MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

Gerolamo Cardano was an influential mathematician from the early 1500’s who pioneered the theory of probability and the solution to the cubic equation. He had a difficult life beginning by being the illegitimate son of Fazio Cardano and Chiara Micheana, who eventually married. They lived in Milan, the same town as Leonardo da Vinci who consulted Fazio several times on geometric questions (Ore 3). Fazio refused to send his son away to school, but eventually Geralamo attended Pavia to study medicine when he was nineteen. As a student, Cardano was frivolous and he described that "gambling was not only his favorite pastime but also his main source of income" (Ore 7), which led to "gathering notes on the subject of chance and probability" (Ore 8). Cardano worked hard and received his doctorate in medicine, which led to him settling down with a practice in Sacco. He only spent six years in Sacco, but this was where he met his wife and started a family, which he "recalled this period as the happiest of his life"(Gliozzi 64). Cardano moved to Milan to teach mathematics and continue his practice of medicine. He soon became a famous physician and accepted a job as the chair of medicine at the University of Pavia. Unfortunately, Cardano faced hardships in his life with his eldest son being beheaded after he poisoned his wife and the Inquisition for the heresy of casting the horoscope of Jesus Christ imprisoned Cardano. From his interest in astrology, he predicted his won life ending at the age of 75, which it did by committing suicide on September 12, 1576 (Biography of Cardano).

Throughout his career Cardano wrote over 200 works on subjects that interested him including medicine, mathematics, physics, philosophy, religion, and music (Gliozzi 65). These works included an autobiography (which is why we have so many details about his life), The Gambling Scholar, Ars Magna, and De Utilitate to name a few. Gambling was one of Cardano’s favorite pastimes, which sparked his interest in the mathematics behind the various games. He produced the probability recipe p = f/c or probability equals favorable outcomes divided by the "circuit" or "the total number of equally likely cases" (Ore 148). Even though Cardano enjoyed gambling he expressed in De Utilitate that "gambling brings loss in two ways, first because a man loses money, secondly because he is led to neglect his business, arts, and studies." This statement is ironic because Cardano frequently gambled and won money.

Cardano was the first to publish the solution to the cubic equation, which revealed a controversy about who should be given credit. Nicolo Tartaglia shared the procedure with Cardano who promised not to publish it, but six years later Cardano learned that Scipione dal Ferro was the one who actually discovered it and gave it to Tartaglia. Therefore, since Cardano completed the other solutions to the cubic equation and Tartaglia did not deserve the credit, he published the solutions in his Ars Magna. Tartaglia then "accused Cardano of perjury and wrote of him in offensive terms" (Gliozzi 65). One of the demonstrations from the Ars Magna can be found here: .

This demonstration can be difficult to follow and construct, because Cardano presents a two dimensional object to represent a three dimensional object. He also introduces negatives, which many struggle to follow. These solutions are difficult because Cardano is creating the solutions geometrically and not algebraically, which is what we use today. Cardano constructs the cube similar to the way Muhammad al-khwarizmi constructed the solution to the quadratic equation. The demonstration above deals with the equation x³ + 6x = 20 and Cardano builds onto three sides of the cube (x^3) which each equals 2x, to acquire the 6x in the equation. He eventually builds the cube up so that there are overlaps in the model and he must use negatives to eliminate those overlaps. For futher reference look at Karen Parshall’s The Art of Algebra.

Cardano was a gifted man who did not follow society’s norms through gambling and being imprisoned, but he was influential. Others may be more well known for greater contributions today, but he "must always be credited with having introduced new ideas that inspired new investigations" (Gliozzi 66).

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