Gerolamo Cardano

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Gerolamo Cardano
Jerôme Cardan.jpg
Gerolamo Cardano
Born (1501-09-24)24 September 1501
Died 21 September 1576(1576-09-21) (aged 74)
Nationality Italian
Fields Mathematics
Alma mater University of Pavia
Known for Founder of several fields and objects
Influences Archimedes, Muḥammad ibn Mūsā al-Khwārizmī, Leonardo Fibonacci
Influenced Blaise Pascal, Pierre de Fermat, Isaac Newton, Gottfried Wilhelm von Leibniz, Maria Gaetana Agnesi, Joseph-Louis Lagrange, Carl Friedrich Gauss

Gerolamo (or Girolamo, or Geronimo) Cardano (Italian: [dʒeˈrɔlamo karˈdano]; French: Jérôme Cardan; Latin: Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian mathematician, physician, astrologer, philosopher and gambler,[1] best known as the earliest founder of probability and the establisher of the binomial coefficients and the binomial theorem, which was comprised in his book, Opus novum de proportionibus. He wrote more than 200 works on medicine, mathematics, physics, philosophy, religion, and music.[2]

Cardano partially invented and described several mechanical devices including the combination lock, the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely, and the Cardan shaft with universal joints, which allows the transmission of rotary motion at various angles and is used in vehicles to this day. He studied hypocycloids, published in de proportionibus 1570. The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.[3]

Today, he is well-known for his achievements in algebra. Cardano was the first mathematician to make systematic use of numbers less than zero.[4] He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna. The solution to one particular case of the cubic equation ax^3+bx+c=0[5] (in modern notation), was communicated to him by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long dispute) in the form of a poem.[6] The quartic was solved by Cardano's student Lodovico Ferrari. Both were acknowledged in the foreword of the book, as well as in several places within its body. In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties, described for the first time by his Italian contemporary Rafael Bombelli. In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem.

Early life and education[edit]

De propria vita, 1821

He was born in Pavia, Lombardy, the illegitimate child of Fazio Cardano, a mathematically gifted jurist and lawyer, who was a friend of Leonardo da Vinci. In his autobiography, Cardano claimed that his mother had attempted to abort him. Shortly before his birth, his mother had to move from Milan to Pavia to escape the Plague; her three other children died from the disease.

After a difficult childhood, saddened by frequent illnesses and the harsh upbringing by his overbearing father, in 1520, Cardano entered the University of Pavia against his father's wish, who wanted his son to undertake studies of law, but Girolamo felt more attracted to philosophy and science. During the ongoing war between France and Spain, the authorities in Pavia were forced to close the university, Cardano resumed his studies at the University of Padua, where he graduated in medicine in 1526. His eccentric and confrontational style did not earn him many friends and he had a difficult time finding work after his studies had ended. In 1525, Cardano repeatedly applied to the College of Physicians in Milan, but was not admitted owing to his combative reputation and illegitimate birth.

Eventually, he managed to develop a considerable reputation as a physician and his services were highly valued at the courts. He was the first to describe typhoid fever. In 1553 he cured the Scottish Archbishop of St Andrews of a disease that had left him speechless and was thought incurable. The diplomat Thomas Randolph recorded the "merry tales" rumoured about his methods still current in Edinburgh nine years later.[7] Cardano himself wrote that the Archbishop had been short of breath for ten years, and after the cure was effected by his assistant, he was paid 1,400 gold crowns.[8]


Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews.

Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae ("Book on Games of Chance"), written around 1564,[9] but not published until 1663, contains the first systematic treatment of probability,[10] as well as a section on effective cheating methods. He used the game of throwing dice to understand the basic concepts of probability. He demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes[11]). He was also aware of the multiplication rule for independent events but was not certain about what values should be multiplied.[12]

Cardano's philosophy is heavily influenced by characteristic trends of late scholastic Aristotelianism, with a strong penchant for Averroist interpretations. Cardano shows a great interest in Averroes' opinion that one intellect would perform intellective functions for all human beings. However, he tends to provide a historicized version of this radical view, in that he looks at the one intellect as the varying amount of learning accumulated by mankind throughout the centuries rather than simply justifying it from a purely epistemological point of view (seen as the one intellective power that actualizes the life and knowledge of the sublunary world as a whole). Cardano's philosophy also displays clear traces of Platonic influences, absorbed through the reading of Marsilio Ficino's recent translations and commentaries, especially Plotinus and Iamblichus. Together with his impressive knowledge of astrological and medical literature, both scholasticism and Platonism give a characteristically vitalistic slant to his cosmological views. Cardano's philosophy has often been described as suggestive and rich in original intuitions, but cluttered and inconsistent as a whole. In fact, his philosophical work is yet another example, common during the Renaissance, of how different philosophical traditions could converge into one composite but coherent picture. Throughout his life, from his early endeavours in the 1540 to the last philosophical attempt, Cardano demonstrated a distinctive commitment to a certain number of philosophical issues: the relationship between oneness and multiplicity, with the notable corollaries dealing with order and disorder, determinism and chance, life and decay; the view of the intellect as the ultimate principle of reality and knowledge; a general theory of celestial heat, described as the main formative agent in nature; the interplay of nature and the soul in the organization of the universe; a general doctrine of the immortality of the soul, seen as the foundation of both cognitive clarity and moral certitude. As a whole, the originality of Cardano's eclecticism lies in the unique way in which he characterizes the interdependence of life, knowledge and matter, in which a pronounced sense of reality and truth is constantly being questioned and jeopardized by a realistic view of human nature, mercilessly presented as prone to fear, delusion and deceit.

His work with hypocycloids led him to the "Cardan gear mechanism," in which a pair of gears with the smaller being one-half the size of the larger gear is used converting rotational motion to linear motion, with greater efficiency and precision than a Scotch yoke, for example.[13]

Cardano made several contributions to hydrodynamics and held that perpetual motion is impossible, except in celestial bodies. He published two encyclopedias of natural science which contain a wide variety of inventions, facts, and occult superstitions. He also introduced the Cardan grille, a cryptographic tool, in 1550.

Someone also assigned to Cardano the credit for the invention of the so-called Cardano's Rings, also called Chinese Rings, but it is very probable that they predate Cardano.

Significantly, in the history of education of the deaf, he said that deaf people were capable of using their minds, argued for the importance of teaching them, and was one of the first to state that deaf people could learn to read and write without learning how to speak first. He was familiar with a report by Rudolph Agricola about a deaf mute who had learned to write.

Knowledge and Nature[edit]

A portrait of Gerolamo Cardano

Cardano's cosmological views belong to a long-established system of astro-biological doctrines whose origins go back to Aristotelian physics, Hippocratic vitalism, and fundamental assumptions underlying the tradition of astrological and meteorological learning, reshaped through a series of Hebrew and Arabic mediations. His account of the supralunary world combines elements from Neoplatonic philosophy and Christian theology. In line with many of his contemporaries, Cardano maintains that there is a clear division between the supralunary and sublunary world. The life of the universe is the result of varying degrees of celestial energy overflowing from the One, i.e., God. From God to matter, cohorts of the most disparate souls mediate between these two extremes. From a material point of view, the connective element between heaven and earth is celestial heat. Cardano's cosmological views belong to a long-established system of astro-biological doctrines whose origins go back to Aristotelian physics, Hippocratic vitalism, and fundamental assumptions underlying the tradition of astrological and meteorological learning, reshaped through a series of Hebrew and Arabic mediations. His account of the supralunary world combines elements from Neoplatonic philosophy and Christian theology. In line with many of his contemporaries, Cardano maintains that there is a clear division between the supralunary and sublunary world. The life of the universe is the result of varying degrees of celestial energy overflowing from the One, i.e., God. From God to matter, cohorts of the most disparate souls mediate between these two extremes. From a material point of view, the connective element between heaven and earth is celestial heat. The principal constituents of the sublunary world are matter (earth, water and air), celestial heat, and a wide variety of souls (spanning from demonic minds to substantial forms understood as specific principles of life).

In line with the principles of Greek ontology, Cardano maintains that nothing comes out of nothing; rather, all things derive from something, and this something cannot be infinite. Aristotle called this something “hyle or prime matter,” but Cardano prefers to discard this notion of an intermediate entity between being and non being, replacing it with the view of the elements as the material starting point and celestial heat as the efficient active principle, “for otherwise the elements would be completely redundant if there were prime matter”. The elements, which represent the first level of organization in matter, are three (and not four as demanded by Aristotelian and scholastic physics): earth, water and air. As for fire, Cardano considers this to be a product of celestial heat, which is one of the various streams of vital energy flowing from the supralunary sources of life and knowledge and which pervade the universe as one living organism. Innate heat of a celestial origin is the active element that mediates between the state of utter immobility which characterizes intelligible substances and the incessant mobility that defines the life of material beings. Through an exercise in introspective analysis, Cardano enumerates three principles that regulate both our inner life and the life of all created things: one “is moved and does not move, resulting from the heavy elements;” another “moves, and it is not moved, that is, the soul;” the third, finally, “is moved by the soul” and moves the body, i.e., the innate heat. To the question of whether the soul can be identified with celestial heat, Cardano replies that, unlike the latter, the former is incorporeal, does not occupy any place and therefore is never in motion. Also, that which is in motion does not have that level of self-stability that is necessary for a being to be able to perceive or to think .

In Cardano's metaphysics, matter and form are complementary, in that in nature there cannot be matter without form, and forms are always with a body. Forms represent the primordial stage in the process through which the created universe becomes one living being. The difference between souls and forms is that souls, albeit involved in the animation of bodies, remain nevertheless unaffected by corporeal reality. Up on a higher level, minds are souls that are completely independent of matter, bodies and motion. However, even within the ontological sphere of the minds, there are varying degrees of embodiment. While the highest celestial intelligences are wholly separated from the material cosmos, demonic substances, albeit incorporeal, can affect the corporeal world through forces and influences of various kind. One of these is “that force which is connected to demons, regardless of whether this power is corporeal and depends on humours, or it is incorporeal.” Cardano maintains that it is through forces of this nature that “the parts of the universe are aroused by demons, stars or some other hidden cause” . The principle that in a way collects and administers all these currents of celestial energy is the soul of the world. In Cardano's system of astro-biological determinism, the universal soul keeps the whole cosmos together and performs paramount operations in accordance with the original plan devised by God and implemented by the planetary intelligences, for “all things are influenced by the higher heaven and are moved at the command of the soul of the world.” The soul of the world, which “cannot be understood without God,” directs the work of nature, and, “in the process of generating things, produces supercelestial lives and multiplicity”. In the sublunary world, the major operations of life and generation are performed by nature, understood as a source of teleological activity supervised by the intellect and the soul of the world . The complex relationship between soul and nature, and the role played by celestial forces, is a crucial point in Cardano's philosophy.

De Subtilitate (1552)[edit]

As quoted from Charles Lyell's Principles of Geology:

The title of a work of Cardano's, published in 1552, De Subtilitate (corresponding to what would now be called transcendental philosophy), would lead us to expect, in the chapter on minerals, many far fetched theories characteristic of that age; but when treating of petrified shells, he decided that they clearly indicated the former sojourn of the sea upon the mountains.[14]

Later years[edit]

Cardano's eldest and favorite son was executed in 1560 after he confessed to having poisoned his cuckolding wife. His other son was a gambler, who stole money from him. He allegedly cropped the ears of one of his sons. Cardano himself was accused of heresy in 1570 because he had computed and published the horoscope of Jesus in 1554. Despite numerous stories to the contrary, it is not true that his own son contributed to the prosecution after being bribed by Tartaglia, as Tartaglia died 13 years previously.[15] He was arrested, had to spend several months in prison and was forced to abjure his professorship. He moved to Rome, received a lifetime annuity from Pope Gregory XIII (after first having been rejected by Pope Pius V) and finished his autobiography. It appears that he was still practicing medicine up to his death in 1576.[2] The date of his death is disputed, but a death year is given as 1576.[16]

References in literature[edit]

The seventeenth century English physician and philosopher Sir Thomas Browne once possessed the ten volumes of the Leyden 1663 edition of the complete works of Cardan in his library.[17]

Browne critically viewed Cardan as-

that famous Physician of Milan, a great Enquirer of Truth, but too greedy a Receiver of it. He hath left many excellent Discourses, Medical, Natural, and Astrological; the most suspicious are those two he wrote by admonition in a dream, that is De Subtilitate & Varietate Rerum. Assuredly this learned man hath taken many things upon trust, and although examined some, hath let slip many others. He is of singular use unto a prudent Reader; but unto him that only desireth Hoties, or to replenish his head with varieties; like many others before related, either in the Original or confirmation, he may become no small occasion of Error.[18]

Richard Hinckley Allen tells of an amusing reference made by Samuel Butler in his book Hudibras:

Cardan believ'd great states depend
Upon the tip o'th' Bear's tail's end;
That, as she wisk'd it t'wards the Sun,
Strew'd mighty empires up and down;
Which others say must needs be false,
Because your true bears have no tails.

Alessandro Manzoni's novel I Promessi Sposi portrays a pedantic scholar of the obsolete, Don Ferrante, as a great admirer of Cardano. Significantly, he values him only for his superstitious and astrological writings; his scientific writings are dismissed because they contradict Aristotle, but excused on the ground that the author of the astrological works deserves to be listened to even when he is wrong.

English novelist E. M. Forster's Abinger Harvest, a 1936 volume of essays, authorial reviews and a play, provides a sympathetic treatment of Cardano in the section titled 'The Past'. Forster believes Cardano was so absorbed in "self-analysis that he often forgot to repent of his bad temper, his stupidity, his licentiousness, and love of revenge" (212).


  • De malo recentiorum medicorum usu libellus, Venice, 1536 (on medicine).
  • Practica arithmetice et mensurandi singularis, Milan, 1577 (on mathematics).
  • Artis magnae, sive de regulis algebraicis (also known as Ars magna), Nuremberg, 1545 (on algebra).[19]
  • De immortalitate (on alchemy).
  • Opus novum de proportionibus (on mechanics) (Archimedes Project).
  • Contradicentium medicorum (on medicine).
  • De subtilitate rerum, Nuremberg, Johann Petreius, 1550 (on natural phenomena).
  • De libris propriis, Leiden, 1557 (commentaries).
  • De varietate rerum, Basle, Heinrich Petri, 1559 (on natural phenomena).
  • Neronis encomium, Basle, 1562.
  • De Methodo medendi, 1565
  • Opus novum de proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum. Item de aliza regula, Basel, 1570.
  • De vita propria, 1576 (autobiography); a later edition, De Propria Vita Liber, Amsterdam, (1654)
  • Liber de ludo aleae, ("On Casting the Die"),[20] posthumously published in 1663 (on probability).
  • De Musica, ca 1546 (on music theory), posthumously published in Hieronymi Cardani Mediolensis opera omnia, Sponius, Lyons, 1663
  • De Consolatione, Venice, 1542
  • HIERONY-||MI CARDANI ME=||DIOLANENSIS MEDICI,|| DE RERVM VARIETATE, LI-||BRI XVII. Iam denuò ab in numeris || mendis summa cura ac studio repur-||gati, & pristino nito-||ri restituti.|| ADIECTVS EST CAPITVM, RE-||rum & sententiarum … || INDEX utilissimus.||, Basel, 1581 Digital edition by the University and State Library Düsseldorf
  • Synesiorum somniorum omnis generis insomnia explicantes (Book of Dreams)

See also[edit]


  1. ^ Patty, Peter Fletcher, Hughes Hoyle, C. Wayne (1991). Foundations of Discrete Mathematics (International student ed.). Boston: PWS-KENT Pub. Co. p. 207. ISBN 0-534-92373-9. Cardano was a physician, astrologer, and mathematician.... [He] supported his wife and three children by gambling and casting horoscopes. 
  2. ^ a b Westfall, Richard S. "Cardano, Girolamo". The Galileo Project. Archived from the original on 2012-07-19. Retrieved 2012-07-19. 
  3. ^ Jerome Cardan: A Biographical Study. Dodo Press. 2009-01. ISBN 9781409959595.  Check date values in: |date= (help)
  4. ^ Isaac Asimov, Asimov On Numbers, published by Pocket Books, a division of Simon & Schuster, 1966, 1977, page 119.
  5. ^ Burton, David. The History of Mathematics: An Introduction (7th (2010) ed.). New York: McGraw-Hill. 
  6. ^ Katz, Victor J. A History of Mathematics: An Introduction. 3rd ed. Boston: Pearson Education, 2009. Print.
  7. ^ Calendar State Papers Scotland, vol.1 (1898), p.592: Melville, James, Memoirs of his own life, Brookman, (1833), 21, 73
  8. ^ Cardanus, Gerolamo, De Propria Vita Liber: His Own Life, Amsterdam, (1654), pp.136-7, (Latin)
  9. ^ In Chapter 20 of Liber de Ludo Aleae he describes a personal experience from 1526 and then adds that "thirty-eight years have passed" [elapsis iam annis triginta octo]. This sentence is written by Cardano around 1564, age 63.
  10. ^ Katz, ibid., p. 488
  11. ^ Some laws and problems in classical probability and how Cardano anticipated them Gorrochum, P. Chancemagazine 2012
  12. ^ Katz, ibid., p. 488
  13. ^ "How does a Cardan gear mechanism work?". Seyhan Ersoy. Retrieved 1 April 2015. 
  14. ^ Charles Lyell, Principles of Geology, 1832, p.29
  15. ^ Tony Rothman, Cardano v Tartaglia: The Great Feud Goes Supernatural.
  16. ^ Katz, ibid., p. 401
  17. ^ A Facsimile of the 1711 Sales Auction Catalogue of Sir Thomas Browne and his son Edward's Libraries. Introduction, notes and index by J.S. Finch (E.J. Brill: Leiden, 1986)
  18. ^ Pseudodoxia Epidemica Bk 1: chapter 8 no. 13
  19. ^ [1] An electronic copy of his book Ars Magna (in Latin)
  20. ^ p. 963, Jan Gullberg, Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 0-393-04002-X ISBN 978-0-393-04002-9


  • Cardano, Girolamo, Astrological Aphorisms of Cardan. Edmonds, WA: Sure Fire Press, 1989.
  • Cardano, Girolamo, The Book of My Life. trans. by Jean Stoner. New York: New York Review of Books, 2002.
  • Cardano, Girolamo, Opera omnia, Charles Sponi, ed., 10 vols. Lyons, 1663.
  • Cardano, Girolamo, Nero: an Exemplary Life Inckstone 2012, translation in English of the Neronis Encomium.
  • Dunham, William, Journey through Genius, Chapter 6, 1990, John Wiley and Sons. ISBN 0-471-50030-5. Discusses Cardano's life and solution of the cubic equation.
  • Ekert, Artur, "Complex and unpredictable Cardano". International Journal of Theoretical Physics, Vol. 47, Issue 8, pp. 2101–2119. arXiv e-print (arXiv:0806.0485).
  • Giglioni, Guido, "'Bolognan boys are beautiful, tasteful and mostly fine musicians’: Cardano on male same-sex love and music", in: Kenneth Borris & George Rousseau (curr.), The sciences of homosexuality in early modern Europe, Routledge, London 2007, pp. 201-220.
  • Grafton, Anthony, Cardano's Cosmos: The Worlds and Works of a Renaissance Astrologer. Harvard University Press, 2001.
  • Morley, Henry, The life of Girolamo Cardano, of Milan, Physician 2 vols. Chapman & Hall, London 1854.
  • Ore, Øystein, Cardano, the Gambling Scholar. Princeton, 1953.
  • Rutkin, H. Darrel, "Astrological conditioning of same-sexual relations in Girolamo Cardano’s theoretical treatises and celebrity genitures", in: Kenneth Borris & George Rousseau (curr.), The sciences of homosexuality in early modern Europe, Routledge, London 2007, pp. 183-200.
  • Sirasi, Nancy G., The Clock and the Mirror: Girolamo Cardano and Renaissance Medicine, Princeton University Press, 1997.

External links[edit]