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Jacob Bernoulli

b. 27 December 1654 Basel, Switzerland
d. 16 August 1705 Basel

Jacob I Bernoulli earned his master of arts in philosophy in 1671 and a licentiate in theology in 1676. Against the wishes of his father, he began studying mathematics. In 1687 Bernoulli became professor of mathematics at Basel. During this time he mastered the new calculus of Leibnitz and published extensively in it.

The collected works are published by Birkhäuser Verlag as Die Werke von Jacob Bernoulli in seven volumes. These are

Der Briefwechsel von Jacob Bernoulli, 1993

and

1. Astronomie, Philosophia naturalis (2nd edition forthcoming)

2. Elementarmathematik, 1989

3. Wahrscheinlichkeitsrechnung, 1975

4. Reihentheorie, 1993

5. Differentialgeometrie, 1999

6. Mechanik (in preparation)

His most important work, for our purposes that is, is the Ars Conjectandi. His interest in the theory of probability began no later than 1684. Jacob had certainly studied the short work of Huygens, De ratiociniis in ludo aleae. He proposed two problems in probability which were published in the Journal des Sçavans the following year. Based upon his own journals, it is possible to conclude that Jacob had found a proof of what is called today Bernoulli's Theorem or, rather, the Weak Law of Large Numbers sometime between 1687 and 1689. In fact, Jacob informed his brother Johann that he had achieved such a result in 1691. Publications of Jacob Bernoulli concerning Probability

• Parallelismus ratiocinii logici et algebraici, Basel, 1685.

• In the Journal des Sçavans 1685 (26.VIII), 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 (May), pp. 219-223 in the article Quaestiones nonnullae de usuris, cum solutione Problematis de Sorte Alearum. The solution is computed here. 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 Ars Conjectandi published posthumously in 1713.

• 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. The rules are descibed in these selections from the Encyclopedia. Nicholas Bernoulli challenged Montmort with some exercises derived from this work. The progress of Montmort can be traced in the correspondence between them during 1711 and 1712.

The Ars Conjectandi

Contents

The Ars Conjectandi consists of four parts. These are

Part I   The Treatise of Huygens on Reasoning in Games of Chance with Annotations
Part II The Theory of Permutations and Combinations
Part III The Use of the preceding Theory in various random Events and Games of Chance
Part IV The Use and Application of the previous Theory in Politics, Law and Economics.

Part I was translated into French by G. F. Vastel as L'Art de Conjecturer, published in 1801.
Part II was translated into English as The Doctrine of Permutations and Combinations, published by Francis Maseres in 1795.

Part IV is the most original section of the treatise. It contains philosophical thoughts on probability, a definition of moral certainty and the centerpiece of the work, this being a proof of what is now known as Bernoulli's theorem. Either because of ill-health or because he did not know how to do so, Bernoulli never completed Part IV. The theorem, for which see below, had certainly been proved prior to 1690. It is suspected that Bernoulli was hesitant to publish because he felt that his treatment of the applications to politics, law and economics were inadequate. His correspondence with Leibniz frequently refers to efforts to obtain statistics he might use to create applications. As he was unable to ever satisfy this need, the work was never published in his lifetime and it ultimately appeared only after much difficulty in 1713. Due to this delay, it never had the impact it merited since it was anticipated by the works of Montmort and Moivre.

The Latin title of the Port-Royal Logic of Arnauld and Nicole is the Ars Cogitandi. Bernoulli likely chose his title in direct imitation and saw his work as a natural extension of it. Throughout the first three chapters of Part IV of the Ars Conjectandi, Bernoulli considered the problem of testimony, a topic addressed in the Logic. In fact, there is fairly clear evidence in chapter 3 that he had the Port-Royal Logic before him.

Bernoulli, being a mathematician, approached the problem of evidence from a quantitative viewpoint. He wanted to compute contingencies. In this way he could deduce the most probable decision simply by comparison of numbers.

If one cannot have absolute certainty in a decision, it may be possible to have a very high degree nonetheless. A morally certain event is an event whose probability is nearly that of a certain event. Therefore, it is difficult to conceive that a morally certain event not happen. But determination if an event is morally certain is necessarily subjective. Bernoulli suggested if the probability of an event is at least 999/1000 then it is morally certain.

There is now a complete translation into English of the Ars Conjectandi by Edith Sylla. Previous renderings into English are limited. Francis Maseres published a translation of Part II at London in 1795 entitled The Doctrine of Permutations and Combinations, being an essential part of the Doctrine of Chances, as it is delivered by M. James Bernoulli in his excellent treatise on the Doctrine of chances intitled 'Ars conjectandi' and by ... Dr. John Wallis, ... as part of a collection of reprints of various works on the same theme. Bing Sung made a translation of Part IV and relevant portions of the letters exchanged between Bernoulli and Leibniz which is in the public domain.

Letters exchanged between Jacob Bernoulli and Gottfried Wilhelm Leibniz

In 1697, Johann Bernoulli, brother of Jacob, exchanged letters with Leibniz in which Johann informed Leibniz that Jacob had been writing a treatise on probability called the Ars Conjecturandi and which concerned applications to life beyond the study of games. Leibniz replied that he himself had once considered such things and hoped that mathematicians would pursue this line of inquity.

Six years later Leibniz returns to the subject in a letter to Jacob Bernoulli. Within the two year period of April 1703 and April 1705 Jacob Bernoulli and Leibniz exchanged a number of letters regarding topics relevant to the Ars Conjectandi.

 No. City Date Comment XI Leibniz to Bernoulli Berlin April 1703 Leibniz indicates he has heard of Bernoulli's work. XII Bernoulli to Leibniz Basel 3 October 1703 Bernoulli gives a description of his Theorem and asks for de Witt's pamphlet. XIII Leibniz to Bernoulli Hannover 26 Nov./3 Dec.1703 Leibniz says he has difficulty with Bernoulli's assertion. XIV Bernoulli to Leibniz Basel 20 April 1704 Bernoulli offers another explanation of the Theorem. He asks again for de Witt's pamphlet. - Leibniz to Bernoulli prior August 1704 Mentioned in Letter XV. - Leibniz to Bernoulli prior August 1704 Mentioned in Letter XV. XV Bernoulli to Leibniz Basel 2 August 1704 Bernoulli again requests de Witt's pamphlet. He also reports more of the contents of the Ars Conjectandi. - Leibniz to Bernoulli prior 15 Oct. 1704 Mentioned in Letter XVI. XVI Bernoulli to Leibniz Basel 15 Oct. (Nov.?) 1704 Bernoulli expects de Witt's pamphlet. XVII Leibniz to Bernoulli Berlin 28 Nov. 1704 Leibniz promises to send de Witt's pamphet. He makes remarks regarding utility of the theory of probability. XVIII Bernoulli to Leibniz Basel 28 Feb. 1705 Bernoulli begs for de Witt's pamphlet. He asserts that his Theorem will be pleasing to Leibniz. XIX Leibniz to Bernoulli Hannover April 1705 Leibniz cannot find de Witt's pamphlet. He emphasizes that there is nothing worthwhile in it.

Correspondence related to the Ars Conjectandi and its Publication

Nearly all correspondence presented below has been translated from the Latin letters contained in Leibnizens mathematische Schriften, Bands III and IV as edited by Gerhardt and in Montmort's Jeux de Hazard. The remaining letters (identified as unpublished correspondence) have not appeared except as excerpts in the Commentary 3 by Kohli which appears in Die Werke von Jacob Bernoulli, Band 3, 1973. These have been translated from their corresponding French or German text.

4 April 1705 to 17 July 1706. Jacob Hermann, like Nicolaus Bernoulli, was a pupil of Jacob Bernoulli. He reported in April 1705 to Leibniz that the Ars Conjectandi had been completed to the central proposition, namely what is now called Bernoulli's Theorem. After learning of the contents of the manuscript, and with the death of Jacob, Leibniz became concerned that work might be lost. In July 1706, Leibniz is informed by Hermann that Nicolaus the Painter, son of Jacob, has been entrusted with the work.

6 September 1705, 28 October 1705 through 1706. Fontenelle, Saurin and Hermann each prepared an Eloge for Jacob Bernoulli published in the scholarly journals Histoire de l'Académie Royale des Sciences (1705), Journal des Sçavans (1706), and Acta Eruditorum (1706) respectively. As a result the contents of the Ars Conjectandi as well as the central proposition become more widely known.

26 February 1707 to 8 May 1708. Johann engaged in correspondence with Pierre Varignon. The Eloge of Saurin had urged Johann Bernoulli, the younger brother of Jacob, to undertake the completion of the Ars Conjectandi. This, of course, was impossible due the rivalry between Jacob and Johann. Meanwhile, Pierre de Montmort has had published his work, Essay d'analyse sur les jeux de hazard, in 1708. The Preface to this edition contains the following:

"It is a longtime that the Geometers brag to be able by their methods to discover in the natural Sciences, all the truths which are in the range of the human mind; & it is certain that by the marvelous alloy that they have made since fifty years of Geometry with Physics, they have forced men to recognize that that which they say to the advantage of Geometry is not without foundation. What glory would there be for Science if it could again serve to rule the judgments & conduct of men in the practice of the things of life!

"The eldest of the Messers Bernoulli, both so known in the scholarly world, has not believed that it was impossible to carry Geometry to this point."

27 June 1708 to 6 September 1709. At this time, Leibniz is concerned that further delay will lessen the significance of the publication of the Ars Conjectandi. Johann cannot offer assistance because of family rivalries. At the same time, Nicolaus the Painter, who has been charged with it, is inadequate to the task.

15 September 1709 to 26 May 1711.  Pierre de Montmort wrote twice to Johann Bernoulli that he will print the Ars Conjectandi at his own expense. Johann suggested rather that Montmort expand his work to include the topics in Jacob's work because of the problems with the heirs. Nonetheless Nicolaus asked Hermann to plead with the heirs. Nicolaus wrote to Montmort of the efforts made to pass the manuscript to him and also to Hermann that his cousin, Nicolaus the Painter, communicated to him that he would prefer to pass it to Montmort.

In the meantime, Johann received a copy of Jeux de Hazard and, at the request of the author, commented on the work. He then reported to Leibniz that the Ars Conjectandi is superior.

15 July 1712 to 25 December 1713. During this period, Nicolaus the Painter sees the manuscript through its printing. At the last minute, his cousin, Nicolaus I, was asked to produce a list of errata and to write a preface. During this same time, 23 January 1713 to be precise, Nicolaus I wrote a letter to Montmort in which he provided a proof of Bernoulli's Theorem. Here are some further comments as printed in the Jeux de Hazard, second edition of 1713, pp. 388-393:

"I myself remember that my late Uncle has demonstrated a similar thing in his Treatise De Arte conjectandi, which is printed at present at Basel, namely, that one wishes to discover by often repeated experiences the number of cases by which a certain event can arrive or not, one can increase the observations in such manner that finally the probability that we have discovered the true ratio that there is between the numbers of the cases, is greater than a given probability. When this Book will appear we will see if in these sorts of matters I have found an approximation as exact as he."

Bernoulli's Theorem

Sometime between 1688 and 1690, Jacob Bernoulli was able to find an estimate of the sample size required to be morally certain that a sample proportion lies within a prescribed distance of the true proportion. In other words, he was able to guarantee that the sample proportion would fall within a narrow radius of the true value with high probability. The theorem which gives this result is proved in Chapter V of the Ars Conjectandi. In essence, Bernoulli obtains an estimate of the sum of the central terms of a binomial.

Let moral certainty be defined by the ratio c:1. That is, an event is morally certain if there are c cases for the event and 1 case against. We have

Bernoulli's Theorem. Suppose every trial results in either a Success or a Failure in ratio r:s. Let p denote the sample proportion of successes in a total of n independent trials. The number of trials n required so that

$\Pr\left\{\Big|p-\frac{r}{r+s}\Big|\leq\frac{1}{r+s}\right\}>\frac{c}{c+1}$

is given as follows:

1. Let m1 be the least integer greater than [(ln[c(r - 1)])/( ln[(s+1)/s])] and let k1 be the least integer greater than [(m1(r+s+1) - r)/( s+1)].

2. Let m2 be the least integer greater than [(ln[c(s - 1)])/( ln[(r+1)/r])] and let k2 be the least integer greater than [(m2(r+s+1) - s)/( r+1)].

Then n = max{k1,k2}(r+s).

Selected Literature

Pearson, Karl. "James Bernoulli's Theorem," Biometrika, Vol. 17, Issue 3/4 (Dec. 1925), pp. 201-210.

Hacking, Ian. "Jacques Bernoulli's Art of Conjecturing," British Journal for Philosophy of Science, Vol. 22, No. 3, (Aug. 1971), pp. 209-229.

Yushkevich, A. P., "Nicholas Bernoulli and the Publication of James Bernoulli's Ars Conjectandi," Theory Probab. Appl., Vol. 31, No. 2, (June 1987), pp. 286-303.

Sylla, Edith Dudley, "The Emergence of Mathematical Probability from the Perspective of the Leibniz-Jacob Bernoulli Correspondence," Perspectives on Science, Vol. 6.1 & 2, 1998, pp. 41-76.

Musnier, Norbert. "Quelques échanges?", Journ@l Electronique d'Histoire des Probabilités et de la Statisque, Vol. 2, No. 1 (June 2006)

Schneider, Ivo. "Direct and indirect influences of Jacob Bernoulli's Ars Conjectandi in 18th century Great Britain," Journ@l Electronique d'Histoire des Probabilités et de la Statisque, Vol. 2, No. 1 (June 2006)

del Cerro, Jesus Santos. "L'ars conjectandi: La géométrique du hasard versus le probabilisme moral," Journ@l Electronique d'Histoire des Probabilités et de la Statisque, Vol. 2, No. 1 (June 2006)

Sylla, Edith Dudley, The Art of Conjecturing, Johns Hopkins University Press, 2006. This work includes a translation of  the Ars Conjectandi, Letter à un amy sur les parties du jeu de paume and extracts from Theses logicae de conversione et oppositione enunciationum.