Etienne B\'ezout on Elimination Theory

B\'ezout's name is attached to his famous theorem. B\'ezout's Theorem states that the degree of the eliminand of a system a $n$ algebraic equations in $n$ unknowns, when each of the equations is generic of its degree, is the product of the degrees of the equations. The eliminand is, in the terms of XIXth century algebra, an equation of smallest degree resulting from the elimination of $(n-1)$ unknowns. B\'ezout demonstrates his theorem in 1779 in a treatise entitled "Th\'eorie g\'en\'erale des \'equations alg\'ebriques". In this text, he does not only demonstrate the theorem for $n>2$ for generic equations, but he also builds a classification of equations that allows a better bound on the degree of the eliminand when the equations are not generic. This part of his work is difficult: it appears incomplete and has been seldom studied. In this article, we shall give a brief history of his theorem, and give a complete justification of the difficult part of B\'ezout's treatise.