The casus irreducibilis in Cardano's Ars Magna and De Regula Aliza

摘要

In Cardano's classification in the Ars Magna (1545, 1570), the cubic equations were arranged in thirteen families. This paper examines the well-known solution methods for the families and and then considers thoroughly the systematic interconnections between these two families and the remaining ones and provides a diagram to visualize the results clearly. In the analysis of these solution methods, we pay particular attention to the appearance of the square roots of negative numbers even when all the solutions are real-the so-called casus irreducibilis. The structure that comes to light enables us to fully appreciate the impact that the difficulty entailed by the casus irreducibilis had on Cardano's construction in the Ars Magna. Cardano tried to patch matters first in the Ars Magna itself and then in the De Regula Aliza (1570). We sketch the former briefly and analyze the latter in detail because Cardano considered it the ultimate solution. In particular, we examine one widespread technique that is based on what I have called splittings.