Cardano vs. Tartaglia: Duels and Betrayals on the Road to Solving Cubic Equations
The story of solving cubic equations is not just a mathematical challenge—it’s a real-life drama filled with betrayals, duels, and scientific discoveries that shaped modern mathematics. At the heart of the action are two 16th-century Italian mathematicians: Gerolamo Cardano and Niccolò Fontana, better known as Tartaglia (nicknamed “the Stammerer”). Their rivalry can be compared to some of history’s most famous feuds, like Edison and Tesla or Tupac and Biggie. It all began with the desire to crack one of the greatest mysteries of the time: finding a solution to cubic equations.
Today, high school students know how to solve quadratic equations using a formula that finds the roots of equations like ax2 + bx + c = 0. But in the 16th century, mathematicians were searching for similar solutions for higher-degree equations, especially cubic equations. Despite lacking modern symbolic algebraic language, scholars like Cardano and Tartaglia worked to solve these problems using geometric and verbal descriptions.
Algebra as we know it began to take shape only in the 17th century, but by then, many preliminary discoveries had been made over previous centuries. In the 16th century, equations were still expressed rhetorically, and negative numbers were not accepted. Cubic equations were written as “a cube and something equals a number,” which made them much harder to understand and solve. Mathematicians approached these equations geometrically, trying to break down cubic expressions into simpler forms.
The First Breakthrough: Scipione del Ferro
The first major step in solving cubic equations was made by Scipione del Ferro, a professor at the University of Bologna. He found a solution for equations of the form x3 + cx = d, where all coefficients are positive. These became known as “reduced cubic equations.” At the time, it was common practice to keep mathematical discoveries secret to use them in “mathematical duels”—competitions where scholars challenged each other with tough problems, and the winner was the one who solved the most. As a result, discoveries were often kept hidden for future advantage.
We know that del Ferro could solve reduced cubic equations because he passed his knowledge to his student Antonio Fior, who later boasted about his ability to solve such problems. Meanwhile, Tartaglia, a self-taught mathematician, found a solution to another type of cubic equation, one without a linear term. This led to a mathematical duel between Fior and Tartaglia in 1535. They exchanged thirty problems each, with a deadline of one and a half months. Fior, confident in his advantage, sent Tartaglia thirty reduced cubic equations. But just days before the deadline, Tartaglia cracked the solution and solved all the problems in two hours, while Fior couldn’t solve any of Tartaglia’s, leading to Fior’s defeat and loss of reputation.
Cardano’s Obsession and the Breaking of a Vow
Tartaglia’s discovery shocked the scientific community, as it was previously believed that solving cubic equations was impossible. Cardano, a successful physician and talented mathematician, was amazed by Tartaglia’s success and desperately tried to uncover his secret. For years, Cardano pleaded with Tartaglia to share his method for solving cubic equations, even swearing on the Bible to keep it secret. Finally, in 1539, Tartaglia gave in and revealed his method to Cardano, but did not provide proof of its correctness.
For Cardano, this was enough. Understanding the principle behind the method, he soon found a way to solve all types of cubic equations by transforming them into the reduced form. This was a major mathematical breakthrough, and Cardano, realizing the significance of the discovery, wanted to publish the results. However, he was bound by his oath to Tartaglia.
The situation changed in 1543, when Cardano discovered that del Ferro had solved reduced cubic equations long before Tartaglia. Cardano believed this freed him from his oath. In 1545, he published his work Ars Magna, where he detailed methods for solving cubic and even quartic equations, developed with his student Lodovico Ferrari.
The Fallout: Accusations and a Final Duel
Tartaglia was furious. He accused Cardano of breaking his oath and stealing his discoveries. Although Cardano acknowledged Tartaglia’s contribution in his book, it didn’t save him from public accusations. Cardano left the defense of his reputation to his student Ferrari, leading to another mathematical duel—this time between Tartaglia and Ferrari. The duel took place in Milan, Ferrari’s hometown, and ended with Tartaglia’s defeat. Tartaglia left the city, his reputation ruined.
While Cardano and Ferrari gained fame, Tartaglia died in poverty and obscurity, despite his many achievements in mathematics. The publication of Ars Magna is considered the beginning of modern algebra, and this discovery forever changed the mathematical world.
The Legacy: Beyond Cubic Equations
The story of cubic equations didn’t end with Cardano. Mathematicians continued to study higher-degree equations, but soon hit an insurmountable barrier. In the 19th century, Norwegian mathematician Niels Abel proved that general fifth-degree equations cannot be solved by radicals. His work was continued by Évariste Galois, who provided clear criteria for solving equations of any degree. Galois died in a duel at age 20, but his contribution to mathematics was immense and continues to influence research in the field.
Thus, the struggle to solve cubic equations played a key role in the development of mathematics, paving the way for modern methods of algebraic analysis and the study of higher-degree equations.