  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.
