# Who was the first person to prove the invariance of the Euler characteristic under triangulations?

Given a compact orientable surface $$S$$ and any triangulation where $$F$$ denotes the amount of triangles, $$E$$ denotes the amount of edges, and $$V$$ denotes the amount of vertices, we know that the Euler-characteristic of $$S$$ can be defined as $$\chi(S)=F−E+V$$

(in the sense that it doesn't matter what triangulation you give, the equation will still hold). My question is who the first person is that proved the invariance. I've tried look it up but couldn't find any definitive answers.

On wikipedia I saw that Euler's formula was proven a lot of times including a proof by Cauchy in 1811, however the formula doesn't directly proof the invariance of triangulations (I think). I've also read through a textbook proving it using the Poincare-Hopf theorem, but my guess will be that someone else proved this before the author of the textbook did (the textbook is "Algebraic topology by William Fulton, the proof is on page 113).

Does anyone know who the person was that proved this? Thanks in advance!

• It depends on what you exactly qualify as a "proof". The standards of rigor changed with time. A COMPLETE answer to your question occupies a whole book, and such book exists: I. Lakatos, Proofs and refutations. He analyses the proofs of this very theorem, as they developed. – Alexandre Eremenko Feb 26 at 23:53