Jakob Bernoulli's Ars Conjectandi is one of the most famous books in the history of mathematics. It was really the first comprehensive book on the theory of probability, and the first book to make the attempt to move the theory from its initial focus on games of chance to something much broader and more ambitious. It was in this book that the idea that one could apply the "doctrine of probabilities" to "civil, moral, and economic matters," a move that has changed the way we understand the world. There is no doubt, then, that this is a book of great historical importance.
The Art of Conjecturing was published posthumously by Nikolaus Bernoulli, who does not seem to have attempted to complete his uncle's work. The first part of the published book contains Christiaan Huygens' tract on games of chance with commentary by Bernoulli. The second is an introduction to the theory of permutations and combinations. The third part applies this combinatorics to the study of probabilities related to games of chance and the casting of lots. The fourth generalizes these ideas, as indicated above, to "civil, moral, and economic matters."
Part IV begins with a philosophical and theological discussion that attempts to carve up room for probability within a theistic worldview. Noting that we do not have any direct way to compute the probabilities of such events, Bernoulli goes on to argue that one can determine these probabilities by observing many trials. This leads to a formulation of a kind of "law of large numbers": relative frequencies over many observations approach, with high probability, the true probability of the event. Thus, if we observe enough trials, the relative frequency of the event we are studying should be quite close to the true probability of the event. (Bernoulli assumes that the phrase "the true probability of the event" has intrinsic meaning.)
The final computation is an explicit example. He considers a situation where the true probability is 3/5, and estimates that if we want to have a probability better than 999/1000 that the relative frequency is between 29/50 and 31/50, it is sufficient (!!) to make 25,550 observations. This very large number — much larger than really necessary — is a result of Bernoulli having used a rather coarse approximation in his computation. I have always had a fondness for the conjecture that Bernoulli never finished the book because of his disatisfaction with this estimate.
The original edition included two appendices. The first, on infinite series, seems to be an independent treatise with no direct connection to the rest of the book. It is not included in this edition. The second, on "court tennis" (jeu de paume), is included under the title "Letter to Friend on Sets in Court Tennis."
This is not the first English edition of Bernoulli's book. A translation of parts I to III was published by Francis Maseres in 1795. It is, of course, now very hard to find. An English translation of part IV was done by Bing Sung in 1966 and published as a Technical Report from the Harvard Department of Statistics. This circulated in samizdat form among some historians of mathematics, but was never generally available.
In this new edition, the text was translated by Edith Dudley Sylla, who also provides an extensive and very useful introduction and a small number of notes. Though I cannot compare her English to the Latin original, I can say that the translation reads well and that it does not seem to import any anachronisms into Bernoulli's texts (always a danger in such translations). In particular, as Sylla explains, she has tried to make judicious choices when dealing with mathematical notation and technical terms. Readers using this book should look carefully at the part of the introduction in which the translation is discussed.
There is much here to interest any historian of mathematics, and also any mathematician interested in the origins of probability theory. Those of us who teach elementary combinatorics and probability might even be able to use some passages with our classes. Finally, those who are interested in the philosophical questions surrounding probability theory and in the history of its application to the "real world" will want to read and ponder part IV.
One more great mathematical classic is now available in English. I'm delighted to have it.
Fernando Q. Gouvêa, who is professor of mathematics at Colby College and the editor of MAA Reviews, is glad not to have to use his copy of the 1966 translation of Part IV of this book anymore.