Intuitive reason why the Euler characteristic is an alternating sum? The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups.  Since homeomorphic spaces have isomorphic homology groups, however, even the non-alternating sum of the ranks of the homology groups is an invariant.  So, is there an intuitive reason why the Euler characteristic should be defined as an alternating sum instead of a non-alternating sum -- aside from the fact that other theorems, such as Gauss-Bonnet, would break (or at least need to be re-worked)?  Is this related to rank being additive?  If so, then a historical question: what would have been the original motivation of Euler to consider an alternating sum?

The Euler characteristic can be computed from the number of cells of each dimension. The non-alternating sum (or other functions of the ranks) cannot.
The Euler characteristic has nice formulas when $X=X_1\cup X_2$, or when $X=X_1\times X_2$, or when $X$ is a bundle etc. The non-alternating sum (or other functions of the ranks) have not.
