The Euler Characteristic V-E+F has an interesting history. It was initially stated that, for all polyhedra,
$$V(ertices)-E(dges)+F(aces)=2$$
and its proof was widely accepted, until people found counter-examples.
Imre Lakatos' book Proofs and Refutations has an imagined dialogue between teacher and student giving arguments and counter-examples leading to the correct formulation, which, he explains in his footnotes, traces the actual historical development of the statement and proof of the theorem.
