  Because the problem of Apollonius is generally considered over the reals, it
suffers from variance of number: there are at most eight circles simultaneously
tangent to a given trio of circles, but some configurations have fewer than
eight tangent circles. This issue arises over other non-closed fields as well.
Using the tools of enriched enumerative geometry, we give two different ways to
count the circles of Apollonius such that invariance of number holds over any
field of characteristic not 2. We also pose the geometricity problem for local
indices in enriched enumerative geometry.
