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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7$\begingroup$ Yes, there was this thing about the intersection of circles. $\endgroup$– Asaf Karagila ♦Jun 24, 2014 at 1:36
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5$\begingroup$ The list of axioms is highly incomplete. $\endgroup$– André NicolasJun 24, 2014 at 1:46
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1$\begingroup$ What's this about the intersection of circles? $\endgroup$– user7530Jun 24, 2014 at 1:47
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$\begingroup$ Don’t forget Pasch’s Axiom! $\endgroup$– Berrick Caleb FillmoreJun 24, 2014 at 3:50
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1$\begingroup$ In his history Mathematical Thought, from Ancient to Modern Times, Morris Kline discusses this at some length. $\endgroup$– David MitraJun 24, 2014 at 15:48
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2 Answers
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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.
answered Jun 24, 2014 at 1:49
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$\begingroup$ 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. $\endgroup$ Jun 24, 2014 at 3:12
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$\begingroup$ Yes thats very true. Euclid was a giant, we must not forget that. $\endgroup$ Jun 24, 2014 at 3:19
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1$\begingroup$ Just 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. $\endgroup$ Jun 24, 2014 at 3:22
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2$\begingroup$ Sorry I missed that entirely, how Sheldon Cooper of me. $\endgroup$ Jun 24, 2014 at 3:24
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1$\begingroup$ What was it Newton said about giants...I can quite remember. $\endgroup$ Jul 6, 2014 at 1:54
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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.
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$\begingroup$ Actually thats something i have always wondered about, can you show that without using completeness ? $\endgroup$ Jun 24, 2014 at 2:29
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2$\begingroup$ @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. $\endgroup$ Jul 28, 2014 at 3:24
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$\begingroup$ Yes 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 . $\endgroup$ Jul 1, 2017 at 21:00