It would be incomplete to simply list Girolamo Cardano as an Italian physician and mathematician. To reflect the true character of his life, one would have to add that he was the illegitimate son of a noted lawyer, a compulsive gambler, a popular astrologer, a one-time prisoner of the Inquisition, and the father of a convicted murderer. Despite the sordid nature of many aspects of his life, Cardano was also a brilliant physician and mathematician, as well as the author of more than 200 works on human medicine, mathematics, physics, philosophy, religion, natural science, and music. His contributions to mathematics were mainly in the area of algebraand included the first generalized solution to cubic (third degree polynomial) equations as well as the solution to certain quartic (fourth degree polynomial) equations, even though some of the solutions had been borrowed from others. Cardano was also one of the first to recognize the existence of the square root of negative numbers, now called imaginary numbers, although he did not know how to deal with them.
Girolamo Cardano was born in Pavia, Italy, on September 24, 1501. His father was Fazio Cardano, a well-known lawyer and a friend of Leonardo da Vinci. His mother was Chiara Micheri. Cardano's parents were not married at the time he was born, and the stigma of being an illegitimate child followed him through life. As a youth, Cardano was often sick and mistreated. Although his parents did eventually marry, his father did not live with the family until Cardano was seven years old.
When Cardano became of age, his father encouraged him to study the classics, mathematics, and astrology. He entered the University of Pavia, where he completed his undergraduate studies. In 1524, his father died and left him a house and a small inheritance. Cardano returned to study in Pavia, where he earned a doctorate in medicine in 1526. Shortly thereafter he took up a medical practice in Saccolongo near Padua, where he married Lucia Bandareni in 1531. In time they had two sons and a daughter.
In 1534, friends of his father helped Cardano obtain a teaching position in mathematics at a school in Milan. He continued to practice medicine in addition to his teaching duties, and his success in treating several influential patients soon made him the most sought-after physician in Milan. By 1536, Cardano's thriving medical practice allowed him to leave his job as a teacher, although his interest in mathematics continued.
Cardano published his first book on mathematics, Practica arithmetice et mensurandi singularis, in 1539. In this book, he first demonstrated his superior mathematical skills in approaching problems in algebra. In one example, he was able to solve a specific cubic equation by manipulating the terms to reduce the problem to a second-degree quadratic equation that could be solved.
An algebraic solution to cubic and quartic equations had long eluded mathematicians, and Cardano's work in his Practica arithmetice, although limited to only certain cubic equations, was impressive. Cardano felt little satisfaction in his accomplishments, however, because he knew that fellow mathematician Nicolò Fontana, also known as Tartaglia, had achieved solutions to even more difficult cubic equations four years earlier, but had refused to divulge his methods. Cardano had repeatedly beseeched Tartaglia for his secret to no avail. Finally, just as Cardano's book was being published in 1539, Tartaglia agreed to share his methods, but only on the condition that Cardano take an oath that he would not reveal them until Tartaglia had written his own book.
At this point fate smiled on Cardano. A young man named Ludovico Ferrari came asking for a job as a servant. Cardano took him in and quickly discovered that his new servant possessed a brilliant mind. Eager to share his enthusiasm for mathematics, Cardano taught Ferrari algebra and eventually revealed Tartaglia's secret to him. One of the cubic equations Tartagliahad learned to solve was the so-called "depressed cubic," which lacked a second-power term. Working together, Cardano and Ferrari discovered a method to reduce any generalized cubic equation to a depressed cubic. Using Tartaglia's methods, they could then solve the equation. Their elation in making this monumental advancement in algebra was dampened by the knowledge that Cardano had given his oath not to reveal Tartaglia's methods. What was worse, Tartaglia seemed in no hurry to publish his long-promised book, which would have freed Cardano from his oath.
In 1543, Cardano and Ferrari traveled to Bologna where they went over the papers of another mathematician, Scipione dal Ferro, who had died in 1526. They discovered that dal Ferro had solved the depressed cubic equation in 1515, but had kept it secret until just before his death, when he revealed it to his student, Antonio Fior. Fior had foolishly used his new-found knowledge to challenge Tartaglia to a contest, only to have Tartaglia discover the method for himself during the competition.
Armed with the knowledge that it was dal Ferro who had originally solved the depressed cubic equation, Cardano felt his obligation to Tartagliawas removed. Thus, in 1545, Cardano published his second book on mathematics entitled Artis magnae sive de regulis algebraicis liber unus, commonly called Ars magna, ("Great Art"). In it, he revealed not only the solution to the generalized cubic equation, but also the solution to the biquadratic quartic equation, which had been developed by Ferrari.
Although Cardano gave full credit to dal Ferro, Fior, and Tartaglia for their work on cubic equations, Tartaglia was outraged. He claimed Cardano had broken his oath, and he began a long and vicious letter-writing campaign denouncing Cardano as a scoundrel. Despite this dispute, Cardano's book was widely acclaimed. Besides the solutions to cubic and quartic equations, Cardano presented many other new ideas in algebra that became the basis for the theory of algebraic equations.
In 1546, Cardano's wife died at the age of 31, leaving him with three children. His eldest son, Giambattista, was a promising scholar and it appeared he would become a successful physician like his father. Then, in 1557, Giambattista, poisoned his wife. Despite his father's appeals and influence, Giambattista was convicted of murder and was beheaded in 1560. In 1562, Cardano left Milan and took a position teaching medicine at the University of Bologna. Tragedy struck again in 1565, when his devoted student and collaborator Ferrari was poisoned.
In 1570, Cardano was accused of heresy for having cast the horoscope of Jesus Christ and for attributing the events of Christ's life to the influence of the stars. He was imprisoned by the Inquisition for several months before he was released on the condition that he abandon teaching. At the advice of friends, Cardano moved to Rome in 1571 and asked for protection from the Pope. Pope Pius V refused to help him because of his conviction for heresy, but Pope Gregory XIII was more lenient and granted Cardano a lifetime pension in 1573.
Cardano revised his Ars magna in 1570 to include a section dealing with solutions to the cubic equation that involve the square roots of negative numbers. Today, these numbers are known as imaginary numbers. Cardano had no such knowledge of imaginary numbers, but the fact that he recognized their existence led to further work by others. His work on solving cubic and quartic equations stimulated work by Thomas Harriot, Leonhard Euler, René Descartes, and others over the next several hundred years.
Cardano also wrote several books that presented new ideas in many other disciplines. He studied the physicsof projectiles in motion and correctly observed that their trajectories resembled a parabolic curve. Cardano was the first to deduce that the ratio of the distances of a projectile shot through air and water is inversely proportional to the ratio of the densities of the two mediums. In hydrodynamics, he observed and measured the flow of streams and stated that the velocity of the water was greater at the surface than near the bottom, contrary to what most people believed. Cardano's work in geology was also influential.
Cardano retired on his pension and lived quietly in Rome. He died on September 21, 1576, just three days short of his 75th birthday. A century after Cardano's death, Gottfried Leibniz summed up Cardano's turbulent life when he wrote "Cardano was a great man with all his faults; without them, he would have been incomparable."
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