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‘Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.’
Girolamo Cardano, De Vita Propria Liber, translated by Jean Stoner as The Book of My Life (E. P. Dutton, New York, 1930), p. 246.
Girolamo Cardano, Opera omnia (Lyon, 1663), vol 1, portrait.
Girolamo Cardano, born 1501 in Pavia, was a true renaissance man. He is famous today for his works in mathematics but he is also highly regarded in many other fields such as medicine, physics, biology, chemistry, philosophy, astrology and even gambling. Cardano was the son of another mathematician and lawyer, Fazio Cardano, who was reported to be a good friend of Leonardo Da Vinci and who consulted Leonardo with any problems he may have had with geometry. It was from his father that Cardano first learned mathematics and gained his appreciation for it. His father, however, wanted him to study law but Cardano had more of a fondness for academia and, after an argument with his father, he went on to study at the University of Pavia and subsequently Padua from where he received his medical degree. After this, he lived Milan in relative poverty and was known to make money by gambling, using his knowledge of probability to gain an advantage over his opponents in games such as cards, dice and chess. He even wrote a book in 1564 on probability called Liber de Ludo Aleae (The Book on Games of Chance) which was one of the first books to show the systematic computations of probabilities. He eventually went on to become a mathematics lecturer in the Piatti Foundation in Milan. This gave him time to practice his medicine. After saving the lives of some prominent figures while practising medicine, he gained wealth and influence through many of his appreciative patients.
In 1539, Cardano met Tartaglia another famous mathematician who had gained fame after solving certain cubic equations. Cardano asked Tartaglia to tell him how he solved the cubic, and Tartaglia agreed on the condition that Cardano would not publish it until he himself had done so. Cardano swore an oath in agreement. Cardano worked intensively on solving both cubic and quartic equations, surpassing what had been achieved by his rival, Tartaglia, giving rise to a notorious controversy.
Girolamo Cardano, Opera omnia (Lyon, 1663), vol. 4, title page.
The Edward Worth Library is lucky enough to have Cardano’s Opera Omnia (complete works) in ten volumes. In volume 4, for example, Cardano guides the reader through addition, subtraction, multiplication and division.
Girolamo Cardano, Opera omnia (Lyon, 1663), vol 4, p. 20.
In this example of multiplication, the product of 79,507,864 and 73,946 is shown to be 5,879,288,511,344. The method he uses here is the traditional long multiplication, taught in primary schools: he multiplies 79,507,864 by each digit of 73,946, in turn (beginning from the right). Each time he multiplies by one digit he writes the answer below and moves the next answer one digit to the left. This results in five rows of digits which are then added together to give the final answer. There are also examples of how to divide using the galley division method widely used as far back as al-Khwārizmī’s time, c. 800.
Such arithmetic was, of course, quite elementary at the time. Of more significance is Cardano’s Ars Magna (The Great Art, included in his Opera Omnia). This is where the solutions of cubic and quartic equations were first published. It was also where complex numbers first appeared, however Cardano failed to develop fluency in the use of these numbers, a task that was left to his fellow Italian, Rafael Bombelli (1526-1572).
J J O’Connor and E F Robertson, Girolamo Cardano, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 1998).
Richard W. Feldman, ‘The Cardano-Tartaglia Dispute’ in Frank J. Swetz (ed.) From five fingers to infinity: A journey through the history of mathematics (Chicago: Open Court, 1994), 364-368.
Girolamo Cardano entry in Encyclopædia Britannica (2015).
Text: Conor Halton and Maurice OReilly.