Cardano’s lists

Many mathematicians have heard of Girolamo Cardano (sometimes called Jerome Cardan). His book Ars Magna (The Great Art), published in 1545, presents the first advances in algebra made in Europe since classical times, in particular the solution of a general cubic equation, using complex numbers. This was also the occasion of his bitter dispute with Niccolo Tartaglia, who had showed his solution to Cardano after making him swear not to reveal it to anyone. (Cardano later claimed he had seen it in a book by Scipione del Ferro and was therefore absolved from his oath.)

It is a bit of a surprise to learn that Cardano’s profession was not mathematician, but physician. Among his famous patients was Archbishop Hamilton of St Andrews.

He was also a very strange character. He wrote many books; the last was an autobiography, The Book of my Life (not his only autobiographical writing; he clearly found himself a fascinating subject). How many of us would begin Chapter 2 of their autobiography as Cardano does, after some unreliable details about his ancestors in Chapter 1, with the statement

Although various abortive medicines—as I have heard—were tried in vain, I was normally born on the 24th day of September in the year 1500, when the first hour of the night was more than half run, but less than two-thirds.

(The curious precision was necessary for the demands of astrology, another of Cardano’s professions: he cast his own horoscope several times during his life. He also cast the horoscope of Edward VI of England, but got it badly wrong; by the time the horoscope was published, the boy king had already died.) He was in fact illegitimate, although his parents lived together apart from a short break—but so were many others at the time, including Galileo’s three children.

The book is not a narrative, but arranged by topic, and within each topic it consists mainly of a list. The bulleted list had not been invented in those days (I guess), so he numbered the list items within paragraphs of text. Thus, in the chapter “Successes in my practice”, he lists 40 of his successful cases, annotated to show how he had succeeded where others had failed.

Other lists include “Five unique characteristics by which I am helped”, “Books written by me” (165 published books, on topics ranging from dialectics to dreams, as well as 95 unpublished manuscripts), “Things absolutely supernatural” (all of which happened to him), “Guardian angels” (he only had one), “Rare circumstances of my life: the avenging of my son” (a list of the terrible fates suffered by the people who had a hand in having Cardano’s son executed for murder), “Perils, accidents, and manifold, diverse and persistent treacheries”, … you get the idea.

Like me you are probably interested in what he had to say, in his summing up of his life, about mathematics. There are only two references to his work. In “Things of worth which I have achieved in various studies”, he says,

In arithmetic I advanced almost the whole field of the science including the sections treating, as they call it, of algebra; my discoveries dealt also with properties of numbers, especially of those having similar ratios between themselves. I also expounded the numerical functions already discovered, showing either a simplified treatment or some uncommon formula method, or both. In geometry I dealt with confused and reflex proportions, and the treatment of infinity with finite numbers and through finite, although it was first discovered by Archimedes.

(No, I don’t know either what most of this means. The translation is by Jean Stonier, and the editor, Anthony Grafton, says in the introduction, “Stonier worked from a corrupt base text, and produced an imperfect version. But her translation reads well, gives a clear impression of both the author and his book, and has the great merit of existing.” But perhaps the translator was a bit at sea with mathematics.)

Later, in the chapter listing his books, he comments,

From [On Subtlety] I turned to the Book of the Great Art which I composed while Giovanni da Colla was contending with me, and also Tartaglia, from whom I had received the first chapter. He preferred to find in me a rival, and a better man at that, than an associate bound to him by gratitude, and of all men most devotedly his friend.

The Grumpy Old Man is not just a recent phenomenon!

Let us give Cardano the last word (from his epilogue):

Well, then, I am no longer troubled by imputation of falsehood, and am as one who has grown old in the love of truth, with which the love of God, the hope of immortal life, the possession of so much distinction and the advantage of wisdom are closely joined: which united good I am loath to confound with one wrong move. Let us leave that to those who are deceived in their ignorance, who rejoice in lying words, who exaggerate all they have heard, read or even seen, hoping thereby to be able to deceive. But will they, relying on some hope, make question concerning those things to the evidence of which they will subscribe not one jot of credence—though there be a thousand testimonies of one good man, or a thousand bearing witness?

About Peter Cameron

I count all the things that need to be counted.
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1 Response to Cardano’s lists

  1. Klas says:

    There is also a biography about Cardano written by Oystein Ore

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