Dictionary of Scientific Biography

solving third-degree equations. He succeeded in obtaining this information by promising, possibly under oath, not to reveal it. After having kept the promise for six years, he considered himself released from it when he learned that the credit for the discovery actually belonged to Scipione dal Ferro. He therefore published the method in his Artis magnae sive de regulis algebraicis liber unus (1545), commonly called Ars magna, his greatest work in mathematics. Its publication angered Tartaglia, who in his Quesiti et inventioni diverse (1546) accused Cardano of perjury and wrote of him in offensive terms that he repeated in General trattato di numeri et misure (1556-1560). The latter work was well known among mathematicians and thus contributed greatly to posterity's low opinion of Cardano.

In 1543 Cardano accepted the chair of medicine at the University of Pavia, where he taught until 1560, with an interruption from 1552 to 1559 (when the stipend was not paid). In 1560 his elder son, his favorite, was executed for having poisoned his wife. Shaken by this blow, still suffering public condemnation aroused by the hatred of his many enemies, and embittered by the dissolute life of his younger son, Cardano sought and obtained the chair of medicine at the University of Bologna, to which he went in 1562.

In 1570 Cardano was imprisoned by the Inquisition. He was accused of heresy, particularly for having cast the horoscope of Christ and having attributed the events of His life to the influence of the stars. After a few months in prison, having been forced to recant and to abandon teaching, Cardano went in 1571 to Rome, where he succeeded in obtaining the favor of Pope Pius V, who gave him a lifetime annuity. In Rome, in the last year of his life, he wrote De propria vita, an autobiography—or better, an apologia pro vita sua—that did not shrink from the most shameful revelations. The De propria vita and the De libris propriis are the principal sources for his biography.

Cardano wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music. Although he was insensitive to the plastic arts, his was the universal mentality to which no branch of learning was inaccessible. Even his earliest works show the characteristics of his highly unstable personality: encyclopedic learning, powerful intellect combined with childlike credulity, unconquerable fears, and delusions of grandeur.

Cardano's fame rests on his contributions to mathematics. As early as the Practica arithmetice, which is devoted to numerical calculation, he revealed uncommon mathematical ability in the exposition of many original methods of mnemonic calculation and in the confidence with which he transformed algebraic expressions and equations. One must remember that he could not use modern notation because the contemporary algebra was still verbal. His mastery of calculation also enabled him to solve equations above the second degree, which contemporary algebra was unable to do. For example, taking the equation that in modern notation is written 6x3 - 4x2 = 34x + 24, he added 6x3 + 20x2 to each member and obtained, after other transformations,

4x2(3x + 4) = (2x2 + 4x + 6)(3x + 4),

divided both members by 3x + 4, and from the resulting second-degree equation obtained the solution x = 3.

His major work, though, was the Ars magna, in which many new ideas in algebra were systematically presented. Among them are the rule, today called “Cardano's rule,” for solving reduced third-degree equations (i.e., they lack the second-degree term); the linear transformations that eliminate the second-degree term in a complete cubic equation (which Tartaglia did not know how to solve); the observation that an equation of a degree higher than the first admits more than a single root; the lowering of the degree of an equation when one of its roots is known; and the solution, applied to many problems, of the quartic equation, attributed by Cardano to his disciple and son-in-law, Ludovico Ferrari. Notable also was Cardano's research into approximate solutions of a numerical equation by the method of proportional parts and the observation that, with repeated operations, one could obtain roots always closer to the true ones. Before Cardano, only the solution of an equation was sought. Cardano, however, also observed the relations between the roots and the coefficients of the equation and between the succession of the signs of the terms and the signs of the roots; thus he is justly considered the originator of the theory of algebraic equations. Although in some cases he used imaginary numbers, overcoming the reluctance of contemporary mathematicians to use them, it was only in 1570, in a new edition of the Ars magna, that he added a section entitled “De aliza regula” (the meaning of aliza is unknown; some say it means “difficult”), devoted to the “irreducible case” of the cubic equation, in which Cardano's rule is extended to imaginary numbers. This was a recondite work that did not give solutions to the irreducible case, but it was still important for the algebraic transformations which it employed and for the presentation of the solutions of at least three important problems.

His passion for games (dice, chess, cards) inspired