  We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g
\in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The
usual addition law for elliptic curves and composition give rise to two binary
operations on the set of solutions. We show that a particular subset of the set
of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.
