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 plane curve Q defined by f is
non-singular, then f has exactly 8 such representations, up to
equivalence. They correspond to those real 2-torsion points of the
Jacobian of the Q which are not represented by a
conjugation-invariant divisor on Q.