The time of the Renaissance resonates with the names of individuals who have made an impact in the development of mathematics. One name, however, echoes more loudly than the others: Gerolamo Cardano.
A portrait of Gerolamo Cardano. Wellcome Images, CC BY-SA 4.0
We know so much about Cardano’s life and personal opinions because of his candid and forthright account in De Vita Propria Liber (The book of my life). The memoir gives us unparalleled insight into the mind of this polymath as well as detailed information regarding his accomplishments during his lifetime. An analysis of the chapter “Things of worth which I have achieved” certainly gives a great starting point for research.
Gerolamo Cardano was born in what would later become Northern Italy in September 1501. It was a lively period in the development of Europe and the world: the time of the Borgias, Copernicus, Michelangelo and Christopher Columbus.
Cardano was the illegitimate son of Fazio, a lawyer, and Chiara Micheri, who suffered terribly during a three-day labour before the boy was born. The baby Cardano had to be “revived in a bath of warm wine” shortly after birth—the whole story of the incident laid bare in the second chapter of Cardano’s autobiography. Cardano attributes his survival to the alignment of the planets, describing his birth horoscope in intricate detail.
Fazio employed his son as a page for much of the boy’s childhood and ensured that Cardano had a good grounding in mathematics. Fazio was an accomplished mathematical thinker who had been cited in Codex Atlanticus, a collection of papers and drawings produced by Leonardo da Vinci between 1478 and 1519. Cardano talks of his education from his father, which began with basic arithmetic and moved on to geometry:
After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself.
Cardano studied at the University of Pavia, which had been founded in 1361, a school for philosophical and legal thinking. He transferred to the University of Milan when Pavia’s establishment was forced to close due to sieges from the French. It was there that he graduated as a physician.
The university of Pavia. Image: Giovvani Dall’Orto
Cardano spent time in prison, during the prominence of the Inquisition, for casting the horoscope of Jesus. Horoscopes were held in high regard by Cardano, who explains the alignment of stars and planets at many momentous points in his life. This obsession meant that Cardano also used the zodiac to foretell the exact date of his death. He was correct in his prediction, although it is hypothesised that he took matters into his own hands to ensure that his forecast came to pass.
Thanks to the memoir, we know some very personal details of Cardano’s life. We know what he looked like, what he enjoyed eating, the disappointment that he had in his sons, and with surprising intimacy, the fact that he was impotent for much of his young adulthood. Cardano dedicates a whole chapter to “a meditation of the perpetuation of my name”—he was determined to be remembered. This column will focus in on three areas of his mathematical legacy.
Algebra
In 16th century Italy, a popular activity of the top thinkers of the time was to take part in equation solving competitions. Public contests would be held where those competing would challenge each other to solve increasingly difficult equations. The mathematicians were highly secretive, not wanting to give away their methods of working for fear of giving others an advantage.
The solving of cubic equations was classed as an ultimate challenge. One mathematician, Niccolo Fontana (also known as Tartaglia the Stammerer) figured out an algorithm for solving third order polynomials, winning many competitions where the other competitors could only find approximate solutions. Cardano, somehow, managed to convince Tartaglia to tell him his method, and promised not to share it.
Cardano kept his word and did not tell of Tartaglia’s method. However, in 1543 on a visit to Bologna University, Cardano and his student Ferrari met with the son-in-law of Scipione del Ferro, Hannival Nave. Del Ferro, who had died in 1526, was the lecturer for arithmetic and geometry at Bologna University, and Nave had taken over the role and inherited his father-in-law’s notebooks. It was in these notebooks that Ferrari and Cardano found a version of Tartaglia’s method which pre-dated the Stammerer’s work. With this discovery, Cardano felt it was fair to publish the technique himself.
`Cardano’s Rule’ is still used today to solve cubic polynomials. It was published in Cardano’s Ars Magna in 1545 along with the work of Ferrari, which reduced quartics down to cubics. This meant that, for the first time, equations which had no basis in the `real world’ could be solved. Ars Magna is an exceptional legacy. The work on cubics and quartics stimulated further research by others: abstract algebra had been born.
This legacy came at a price however. Tartaglia was very unhappy with the publication and the pair were locked in an acrimonious dispute for decades.
Imaginary numbers
Ars Magna, and the solving of cubics and quadratics lead to another `discovery’–that of imaginary numbers. Cardano posed this problem in his book: “divide 10 into 2 parts so that that product of those parts is 30 or 40”.
Cardano states that this problem is seemingly impossible, but then shows how he can solve it with a bit of imagination…
Image: Cardano archive
Or,
\begin{align}
5 &+ \sqrt{-15} \\
5 &- \sqrt{-15} \\
25 &- (- 15) = 40.
\end{align}
Cardano did not take this discovery further, only using it as a “folly”, but it appears to be the first time that the square root of a negative number is used in such a way. This use of an imaginary number was eventually given prominence by Leonhard Euler and Carl Friedrich Gauss in the eighteenth century. Complex numbers, the mix of imaginary and real numbers, have applications today in areas such as fluid mechanics and electrical engineering.
Probability
A century before the correspondence between Pascal and Fermat, considered by many to be the origin of probability theory, Cardano wrote Liber de Ludo Aleae (The book on games of chance) in which he calculated the different possibilities and outcomes of the gambling games of which he was so fond. He introduced the notion that the probability of an event could be regarded as a value between 0 and 1. Cardano tells us, “I was inordinately addicted to the chessboard and the dicing table” in his autobiography, partaking in some form of betting every day, which is even more enlightening when we discover that one of the chapters in the book is dedicated to methods of cheating!
Dice players in the 16th century. Image: The Walters.
Unfortunately, although the work was completed in 1564, it was not published until 1663. As a result, it did not have the impact of Pascal’s and Fermat’s letters. Five of the letters between the pair covered the same games of chance that Cardano had examined in his book. Cardano also investigated a way to distribute the winnings of a game of chance if the game was interrupted and not able to be completed—known as the `problem of points’. His answer was based on the points already won, while Pascal and Fermat based their solution on the probability that each would win if the game had properly concluded.
Of all his accomplishments, Cardano seems most proud of the reputation that he forged for himself. Parts of his autobiography are solely concerned with the “testimony of illustrious men”, which gives numerous quotations about Cardano’s character, as well as an extensive list of other publications which have cited his work. Two passages stand out:
Julius Caesar Scaliger ascribed more titles to me than I should have thought of arrogating to myself, calling me `ingenium profundissimum, felicissimum et incomparabile’ [A man of most profound, most favoured, and incomparable genius]
Of problems solved or investigated I shall leave something like forty thousand, and of minutiae two hundred thousand, and for this that great light of our country used to call me ‘the man of discoveries’.
Cardano was determined, throughout his life, to be discovered himself. It seems appropriate to now rediscover his legacy, and to herald his achievements for all to hear.
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