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History of Science and Mathematics

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    $\begingroup$ You can see in this post a similar discussion. $\endgroup$ Commented Dec 18, 2014 at 13:14
  • $\begingroup$ Many people did try to prove the parallel postulate from the other four. $\endgroup$ Commented Dec 19, 2014 at 15:04
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    $\begingroup$ @Jack M This is due to sloppy writing in elementary textbooks. Hartshorne in Euclid and Beyond gives detailed account of many attempted proofs, not one of them attempts to prove "from the other four". Most authors explicitly introduce extra axioms, even those who don't, like Saccheri, use first 28 propositions of Elements with all the synthetic inferences "from the diagram" about congruence, betweenness and intersections in their proofs. $\endgroup$ Commented Dec 21, 2014 at 2:04
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    $\begingroup$ @Willemien Since Ptolemy it was known that Euclid's mouthful can be replaced with "two lines parallel to a third are parallel to each other". They could replace it with that and put the matter to rest, but no. Saccheri showed that existence of a single rectangle, no matter how small, implies the parallel postulate. If that is not "self-evident" then neither is any of the 4 postulates. $\endgroup$ Commented Dec 21, 2014 at 2:09
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    $\begingroup$ I think Euclid is happy to infer from the diagram if it is intuitively clear that a small change in the diagram wouldn't change the relevant property (i.e. two circles that meet will still meet if one of them is moved slightly). But if an arbitrarily small change in the configuration (i.e. one of two parallel lines being 'turned' ever-so-slightly) could change the property (e.g. make them meet), then he is unwilling to infer from the diagram and wants to deduce (or assume) explicitly and verbally. $\endgroup$ Commented Sep 30, 2017 at 22:39