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The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations

Sara Confalonieri
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[Reviewed by
Joel Haack
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Sara Confalonieri’s The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations: Gerolamo Cardano’s De Regula Aliza, based on the first part of her 2013 dissertation, is a welcome addition to the literature on Gerolamo Cardano. Its focus is on the portions of one of Cardano’s lesser known works, De Regula Aliza, that are related to Cardano’s publication of an algebraic procedure and formula to solve the general cubic equation. Of concern to Cardano and many other mathematicians since his time, this formula can (and, we know now, must) present the real roots to some cubic equations in terms of complex numbers, the so-called casus irreducibilis. (Cardano did not use this term; the earliest use which Confalonieri has found is in a 1781 contest of the Accademia in Padova.)

To give context for her discussion of the Aliza, Confalonieri provides a brief biography of Cardano, the famous story of the controversy with Tartaglia over the solution of the cubic (including the poem in which Tartaglia provided hints to Cardano regarding the solution), and a summary of Cardano’s mathematical writing. She also provides the mathematical context of solutions of quadratic, cubic, and quartic equations and a discussion of Galois theory that shows that the casus irreducibilis is unavoidable. (Confalonieri promises a study of the history of an understanding of this unavoidability in a forthcoming work.)

To appreciate Cardano’s work in the Aliza, Confalonieri presents first a summary of three more of Cardano’s mathematical works. In fact, the discussion here of the Ars Magna (first published in 1545), Cardano’s best known work, can serve as a useful commentary on that work. Her focus is naturally on the solutions of cubic and quartic equations, with chapters 1, 4–8, 11–23, 25, 37, and 39 considered in some detail. Of course, Cardano considers numerous cases of cubic equations, as he requires the coefficients to be non-negative.

Two notions that Cardano employs to solve cubic equations are splittings and transformations. To illustrate the first of these, we can take a look at a key notion in the development of the formula to solve, for example, \(x^3+a_1x=a_0\), where the coefficients are positive. Write \(x=y-z\), then expand the equation and rewrite it to obtain \((y^3-z^3)-3yz(y-z)=a_0-a_1(y-z)\). Then split the equation, asking that simultaneously \(y^3-z^3=a_0\) and \(3yz(y-z)=a_1(y-z)\). Solving this system of equations is possible because it leads to a quadratic equation whose roots are \(y^3\) and \(z^3\), and \(x\) can then be recovered from the difference of \(y\) and \(z\).

An example of one of Cardano’s transformations is to convert an equation of the form \(x^3=a_2x^2+a_0\) to one of the form we had just solved. The substitution \(x=\sqrt[3]{a_0}^2/y\) leads to the equation \(y^3 + \sqrt[3]{a_0}a_2y=a_0\), which we can solve from our previous work. Then we recover \(x\) from \(y\).

Confalonieri then looks at some of Cardano’s earlier work on solving cubic equations, before he has the full solution in hand, beginning with a description of the contents of the Practica Arithmeticæ (first published in 1539), with attention especially to chapter LI. This work appeared before Cardano received Tartaglia’s poem; it is interesting to see Cardano’s understanding of the solution of cubics at this early period. She then turns to a more lengthy discussion of the Ars Magna Arithmeticæ (first published in Cardano’s Opera Omnia in 1663, but which would seem to be written between 1539 and 1542, before the Ars Magna), especially chapters XVIII–XXXVII and XXXIX.

De Regula Aliza (first published in 1570) is difficult to understand, being something of a miscellany of topics and including some of the more obscure writings of Cardano. (He could be obscure for several reasons: his Latin is very awkward in certain chapters, the proofreading of the text was apparently minimal, and the mathematics he discussed was sometimes left inadequately explained.) The Aliza seems to have been written over an extended period of time, perhaps the 1530s through at least 1550. It seems to have been published without the editing that could make it a logical whole and less episodic.

Confalonieri provides a summary of each of the sixty chapters of the Aliza. She is successful in identifying a number of themes that appear in different places in the Aliza and using this to sort the chapters. She naturally chooses to focus on those that relate to solutions of cubic equations, especially Cardano’s necessarily failed attempt to avoid the casus irreducibilis. With Confalonieri’s guidance, we watch Cardano unsuccessfully attempt to find other splittings and transformations that might avoid the need for imaginary numbers.

There are several other topics discussed by Confalonieri, which are important in their own right, but it would take too much space to summarize them appropriately here. Cardano consideres the possible form of solutions for cubic equations, looking for a classification like that of Book X of Euclid. He was eager to find general solutions rather than merely particular ones. Confalonieri discusses the idea of “general” versus “particular,” while showing that Cardano is not himself particularly clear or consistent on their usage. She also considers how Cardano uses geometry to justify his calculations. We even see in the Aliza a passage in which Cardano considers the possibility of redefining multiplication so that a negative number times a negative number is negative.

Another feature of the book are the tables. These include a 17-page list of all the numerical cubic equations considered in Practica Arithmeticæ, Ars Magna Arithmeticæ, Ars Magna, and De Regula Aliza. Other tables summarize the substitutions in cubic equations in Ars Magna, the shapes of irrational solutions to cubic equations considered in Ars Magna Arithmeticæ, the correspondences between Ars Magna Arithmeticæ and Ars Magna in the treatment of cubic equations, and the possible splittings of cubic equations in the Aliza.

There is much to absorb and study in this treatise. And I look forward to reading the results of Confalonieri’s future work related to the subjects presented here.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.