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?



Did ancient mathematicians know Euler's characteristic for convex polyhedra?
there is no doubt the answer to your question is "no"; for a wonderful and scholarly recent book on the whole story, see [Euler's Gem: The Polyhedron Formula and the Birth of Topology](http://books.google.com/books/p/princeton?id=KUYLhOVkaV4C&printsec=frontcover&source=gbs_ViewAPI&hl=en) by David Richeson.



> 
> They all missed it. The ancient Greeks -- mathematical luminaries 
>  such as Phythagoras, Theaetetus, Plato, Euclid, and
>  Archimedes, who where infatuated with polyhedra -- missed it. 
>  Johannes Kepler, the great astronomer, so in awe of the beauty of polyhedra that he based an early model of the solar system on them, missed it.
>  In his investigation of polyhedra the mathematician and philosopher René
>  Descartes was but a few logical steps away from discovering it, yet he
>  too missed it. These mathematicians, and so many others, missed a
>  relationship that is so simple that it can be explained to any
>  schoolchild, yet is so fundamental that it is part of the fabric of
>  modern mathematics.
> 
> 
> The great Swiss mathematician Leonhard Euler did not miss it. On
>  November 14, 1750, in a letter to his friend, the number theorist
>  Christian Goldbach, Euler wrote, "It astonishes me that these general
>  properties of stereometry have not, as far as I know, been noticed by
>  anyone else".
> 
> 
> 


Centuries later, we remain astonished.


