  Magnitude is a numerical invariant of metric spaces introduced by Leinster,
motivated by considerations from category theory. This paper extends the
original definition for finite spaces to compact spaces, in an equivalent but
more natural and direct manner than in previous works by Leinster, Willerton,
and the author. The new definition uncovers a previously unknown relationship
between magnitude and capacities of sets. Exploiting this relationship, it is
shown that for a compact subset of Euclidean space, the magnitude dimension
considered by Leinster and Willerton is equal to the Minkowski dimension.
