TITLE: Did ancient mathematicians know Euler's characteristic for convex polyhedra?
QUESTION [32 upvotes]: The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid proof). To have a concrete question: Is there any reference to this formula in ancient mathematics?

REPLY [13 votes]: Once I was lecturing to high school students about this theorem. My proof began with the words: "suppose that the net is drawn on a surface of a rubber ball...".
One student asked: "Did rubber exist at the time of Euler?"
I think the moral of this story is that the very statement of the question was foreign to
ancient Greek mathematics. Did they have a definition of ARBITRARY polytope? Or even an
arbitrary convex polytope? I doubt it. (See this related post).
Descartes proof was earlier than that of Euler, and his statement was in terms of
solid angles (which is equivalent to the Gauss-Bonnet theorem, but stated
in terms of elementary geometry). I think it was Euler who recognized for the first time
that we have a topological fact here. And this was a great insight. Like the problem of
the same Euler about the Konigsberg bridges. It is not difficult, but before Euler these
things apparently were not considered part of mathematics. That's why these questions
were not raised in antiquity. 
EDIT. Many things were not discovered for the simple reason that no one was looking for these things. Let me give a more recent example. Could Fermat, Descartes, Huygens, Euler,
etc. discover linear programming and simplex method? There is no doubt they could. After all, this is elementary mathematics which can be easily taught in high school.
But look what had really happen. Fourier had a paper on systems of linear inequalities
(where he invented what is called Fourier-Motzkin elimination nowadays). And in the end
of XIX century (!) the editor of Complete works of Fourier had to APOLOGIZE for including this paper: he wrote that "every work of such a grand master has to be included", where
"every" evidently means "even such trivial nonsense":-)
And linear programming became a part of mainstream mathematics only during WWII, when it
was really needed.