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ON 8 September 1526, the Blessed Virgin’s birthday, Jerome Cardano was in Venice. While others were praying, Cardano was playing cards at the house of senator Thomas Lezun. He was confident that his recent invention – the mathematics of probability – was about to pay off. As well as money, he was hoping to win a night with a beautiful prostitute. Such an experience, he thought, might be just the thing to end his four-year streak of impotence.

He was nearly 25, and had recently graduated from medical school. Though Cardano would later become renowned across Europe for his skills as a physician, as well as a celebrated author, astrologer and mathematician, right now what he needed was money. The Milanese College of Physicians had denied him a licence to practice, possibly because of his illegitimate birth, although his rude, confrontational personality certainly didn’t help. It was this need for funds that drove him to gamble – and sparked his interest in all things mathematical.

Cardano grew up in Renaissance Milan, the son of a lawyer who counted Leonardo da Vinci among his associates. As a child, Cardano sometimes sat on da Vinci’s floor while the adults talked philosophy, law and culture. He was even taken to see The Last Supper, freshly painted on the wall of the convent of Santa Maria delle Grazie; when he saw it again years later, he was amazed at how “blurred and colourless” the once-vivid fresco had become.

It was as a student, during one of many nights in the local tavern playing dice and cards, that Cardano realised his time could be spent much more lucratively if he thought about stakes and the likelihood of certain numbers coming up when rolling several dice at once. Especially since everyone else was working under the assumption that dice rolls were determined by the Almighty and thus couldn’t be predicted.

In his spare time, Cardano began to jot down his insights. Later in life, he gathered these writings into The Book on Games of Chance. This told the reader how to work out various probabilities, such as the likelihood of any particular outcome of a dice roll. Cardano also developed what we now call the law of large numbers, showing that 1000 flips of a coin should turn up almost exactly 500 heads and 500 tails.

Gambling didn’t provide enough income, however, especially as Cardano’s grasp of probability wasn’t strong enough to make him unbeatable, or to make it clear exactly when he should stop. He also yearned for respectability, and took lecturing jobs in mathematics while studying philosophy and astronomy. The properties of the universe were a constant source of fascination. He speculated about the nature of light and the character of time, which he considered to be something that only flows within our universe. In the region outside, it “remains eternal”, he wrote in On Subtlety, his “complete account of the universe”.

He was granted a medical licence eventually, and soon gained a reputation as a skilled and innovative physician. His mathematical skills meant he was repeatedly offered work in military research, though he always turned it down, and his texts teaching the basics of astronomy sold in significant numbers. Such was the demand that European publishing houses sometimes pirated his works. Shakespeare scholars even suggest that much of Hamlet’s “To be or not to be” speech is Consolation, a lament on the death of his eldest son.

Cardano lived at a time when the works of great Islamic mathematicians such as Omar Khayyam had recently become widely available in Latin translation. He was entranced by them, and became obsessed with creating a guide to algebra – the Great Art, as he called it. It would explain how to solve quadratic equations (containing x² terms), as well as cubic (x³), quartic and quintic equations. These weren’t only of interest to mathematicians: they had applications in the military and financial sectors, and solutions were highly prized – and carefully guarded.

Cardano’s big problem was that the books available only provided a method for solving quadratic equations. Eventually (by somewhat questionable means) he “borrowed” a solution for the cubic equation and used this to develop solutions for the quartic and quintic equations. Along the way, Cardano discovered a puzzling phenomenon even more abstract than probability: imaginary numbers.

The most basic imaginary number, now denoted as i, is the square root of -1. It was even more alien and confounding then than it is to countless schoolchildren today. Negative numbers were themselves still a somewhat suspicious concept, and zero had only just become accepted as a mathematical object.

“Cardano realised he could win more often at dice if he thought about probability”

Cardano encountered square roots of negative numbers halfway through some of his algebraic workings. It didn’t really matter: he could keep them in, and since they were squared later in the process, the problem disappeared. But he found their existence curious, labelling them “impossible quantities”. In The Great Art, he declares they are neither positive nor negative, but “some recondite third sort of thing”.

These days, they are far from recondite. Engineers use i to develop electronic circuits, compression algorithms and myriad other facets of 21st-century life. Together with probability theory, i is also essential to our manipulations of the Schrödinger equation of quantum theory. Cardano’s two major mathematical finds have turned out to be the supporting pillars for our best explanation of how everything in the universe works.

Cardano’s lack of fame today may have something to do with his arrest by the Inquisition in 1570. The most likely reason is because he had presented a previous pope with a horoscope of the Son of God. At the time, astrology was widely accepted, and though Paul III had welcomed this gift from someone regarded as a talented astrologer, the papacy had since passed to Pius V, who had outlawed the practice. A horoscope of Christ was viewed by many in Pius’s court as an attempt to subjugate the Creator to his creation: if the stars foretell the life of Christ, that leaves no room for God to act as He chooses.

After a few months of incarceration, Cardano was released to house arrest. But he was forbidden to teach, publish books or even talk about why he had been arrested. He had to pay a significant lump sum to the church for this “freedom”, from which he received a meagre monthly income. With the smell of the Inquisition’s bonfires hanging around him, none of his former associates were comfortable in his company ever again, and his fame and esteem quickly waned.

It was during this time that Cardano wrote the autobiography that tells, among other tales, of that night’s gambling in Venice. He used his sense for probability to guess that his opponent was cheating by using marked cards. Having carefully won back the money he had lost, he drew his dagger and slashed Senator Lezun’s cheek in retribution before vanishing into the night. He didn’t mention what happened about the prostitute.

This article appeared in print under the headline “The original chancer”