Ars Conjectandi
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.
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 (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance.
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.
However, his actual influence on mathematical scene was not great; he wrote only one light tome on the subject titled Liber de ludo aleae (Book on Games of Chance), which was published posthumously in 1663.
[2][3] 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 fruits of Pascal and Fermat's correspondence interested other mathematicians, including Christiaan Huygens, whose De ratiociniis in aleae ludo (Calculations in Games of Chance) appeared in 1657 as the final chapter of Van Schooten's Exercitationes Matematicae.
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.
[6][7] 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.
Later, Johan de Witt, the then prime minister of the Dutch Republic, published similar material in his 1671 work Waerdye van Lyf-Renten (A Treatise on Life Annuities), 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.
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.
[13] From the outset, Bernoulli wished for his work to demonstrate that combinatorics and probability theory would have numerous real-world applications in all facets of society—in the line of Graunt's and de Witt's work— and would serve as a rigorous method of logical reasoning under insufficient evidence, as used in courtrooms and in moral judgements.
It was in this part that two of the most important of the twelvefold ways—the permutations and combinations that would form the basis of the subject—were fleshed out, though they had been introduced earlier for the purposes of probability theory.
[27] After these four primary expository sections, almost as an afterthought, Bernoulli appended to Ars Conjectandi a tract on calculus, which concerned infinite series.
"[32] Perhaps most recently, notable popular mathematical historian and topologist William Dunham called the paper "the next milestone of probability theory [after the work of Cardano]" as well as "Jakob Bernoulli's masterpiece".
[16] To achieve this De Moivre developed an asymptotic sequence for the factorial function —- which we now refer to as Stirling's approximation —- and Bernoulli's formula for the sum of powers of numbers.
[16] Both Montmort and de Moivre adopted the term probability from Jacob Bernoulli, which had not been used in all the previous publications on gambling, and both their works were enormously popular.
Indeed, in light of all this, there is good reason Bernoulli's work is hailed as such a seminal event; not only did his various influences, direct and indirect, set the mathematical study of combinatorics spinning, but even theology was impacted.
