Article ID: CBB001553270

The casus irreducibilis in Cardano's Ars Magna and De Regula Aliza (2015)


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 x3+a1x=a0 and x3=a1x+a0 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.

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Authors & Contributors
Cardano, Girolamo
Confalonieri, Sara
Breathnach, C. S.
Canziani, Guido
Baldi, Marialuisa
Nenci, Elio
Bollettino di Storia delle Scienze Matematiche
Isis: International Review Devoted to the History of Science and Its Cultural Influences
Journal of Medical Biography
Studies in History and Philosophy of Science
Physis: Rivista Internazionale di Storia della Scienza
British Society for the History of Mathematics Bulletin
Franco Angeli Editore
European Mathematical Society
Scribe Publications
Edizioni dell'Orso
ACMRS (Arizona Center for Medieval and Renaissance Studies)
Natural philosophy
Primary literature (historical sources)
Equations and formulae
Probability and statistics
Cardano, Girolamo
Stifel, Michael
Galilei, Galileo
Vanini, Giulio Cesare
Faulhaber, Johann
Briggs, Henry
Time Periods
16th century
17th century
15th century
18th century
Early modern
Milan (Italy)

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