(b. Pavia, Italy, 24 September 1501; d. Rome, Italy, 21 September 1576),
medicine, mathematics, physics, philosophy.
Cardano was the illegitimate son of Fazio Cardano and Chiara Micheri, a widow of uncertain age who was both ignorant and irascible. The early years of his life were characterized by illness and mistreatment. Encouraged to study the classics, mathematics, and astrology by his father, a jurist of encyclopedic learning and a friend of Leonardo da Vinci, Cardano began his university studies in 1520 at Pavia and completed them at Padua in 1526 with the doctorate in medicine. Almost immediately he began to practice his profession in Saccolongo, a small town near Padua, where he spent nearly six years; he later recalled this period as the happiest of his life. Having been cured of impotence, which had afflicted him throughout his youth, he married Lucia Bandareni in 1531; they had two sons and a daughter.
In 1534, sponsored by noblemen who were friends of his father, Cardano became a teacher of mathematics in the “piattine” schools of Milan. (These schools, founded by a bequest of Tommaso Piatti [d. 1502], taught Greek, dialectics, astronomy, and mathematics.) He simultaneously practiced medicine, achieving such success that his colleagues became envious. His first work, De malo recentiorum medicorum usu libellus (Venice, 1536), was directed against them. Within a few years Cardano became the most famous physician in Milan, and among the doctors of Europe he was second only to Vesalius. Among his famous patients was John Hamilton, archbishop of Edinburgh, who suffered from asthma. Cardano remained in Scotland for most of 1552 in order to treat the archbishop and a number of other English noble-men.
In 1539, while awaiting the publication of Practica arithmetice, his first book on mathematics, Cardano learned that Nicolò Tartaglia knew the procedure for 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 Plus V, who gave him a lifetime annuity. In Rome, in the last year of his life, he wrote De propriavita, an autobiography—or better, an apologia pro vitasua—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, headded 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 Cardano to write the Liber de tudo aleae, which he completed in his old age, perhaps during his stay at Bologna; it was published posthumously in the Opera omnia. The book represents the first attempt at a theory of probability based on the philosophical premise that, beyond mere luck, laws and rules govern any given case. The concept of probability was introduced, expressed as the ratio of favorable to possible cases; the law of large numbers was enunciated; the so-called “power law” (if ρ is the probability of an event, the probability that the event will be repeated n limes is ρn) was presented; and numerous problems relating to games of dice, cards, and knucklebones were solved. The book was published, however, subsequent to the first research into the theory of games developed in the correspondence between Fermat and Pascal in 1654; it had no influence on the later development of the field.
Cardano published two encyclopedias of natural science: De subtilitate libri XXI (1550) and De rerumvarietaie (1557), a supplement to De subtilitate. The two works, written in an elliptical and often obscure Latin, contain a little of everything: from cosmology to the construction of machines; from the usefulness of natural sciences to the evil influence of demons; from the laws of mechanics to cryptology. It is a mine of facts, both real and imaginary; of notes on the state of the sciences; of superstition, technology, alchemy, and various branches of the occult. The similarities between the scientific opinions expressed by Cardano in these two works and those of Leonardo da Vinci, at that time unpublished, has led some historians, particularly Pierre Duhem, to suppose that Cardano has used Leonardo’s manuscript notes; others insist that the similarity is entirely coincidental. Be that as it may, Cardano must always be credited with having introduced new ideas that inspired new investigations. In the sixteenth century there were five editions of De rerum varietate and eight of De suhtilitate, as well as seven editions of the French translation of the latter.
Cardano reduced the elements to three (air, earth, water), eliminating fire, which he considered a mode of existence of matter; and he reduced the four qualities to two (hot and moist). His magic was, above all, an attempt to interpret natural phenomena in terms of sympathy and antipathy.
In mechanics, Cardano was a fervent admirer of Archimedes. He studied the lever and the inclined plane in new ways and described many mechanical devices, among them “Cardano’s suspension,” known in classical antiquity, which he attributed to a certain Jannello Turriano of Cremona. Cardano followed a middle road between the partisans of the theory of impetus and the supporters of the Aristotelian theory, who attributed the movement of projectiles to pushing by the air: he favored the idea that at the beginning of its trajectory the projectile was moved by the impetus of the firing mechanism but subsequently was accelerated by the movement of the air. Notable is his observation that the trajectory described by a projectile is not rectilinear at the center, but is a line “which imitates the form of a parabola.” Cardano’s chief claim to fame, however, was his affirmation of the impossibility of perpetual motion, except in heavenly bodies.
Cardano’s contributions to hydrodynamics are important: counter to contemporary belief, he observed that in a conduit of running water, the water does not rise to the level from which it started, but to a lower level that becomes lower as the length of the conduit increases. He also refuted the Aristotelian “abhor-rence of a vacuum,” holding that the phenomena attributed to this abhorrence can be explained by the force of rarefaction. Cardano investigated the measurement of the capacity of streams and stated that the capacity is proportional to the area of the cross section and the velocity. He observed that a stream presses against its banks and, counter to contemporary opinion, he held that the upper levels of moving water move faster than the lower levels.
In his Opus novum de proportionibus, Cardano turned to problems of mechanics, with the principal aim of applying quantitative methods to the study of physics. His use of the concept of moment of a force in his study of the conditions of equilibrium in balance and his attempt to determine experimentally the relation between the densities of air and water are noteworthy. The value that he obtained, 1:50, is rough; but it is the first deduction to be based on the experimental method and on the hypothesis that the ratio of the distances traveled by bullets shot from the same ballistic instrument, through air and through water, is the inverse of the ratio between the densities of air and water.
Geology is indebted to Cardano for several theories: that the formation of mountains is often due to erosion by running water; that rise of the ocean floor is indicated by the presence of marine fossils in land that was once submerged; and the idea—then novel—that streams originate from rainwater, which run back to the sea and evaporates from it, to fall back to earth as rain, in a perpetual cycle.
The many editions of Cardano’s works and the citations of them by writers of the second half of the sixteenth century demonstrate their influence, especially as a stimulus to the study of the particular and the concrete.
I. Original Works. Nearly all of Cardano’s writings are collected in the Opera omnia, Charles Sponi, ed., 10 vols. (Leiden, 1663). The published works to which scholars most often refer are Practica arithmetice et mensurandi singularis (Milan, 1539); Art is magnae, sive de regulis algebraicis liber unus (Nuremberg, 1545); De subtilitate liber XXI (Nuremberg, 1550; 6th ed., 1560), trans. by Richard Le Blanc as De la subtilité… (Paris, 1556; 9th ed., 1611), and bk. 1, trans., with intro. and notes, by Myrtle Marguerite Cass (Williamsport, Pa., 1934); Liber de libris propriis (Leiden, 1557); De rerum varietate libri XVII (Basel, 1557; 5th ed.,1581); De subtilitate… cum additionibus. Addita insuper Apologia adversus calumniatorem (Basel, 1560; 4th ed., 1611); and Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum…. Item de aliza regula liber (Basel, 1570). The autobiography was published by Gabriel Naudé as De propriavita liber… (Paris, 1643; 2nd ed., Amsterdam, 1654); it was translated into Italian (Milan, 1821, 1922; Turin, 1945); German (Jena, 1914); and English (New York, 1930). The French translation by Jean Dayre (Paris, 1936) includes the Latin text with the variants of a 17th-century MS preserved in the Biblioteca Ambrosiana in Milan. The Liberde ludo aleae was first published in the Opera omnia and translated into English by Sidney Henry Gould as The Book on Games of Chance (New York, 1961).
II. Secondary Literature. On Cardano himself, the following works contain many bibliographic references: Angelo Bellini, Girolamo Cardano e it suo tempo (Milan, 1947); and Henry Morley, The Life of Girolamo Cardano of Milan, Physician, 2 vols. (London, 1854). His mathematical work is analyzed in Ettore Bortolotti, I contribute del Tartaglia, del Cardano, del Ferrari e della scuola matematica bolognese alla teoria algebrica delle equazioni cubiche, no. 9 in the series Studi e Memorie per la Storia dell’Università di Bologna (Bologna, 1926), pp. 55–108, and I cartelli di matematica disfida, no. 12 in the series Studi e Memorie per la Storia dell’Università di Bologna (Bologna, 1935), pp. 3–79; Moritz Cantor, Vorlesungen über Geschkhte derMaihemalik, 2nd ed. (Leipzig, 1899), II, 484–510, 532– 541; and Pietro Cossali, Origine e trasporto in Italia dell’algebra, II (Parma, 1797), 159–166, 337–384. The most profound study of Cardano’s contribution to the theory of games is Oystein Ore, Cardano the Gambling Scholar (Princeton, 1953), which concludes with Gould’s translation of the Liber de ludo aleae.
Cardano’s physics is presented in Raffaello Caverni, Storia del metodo sperimentale in Italia, I (Florence, 1891), 47–50, and IV (Florence, 1895), 94–95, 197–198. 385– 386(entire work repr. Bologna, 1969). The hypothesis of his intellectual debt to Leonardo is defended by Pierre Duhem in Les origines de la statique, I (Paris, 1895), 237–238, 242; and Études sur Leonard de Vinci, I (Paris, 1906), 223–245. On Cardano’s work in magic, alchemy, and the arts of divination, see Lynn Thorndike, A History of Magic and Experimental Science, V (New York, 1951), 563–579; on his contributions to cryptology, see David Kahn, The Codebreakers (London, 1967).
(b. Pavia, Italy, 24 September 1501;
d. Rome, Italy, 1576), medicine, mathematics, physics, philosophy. For the original article on Cardano see DSB, vol. 3.
Some of the details of Cardano’s life given in the first edition of the DSB require modification. After completing his doctorate in 1526, Cardano practiced for about a decade as a doctor in two Italian towns (Saccolongo, where he married, and Gallerate). Falling on hard times, he returned in penury to his native Milan to find himself excluded from the College of Physicians and hence from public medical practice; but he was able to obtain through the first of a line of ecclesiastical patrons, Filippo Archinto, an ill-paid post as a public teacher of a variety of arts subjects, including mathematics, in Milan. He tried to make money as an author of prognostications and astrological works, the latter of which were published in the hope that they would attract the attention of Pope Paul III. At about the same time, Ottaviano Scoto, a Venetian publisher friend, generously printed for him two aggressively polemical works on medicine (De malo recentiorum medicorum medendi usu; De simplicium medici-narum noxa, both in 1536) and later brought out a work on a moral theme (De consolatione ). Cardano then found a local sponsor to support the publication of a work of mathematics (Practica arithmetice ), to which he took the precaution of appending a privilege to protect a list of thirty-four of his as-yet unpublished writings, thereby advertising their existence to the wider community of scholars and publishers.
This list inaugurated the second period of his life, in which he made himself an international reputation as an innovative thinker and a solid career as a practicing doctor and a teacher of medicine. A famous publisher in Nürnberg produced several of his writings, notably his mathematical Ars magna (1545). He made contact with the foremost scholarly publisher in Lyon, who brought out philosophical and medical works by him; these include works written explicitly to supersede the writings of Pietro Pomponazzi (the De animorum immortalitate of 1545 and the section of the Contradicentia medica on incantations that appeared first in 1548). Other unpublished works (the De fato and the De arcanis aeternitatis) are witness to his incautious interest in religious subjects that eventually brought him the unwelcome attentions of the ecclesiastical authorities. Later visits to Paris and Basel secured him further publishing outlets, making him better known abroad than in Italy. He was finally elected to the College of Physicians of Milan in 1539 and appointed to a chair of medicine at the University of Pavia shortly thereafter. His medical writings attracted the attention of the archbishop of St. Andrews, who invited him to Scotland. This allowed him to travel widely in Europe, and to return home triumphantly at the beginning of 1554. For the next six years, he enjoyed both financial security and public and professional esteem. He came to be known during this time as a mathematician, a writer on astrology, a medical authority, a natural philosopher, and a writer on moral issues.
The third phase of his life was inaugurated by the execution of his son in 1560, which caused him to lose reputation and support in Milan. Shortly after, scurrilous rumors concerning his alleged sexual misconduct started circulating in Pavia, and a challenge to his professorship was issued on the grounds of his anti-Galenic teaching. He was forced to resign his chair on 11 June 1562. Powerful ecclesiastical patrons obtained for him the chair of medicine at Bologna, which he took up in October 1562. He began to earn money again from consultations, was elected a citizen of Bologna on 26 May 1563, and set about publishing on a wide range of topics. He put into effect his project to write commentaries on the whole Hippocratic corpus. Four of these were published before 1570. He did not neglect mathematics: a large tome on proportion and on the relationship of geometry to algebra appeared in 1570.
The fourth phase of his life began with the disaster of his imprisonment by the Bologna inquisitors on 6 October 1570. As a result, he lost his professorship and the right to teach in the Papal States, and was, moreover, denied the right to publish. His protector, Cardinal Giovanni Morone, probably induced him to move to Rome, where he arrived on 7 October 1571. There he occupied himself with clearing his own name of heresy and purging his own writings, in the hope of getting the restrictions on teaching and publishing lifted, and the condemnation of his books revoked after their correction. Eventually he received a pension from the newly elected pope Gregory XIII, a qualified publishing license for his existing medical publications on 29 October 1572, the right to publish a new work for the first time since his abjuration on 14 May 1574, and the right to return to Bologna to take up his teaching again on 5 January 1576. He was received into the Roman College of Physicians in September 1574. Cardano’s last known will was written on 21 August 1576, and the last date to which he refers in the writings is 1 October 1576. It is unlikely that he survived very long after this date. It is not known where he is buried.
Medical Views . Cardano was a self-proclaimed polymath, whose collected works amount to more than four million words. Whereas it is true to claim that his most enduring contribution to science lies in the field of mathematics, he saw his medical writings as worthy of as much consideration. From his practice in Saccolongo onward, he accumulated insights into pathology and therapy, which he published under the title Contradicentia medica, and was among the first to recognize the importance of Andreas Vesalius’s anatomical work. His growing confidence as a doctor led him to challenge Galen’s supremacy as a medical authority, and to supplant it with Hippocrates. Cardano set out to write commentaries on all of Hippocrates’s works that he deemed authentic: four of these appeared in the course of the 1560s. He was led also to a new approach to the etiology of disease by his interest in natural philosophy.
In the De subtilitate of 1550, he set out a radical new general theory of nature in contradiction to that of Aristotle, but recognized that his new explanations led to an appreciation of its diversity and nonuniformity (hence the title of the work— De rerum varietate—written to complete the De subtilitate). This progression in his work from universal theory to an awareness of unexplained
residues is characteristic of much of his thought. Another strand of his inquiry into nature concerns the nature of human perception and the human mind, which he links to his investigation of the immortality of the soul (De animorum immortalitate ). His Dialectica (1562), which purported to give a succinct account of all logical methods, and his work on natural magic titled De secretis, were both left at his death in forms that evince the incessant revisions to which he subjected his own encyclopedic enterprises.
He was also very well known during his lifetime for writings on astrology, his horoscopes, and his dream interpretation. These were not as original in method as he claimed, and their predictive power was very poor (he told the King Edward VI of England, who died at the age of sixteen in 1553, that he would have a long life). But the impulsion behind these essays was not as irrational as has often been claimed. He set out to ask how both the future and the hidden secrets of nature could be known (in medical terms, this translated itself into a close study of prognosis). His interest in human affairs through moral and historical inquiry was also connected to his investigations into the unfurling of events around him at both an individual and a universal level. Even his analysis of chance in various games was linked to his desire to measure and to quantify the likelihood of outcomes. One of his very last works, left unfinished at his death, the De prudentia eximia, sketched out a general theory that was tested against the course of his own life from 1570. This recourse to himself characterized also his medical and moral writings, which abound in references to his own practice and experiences. It reflects itself in his various autobiographical writings, of which the last and most famous was published posthumously by Gabriel Naudé in 1643. It was through the work of Naudé that his Opera omnia in ten volumes was published in 1663; but his reputation as a radical thinker had been sustained not by a strong positive interest in his novel theories of nature, but negatively by Julius Caesar Scaliger’s systematic attack on his De subtilitate (the Exotericae exercitationes, which were used as a textbook in a number of German universities), and by unfounded rumors about his atheism that were propagated by antilibertine tracts of the early 1600s.
WORKS BY CARDANO
Contradicentia medica. These are versions produced from 1545 to 1663, reproduced on a DVD ROM by the Progetto Cardano, n.d.
Writings on Music. Translated by Clement A. Miller. Rome: American Institute of Musicology, 1973.
Pronostico of 1534. Edited by Germana Ernst. In Girolamo Cardano: le opere, le fonti, la vita. Edited by Marialuisa Baldi and Guido Canziani. Milan, Italy: Franco Angeli, 1999, pp. 457–476.
Gerolamo Cardano nel quinto centenario della nascita. Pavia, Italy: Cardano, 2001.
Liber de orthographia. Edited by Raffaele Passarella. In Cardano e la tradizione dei saperi, edited by Marialuisa Baldi and Guido Canziani. Milan, Italy: Franco Angeli, 2003, pp. 525–618.
De libris propriis. Edited by Ian Maclean. Milan, Italy: Franco Angeli, 2004.
De subtilitate. Books 1–7. Edited by Elio Nenci. Milan, Italy: Franco Angeli, 2004.
De immortalitate animorum. Edited by José Manuel García Valverde. Milan, Italy: Franco Angeli, 2006.
Progetto Cardano. Available from http://filolinux.dipafilo.unimi.it/cardano A reliable bibliography of Cardano’s published works can be found on this Web site, which also contains a complete list of translations of Cardano’s works.
Albé, Patrizia. Girolamo Cardano nel suo tempo . Pavia, Italy: Istituto di studi superiori dell’Insubria Gerolamo Cardano 4, 2003.
Baldi, Marialuisa, and Guido Canziani, eds. Girolamo Cardano: le opere, le fonti, la vita. Milan, Italy: Franco Angeli, 1999.
———. Cardano e la tradizione dei saperi. Milan, Italy: Franco Angeli, 2003.
Fierz, Markus. Girolamo Cardano (1501–1576), Arzt, Naturphilosoph, Mathematiker, Astronom und Traumdeuter. Basel, Switzerland: Birkhäuser, 1977. English translation. Boston: Birkhäuser, 1983.
Grafton, Anthony. Cardanos Kosmos: die Welten und Werke eines Renaissance-Astrologen. Berlin: Berlin Verlag, 1999. English translation. Cambridge, MA: Harvard University Press, 2000.
Ingegno, Alfonso. Saggi sulla filosofia di Cardano. Florence, Italy: La nuova Italia, 1980.
Kessler, Eckhard, ed. Girolamo Cardano: Philosoph, Naturforscher, Arzt. Wiesbaden, Germany: Harrassowitz, 1994.
Milano, Mino. Gerolamo Cardano: mistero e scienza nel cinquecento. Milan, Italy: Camunia, 1990.
Schütze, Ingo. “Bibliografia degli studi su Girolamo Cardano dal 1850 al 1995.” Bruniana e Campanelliana 4 (1998): 449–467.
———. Die Naturphilosophie in Girolamo Cardanos De Subtilitate. Munich, Germany: Wilhelm Fink, 2000.
Siraisi, Nancy G. The Clock and the Mirror: Girolamo Cardano and Renaissance Medicine. Princeton, NJ: Princeton University Press, 1997.
Cardan, Jerome (1501-1576)
Cardan, Jerome (1501-1576)
Italian mathematician, physician, and astrologer, reputed to be a magician. He was a contemporary of Faustus and Paracelsus. He left in his Memoirs a frank and detailed analysis of a curiously complicated and abnormal intellect, sensitive, intense, and not altogether free from the taint of insanity. He declared himself subject to strange fits of abstraction and exaltation, the intensity of which became at length so intolerable that he inflicted on himself severe bodily pain as a means of banishing them.
Cardan described three personal peculiarities to which he was prone. The first was the faculty of projecting his spirit out-side his body, to the accompaniment of strange physical sensations. The second was the ability to perceive through his senses anything he desired to perceive; as a child, he explains, he saw these images involuntarily and without the power of selection, but when he reached manhood he could control them to suit his choice. The third peculiarity was that before every important event in his life he had a dream that warned him about it. Indeed, he had written a commentary of considerable length on Synesius's treatise on dreams, in which he advanced the theory that any virtuous person can acquire the faculty of interpreting dreams. In fact, he believed anyone can draw up for himself a code of dream interpretations by merely studying carefully his own dreams.
In one instance, at least, Cardan's prediction was not entirely successful. He foretold the date of his own death, and, at age 75, was obliged to abstain from food in order to die at the time he had predicted.
He published books on mathematics, astronomy, astrology, rhetoric, and medicine, including Ars Magna (1545), De Subtili-tate Rerum (1551), and De Rerum Varietate (1557).
Morley, H. Jerome Cardan. London, 1854. Waters, W. G. Jerome Cardan. London, 1898.