Abstract

In the framework of the De Regula Aliza, Cardano paid much attention to the so-called splittings for the family of equations $$x^3 = a_1x + a_0$$ x3=a1x+a0 ; my previous article deals at length with them and, especially, with their role in the Ars Magna in relation to the solution methods for cubic equations. Significantly, the method of the splittings in the De Regula Aliza helps to account for how Cardano dealt with equations, which cannot be inferred from his other algebraic treatises. In the present paper, this topic is further developed, the focus now being directed to the origins of the splittings. First, we investigate Cardano’s research in the Ars Magna Arithmeticae on the shapes for irrational solutions of cubic equations with rational coefficients and on the general shapes for the solutions of any cubic equation. It turns out that these inquiries pre-exist Cardano’s research on substitutions and cubic formulae, which will later be the privileged methods for dealing with cubic equations; at an earlier time, Cardano had hoped to gather information on the general case by exploiting analogies with the particular case of irrational solutions. Accordingly, the Ars Magna Arithmeticae is revealed to be truly a treatise on the shapes of solutions of cubic equations. Afterwards, we consider the temporary patch given by Cardano in the Ars Magna to overcome the problem entailed by the casus irreducibilis as it emerges once the complete picture of the solution methods for all families of cubic equations has been outlined. When Cardano had to face the difficulty that appears if one deals with cubic equations using the brand-new methods of substitutions and cubic formulae, he reverted back to the well-known inquiries on the shape of solutions. In this way, the relation between the splittings and the older inquiries on the shape of solutions comes to light; furthermore, this enables the splittings to be dated 1542 or later. The last section of the present paper then expounds the passages from the Aliza that allow us to trace back the origins of the substitution $$x = y + z$$ x=y+z, which is fundamental not only to the method of the splittings but also to the discovery of the cubic formulae. In this way, an insight into Cardano’s way of dealing with equations using quadratic irrational numbers and other selected kinds of binomials and trinomials will be provided; moreover, this will display the role of the analysis of the shapes of solutions in the framework of Cardano’s algebraic works.