Gerolamo Cardano was an Italian expert in many areas, including mathematics, medicine, biology, physics, chemistry, astrology, astronomy, philosophy, music theory, writing, and gambling. He was one of the most important mathematicians during the Renaissance and helped start the study of probability. He introduced the binomial coefficients and the binomial theorem to the Western world. He wrote more than 200 works on science.

Cardano partially invented and described several mechanical devices, such as the combination lock, the gimbal (a device with three rings that let a compass rotate freely), and the Cardan shaft with universal joints (a part that helps transfer motion in vehicles). He studied shapes called hypocycloids and wrote about them in 1570. The circles used to make these shapes were later called "Cardano circles" and helped create early high-speed printing presses.

Today, Cardano is known for his work in algebra. In his 1545 book Ars Magna, he first used negative numbers in Europe. He shared solutions for cubic and quartic equations from other mathematicians and recognized the existence of imaginary numbers.

Early life and education

Girolamo Cardano was born on September 24, 1501, in Pavia, Lombardy. He was the illegitimate child of Fazio Cardano, a jurist and lawyer who had a strong interest in mathematics. Fazio was a close friend of Leonardo da Vinci. In his own writings, Cardano described how his mother, Chiara Micheri, used medicines to end her pregnancy before he was born. He wrote, "I was taken from my mother by force; I was nearly dead." Chiara gave birth after three days of labor. Before his birth, she moved from Milan to Pavia to avoid the Plague, which caused the deaths of her other three children.

Cardano had a difficult childhood marked by frequent illnesses and a strict upbringing by his father. In 1520, he began studying at the University of Pavia. His father wanted him to study law, but Cardano was more interested in philosophy and science. However, during the Italian War of 1521–1526, the university in Pavia closed in 1524. Cardano then continued his studies at the University of Padua, where he earned a doctorate in medicine in 1525. His unusual and argumentative behavior made it hard for him to gain friends or find work after graduation. In 1525, he applied many times to join the College of Physicians in Milan but was not accepted because of his difficult reputation and his status as an illegitimate child. Despite this, many members of the College of Physicians sought his advice due to his strong intellect.

Early career as a physician

Cardano wished to practice medicine in a large, wealthy city such as Milan, but he was not allowed to do so. Instead, he moved to the town of Piove di Sacco, where he practiced medicine without a license. There, he married Lucia Banderini in 1531. Before her death in 1546, the couple had three children: Giovanni Battista (born 1534), Chiara (born 1537), and Aldo Urbano (born 1543). Cardano later said those years were the happiest of his life.

With the support of several noblemen, Cardano earned a teaching position in Milan for mathematics. After finally receiving his medical license, he worked as both a mathematician and a doctor, treating some important patients. Because of this, he became one of Milan’s most popular doctors. By 1536, he no longer needed to teach, though he remained interested in mathematics. His reputation in medicine was so strong that members of the aristocracy tried to persuade him to leave Milan. Cardano later wrote that he refused offers to work for the kings of Denmark and France and the queen of Scotland.

Mathematics

Gerolamo Cardano was the first European mathematician to use negative numbers in an organized way. In his 1545 book Ars Magna, he shared the solution to the cubic equation, which was discovered by Scipione del Ferro, and the solution to the quartic equation, found by his student Lodovico Ferrari. These solutions were important contributions to algebra. In 1539, Niccolò Fontana Tartaglia told Cardano a solution to a specific type of cubic equation in the form of a poem. However, del Ferro had solved this equation earlier than Tartaglia. Cardano recognized the existence of numbers now called imaginary numbers, though he did not fully understand their properties. These properties were later explained by Rafael Bombelli. In his work Opus novum de proportionibus, Cardano introduced binomial coefficients and the binomial theorem.

Cardano often struggled with money and stayed financially stable by being skilled at gambling and playing chess. His book Liber de ludo aleae ("Book on Games of Chance"), written around 1564 but published in 1663, provided the first organized explanation of probability. It also included a section about methods to gain an advantage in games. Cardano used dice games to study probability basics. He showed how to calculate odds by comparing favorable outcomes to unfavorable ones, which relates to the probability of an event being the ratio of favorable outcomes to total possible outcomes. He also understood the multiplication rule for independent events but was unsure which values to multiply.

Other contributions

Cardano was a music expert who studied music privately in Milan during his youth. He wrote two books about music, both titled De Musica. The first was published in his 1663 work Hieronymi Cardani Mediolanensis Opera Omnia. Scholars interested in the history of woodwind instruments find this book valuable because it discusses instruments from that family. The second book was published in 1574, and a copy is kept in the Vatican Library. This work is important for studying harmony because it explains the use of microtones. It is also useful for researchers studying 16th-century music performance practices because it includes details about how music was performed at that time. A later book titled Della natura de principii et regole musicali, sometimes credited to Cardano, is likely not his work, according to The New Grove Dictionary of Music and Musicians. Cardano also composed music, including a piece called Beati estis, which is written for 12 voices and features four overlapping canons.

Cardano’s research on hypocycloids led to the invention of the Cardan gear mechanism, which uses two gears—one half the size of the other—to convert rotational motion into linear motion more efficiently than other methods, such as the Scotch yoke. He is also credited with inventing the Cardan suspension, also known as a gimbal.

Cardano contributed to the study of hydrodynamics and believed that perpetual motion is impossible, except in celestial bodies. He wrote two encyclopedias of natural science that include many inventions, facts, and descriptions of superstitions. He also introduced the Cardan grille, a tool used for writing secret messages, in 1550.

In the history of education for the deaf, Cardano argued that deaf individuals could use their minds and should be taught. He was among the first to suggest that deaf people could learn to read and write without first learning to speak. He was familiar with a report by Rudolph Agricola about a deaf person who learned to write.

Cardano’s medical writings included a commentary on Mundinus’s anatomy and Galen’s medicine, as well as several treatises: Delle cause, dei segni e dei luoghi delle malattie, Picciola terapeutica, Degli abusi dei medici, and Delle orine, libro quattro.

Cardano is sometimes credited with inventing the Chinese Rings, also called Cardano’s Rings, but these likely existed before his time. The universal joint, sometimes called the Cardan joint, was not described by Cardano.

Scotland and Archbishop Hamilton

In 1552, Cardano journeyed to Scotland with the Spanish physician William Casanatus, passing through London, to treat the Archbishop of St Andrews. The Archbishop had been unable to speak and was believed to have a disease that could not be cured. The treatment was successful, and diplomat Thomas Randolph wrote that stories about Cardano's methods were still being shared in Edinburgh in 1562. Cardano and Casanatus disagreed about who was responsible for the Archbishop's recovery. Cardano stated that the Archbishop had struggled to breathe for ten years, and after his assistant performed the cure, Cardano received 1,400 gold crowns as payment.

Later years and death

Two of Cardano's children—Giovanni Battista and Aldo Urbano—died in dishonorable ways. Giovanni Battista, Cardano's oldest and most loved son, was arrested in 1560 for poisoning his wife after he learned their three children were not his. Giovanni was put on trial, and because Cardano could not pay the money owed to the victim's family, he was sentenced to death and beheaded. Cardano's other son, Aldo Urbano, was a gambler who stole money from his father. Because of this, Cardano removed Aldo from his family inheritance in 1569.

Cardano moved from Pavia to Bologna partly because he believed the decision to execute his son was influenced by his conflicts with the academic community in Pavia and the jealousy of his colleagues over his scientific work. He also faced accusations of inappropriate relationships with his students. In Bologna, he became a professor of medicine at the University of Bologna.

In 1570, Cardano was arrested by the Inquisition after an accusation of heresy by the Inquisitor of Como, who focused on Cardano's book De rerum varietate (1557). The inquisitors criticized Cardano's writings on astrology, especially his claim that the actions of martyrs and heretics were influenced by the stars. In his 1543 book De Supplemento Almanach, a commentary on the astrological work Tetrabiblos by Ptolemy, Cardano also published a horoscope of Jesus. Cardano was imprisoned for several months and lost his position at the University of Bologna. He renounced his beliefs and was released, likely with help from powerful church leaders in Rome. All of his non-medical works were banned and added to the Church's list of prohibited books.

Cardano moved to Rome, where he received a lifetime payment from Pope Gregory XIII (after being initially rejected by Pope Pius V, who died in 1572). He completed his autobiography there. He joined the Royal College of Physicians and continued practicing medicine and studying philosophy until his death in 1576.

References in literature and culture

In the 1600s, Sir Thomas Browne, an English doctor and thinker, had ten books from the Lyon 1663 edition of the complete works of Cardan in his library.

Browne had a critical view of Cardan.

Richard Hinckley Allen wrote about a funny mention made by Samuel Butler in his book Hudibras.

In Alessandro Manzoni’s novel The Betrothed (I promessi sposi), a character named Don Ferrante is shown as a fan of Cardano’s work. However, Don Ferrante only respects Cardano’s writings about superstition and astrology. He ignores Cardano’s scientific writings because they go against the ideas of Aristotle. Still, he believes Cardano’s astrology-related works are worth listening to, even if they are not correct.

In E. M. Forster’s 1936 book Abinger Harvest, which includes essays, reviews, and a play, Cardano is described kindly in the section titled "The Past." Forster writes that Cardano was so focused on analyzing himself that he often forgot to apologize for his bad temper, his lack of intelligence, his inappropriate behavior, and his desire for revenge.

Works

• A book about the bad practices of recent doctors, Hieronymus Scotus, Venice, 1536 (on medicine).
• A special practice in arithmetic and measurement (on mathematics), Io. Antoninus Castellioneus/Bernadino Caluscho, Milan, 1539.
• Three books titled De Consolatione, Hieronymus Scotus, Venice, 1542. Translated into English by T. Bedingfield in 1573.
• Two books: De Supplemento Almanach; De Restitutione temporum et motuum coelestium; Item Geniturae LXVII insignes casibus et fortuna, cum expositione, Iohan. Petreius, Nuremberg, 1543.
• Five books titled De Sapientia, Iohan. Petreius, Nuremberg, 1544 (with a reprint of De Consolatione and De Libris Propriis, book I).
• De Immortalitate animorum, Henric Petreius, Nuremberg 1544/Sebastianus Gryphius, Lyons, 1545.
• Contradicentium medicorum (on medicine), Hieronymus Scotus, Venice, 1545.
• Artis magnae, sive de regulis algebraicis (on algebra: also known as Ars magna), Iohan. Petreius, Nuremberg, 1545. Translated into English by D. Witmer in 1968.
• Della Natura de Principii e Regole Musicale (on music theory: in Italian), ca. 1546 (posthumously published). (Most likely falsely attributed to Cardano).
• De Subtilitate rerum (on natural phenomena), Johann Petreius, Nuremberg, 1550. Translated into English by J.M. Forrester in 2013.
• Metoposcopia libris tredecim, et octingentis faciei humanae eiconibus complexa (on physiognomy), written 1550 (published posthumously by Thomas Jolly, Paris (Lutetiae Parisiorum), 1658).
• In Cl. Ptolemaei Pelusiensis IIII, De Astrorum judiciis… libros commentaria: cum eiusdem De Genituris libro, Henrichus Petri, Basel, 1554.
• Geniturarum Exemplar (De Genituris liber, separate printing), Theobaldus Paganus, Lyons, 1555.
• Ars Curandi Parva (written c. 1556).
• De Libris propriis (about the books he has written, and his successes in medical work), Gulielmus Rouillius, Leiden, 1557.
• De Rerum varietate, Libri XVII (on natural phenomena); (Revised edition), Matthaeus Vincentius, Avignon 1558. Also Basel, Henricus Petri, 1559.
• Actio prima in calumniatorem (reply to J.C. Scaliger), 1557.
• De Utilitate ex adversis capienda, Libri IIII (on the uses of adversity), Henrich Petri, Basel, 1561.
• Theonoston, seu De Tranquilitate, 1561. (Opera, Vol. II).
• Somniorum synesiorum omnis generis insomnia explicantes, Libri IIII (Book of Dreams: with other writings), Henricus Petri, Basel 1562.
• Neronis encomium (a life of Nero), Basel, 1562. Translated into English by A. Paratico in 2012.
• De Providentia ex anni constitutione, Alexander Benaccius, Bologna, 1563.
• De Methodo medendi, Paris, In Aedibus Rouillii, 1565.
• De Causis, signis ac locis morborum, Liber unus, Alexander Benatius, Bologna, 1569.
• Commentarii in Hippocratis Coi Prognostica, Opus Divinum; Commentarii De Aere, aquis et locis opus, Henric Petrina Officina, Basel, 1568/1570.
• Opus novum, De Proportionibus numerorum, motuum, ponderum, sonorum, aliarumque rerum mensurandarum. Item de aliza regula, Henric Petrina, Basel, 1570.
• Opus novum, cunctis De Sanitate tuenda, Libri quattuor, Sebastian HenricPetri, Basel, 1569.
• De Vita propria, 1576 (autobiography). Translated into English by J. Stoner in 2002.
• Liber De Ludo aleae ("On Casting the Die"; on probability): posthumously published. Translated into English by S.H. Gould in 1961.
• Proxeneta, seu De Prudentia Civili (posthumously published: Paulus Marceau, Geneva,