  There is a general notion of the magnitude of an enriched category, defined
subject to hypotheses. In topological and geometric contexts, magnitude is
already known to be closely related to classical invariants such as Euler
characteristic and dimension. Here we establish its significance in an
algebraic context. Specifically, in the representation theory of an associative
algebra A, a central role is played by the indecomposable projective A-modules,
which form a category enriched in vector spaces. We show that the magnitude of
that category is a known homological invariant of the algebra: writing chi_A
for the Euler form of A and S for the direct sum of the simple A-modules, it is
chi_A(S, S).
