17 at 13:42
  • $\begingroup$ How do you prove SSA congruence using Euclid's fifth postulate, and more easily than Elements I, 26? $\endgroup$ – Edward Porcella Oct 24 '17 at 16:01
  • $\begingroup$ SAA, not SSA! 1. Prove that sum of the angles of any triangle is 180 degrees. 2. Use 1 to reduce SAA to ASA. QED! $\endgroup$ – user2321323 Oct 25 '17 at 7:20
  • $\begingroup$ Thanks, SSA was a typo. So Euclid would have had to delay SAA until after I, 32-- the sum of angles in a triangle is two right angles. I suspect he wanted to wrap up triangle congruence before moving on to parallelograms and quadrature of rectilinear figures. Even so, he achieves the latter not in Bk 1 (the great I, 47 is kind of a consolation prize), but only at the end of Bk 2. But can you say exactly how you use I, 32 to reduce SAA to ASA? $\endgroup$ – Edward Porcella Oct 25 '17 at 16:27
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If he avoided use of that postulate where he saw that it is possible to do like that he did it most likely in the spirit of proving results with minimal assumptions, so that result proven in such a way will hold for some axiom systems different than the one used by him in his books, and different in the sense that those alternative axiom systems will contain some of his axioms but not all, so, if he proved that something is true without the usage of parallel postulate then he proved also that it is true with the usage of parallel postulate, so he most likely did it all in the spirit of trying to be as general as possible, and aware of the fact that other axioms systems and geometries are also possible.

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    $\begingroup$ Is there any evidence that Euclid or other mathematicians of his age have ever thought of any other axiomatic system? You're suggestion seems so "modern" to me. $\endgroup$ – Behzad Oct 23 '17 at 18:52
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    $\begingroup$ @Behzad Haha, yes, it looks modern, but "ancient" and "modern" mathematicians, especially "big ones", are not much different as a people (modulo global changes in society), so the spirit is almost the same, there is no reason to think that Euclid did not see that other axiomatic approaches are possible. $\endgroup$ – Antoine Pal Adeen Oct 23 '17 at 18:57
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    $\begingroup$ If Euclid and other ancient mathematicians thought of other axiom systems, why did it take another 2000 years before non-Euclidean geometries were developed? $\endgroup$ – Qudit Oct 23 '17 at 19:51
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    $\begingroup$ @Quidit: the vast majority of ancient western scientific literature was deliberately and permanently destroyed more than 1000 years ago. So the claim that it took "another 2000 years" to develop a non-Euclidean geometry is already making a statement about history whose truth is unknown. (I suspect Archimedes would have just said "yes, obviously there are non-Euclidean geometries" and would then have drawn you a picture of the Poincare disc.) For all we know Diophantus could have made what we now call Fermat's conjecture in a lost work. So the inference implied by your comment is vacuous. $\endgroup$ – Rob Arthan Oct 23 '17 at 20:15
  • $\begingroup$ From my philosophy classes, my understanding was not an awareness of other axiom systems, but a general opinion that the fifth postulate stood out from the rest as simply not being "obviously" true. $\endgroup$ – user2069 Oct 23 '17 at 21:13
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I cannot answer with confidence as I havent extensively studied his work. I can think of a number of possible reasons, however.

  1. Maybe he didnt think of that solution. Genius or not, no one sees everything.
  2. Maybe that solution required a theorem he had yet to prove, and he thought it unnecessary to go back and change it. Its easy to judge in hindsight bias when you already know the theorem.

    • Maybe unbeknownst to you, the theorem you would have to invoke on some level requires the theorem you would like to prove with it, in order to itself be proven, and it would be circular reasoning to do it your way.
  3. To avoid unnecessary assumption. As Antoine pointed out, the whole point of his work is to minimize assumption.

  4. Maybe he foresaw that if one postulate were proven wrong at a later date, it would undermine your "simpler proof", whereas taking an alternate approach prevented this possibility.
  5. Maybe he didnt have the utmost confidence in that postulate.
  6. Maybe he wanted to keep things as generalizable as possible so that it could be applicable to a different set of axioms and the like.
  7. Maybe he wanted to exercise his genius instead of taking the "easy path".
  8. Maybe he wanted to demonstrate the approach/thinking/technique for later scholars benefit, rather than just arrive at a solution.
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  • $\begingroup$ Just about your seventh reason: please see my comment for @Antoine. $\endgroup$ – Behzad Oct 23 '17 at 18:55
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    $\begingroup$ My reasons are explicitly numbered. Why are you being deliberately vague? You couldnt have glanced up to find the number? Make comments, dont cite them from elsewhere. I dont much care to circumnavigate the internet to find a comment you cant be bothered to make. Why should I be bothered to respond? $\endgroup$ – CogitoErgoCogitoSum Oct 23 '17 at 19:50
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    $\begingroup$ Maybe Euclid HAD thought about other axioms. He created them, after all, he should have been able to recognize the fact that he could have written anything down. Logical self-consistency was essential to his work. He wanted to capture reality as he knew it, which might be why he didnt discuss alternatives. If he had thought of other axiomatic systems, he might have said they were absurdities. He could have - and maybe he did - derive other systems he concluded didnt reflect reality, and rejected them. To say he didnt think of it at all seems even more absurd to me, a man of his genius. $\endgroup$ – CogitoErgoCogitoSum Oct 23 '17 at 19:54
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    $\begingroup$ It seems perfectly natural to me that if youre going to invent math, making up your own axioms, youre going to be taken down a multitude of paths that either lead to contradictions or lead to conclusions that seem unreasonable. Do you think he should have published these "mistakes"? $\endgroup$ – CogitoErgoCogitoSum Oct 23 '17 at 20:00
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    $\begingroup$ I'm surprised and shocked how an unintended small mistake can makes someone so annoyed. I honestly think that wasn't any big deal. $\endgroup$ – Behzad Oct 23 '17 at 20:01
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Euclid does not call on his fifth postulate until $I, 29$, where he cannot do without it. It is not needed until the treatment of parallels, which begins at $I, 27$. The last of the triangle congruence theorems is $I, 26$. Euclid had some dramatic sense: it would be premature to bring postulate five onstage needlessly, and just moments before the scene that really requires it.

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Yes. Somebody calls "Absolute geometry" the set of propositions that are provable without the $5$-th postulate.

IMHO this is an because it seems less obvious than the others, at least as we learnt at school the version "In a plane given a line and a point outside the line, there exists one and only one line passing through the given point and parallel to the given line".

Which is honestly quite hard to understand because it implies an impossible infinite producing of the parallel line to "verify" it does not intersect the given one.

But Euclid does not use these words! He says, very smartly here

"Let the following be postulated: [...]

  1. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

This is much easier to digest: at least the producing stops at a certain point.

Hope it is useful

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  • $\begingroup$ -1: IMHO, your opinion is utterly devoid of value as it based on no knowledge whatsoever of the extensive criticism that Euclid's Elements have received from commentators since shortly after the books were written. You can't do useful history of mathematics by speculation. $\endgroup$ – Rob Arthan Oct 23 '17 at 19:17
  • $\begingroup$ ... so, looking no further than en.wikipedia.org/wiki/Parallel_postulate for some actual information rather than your opinions, how can you argue that Proclus in the 5th century and Omar Khayyam in the 11th century are "relatively recent" sources concerned with a "legendary" problem. $\endgroup$ – Rob Arthan Oct 23 '17 at 19:39

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