Amathematician and physician, Girolamo or Geronimo Cardano lived a turbulent personal and professional life, and became embroiled in a conflict over cubic equations so full of drama and surprises that it sounds more like a movie script than an incident from the history of mathematics. He was also one of the first mathematicians to conceive the idea that negative numbers have square roots, but lacked the conceptual framework for understanding these imaginary numbers.
Born in Pavia, Italy, on September 24, 1501, Cardano was the illegitimate son of Fazio Cardano and Chiara Micheri. The father was a successful lawyer and friend of Leonard da Vinci (1452-1519), but the fact that his parents did not marry until after his birth—and the father did not begin living with the family until Cardano was seven years old—provided a constant source of stigma for the young Cardano.
Cardano studied mathematics, astrology, and the classics at the University of Pavia, and went on to earn his doctorate in medicine there in 1526. He then began his medical practice in Saccolongo, a town near Padua, and in 1531 married Lucia Bandareni, with whom he had two sons and a daughter. Cardano supplemented his income by teaching mathematics at a school in Milan from 1534 to 1536, but soon his medical practice had become so successful that he was able to devote himself fully to it. His interest in mathematics continued for the rest of his life, however, and in his first mathematical work, Practica arithmetice et mensurandi singularis (1539), he proved himself adept at solving cubic equations.
His interest in the latter led Cardano into a fascinating entanglement with Nicolò Fontana, a.k.a. Tartaglia (1499-1557). Tartaglia had solved even more difficult cubic equations than those in Practica arithmetice, but refused to explain how he had done so, yet after much pressure from Cardano, he agreed to share his secret—provided that Cardano would not pass it on to anyone until Tartaglia had published it. Soon afterward, however, Cardano took as his servant a promising young mathematician, Ludovico Ferrari (1522-1565), and shared Tartaglia's methods with him. Ferrari had learned how to solve a type of cubic equation called a "depressed cubic," which lacks a second-power term, and the two men discovered a method for reducing any generalized cubic equation to a depressed cubic. With the use of Tartaglia's methods, they could solve the equation.
They could not share this information with the mathematical community, however, because Tartaglia had yet to publish his method, and the prospects of him doing so any time soon appeared dim. It so happened, however, that in 1534 Cardano and Ferrari were examining the papers of the late Sciopione dal Ferro (1465-1526) when they discovered that he had solved the depressed cubic equation two decades earlier. Practically on his deathbed, dal Ferro had explained the secret to his student, Antonio Fior, who later foolishly challenged Tartaglia to a mathematical contest. As a result, Tartaglia had been able to discover the method that he later claimed as his own.
In 1545, Cardano—who reasoned that he was no longer under any obligation to Tartaglia—published his findings in his Ars magna. Tartaglia was incensed, and began a letter-writing campaign against Cardano, which coincided with a series of other misfortunes spanning nearly two decades in the latter's life: in 1546, his wife died; in 1560, his son Giambattista was executed for murdering his own wife; in 1565, Ferrari died of poisoning; and in 1570, Cardano was accused of heresy by the Inquisition. The charge, that he had claimed astrological causes and not divine intervention as the force behind events in the life of Jesus Christ, was enough to land him in jail. He was released upon agreeing to stop teaching, and only in 1573, when Pope Gregory XIII granted him a lifetime pension, did his world return to some semblance of normalcy.
Cardano had only a few years left, but even in the midst of the turmoil resulting from the heresy charge, he had revised his Ars magna by adding a section on what are now known as imaginary numbers. He did not recognize them as such, however, and regarded them as a mere mathematical novelty. He died in Rome on September 21, 1576.