Gerolamo Cardano

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Gerolamo Cardano
Jerome Cardan.jpg
Gerolamo Cardano
Born24 September 1501
Died21 September 1576(1576-09-21) (aged 74)
Italy
NationalityItalian
Alma mater University of Pavia
Known forPolymath, founder of various fields and inventor of several machines
Scientific career
FieldsScience, maths, philosophy, and literature
Influences Archimedes, Muḥammad ibn Mūsā al-Khwārizmī, Leonardo Fibonacci
Influenced Blaise Pascal, [1] François Viète, Pierre de Fermat, [1] Isaac Newton, Gottfried Wilhelm von Leibniz, Maria Gaetana Agnesi, Joseph-Louis Lagrange, Carl Friedrich Gauss

Gerolamo (or Girolamo, [2] or Geronimo [3] ) Cardano (Italian:  [dʒeˈrɔlamo karˈdano] ; French: Jérôme Cardan; Latin : Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler. [4] He was one of the most influential mathematicians of the Renaissance, and was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficients and the binomial theorem in the Western world. He wrote more than 200 works on science. [5]

Polymath Individual whose knowledge spans a significant number of subjects

A polymath is an individual whose knowledge spans a significant number of subjects, known to draw on complex bodies of knowledge to solve specific problems. The term entered the lexicon in the 20th century and has now been applied to great thinkers living before and after the Renaissance.

Physician professional who practices medicine

A physician, medical practitioner, medical doctor, or simply doctor, is a professional who practises medicine, which is concerned with promoting, maintaining, or restoring health through the study, diagnosis, prognosis and treatment of disease, injury, and other physical and mental impairments. Physicians may focus their practice on certain disease categories, types of patients, and methods of treatment—known as specialities—or they may assume responsibility for the provision of continuing and comprehensive medical care to individuals, families, and communities—known as general practice. Medical practice properly requires both a detailed knowledge of the academic disciplines, such as anatomy and physiology, underlying diseases and their treatment—the science of medicine—and also a decent competence in its applied practice—the art or craft of medicine.

Biologist Scientist studying living organisms

A biologist is a scientist who has specialized knowledge in the field of biology, the scientific study of life. Biologists involved in fundamental research attempt to explore and further explain the underlying mechanisms that govern the functioning of living matter. Biologists involved in applied research attempt to develop or improve more specific processes and understanding, in fields such as medicine and industry.

Contents

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 made significant contributions to hypocycloids, published in De proportionibus, in 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. [6]

Combination lock

A combination lock is a type of locking device in which a sequence of symbols, usually numbers, is used to open the lock. The sequence may be entered using a single rotating dial which interacts with several discs or cams, by using a set of several rotating discs with inscribed symbols which directly interact with the locking mechanism, or through an electronic or mechanical keypad. Types range from inexpensive three-digit luggage locks to high-security safes. Unlike ordinary padlocks, combination locks do not use keys.

Gimbal support allowing rotation

A gimbal is a pivoted support that allows the rotation of an object about a single axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of the rotation of its support. For example, on a ship, the gyroscopes, shipboard compasses, stoves, and even drink holders typically use gimbals to keep them upright with respect to the horizon despite the ship's pitching and rolling.

Compass direction finding and navigation instrument

A compass is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions. Usually, a diagram called a compass rose shows the directions north, south, east, and west on the compass face as abbreviated initials. When the compass is used, the rose can be aligned with the corresponding geographic directions; for example, the "N" mark on the rose points northward. Compasses often display markings for angles in degrees in addition to the rose. North corresponds to 0°, and the angles increase clockwise, so east is 90° degrees, south is 180°, and west is 270°. These numbers allow the compass to show magnetic North azimuths or true North azimuths or bearings, which are commonly stated in this notation. If magnetic declination between the magnetic North and true North at latitude angle and longitude angle is known, then direction of magnetic North also gives direction of true North.

Today, he is well known for his achievements in algebra. He made the first systematic use of negative numbers in Europe, published with attribution the solutions of other mathematicians for the cubic and quartic equations, and acknowledged the existence of imaginary numbers.

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.

Early life and education

De propria vita, 1821 Cardano - De propria vita, 1821 - 698063 F.jpg
De propria vita, 1821

He was born in Pavia, Lombardy, the illegitimate child of Fazio Cardano, a mathematically gifted jurist, lawyer, and close personal friend of Leonardo da Vinci. In his autobiography, Cardano wrote that his mother, Chiara Micheri, had taken "various abortive medicines" to terminate the pregnancy; he was "taken by violent means from my mother; I was almost dead." She was in labour for three days. [7] 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.

Pavia Comune in Lombardy, Italy

Pavia is a town and comune of south-western Lombardy in northern Italy, 35 kilometres south of Milan on the lower Ticino river near its confluence with the Po. It has a population of c. 73,000. The city was the capital of the Kingdom of the Lombards from 572 to 774.

Fazio Cardano was an Italian jurist and mathematician. He was a student of perspective. Cardano was also a professor at the University of Pavia, and was devoted to hermetical science and the world of the occult. He was a friend of Leonardo da Vinci.

Jurist Legal scholar or academic, a professional who studies, teaches, and develops law

A jurist is someone who researches and studies jurisprudence. Such a person can work as an academic, legal writer or law lecturer. In the United Kingdom, Australia, New Zealand, South Africa, and in many other Commonwealth countries, the word jurist sometimes refers to a barrister, whereas in the United States of America and Canada it often refers to a judge.

After a depressing childhood, with frequent illnesses, including impotence, and the rough 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 Italian War of 1521-6, however, the authorities in Pavia were forced to close the university in 1524. [8] Cardano resumed his studies at the University of Padua, where he graduated with a doctorate in medicine in 1525. [9] 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. However, he was consulted by many members of the College of Physicians due to his irrefutable intelligence. [10]

University of Pavia University in Italy

The University of Pavia is a university located in Pavia, Lombardy, Italy. There was evidence of teaching as early as 1361, making it one of the oldest universities in the world. It was the sole university in Milan and the greater Lombardy region until the end of the 19th century.

Early career as a physician

Cardano wanted to practice medicine in a large, rich city like Milan, but he was denied a license to practice, so he settled for the town of Saccolongo, where he practiced without a license. There, he married Lucia Banderini in 1531. Before her death in 1546, they had three children, Giovanni Battista (1534), Chiara (1537) and Aldo Urbano (1543). [7] Cardano later wrote that those were the happiest days of his life.

Saccolongo Comune in Veneto, Italy

Saccolongo is a comune (municipality) in the Province of Padua in the Italian region Veneto, located about 45 kilometres (28 mi) west of Venice and about 9 kilometres (6 mi) west of Padua. As of 31 December 2004, it had a population of 4,538 and an area of 13.7 square kilometres (5.3 sq mi).

With the help of a few noblemen, Cardano obtained a teaching position in mathematics in Milan. Having finally received his medical license, he practiced mathematics and medicine simultaneously, treating a few influential patients in the process. Because of this, he became one of the most sought-after doctors in Milan. In fact, by 1536, he was able to quit his teaching position, although he was still interested in mathematics. His notability in the medical field was such that the aristocracy tried to lure him out of Milan. Cardano later wrote that he turned down offers from the kings of Denmark and France, and the Queen of Scotland. [11]

Mathematics

Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews. Gerolamo Cardano (colour).jpg
Portrait of Cardano on display at the School of Mathematics and Statistics, University of St Andrews.

Cardano was the first mathematician to make systematic use of negative numbers. [12] He published with attribution the solution of Scipione del Ferro to the cubic equation and the solution of his student Lodovico Ferrari to the quartic equation in his 1545 book Ars Magna . The solution to one particular case of the cubic equation [13] (in modern notation), had been communicated to him in 1539 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, [14] but Ferro's solution predated Fontana's. [11] 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.

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, [15] but not published until 1663, contains the first systematic treatment of probability, [16] 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 [17] ). He was also aware of the multiplication rule for independent events but was not certain about what values should be multiplied. [18]

Other contributions

"Oneiron" ("Dream"), reverse of the medallion of Cardano by Leone Leoni, 1550-51. Leone leoni (attr.), medaglia di girolamo cardano, verso con sogno di cardano, 1550-51.JPG
"Oneiron" ("Dream"), reverse of the medallion of Cardano by Leone Leoni, 1550-51.

Cardano's work with hypocycloids led him to the Cardan's Movement or 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. [19] He is also credited with the invention of the Cardan suspension or gimbal.

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 writing 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.

De Subtilitate (1550)

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. [20]

Later years and death

Medallion portrait of Cardano aged 49 by Leone Leoni (1509-1590) Leone leoni, medaglia di girolamo cardano.JPG
Medallion portrait of Cardano aged 49 by Leone Leoni (1509-1590)

Two of Cardano's children — Giovanni Battista and Aldo Urbano — came to ignoble ends. Giovanni Battista, Cardano's eldest and favorite son, was tried and beheaded in 1560 for poisoning his wife, [11] after he discovered that their three children were not his. Aldo Urbano was a gambler, who stole money from his father, and so Gerolamo disinherited him in 1569.

Cardano moved from Pavia to Bologna, in part because he believed that the decision to execute Giovanni was influenced by Gerolamo's battles with the academic establishment in Pavia, and his colleagues' jealousy at his scientific achievements and also because he was beset with allegations of sexual impropriety with his students. [7] Cardano was arrested by the Inquisition in 1570 for unknown reasons, and forced to spend several months in prison and abjure his professorship. He moved to Rome, and received a lifetime annuity from Pope Gregory XIII (after first having been rejected by Pope Pius V) and finished his autobiography. He was accepted in the Royal College of Physicians, and as well as practising medicine he continued his philosophical studies until his death in 1576. [5] [7] Cardano is reported to have correctly predicted the exact date of his own death but it has been claimed that he achieved this by committing suicide. [11] [21]

References in literature

The seventeenth century English physician and philosopher Sir Thomas Browne possessed the ten volumes of the Leyden 1663 edition of the complete works of Cardan in his library. [22]

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." [23]

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).

Works

Collected Works

(A chronological key to this edition is supplied by M. Fierz. [62] )

See also

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References

Citations

  1. 1 2 O'Connor, J.J.; Robertson, E.F. (August 2006). "Étienne Pascal". University of St. Andrews, Scotland . Retrieved 5 February 2010.
  2. Chisholm, Hugh, ed. (1911). "Cardan, Girolamo"  . Encyclopædia Britannica (11th ed.). Cambridge University Press.
  3. Giglioni, Guido (23 April 2013). "Girolamo [Geronimo] Cardano".Stanford Encyclopedia of Philosophy.
  4. 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.
  5. 1 2 Westfall, Richard S. "Cardano, Girolamo". The Galileo Project. rice.edu. Archived from the original on 28 July 2012. Retrieved 2012-07-19.
  6. W.G. Waters, Jerome Cardan, a Biographical Study (Lawrence and Bullen, London 1898), from Internet Archive.
  7. 1 2 3 4 Armando Maggi (1 September 2001). Satan's Rhetoric: A Study of Renaissance Demonology. University of Chicago Press. pp. 181–. ISBN   978-0-226-50132-1.
  8. Angus., Konstam, (1996). Pavia 1525 : the climax of the Italian wars. London: Osprey Military. ISBN   1855325047. OCLC   36143257.
  9. "Cardan biography". MacTutor History of Mathematics archive . Retrieved 30 October 2017.
  10. http://www-history.mcs.st-and.ac.uk/Biographies/Cardan.html
  11. 1 2 3 4 Bruno, Leonard C (2003) [1999]. Math and mathematicians : the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 60. ISBN   0787638137. OCLC   41497065.
  12. Isaac Asimov, Asimov on Numbers, published by Pocket Books, a division of Simon & Schuster, 1966, 1977, page 119.
  13. Burton, David. The History of Mathematics: An Introduction (7th (2010) ed.). New York: McGraw-Hill.
  14. V.J. Katz, A History of Mathematics: An Introduction, 3rd edn. (Boston: Pearson Education, 2009).
  15. 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.
  16. Katz, ibid., p. 488
  17. Some laws and problems in classical probability and how Cardano anticipated them Gorrochum, P. Chancemagazine 2012
  18. Katz, ibid., p. 488
  19. "How does a Cardan gear mechanism work?". Seyhan Ersoy. Retrieved 1 April 2015.
  20. Charles Lyell, Principles of Geology , 1832, p.29
  21. "Girolamo Cardano". www2.stetson.edu. Retrieved 10 October 2017.
  22. J.S. Finch (ed.), A Facsimile of the 1711 Sales Auction Catalogue of Sir Thomas Browne and his son Edward's Libraries, with Introduction, notes and index (E.J. Brill: Leiden, 1986).
  23. Pseudodoxia Epidemica Bk 1: chapter 8 no. 13
  24. 1545 edition, Full text (original page views) at Internet Archive.
  25. Full text (original page views) at Internet Archive.
  26. Full text (original page views) at Google.
  27. T. Bedingfield, Cardanus Comforte, T. Marshe, London 1573. Full text (page views) at Hathi Trust.
  28. Full text (original page views) at Google.
  29. Full text (original page views) at Bayerische StaatsBibliothek; De Sapientia at pp. 1-273.
  30. 1545 edition, Full text (original page views) at Google.
  31. Full text (original page views) at Google.
  32. An electronic copy of his book Ars Magna (in Latin)
  33. Full text (original page views) at Bayerische StaatsBibliothek; another at Internet Archive.
  34. The Rules of Algebra: Ars Magna, Dover Books on Mathematics, translated by Witmer, T. Richard, foreword by Ore, Oystein, Dover Publications, 2007 [1968], p. 304, ISBN   978-0-486-45873-1 CS1 maint: others (link)
  35. C. Sponius (ed.), Hieronymi Cardani Mediolanensis opera omnia (Lyons, 1663), IV, pp. 621-end (Google).
  36. Full text (original page views) at Internet Archive; another at New York Public Libraries. Paris 1550 edition, Michael Fezandat and Robert GranIon, (original page views) at Google.
  37. J.M. Forrester (trans.), The De Subtilitate of Girolamo Cardano, (Arizona Center for Medieval and Renaissance Studies, Tempe 2013).
  38. Full text (original page views) at Google.
  39. 1554 edition, Full text (original page views) at Google; from Bayerische StaatsBibliothek/Münchener DigitalisierungsZentrum. 1578 Heinrich Petri edition, Basle, at Google.
  40. Full text (original page views) at Google.
  41. 1564/66 edition, 2 volumes, HenricPetrini, Basel, Full texts at Google, Vol. I, Vol. II.
  42. Full text (original page views) at Google.
  43. Full text (original page views) at Internet Archive. Another at Google.
  44. D.F. Larder, 'The Editions of Cardanus' "De rerum varietate"', Isis, Vol. 59, No. 1 (Spring, 1968), pp. 74-77 (Jstor - open).
  45. 1581 Basle edition (original page views) at University and State Library, Düsseldorf.
  46. Full text (original page views) at Google.
  47. 1582 edition, Full text (original page views) at Hathi Trust.
  48. Full text (John and Cornelius Blaeu, Amsterdam 1640 edition), (original page views) at Google.
  49. A. Paratico (trans.), Nero: An Exemplary Life, by Girolamo Cardano (Inkstone publications, Chameleon Press, Hong Kong 2012).
  50. Full text (original page views) at Internet Archive; another at Google.
  51. Full text (original page views) at Internet Archive. Another at Google.
  52. Full text (original page views) at Internet Archive; also in Google.
  53. 1568 and 1570 editions, Full text (original page views) at Google. 1570 only, at Google; another at Freiburger historische Beistände.
  54. Full text (original page views) at Münchener DigitalisierungsZentrum/Bayerische Staatsbibliothek or at Google.
  55. Text (incomplete, original page views) at Google. Franciscus Zannettus, Rome 1580, Full text (original page views) at Google.
  56. Full text (page views): Iacobus Villery, Paris 1653, edition at Internet Archive; Amsterdam 1654 edition at Google.
  57. The Book of My Life, New York Review Books Classics, translated by Stoner, Jean, introduction by Grafton, Anthony, NYRB Classics, 2002, p. 320, ISBN   978-1-59017-016-8 CS1 maint: others (link)
  58. C. Sponius (ed.), Hieronymi Cardani Mediolanensis opera omnia (Lyons, 1663), I, pp. 262-76 (Internet Archive).
  59. J. Gullberg, Mathematics from the birth of numbers (W.W. Norton & Company, 1997), p. 963. ISBN   0-393-04002-X ISBN   978-0-393-04002-9
  60. The Book on Games of Chance: The 16th-Century Treatise on Probability, Dover Recreational Math, translated by Gould, Sydney Henry, foreword by Wilks, Samuel S., Dover Publications, 2015 [1961], p. 64, ISBN   978-0-486-79793-9 CS1 maint: others (link)
  61. Full text (original page views) at Google.
  62. M. Fierz (trans. H. Niman), Girolamo Cardano, 1501-1576, Physician, Natural Philosopher, Mathematician, Astrologer and Interpreter of Dreams (Birkhäuser, Boston/Basel/Stuttgart 1983), pp. 32-33 (Google).

Sources

  • 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.