Gerolamo Cardano’s *The Book on Games of Chance: The 16th-Century Treatise on Probability*, in its translation by Sydney H. Gould, has been released by Dover Publications. It had previously appeared as an appendix in Oystein Ore’s *Cardano, the Gambling Scholar*. I recommend Ore’s book for its useful commentary on Cardano; even in translation, Cardano’s book can be difficult to understand. This is especially true in light of Cardano’s habit of leaving his early incorrect attempts to solve a problem in the manuscript even when the problem is later solved correctly. Because the book did not appear in print until almost a century after his death, it is of course possible that he would have revised the work had he sought to publish it himself.

*Liber de ludo aleae*, to give the book its Latin title, was likely written or revised in 1564; he relates a story in Chapter 26 of an incident that occurred when he was 25, and he mentions that 38 years have passed since that event. Thus this work would predate other mathematical treatments of probability given by Galileo in the early seventeenth century and the famous Pascal-Fermat correspondence on the problem of points later that century.

Computations of odds and probabilities occupy approximately ten of the thirty-two chapters. In much of the rest of the book, it is entertaining to read about the ills and the utility of gambling from Cardano’s point of view, as he was himself a life-long gambler. The fifth chapter includes his professed motivation for writing the book: "Gambling ... would seem to be a natural evil. For that very reason it ought to be discussed by a medical doctor [Cardano was himself a physician] like one of the incurable diseases. ... It has been the custom of philosophers to deal with the vices in order that advantage might be drawn from them, as, for example, in the case of anger."

What of the mathematical content? Beginning in Chapter 9, Cardano notes the probabilities with one die. In Chapter 11, he calculates the probabilities involving the throw of two dice, recognizing that the probability of any particular double is half that of the throw of a combination of unequal faces such as 1,2. He successfully computes the probability of throwing at least one 1 in two rolls as 11/36, then notes that the probability of obtaining at least one 1 twice in consecutive rolls of two dice to be between 1/12 and 1/8, making use of the multiplication principle in counting. There is a hint of a statement of the law of large numbers. He miscalculates the probability of obtaining at least one 1 twice in three rolls of two dice.

In Chapter 12, Cardano considers the probabilities when rolling three dice. He again exhibits his understanding of the multiplication principle in counting, the addition principle for mutually exclusive cases, and ordered versus unordered events. He does not make any use of complementation in his counting computations. Again, some results are correct and some incorrect.

In Chapter 13, Cardano gives correct results for probabilities of the sum of the faces of two dice. Ore has deduced some of the rules for the game of Fritillus (in the backgammon family) from the remarks and computations in this chapter.

In Chapter 14, Cardano shows an understanding of the general addition rule, subtracting the number of cases common to two events. He computes some elementary expected values. In Chapter 12 he realizes his mistake in the computation of the probability of obtaining at least one 1 in the roll of three dice. His analysis of some of the numbers of possible outcomes includes the computation of differences and the differences of differences, noting that the differences of the number of outcomes for a situation rolling two dice is constant while it is the second differences that are constant for an analogous situation when rolling three dice.

Computing the probability of an event occurring multiple times in succession, Cardano at first incorrectly considers a multiplicative principle for odds rather than probabilities, but in Chapter 15 he realizes that he was wrong. His first attempt at a correction is again incorrect: he reasons that the correct generalization of \(2^2-1\) should be \(n^2-1\) rather than \(2^n-1\) Recognizing that this answer is incorrect, he then provides the correct generalization, essentially noting that if the probability of an event \(X\) is \(p\), then the probability of \(X\) occurring \(n\) times in a row is \(p^n\).

Chapters 16 and 18 concern the card game Primero, which may be an early form of poker; I would suggest reading Ore to see what can be said about Primero. In Chapter 31 Cardano discusses probabilities for the throw of astragals (four-sided knuckle bones), and in Chapter 32 he offers another statement that can be interpreted as a law of large numbers.

It is good to see this book available in an inexpensive edition, but if you can find a copy of Ore’s book that includes Cardano’s manuscript, I would recommend that instead.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.