  In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with
real coefficients is a sum of three squares of quadratic forms. His proof was
ahead of its time and used advanced methods from topology and algebraic
geometry. Up to now, no elementary proof is known. Here we present a completely
new approach. Although our proof is not easy, it uses only elementary
techniques. As a by-product, it gives information on the number of
representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show
that this number is 8 for generically chosen f, and that it is 4 when f is
chosen generically with a real zero. Although these facts were known, there was
no elementary approach to them so far.
