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CARDANO, GIROLAMO (b. Pavia, Italy, 24 September
1501; d. Rome, Italy, 21 September 1576),
medicine, mathematics, physics, philosophy.
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