  David Hilbert proved that a non-negative real quartic form f(x,y,z) is the
sum of three squares of quadratic forms. We give a new proof which shows that
if the complex plane curve Q defined by f is smooth, then f has exactly 8 such
representations, up to equivalence. They correspond to those real 2-torsion
points of the Jacobian of Q which are not represented by a
conjugation-invariant divisor on Q.
