Gerolamo Cardano

Gerolamo Cardano was a Renaissance polymath whose expertise spanned mathematics, medicine, biology, physics, chemistry, astrology, astronomy, philosophy, music theory, writing, and gambling. He ranks among the most significant mathematicians of his era and played a foundational role in the development of probability theory. He was responsible for bringing binomial coefficients and the binomial theorem to the Western world, and authored over two hundred scientific works.

Cardano is credited with partially inventing and describing several mechanical devices, including the combination lock, a gimbal made of three concentric rings that allows a compass or gyroscope to rotate freely, and the Cardan shaft with universal joints—a mechanism still used in vehicles to transmit rotary motion at various angles. He also made important contributions to the study of hypocycloids, which he published in *De proportionibus* in 1570. The generating circles for these curves, later called Cardano circles, were employed in building the first high-speed printing presses.

Today, Cardano is best remembered for his work in algebra. In his 1545 book *Ars Magna*, he became the first European to systematically use negative numbers. He also published, with proper attribution, the solutions to cubic and quartic equations developed by other mathematicians, and acknowledged the existence of imaginary numbers.

Cardano was born on 24 September 1501 in Pavia, Lombardy, the illegitimate son of Fazio Cardano, a mathematically gifted jurist and lawyer who was a close friend of Leonardo da Vinci. According to Cardano’s autobiography, his mother, Chiara Micheri, tried various abortive medicines to end the pregnancy. He wrote that he was “taken by violent means” from her and was “almost dead”; her labor lasted three days. Shortly before his birth, his mother fled Milan for Pavia to escape the Plague, which had already killed her other three children.

His childhood was marked by depression, frequent illness, and a harsh upbringing by his overbearing father. In 1520, he entered the University of Pavia. Against his father’s wish that he study law, Cardano was drawn to philosophy and science. The university was forced to close in 1524 due to the Italian War of 1521–1526, so he finished his studies at the University of Padua, earning a doctorate in medicine in 1525. His eccentric and confrontational personality made him few friends, and he struggled to find work after graduating. In 1525, he repeatedly applied to the College of Physicians in Milan but was rejected due to his combative reputation and illegitimate birth. Nonetheless, many members of the College consulted him because of his undeniable intelligence.

Cardano wanted to practice medicine in a large, wealthy city like Milan, but lacking a license, he settled in the town of Piove di Sacco, where he practiced without one. In 1531, he married Lucia Banderini. Before her death in 1546, they had three children: Giovanni Battista, Chiara, and Aldo Urbano. Cardano later described those years as the happiest of his life.

With help from several noblemen, he obtained a mathematics teaching position in Milan. After finally receiving his medical license, he worked as both a mathematician and a physician, treating several influential patients. He soon became one of the most sought-after doctors in Milan, and by 1536 he could afford to quit teaching, though he remained interested in mathematics. His medical reputation was so strong that aristocrats tried to lure him away from Milan; Cardano later wrote that he turned down offers from the kings of Denmark and France and the Queen of Scotland.

In 1541, Cardano served as rector of the college of medicine in Milan, where he met the city’s ruler, Holy Roman Emperor Charles V. He was a strong supporter of Charles, viewing him as a defender of Europe against the Ottoman Empire and a patron of culture, yet he also praised the emperor’s enemies, Francis I of France and Sultan Suleiman the Magnificent, as virtuous opponents.

Cardano was the first European mathematician to systematically use negative numbers. In *Ars Magna*, he published, with attribution, Scipione del Ferro’s solution to the cubic equation and Lodovico Ferrari’s solution to the quartic equation. The solution to a specific cubic equation had been communicated to him in 1539 by Niccolò Fontana Tartaglia in the form of a poem (Tartaglia later claimed Cardano had sworn to keep it secret, sparking a decade-long dispute), but del Ferro’s solution predated Tartaglia’s. In his exposition, Cardano acknowledged the existence of what are now called imaginary numbers, though he did not understand their properties—those were first described by his contemporary Rafael Bombelli. In *Opus novum de proportionibus*, he introduced binomial coefficients and the binomial theorem.

Chronically short of money, Cardano supported himself as a skilled gambler and chess player. His book *Liber de ludo aleae* (“Book on Games of Chance”), written around 1564, was not published until 1663.