The study of Guido Zappa reviews the main stages made, in the 19th century, for the determination of the formulae that should solve the general algebraic equations of the 5th and 6th order, special stress being laid on the contributions brought to the field by Francesco Brioschi. Thus, mention is being made in the beginning of the mathematicians Ruffini and Abel (and of the theorem bearing their names), of Galois (who established a general theory for the possible solving of a given equation through radicals) and also of the so-called “gruppo di Galois”. There follows a brief discussion on the elliptical functions (starting from the elliptical integrals), on modular equations, the names of Auguste Chevalier, Ott. Mossotti, Enrico Betti, etc., being here remembered. More ample space has been given to the contributions of Francesco Brioschi, and also to those of E. S. Bring, G. B. Jerrard and Hermite. Brioschi’s important study on the equation of multiplication – published in 1858 in “Annali di Matematica” – is also presented. A special section deals with the hyperelliptical functions as studied by Masche, Klein and Kronecker.