Cardano, The Gambling Scholar

Title:                      Cardano, The Gambling Scholar

Author:                 Øystein Ore

Ore, Øystein (1965). Cardano, The Gambling Scholar. New York: Dover Publications

LCCN:    65017670

Q143.C3 O7 1965


Date Posted:      December 11, 2014

Review by Francis R. Johnson.[1]

Gerolamo Cardano, or Jerome Cardan, has a place in most histories of science—especially in the histories of mathematics, although histories of medicine accord him well-merited notices. In the brief accounts, the author, with a keen eye for a lively story, gives over most of his limited space to reporting the dramatic controversy with Tartaglia over Cardano’s announcement in his Ars Magna (1545) of a method for solving cubic equations.

In this book Professor Ore’s main purpose, however, is to examine analytically and historically Cardano’s little treatise on games of chance, his Liber de Ludo Aleae, and upon the basis of that examination to establish its author’s right to a place of primary importance in this history of the theory of probability. The book includes, as an appendix, and English translation by Professor Sydney Henry Gould of Cardano’s Latin treatise, with annotations by Professor Ore. The merit of the volume, therefore, is not that of a new and definitive biography of Cardano; it presents no important new facts concerning his life, except in so far as its careful analysis of the ideas expounded in the Liber de Ludo Aleae reveals the originality and significance of its author’s thinking upon the problems of computing probability. Nevertheless, it provides the reader interested in the history of science with a lively, well-written account of the stormy dispute-ridden career of a versatile sixteenth-century scholar whose works and fame were known throughout Europe.

Many illustrations, based upon woodcuts in sixteenth-century books and photographs of contemporary paintings, add to the attractiveness of Professor Ore’s volume as does its excellent typography and design. But the original contributions to knowledge are found principally in Chapters 3 and 5. In the first Professor Ore reviews the frequently-told story of the controversy with Tartaglia. The accounts of both participants are preserved in their own works. For the historian of science, therefore, the problem is one of reconciling the opposing stories and arriving at an interpretation that will accord with the admitted facts and be just to both participants. Tartaglia, by the virulence of his accusations, illustrated the soundness of the military maxim that attack is the best defense by causing most later historians to recount the incident from his point of view, to the detriment of Cardano. Professor Ore, however, believes Cardano’s account is less biased; in fact, that is essentially true and justifies his actions. Both disputants agree that Cardano, having heard that Tartaglia had won in a public contest with Scipione de Ferro’s pupil Antonio Mario Fiore, by solving thirty equations proposed by Fiore of the type we would write as

a^3 + ax = b

approached Tartaglia about the year 1539 with a request that he make public the details of his method. Cardano wished to include it, with proper acknowledgement, in his forthcoming treatise on algebra. Tartaglia refused to allow its publication under any circumstances, maintaining that he intended to publish it himself, but told the secret to Cardano, after obliging the latter to swear a solemn oath that he would never publish it. Strictly speaking, Cardano violated his oath when he included the method of solving cubic equations in his Ars Magna in 1545. But his defense was (1) that Tartaglia after six years had not yet published, and (2) that in fact Tartaglia had been anticipated by Scipione del Ferro, who died in 1526, and among whose papers Cardano had found the supposed “secret” recorded. Considering that he was no longer bound by oath not to reveal a “secret” that had never been a secret, Cardano, in publishing the method in his Ars Magna, nevertheless gave full credit to del Ferro as the discoverer and to Tartaglia as the rediscoverer of the method. Answering repeated accusations of perfidy, Cardano was steadfast in his contention that it was the duty of a scientist to make public his discoveries rather than to conceal them indefinitely—that the welfare of society should take precedence over self-glorification. In maintaining this idea he aligns himself with countless other humanist scholars of the sixteenth century and characteristically supports his position by an allusion to a familiar and oft-paraphrased passage in Cicero’s De officiis about a man’s not being born for himself alone but for his friends, his native land, and for the whole human race. Professor Ore maintains that we Cardano with sincerity in his championship of this pervasive Renaissance concept and applaud his defense of freedom of scientific information rather than rally to the side of Tartaglia’s more selfish and less socially responsible attitude.

For the material presented in the chapter we have just discussed, the author records indebtedness to the researches of Italian historians of mathematics during the last four decades, especially those of Ettore Bortolotti. Chapter 5, however, entitled “The Science of Gambling,” is the core of the book. In it Professor Ore subjects Cardano’s treatise on games of chance to a close examination by an expert mathematician. He admits that certain sections of the Liber de Ludo Aleae are badly written and far from clear, but finds that to one who fully understands the subject the meaning emerges after patient and detailed analysis. Part of the difficulty derives from the fact, confessed by Cardano himself, that he wrote certain parts by merely jotting down brief notes from time to time as ideas occurred to him. Furthermore when he found that a previous idea proved to be erroneous he proceeded to add new thoughts without pausing to go back and correct his earlier statements. For example, he might write down three different solutions for the same problem, each time affirming it to be the correct procedure, but not until the third and last attempt arrive at the proper method.

His detailed examination of each statement in the Liber de Ludo Aleae Professor Ore found to be extremely rewarding, because in the end it enabled him to retrace Cardano’s thought and to correct the deprecatory statements which previous historians had made concerning the book, owing to their insufficient study of it and to their failure to grasp the meaning of some of Cardano’s assertions. He concludes that in the Liber de Ludo Aleae the chief principles underlying the mathematical calculation of probability were deduced and formulated more than a century before the correspondence between Pascal and Fermat in 1654, which is customarily considered as the discovery of the probability theory. The translation of the treatise printed as an appendix makes available to the reader the evidence for the high rating given to the originality of Cardano’s achievement.

Professor Ore’s book is an illustration of the important contributions to the history of science that come about when an able scientist starts with the assumption that his predecessors, though lacking some of his modern information, were his equals intellectually and then sets out to acquire the necessary linguistic and historical knowledge to understand one of their works. By so doing he is able to bridge the chasm caused by the changes in ideas and terminology that time has brought about and to penetrate to the true meaning of an early scientist’s words and thought. By detailed studies such as this the most permanent advances are made in establishing solid foundations for a sound history of science.

[1] Frances R. Johnson, Stanford University in Isis (1953).