Liber de ludo aleae (review)

Historians of probability and statistics have long cited Girolamo Cardano's brief work on games of chance as an early landmark in the field. Cardano tinkered endlessly with his numerous writings, amending published works and revising those in manuscript. Many of the latter were published only long after his death in 1663, edited by fellow physician Charles Spon. The Liber de ludo aleae was among this number. This Lyons 1663 edition (with facsimiles in 1966 and 1967) and an English translation by Sydney Henry Gould (which circulated both with Oystein Ore's 1953 study and independently) had long been the only available versions of this work. Tamborini's edition compares the Lyons edition with a Milanese manuscript at the Biblioteca Ambrosiana that predated it; Spon's manuscript does not survive. More substantial discussion of the Ambrosiana manuscript would have been helpful in assessing its significance, since it appears to be the only manuscript copy.

Cardano treats games of chance using dice, astragals (or knucklebones), and playing cards; in the case of cards, he focuses mainly on the game primero. For each type he sets sample spaces with the odds of throwing or drawing the various points and possible outcomes. As he notes, the most important principle in gaming is equality, and all departures from it favor one player or the other. Thus Cardano takes pains to distinguish all such conditions of play. The probability of one or another particular outcome is only one such condition. Others include social settings; physical factors such as visual contrasts; psychological factors; card counting and similar memory feats that allow a player to place more informed bets; and, of course, the many and various ways of cheating. Cardano acknowledges the low esteem in which some have held gambling, and asserts that he hopes to reduce the disadvantages of gaming while also indicating its value as recreation; nonetheless, he observes that it is certainly a widespread pastime with a venerable past.

Tamborini's introduction launches directly into a discussion of dating the work's composition: it was begun early, in 1524 or 1525, when Cardano was still a student of twenty-three. Cardano referred to it in a number of his works over the years, and returned to the text sporadically until nearly 1570; he never completed a full revision, and left a draft with some inconsistencies. Tamborini summarizes the work's structure and surveys briefly Cardano's work in probability in the context of his general mathematical interests.

Cardano's text is presented clearly; each chapter heading includes a reference to the Lyons pagination, helpful for those reading earlier scholarship that would have cited it. The bulk of the volume, and the particular value of the edition, lies in Tamborini's extensive notes to the text. They include the complete passages of Cardano's classical references, modern mathematical notation, references to contemporary texts and authors who address topics similar to that of the passage in question, biographical discussions, and similar material designed to offer a range of contexts for Cardano's text: they constitute something of a commentary on the whole. The notes are placed after the text, and employ a complex and some-what confusing set of numbers to refer back to the passage in question. The awkwardness of the format serves to remind the reader of the value of the formats in use in Cardano's day for commentaries and annotations, in which the publisher situated the text in the center of the page and surrounded it with the commentary distinguished by a second font, in the style of earlier manuscript texts with commentaries.

This volume is the fourth by FrancoAngeli of Cardano's writings. It is to be hoped that the series will continue to make his works more easily accessible to scholars.