  Hesse claimed that an irreducible projective hypersurface in $\PP^n$ defined
by an equation with vanishing hessian determinant is necessarily a cone. Gordan
and Noether proved that this is true for $n\leq 3$ and constructed
counterexamples for every $n\geq 4$. Gordan and Noether and Franchetta gave
classification of hypersurfaces in $\PP^4$ with vanishing hessian and which are
not cones. Here we translate in geometric terms Gordan and Noether approach,
providing direct geometrical proofs of these results.
