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The Secret Formula: How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation

Fabio Toscano
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Daniel J. Curtin
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The story of how the cubic equation came to be solved should interest most mathematicians and many students of mathematics. The solution itself represents a major step forward in the rebirth of mathematics in Europe, springing from important earlier work in the Islamic world. The story, which includes the fraught relations among the16th century-Italians involved, is a fascinating one.
All that is really needed to appreciate the problem and to understand the form of the solution is a familiarity with algebra including the quadratic formula. The challenge, in modern terms, was to find the exact roots of a cubic equation from the coefficients using only the basic operations of arithmetic together with cube and square roots. In the Islamic world and for the Italians, the problem was given much more concretely. In addition, negative coefficients were not considered so our single problem was viewed as a cluster of related problems. In effect, \( x^{3} + ax=b \) and \( x^{3}=ax+b \) required different formulas.
Fabio Toscano, a science writer, lucidly presents not only the solutions of the cubic equation but also the history of the problem and the development of algebra in general from Diophantus, through the Islamic scholars, to Renaissance Italy. He briefly explains how the actual notation was used historically, but his decision to employ modern notation for details keeps his work accessible to non-experts. Similarly, he discusses the need for proofs for the results but generally omits them. This too ensures a broader audience; those wanting a deeper dive can easily find more historical detail is easily found in standard texts on the History of Mathematics.
The drama has two Italian heroes, Girolamo Cardano and Nicholas Tartaglia, and a strong cast of supporting characters. Toscano gives lively capsule biographies of many of these mathematicians.  He discusses the role of mathematical duels, in which each combatant posed problems to his rival, with large sums or even university positions at stake. Such rewards tempted mathematicians to keep their discoveries secret, reserving them as weapons in the next duel.
The conflict over the cubic equation went something like this. Around 1530, Cardano was working in algebra and planning to write a book. On learning that Tartaglia had solved one version of the cubic equation, he asked Tartaglia to allow its inclusion in the book. But Tartaglia refused, intending to publish it himself. After much back and forth Tartaglia gave a big hint in the form of a poem, which Cardano swore never to reveal. Years passed with no publication.  Meanwhile Cardano figured out all cases of the cubic. His student Ferrari used it to solve the fourth degree equation. Acting on new information, Cardano uncovered evidence that another Italian, Scipione del Ferro, had found the solution prior to Tartaglia. Cardano now felt free to print his solutions, with proofs, in his Ars Magna (1545), a work that proved fundamental in the development of algebra. Tartaglia was furious. He, Cardano, and Ferrari exchanged a volley of letters and accusations, each more insulting than the last. The dispute ended in a mathematical duel between Ferrari and Tartaglia. The outcome is still in dispute, but Tartaglia’s career was ruined.  When did Tartaglia solve the equation? How did Cardano find out? What clues led Cardano to deduce that another had done it first? Toscano brings this drama to life through the actual letters of the principles, as they swing from mutual admiration to animosity and beyond. Reading this is both informative and great fun.
Toscano weaves together his sources deftly to make the story as lively and exciting as a novel, with mathematics an organic part of the tale.
Dan Curtin is Professor Emeritus of Mathematics at Northern Kentucky University. His scholarly activity has largely been in the History of Mathematics, especially the development of Algebra from the Renaissance to the rise of the Calculus.