  In 1888 Hilbert showed that every nonnegative homogeneous polynomial with
real coefficients of degree $2d$ in $n$ variables is a sum of squares if and
only if $d=1$ (quadratic forms), $n=2$ (binary forms) or $(n,d)=(3,2)$ (ternary
quartics). In these cases, it is interesting to compute canonical expressions
for these decompositions. Starting from Carath\'eodory's Theorem, we compute
the Carath\'eodory number of Hilbert cones of nonnegative quadratic and binary
forms.
