  A surface that is the pointwise sum of circles in Euclidean space is either
coplanar or contains no more than 2 circles through a general point. A surface
that is the pointwise product of circles in the unit-quaternions contains
either 2, 3, 4, or 5 circles through a general point. A surface in a
unit-sphere of any dimension that contains 2 great circles through a general
point contains either 4, 5, 6, or infinitely many circles through a general
point. These are some corollaries from our classification of translational and
great Darboux cyclides. We use the combinatorics associated to the set of low
degree curves on such surfaces modulo numerical equivalence.
