Gerolamo Cardano

Italian mathematician

• Born: September 24, 1501
• Birthplace: Pavia, Duchy of Milan (now Italy)
• Died: September 21, 1576
• Place of death: Rome, Papal States (now in Italy)

Cardano, best known for a quarrel over intellectual property with fellow mathematician Niccolò Fontana Tartaglia, also helped to transmit the results of a flurry of sixteenth century work in algebra. Cardano also initiated studies in the field of probability theory, which evolved into games of chance in the seventeenth century.

Early Life

Gerolamo Cardano (jay-RAW-lah-moh kahr-DAH-noh) wrote an autobiography describing the activities of his early life, but much subsequent scholarship has clarified his recollections. Cardano denied having been born illegitimately, but his parents, Fazio Cardano and Chiara Michena, apparently were not married at the time of his birth. They did marry, however, in 1524.

Cardano’s father was a distinguished scholar as well as a lawyer, and he encouraged Cardano in his intellectual pursuits. At the time of his death, he left his son a small inheritance to help support him in his studies. In general, Cardano seems not to have enjoyed his childhood, and he accused those who were looking after him of neglect, even with regard to food.

Cardano began his university studies at Pavia in 1520 but proceeded to a medical degree in Padua in 1526. While his medical degree may have been intended to provide a livelihood for Cardano, his attention was not restricted to medical issues. He was perhaps fated to be known more for his mathematical work (which he presented to the public) than for his medicine.

It was his medical income, though, that enabled him to marry in 1531, and he had two sons and a daughter. It can be said that his family life was not a happy one, as one of his sons poisoned his wife and was executed. By 1534, Cardano had become an instructor of mathematics in Milan, where he also practiced medicine, earning a reputation in the process.

Life’s Work

Algebra from the Arabic word al-jabr and Hindu-Arabic numerals were introduced to the West through the work of the Arabic mathematician and astronomer al-Khwārizmī (c. 780-c. 850). His work Kitāb al-jabr wa al-muqābalah (c. 820), which gave the word al-jabr, “algebra,” to the West, means “the book of integration and equation.” In the Western world, even into the last years of the Middle Ages, the influence of Euclid’s Elements guaranteed that mathematics would be approached as a branch of geometry. While this was not much of a handicap when it came to solving algebraic equations in which the highest power of the variable is a second power (called a quadratic equation), it was not helpful in tackling equations in which there was a third power of the variable (called a cubic equation). With al-Khwārizmī’s work, there had been progress in dealing with cubic equations, but this was not immediately known to Western Europe. The end of the Eastern Roman Empire with the fall of Constantinople in 1453 helped to direct a flow of scholarly material from the Middle East to Europe.

Another source of difficulty among Europeans in solving the cubic equation was the lack of a suitable notation. The Roman numeration system was scarcely designed for mathematical work, but it was still in common use in the sixteenth century. Also, the description of mathematical problems and their solutions was usually carried out with words rather than with the symbols later used to put together algebraic equations. The more complicated the problem, the more the lack of a helpful notation was felt, and the leap in level of difficulty between quadratic and cubic equations was substantial.

The first large step toward solving the cubic equation was taken by Scipione del Ferro. While he developed a method for solving a whole class of cubic equations, he did not reveal the method publicly because the method’s secrecy was worth something as a weapon in public disputations. Such disputations helped one build a reputation in the intellectual circles of Italy at that time. Del Ferro passed along the secret to his student Antonio Fiore, who tried using it as a tactic in a public disputation with the mathematician Niccolò Fontana Tartaglia . Tartaglia, however, more so than del Ferro, had managed to solve an even broader class of cubic equations and was able to emerge triumphant from his dispute with Fiore.

Cardano learned of Tartaglia’s success and wanted to profit from his discovery. Tartaglia followed del Ferro in refusing to bring his technique before the public, but he did disclose the technique to Cardano under condition of confidentiality. Cardano’s subsequent actions have been the subject of detailed scrutiny, but he seems to have felt absolved from his vow to Tartaglia for two reasons.

First, he discovered that del Ferro had a version of the formula for solving cubic equations before Tartaglia, even if it was not so general. Then Cardano worked with his own son-in-law, Ludovico Ferrari, who pushed the ideas of Tartaglia even further. Once Ferrari came up with a method for solving an equation with a fourth power of the variable (called a quartic equation), Cardano felt that he owed it to the world to reveal these discoveries.

As a result, Cardano published a volume called Artis magnae, sive de regulis algebraicis (1545; The Great Art: Or, The Rules of Algebra , 1968). In it he detailed the various contributions of his predecessors and the work of Ferrari. Since the solution of the quartic equation depended in part on the solution of the cubic equation, he discussed the cubic equation and gave credit to Tartaglia, but he assumed credit for the quartic equation. Tartaglia was outraged, and his subsequent denunciations of Cardano’s infidelity did a great deal to blacken Cardano’s reputation in the scholarly world. A scholarly community that regards publication as an important part of the process of scientific discovery views Cardano with less distrust than does a community that believes discoveries are the property of their initial discoverer.

The other branch of mathematics to which Cardano made his most notable contributions was the field of probability. There had been a certain amount of discussion of counting cases (the number of possible outcomes for experiments) through the Middle Ages, but there was no basic mathematical formula connecting the ideas of probability from philosophy and religion with the calculation of possible outcomes, as in the work of Raymond Lull (c. 1235-1316). Cardano was the first to offer a definition of the probability of an event: the number of outcomes where that event occurs to the total number of possible outcomes. From this definition he went on to state a form of the law of large numbers. Cardano’s important contributions, however, remained unpublished until after his death and the start of the work of the French mathematicians Pierre de Fermat (1601-1665) and Blaise Pascal (1623-1662) in the seventeenth century, to which are traced most subsequent developments in the field of probability.

Cardano obtained the chair of medicine at the University of Pavia in 1542 and remained there for almost twenty years. His autobiography bears witness to the envy of his contemporaries and also the extent to which his life was embittered by their comments. He proceeded to the chair of medicine at Bologna in 1562 and was involved in public disputations on Galen (the Greek physician) as part of the intellectual life of the city and the university.

In 1570, however, Cardano was arrested by the Inquisition and imprisoned for casting the horoscope of Jesus. Theological objections to Cardano’s act suggested that the events of the life of Jesus were the result of the influence of the stars rather than direct divine intention. That Cardano had worked on various ways of concealing texts of messages probably helped make him an object of suspicion.

After a time in prison, Cardano was forced to recant and abandon teaching. Cardano managed to outwait the ban and proceed to Rome by the time that a new pope had been elected. In this new environment Cardano was able to secure an annuity. Perhaps more important to Cardano was that he was able to use the more tolerant reception for his writings to write his autobiography De propria vita liber (1576; The Book of My Life , 1930).

Significance

Cardano contributed to many areas of scholarship, such as geology, hydrodynamics, and mechanics, and he argued against the continued influence of Aristotle in the physical sciences. His most enduring legacy, however, remains the creation of a discipline of algebra based on the researches of the Italian school to which he belonged. After generations of secrecy, Cardano brought recent advances in mathematics to the scholarly community at large.

Cardano introduced variations on the methods he had learned from others and took seriously the possibility of solutions that involved imaginary numbers. Even though he may have suffered abuse from contemporaries, posterity benefited as much from his arrangement of solution methods as from his own particular discoveries.

Bibliography

Cardan, Jerome. The Book of My Life. Translated by Jean Stoner. New York: E. P. Dutton, 1930. An English translation of Cardano’s autobiography. Includes a brief bibliography.

Cardano, Girolamo. The Great Art: Or, The Rules of Algebra. Translated and edited by T. Richard Witmer. Cambridge, Mass.: M.I.T. Press, 1968. A translation of Cardano’s Artis magnae. Illustrations, bibliographical footnotes.

Eckman, James. Jerome Cardan. Baltimore: Johns Hopkins University Press, 1946. A supplement to a bulletin of the history of medicine, but giving an overall view of Cardano’s life written in English, with Latin chapter titles.

Mankiewicz, Richard. The Story of Mathematics. Princeton, N.J.: Princeton University Press, 2000. Captures the difficulty of trying to do algebra in the absence of suitable notation and vocabulary.

Ore, Oystein. Cardano: The Gambling Scholar. Princeton, N.J.: Princeton University Press, 1953. By a distinguished mathematician, especially devoted to Cardano’s work on probability. Includes a translation of Cardano’s book on games of chance.

Wrixon, Fred B. Codes, Ciphers, and Other Cryptic and Clandestine Communication. New York: Black Dog and Leventhal, 1998. Description of some of Cardano’s contributions to the field that may have led to his facing the Inquisition.