Ars Conjectandi
Ars Conjectandi is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Nicolaus I Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.
Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work.
Background
In Europe, the subject of probability was first formally developed in the 16th century with the work of Gerolamo Cardano, whose interest in the branch of mathematics was largely due to his habit of gambling. He formalized what is now called the classical definition of probability: if an event has a possible outcomes and we select any b of those such that b ≤ a, the probability of any of the b occurring is. However, his actual influence on mathematical scene was not great; he wrote only one light tome on the subject titled Liber de ludo aleae, which was published posthumously in 1663.The date which historians cite as the beginning of the development of modern probability theory is 1654, when two of the most well-known mathematicians of the time, Blaise Pascal and Pierre de Fermat, began a correspondence discussing the subject. The two initiated the communication because earlier that year, a gambler from Paris named Antoine Gombaud had sent Pascal and other mathematicians several questions on the practical applications of some of these theories; in particular he posed the problem of points, concerning a theoretical two-player game in which a prize must be divided between the players due to external circumstances halting the game. The fruits of Pascal and Fermat's correspondence interested other mathematicians, including Christiaan Huygens, whose De ratiociniis in aleae ludo appeared in 1657 as the final chapter of Van Schooten's Exercitationes Matematicae. In 1665 Pascal posthumously published his results on the eponymous Pascal's triangle, an important combinatorial concept. He referred to the triangle in his work Traité du triangle arithmétique as the "arithmetic triangle".
In 1662, the book La Logique ou l’Art de Penser was published anonymously in Paris. The authors presumably were Antoine Arnauld and Pierre Nicole, two leading Jansenists, who worked together with Blaise Pascal. The Latin title of this book is Ars cogitandi, which was a successful book on logic of the time. The Ars cogitandi consists of four books, with the fourth one dealing with decision-making under uncertainty by considering the analogy to gambling and introducing explicitly the concept of a quantified probability.
In the field of statistics and applied probability, John Graunt published Natural and Political Observations Made upon the Bills of Mortality also in 1662, initiating the discipline of demography. This work, among other things, gave a statistical estimate of the population of London, produced the first life table, gave probabilities of survival of different age groups, examined the different causes of death, noting that the annual rate of suicide and accident is constant, and commented on the level and stability of sex ratio. The usefulness and interpretation of Graunt's tables were discussed in a series of correspondences by brothers Ludwig and Christiaan Huygens in 1667, where they realized the difference between mean and median estimates and Christian even interpolated Graunt's life table by a smooth curve, creating the first continuous probability distribution; but their correspondences were not published. Later, Johan de Witt, the then prime minister of the Dutch Republic, published similar material in his 1671 work Waerdye van Lyf-Renten, which used statistical concepts to determine life expectancy for practical political purposes; a demonstration of the fact that this sampling branch of mathematics had significant pragmatic applications. De Witt's work was not widely distributed beyond the Dutch Republic, perhaps due to his fall from power and execution by mob in 1672. Apart from the practical contributions of these two work, they also exposed a fundamental idea that probability can be assigned to events that do not have inherent physical symmetry, such as the chances of dying at certain age, unlike say the rolling of a dice or flipping of a coin, simply by counting the frequency of occurrence. Thus, probability could be more than mere combinatorics.
Development of ''Ars Conjectandi''
In the wake of all these pioneers, Bernoulli produced many of the results contained in Ars Conjectandi between 1684 and 1689, which he recorded in his diary Meditationes. When he began the work in 1684 at the age of 30, while intrigued by combinatorial and probabilistic problems, Bernoulli had not yet read Pascal's work on the "arithmetic triangle" nor de Witt's work on the applications of probability theory: he had earlier requested a copy of the latter from his acquaintance Gottfried Leibniz, but Leibniz failed to provide it. The latter, however, did manage to provide Pascal's and Huygens' work, and thus it is largely upon these foundations that Ars Conjectandi is constructed. Apart from these works, Bernoulli certainly possessed or at least knew the contents from secondary sources of the La Logique ou l’Art de Penser as well as Graunt's Bills of Mortality, as he makes explicit reference to these two works.Bernoulli's progress over time can be pursued by means of the Meditationes. Three working periods with respect to his "discovery" can be distinguished by aims and times. The first period, which lasts from 1684 to 1685, is devoted to the study of the problems regarding the games of chance posed by Christiaan Huygens; during the second period the investigations are extended to cover processes where the probabilities are not known a priori, but have to be determined a posteriori. Finally, in the last period, the problem of measuring the probabilities is solved.
Before the publication of his Ars Conjectandi, Bernoulli had produced a number of treatises related to probability:
- Parallelismus ratiocinii logici et algebraici, Basel, 1685.
- In the Journal des Sçavans 1685, p. 314 there appear two problems concerning the probability each of two players may have of winning in a game of dice. Solutions were published in the Acta Eruditorum 1690, pp. 219–223 in the article Quaestiones nonnullae de usuris, cum solutione Problematis de Sorte Alearum. In addition, Leibniz himself published a solution in the same journal on pages 387-390.
- Theses logicae de conversione et oppositione enunciationum, a public lecture delivered at Basel, 12 February 1686. Theses XXXI to XL are related to the theory of probability.
- De Arte Combinatoria Oratio Inauguralis, 1692.
- The Letter à un amy sur les parties du jeu de paume, that is, a letter to a friend on sets in the game of Tennis, published with the Ars Conjectandi in 1713.
In Bernoulli's own words, the "art of conjecture" is defined in Chapter II of Part IV of his Ars Conjectandi as:
The art of measuring, as precisely as possible, probabilities of things, with the goal that we would be able always to choose or follow in our judgments and actions that course, which will have been determined to be better, more satisfactory, safer or more advantageous.
The development of the book was terminated by Bernoulli's death in 1705; thus the book is essentially incomplete when compared with Bernoulli's original vision. The quarrel with his younger brother Johann, who was the most competent person who could have fulfilled Jacob's project, prevented Johann from accessing the manuscript. Jacob's own children were not mathematicians and were not up to the task of editing and publishing the manuscript. Finally Jacob's nephew Nicolaus, 7 years after Jacob's death in 1705, managed to publish the manuscript in 1713.