A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse

Hesse claimed that an irreducible projective hypersurface in ${\text{PP}}^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\le 3$ and constructed counterexamples for every $n\ge 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in ${\text{PP}}^4$ 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.