Has any error ever been found in Euclid's elements?

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.


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.


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.