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A Modern Vision of the Work of Cardano and Ferrari on Quartics

Harald Helfgott (University of Bristol) and Michel Helfgott (East Tennessee State University)

The Great Art or the Rules of Algebra [1] by Gerolamo Cardano (1501-1576), published in 1545, is one of the great books in the history of mathematics. Most of it deals with cubic equations, whose algebraic solution had never appeared in print before Cardano wrote his masterpiece. Almost at the end of the book (chapter 39) we can find, also for the first time, a systematic approach to the solution of all quartics -- a method developed by Lodovico Ferrari (1522-1565), a disciple of Cardano.

Cardano was a remarkable physician and a first rate mathematician. Besides, as anyone who reads The Great Art (Ars Magna in Latin) will agree, he was an excellent writer. For instance, he wisely discusses quartics only after having analyzed biquadratic equations in chapters 1 and 24, and some special quartics in chapter 26. That is to say, Cardano starts with numerical examples of equations of the type $$x^{4}+ax^{2}+b=0$$ before dealing with numerical examples of the general case $$x^{4}+ax^{3}+bx^{2}+cx+d=0.$$ Indeed, one can understand Ferrari's method better after getting acquainted with the solution of biquadratics.

In Cardano's lifetime complex numbers were not considered bona fide numbers, neither were decimal approximations commonly used; thus, it is interesting to analyze parts of Ars Magna from a modern perspective. We have chosen all of Cardano's examples in chapter 39 related to quartics, dissected them in detail, and examined them through the lens of contemporary mathematics. We have not kept the original order, rather we have chosen to analyze problems I and X first (both lead to biquadratics); thereafter we discuss eight problems about quartics, starting from the ones that require less work.

We have benefited from T.R. Witmer's translation from Latin into English, and from Witmer's use of modern symbolism. Ars Magna was written using many idiosyncratic symbols, a notation somehow at the middle between the rhetorical algebra of Islamic mathematicians and modern day symbolic algebra.

Ars Magna Title Page

Harald Helfgott (University of Bristol) and Michel Helfgott (East Tennessee State University), "A Modern Vision of the Work of Cardano and Ferrari on Quartics," Convergence (February 2010), DOI:10.4169/loci003312