Ars Conjectandi

The cover page of Ars Conjectandi

Ars Conjectandi (Latin for The Art of Conjecturing) is a combinatorial mathematical paper written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, most notably among other combinatorial topics, probability theory: 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 Christian 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 twelvefoldway—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

Christiaan Huygens published the first book on probability

In Europe, the subject of probability was first formally developed in the sixteenth 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 $\begin{smallmatrix}\frac{b}{a}\end{smallmatrix}$. However, his actual influence on mathematical scene was not great; he wrote only one light tome on the subject in 1525 titled Liber de ludo aleae (Book on Games of Chance), 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 Christian Huygens, who in 1657 published De ratiociniis in aleae ludo (Calculations in Games of Chance). 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 (Traits of the Arithmetic Triangle) as the "arithmetic triangle". Later, Johan de Witt 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 sapling branch of mathematics had significant pragmatic applications.

In the wake of all this pioneers, Bernoulli produced Ars Conjectandi, during a fertile mathematical period he had between 1684 and 1689. 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 Huygen's work, and thus it is largely upon these foundations that Ars Conjectandi is constructed. From the outset, Bernoulli wished for his work to demonstrate that combinatorial and probability theory would have numerous real-world applications in all facets of society—in the line of de Witt's work—and thus the title Ars Conjectandi was chosen: a link to the concept of ars inveniendi from scholasticism, which provided the symbolic link to pragmatism he desired. His nephew Niklaus published the manuscript in 1713 after Bernoulli's death in 1705.

Contents

Portrait of Jakob Bernoulli in 1687

Bernoulli's work, originally published in Latin is divided into four parts. It covers most notably his theory of permutations and combinations; the standard foundations of combinatorics today and subsets of the foundational problems today known as the twelvefold way. It also discusses the motivation and applications of a sequence of numbers more closely related to number theory than probability; these Bernoulli numbers bear his name today, and are one of his more notable achievements.

The first part is an in-depth expository on Huygens' De ratiociniis in aleae ludo. Bernoulli provides in this section solutions to the five problems Huygens posed at the end of his work. He particularly develops Huygens' concept of expected value—the weighted average of all possible outcomes of an event. Huygens had developed the following formula:

$E=\frac{p_0a_0+p_1a_1+p_2a_2+\cdots+p_na_n}{p_0+p_1+\cdots+p_n}.$

In this formula, E is the expected value, p_i are the probabilities of attaining each value, and a_i are the attainable values. Bernoulli normalizes the expected value by assuming that p_i are the probabilities of all the disjoint outcomes of the value, hence implying that p₀ + p₁ + ... + p_n = 1. Another key theory developed in this part is the probability of achieving at least a certain number of successes from a number of binary events, today named Bernoulli trials, given that the probability of success in each event was the same. Bernoulli shows through mathematical induction that given a the number of favorable outcomes in each event, b the number of total outcomes in each event, d the desired number of successful outcomes, and e the number of events, the probability of at least d successes is

$P=\sum_{i=0}^{e-d}\binom{e}{d+i}\left(\frac{a}{b}\right)^{a+v}\left(\frac{b-a}{b}\right)^{e-d-i}.$

The first part concludes with what is now known as the Bernoulli distribution.

The second part expands on enumerative combinatorics, or the systematic numeration of objects. 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. He gives the first non-inductive proof of the the binomial expansion for integer exponent using combinatorial arguments. On a note more distantly related to combinatorics, the second section also discusses the general formula for sums of integer powers; the free coefficients of this formula are therefore called the Bernoulli numbers, which have proven to have numerous applications in number theory. Additionally, this part also contains Bernoulli's formula for the sum of powers of integers, which influenced Abraham de Moivre's work later.

In the third part, Bernoulli applies the probability techniques from the first section to the common chance games played with playing cards or dice. Interestingly, he does not feel the necessity to describe the rules and objectives of the card games he analyzes. He presents probability problems related to these games and, once a method had been established, posed generalizations. For example, a problem involving the expected number of "court cards"—jack, queen, and king—one would pick in a five-card hand from a standard deck of 52 cards containing 12 court cards could be generalized to a deck with a cards that contained b court cards, and a c-card hand.

The fourth section continues the trend of practical applications by discussing applications of probability to civilibus, moralibus, and oeconomicis, or to personal, judicial, and financial decisions. In this section, Bernoulli differs from the school of thought known as frequentism, which defined probability in an empirical sense. As a counter, he produces a result resembling the law of large numbers, which he describes as predicting that the results of observation would approach theoretical probability as more trials were held—in contrast, frequents defined probability in terms of the former. Bernoulli was very proud of this result, referring to it as his "golden theorem", and remarked that it was "a problem in which I’ve engaged myself for twenty years". This early version of the law is known today as either Bernoulli's theorem or the weak law of large numbers, as it is less rigorous and general than the modern version.

After these four primary expository sections, almost as an afterthought, Bernoulli appended to Ars Conjectandi a tract on calculus, which concerned infinite series. It was a reprint of five dissertations he had published between 1686 and 1704.

Legacy

Abraham de Moivre's work was built in part on Bernoulli's

Colin Maclaurin

Ars Conjectandi is considered a landmark work in combinatorics and the founding work of mathematical probability. Among others, an anthology of great mathematical writings published by Elsevier and edited by historian Ivor Grattan-Guinness describes the studies set out in the work "[occupying] mathematicians throughout 18th and 19th centuries"—an influence lasting three centuries. Statistician Anthony Edwards praised not only the book's groundbreaking content, writing that it demonstrated Bernoulli's "thorough familiarity with the many facets [of combinatorics]," but its form: "[Ars Conjectandi] is a very well-written book, excellently constructed." 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". It greatly aided what Dunham describes as "Bernoulli's long-established reputation".

Nicolaus Bernoulli assisted in the publication of Jacob Bernoulli's Ars conjectandi. Later Nicolaus edited Jacob Bernoulli's complete works and supplemented it with results taken from Jacob's diary. Bernoulli's work influenced many contemporary and subsequent mathematicians. Even the afterthought-like tract on calculus has been quoted frequently; most notably by the Scottish mathematician Colin Maclaurin. Apart from Nicolaus Bernoulli, who in conjunction with Pierre Rémond de Montmort wrote a book on probability Essay d'analyse sur les jeux de hazard which appeared in 1708, Abraham de Moivre was particularly influenced by Bernoulli's work in probability; he wrote extensively on the subject in The Doctrine of Chances. De Moivre's most notable achievement in probability was the central limit theorem, by which he was able to approximate the binomial distribution, using an asymptotic sequence for the factorial function—which he had developed with James Stirling—and Bernoulli's formula for the sum of powers of numbers.

The refinement of Bernoulli's Golden Theorem, regarding the convergence of theoretical probability and empirical probability, was taken up by many notable later day mathematicians like Poisson, Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. The complete proof of the Law of Large Numbers for the arbitrary random variables was finally provided during first half of 20th century.

A significant indirect influence was Thomas Simpson, who achieved a result that closely resembled de Moivre's. According to Simpsons' work's preface, his own work depended greatly on de Moivre's; the latter in fact described Simpson's work as an abridged version of his own. Finally, Thomas Bayes wrote an essay discussing theological implications of de Moivre's results: his solution to a problem, namely that of determining the probability of an event by its relative frequency, was taken as a proof for the existence of God by Bayes. Finally in 1812, Pierre-Simon Laplace published his Théorie analytique des probabilités in which he consolidated and laid down many fundamental results in probability and statistics such as the moment generating function, method of least squares, inductive probability, and hypothesis testing, thus completing the final phase in the development of classical probability. 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.

See also

• Multinomial distribution
• Bernoulli trial

References

• Bernoulli, Jakob (1713), Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis, Basel: Thurneysen Brothers, OCLC 7073795 
• Bernoulli, Jakob, translated by Edith Sylla (1713/2005), The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis (English translation), Baltimore: Johns Hopkins Univ Press, ISBN 0-8018-8235-4, http://books.google.com/?id=-xgwSAjTh34C&dq=edith+dudley+sylla 
• Bernoulli, Jakob, translated by Oscar Sheynin (1713/2005) (PDF), On the Law of Large Numbers, Part Four of Ars Conjectandi (English translation), Berlin: NG Verlag, ISBN 3-938417-14-5, http://www.sheynin.de/download/bernoulli.pdf 
• Bernoulli, Jakob; Haussner, Robert (translator) (1713/2002), Wahrscheinlichkeitsrechnung (Ars conjectandi) (German translation), Robert Haussner, Frankfurt am Main: Harri Deutsch, ISBN 3-8171-3107-0, http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABZ9501 
• Brakel, J. van (June 1976), "Some Remarks on the Prehistory of the Concept of Statistical Probability", Archive for History of Exact Sciences (Heidelberg) 16 (2): 119, doi:10.1007/BF00349634 
• Dunham, William (1990), Journey Through Genius (1st ed.), John Wiley and Sons, ISBN 0-471-50030-5 
• Dunham, William (1994), The Mathematical Universe (1st ed.), John Wiley and Sons, ISBN 0-471-53656-3 
• Edwards, Anthony W.F. (1987), Pascal's Arithmetical Triangle: The Story of a Mathematical Idea, John Hopkins University Press, ISBN 0-8018-6946-3 
• Elsevier (2005), Grattan-Guinness, Ivor, ed., Landmark Writings in Western Mathematics, 1640–1940, Elsevier 
• Hald, Anders (2005), A History of Probability and Statistics and Their Applications Before 1750, Wiley, ISBN 978-0-471-47129-5 
• Maĭstrov, Leonid (1974), Academic Presstitle=Probability Theory: A Historical Sketch 
• Maseres, Francis; Bernoulli, Jakob; Wallis, John (1798), The Doctrine of Permutations and Combinations, British Critic 
• de Moivre, Abraham (1716/2000), The Doctrine of Chances (3 ed.), New York: Chelsea Publishers, ISBN 978-0821821039 
• Schneider, Ivo (June 2006), "Direct and Indirect Influences of Jakob Bernoulli's Ars Conjectandi in 18th Century Great Britain", Electronic Journal for the history of Probability and Statistics 2 (1) 
• Shafer, Glenn (1996), "The Significance of Jacob Bernoulli’s Ars Conjectandi for the Philosophy of Probability Today" (PDF), Journal of Econometrics 75 (1): 15–32, doi:10.1016/0304-4076(95)01766-6, http://www.glennshafer.com/assets/downloads/articles/article55.pdf 

External links

• Quotations by Jakob Bernoulli