A further analysis of Cardano's main tool in the \textit{De regula aliza}: on the origins of the splittings (Q1643450)

Gerolamo Cardano (1501--1576) was an Italian scientist, whose interests ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler. Cardano was one of the most influential mathematicians of the Renaissance, and was one of the key figures in the foundation of probability and the pioneer of the binomial coefficients and the binomial theorem in the Western world. This paper studies related matters in relationship with the Cardano algebra, quadratic irrational numbers and selected kinds of binomials and trinomials. The subjects are discussed in connection with some of Cardano's favorite strategies in the \textit{Ars magna arithmeticae}, which is mainly devoted to the study of the shape of solutions. The paper also includes some of Cardano's contributions to the study of equations of the type \(x^3=a_1x+a_0\), as well as comments on the origin of the splittings.