Has any error ever been found in Euclid's elements since its publication? Or it is still perfect from the view point of modern mathematics.
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7Yes, there was this thing about the intersection of circles. – Asaf Karagila Jun 24 '14 at 01:36
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5The list of axioms is highly incomplete. – André Nicolas Jun 24 '14 at 01:46
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1What's this about the intersection of circles? – user7530 Jun 24 '14 at 01:47
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Don’t forget Pasch’s Axiom! – Berrick Caleb Fillmore Jun 24 '14 at 03:50
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1In his history Mathematical Thought, from Ancient to Modern Times, Morris Kline discusses this at some length. – David Mitra Jun 24 '14 at 15:48
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It depends on what you mean by error. The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof. Further Euclids definitions, although nice sounding, are never used. We now know that there must be undefined terms in an axiomatic system. Finally Euclid did not treat the issue of order. Hilbert's axioms are a completion of Euclid in that he gives all undefined terms and all axioms necessary for geometry. Ironically, Euclid was right about parallels, the one thing for which he was criticised for centuries.
Rene Schipperus
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It's easier to be right about the big things than the details. Poor Euclid was never told what an order was, but he did know what relationships characterized his geometry. – Ryan Reich Jun 24 '14 at 03:12
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Yes thats very true. Euclid was a giant, we must not forget that. – Rene Schipperus Jun 24 '14 at 03:19
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1Just to be clear, since on the internet it's not always obvious: I was not trying to be sarcastic to you, just a little snarky on the whole. – Ryan Reich Jun 24 '14 at 03:22
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@ReneSchipperus: Giant? More like lucky, his stuff survived. If one compares Archimedes with Euclid's Elements, it is clear that Archimedes is in a different class. Apollonius also, and perhaps Pappus. – André Nicolas Jul 06 '14 at 01:15
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As pointed out by @Asaf, the very first theorem, Book I, Proposition 1, on the construction of an equilateral triangle, assumes two circles intersect but there is no axiom to ensure that.
The book Geometry: Euclid and Beyond by Hartshorne discusses this in section 11.
lhf
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Actually thats something i have always wondered about, can you show that without using completeness ? – Rene Schipperus Jun 24 '14 at 02:29
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3@ReneSchipperus: If you look at the circles $x^2 + y^2 = 1$ and $(x-1)^2 + y^2 = 1$, they intersect at the points $(1/2, \pm \sqrt{3}/2)$, and hence if you just work over $\mathbb Q$ rather than over $\mathbb R$, they have no point in common. So some form of completeness is needed. – user160609 Jul 28 '14 at 03:24
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1Yes you can show I 1 without completeness if you have the parallel postulate. You only need the field to be Pythagorean to have the existence of equilateral triangle on a given base. If you do not have the parallel postulate you can use the circle circle axiom as Euclid implictly did which is weaker then completeness . – Julien Narboux Jul 01 '17 at 21:00
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I ran into such an error in Book VII and asked about it here. The beautiful 2006 Monthly article (summarized in the accepted answer) that explains the details has unfortunately since been moved behind a pay-wall. If someone knows another link to an accessible version, please edit it in.
Vincent
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