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Daniel I Bernoulli

b. Groningen, Netherlands, 8 February 1700
d. Basel, Switzerland, 17 March 1782

Daniel Bernoulli was a son of Johann I Bernoulli. His studies at Basel between 1713 and 1716 included not only the usual philosophy and logic but also mathematics under the tutelage of his father and older brother Niklaus II. He next studied medicine, earning a doctorate in 1721 for his Dissertatio inauguralis physico-medica de respiratione. Bernoulli continued his medical studies in Venice where the 1724 publication of his Exercitationes quaedam mathematicae resulted in an invitation to the St. Petersburg Academy. He and his brother Niklaus II Bernoulli both went to St. Petersburg in 1725. Niklaus died there the following year. While staying at St. Petersburg (1725 to 1733) he also worked with Euler beginning in the year 1727. In 1732 he was offered the chair of anatomy and botany at Basel. From that time on he remained at Basel lecturing in medicine and physics.

The collected works of Daniel Bernoulli are being published by Birkhaüser Verlag, Basel as Die Werke von Daniel Bernoulli in eight volumes. These are

  1. Medizin und Physiologie, Mathematische Jugendschriften, Positionsastronomie, 1996.

  2. Analysis, Wahrscheinlichkeitsrechnung, 1982. Disputations

  3. Mechanik, 1987.

  4. Hydrodynamic I (In preparation.)

  5. Hydrodynamic II, 2002.

  6. Elastizität (In preparation.)

  7. Magnetismus, Technologie I, 1994.

  8. Technologie II, 2004.

The first part of a youthful work, Exercitationes quaedam mathematicae, Venice 1724, contains a discussion of the game of Pharaon (Faro) and correspondence of Bernoulli with the Italian mathematician Rizetti. In the same year, Bernoulli exchanged letters with Christian Goldberg over the controversy. 

The papers of Daniel Bernoulli related to probability and statistics may be partitioned into four classes.

Petersburg Problem

"Specimen theoriae novae de mensura sortis," Commentarii Acad. Petrop. Vol. V, 1730-1 (1738), pp. 175-192. A translation of this has been published in Econometrica, Vol. 22 Issue 1 (Jan. 1954) pp. 23-36. This paper, written while Bernoulli was at St. Petersburg, deals with the famous Petersburg problem of which much has been written. Here is a summary of the correspondence with Nicolas Bernoulli and Montmort on this problem. See also the anonymous work by Windisch-Grätz.

Orbital plane of the planets

"Recherches physiques et astronomiques sur le problème proposé pour la second fois par l'Académie Royale des Sciences de Paris," Recueil des pieces qui ont remporté le prix de l'Académie Royale des Sciences, T. 3, 1734.

Smallpox inoculation

A popular exposition concerning the advantage of inoculation against smallpox appeared as "Reflexions sur les avantages de l'inoculation, par M. Daniel Bernoulli." Mercure de France (June 1760), pp. 173-190. In this Bernoulli mentions that he intends at first opportunity to study the problem in depth. In fact, the publisher inserted a note at the end of the article indicating that the promised paper had been written and submitted to the Academy of Sciences in Paris.

The aforementioned paper is "Essai d'une nouvelle analyse de la mortalité causée par la petite verole, et des advantages de l'inoculation pour la prévenir," Hist. et Mém. de l'Acad. Royale des Sciences de Paris, 1760 (1766) pp. 1-45. This paper, and a companion piece by d'Alembert, have been translated by L. Bradley and published as Smallpox Inoculation: An Eighteenth Century Controversy, Adult Education Department, University of Nottingham, 1971.

Bernoulli will also refer to this paper in "De duratione matrimoniorum media pro quacunque coniugum aetate, aliisque quaestionibus affinibus," which is taken up in the section on Urn problems immediately below.

Urn  problems

Here are grouped five papers which describe problems that may be modeled by the withdrawal and replacement of objects from urns. Common to them is the use of the infinitesimal calculus to produce simple results applicable to situations involving large numbers of observations. Four of the papers are presented as pairs - each pair presenting theory and then application of the theory.

1 & 2. "De usu algorithmi infinitesimalis in arte coniectandi specimen," Novi Commentarii Acad. Petrop. Vol. XII, 1766-7 (1768), pp. 87-98. This is followed immediately by "De duratione matrimoniorum media pro quacunque coniugum aetate, aliisque quaestionibus affinibus. "Novi Commentarii Acad. Petrop. Vol. XII, 1766/7 (1768), pp. 99-126.

The first paper introduces the probability theory which is the basis of the analysis found in the second. Here Bernoulli considers the following problems:

An urn contains n white-black pairs of tickets, 2n tickets in all. Tickets are drawn consecutively without replacement. In so doing, pairs are broken. What is the expected number of pairs to be found in the urn when r tickets remain there?

If the rate by which the black tickets are extracted differs from that of the white, again the expected number of pairs to be found in the urn is asked.

When the number of tickets in the urn is very large, Bernoulli observes that infinitesimal techniques may replace discrete calculations. Available here are notes to accompany this paper.

The theory developed in the first paper is now applied in the second to the problem of the duration of marriages. Bernoulli considers the following problems:

Suppose n marriages are contracted by an equal number of men and women of the same age. One same mortality table describes the rate at which each sex dies. What is the expected number of intact marriages remaining each successive year? What is the expected duration of such marriages?

Suppose now the men and women who marry are of different ages. In this case, men and women who have married now die at different rates. Again the same questions are asked.  

Bernoulli bases his calculations on the mortality table, the Breslau Table, of Halley. Available here are notes which clarify some points in the paper. Given Halley's Breslau Table, it is very easy to create a spreadsheet which replicates Bernoulli's work. However, Bernoulli's numbers will differ by 1 or 2 from the computed values because he has smoothed the data. 

The second paper was criticized by Jean Trembley in "Observations sur les calculs relatifs à la durée des mariages et au nombre des époux subsistans." Mémoires de l'Académie de Berlin 1799-1800, pp. 110-130.

3. "Disquisitiones analyticae de novo problemate coniecturali," Novi Commentarii Acad. Petrop. Vol. XIV, 1769, pars prior (1770), pp. 3-25. The main content of the paper concerns the following mixing problem:

Three urns are given of which the first contains n white tickets, the second n black and the third n red. A ticket is drawn at random from the first urn, and then deposited into the second. Next one is drawn from the second and deposited into the third. Finally one is drawn from the third and deposited into the first. The expected number of white tickets in each urn is desired after any number of such cycles of extractions and deposits.

The first sections of the paper (2 - 6) examine a similar problem involving only two urns. The solution here is used to motivate the discovery of the pattern for the case of three urns.

From the recurrence relations which describe the expectations, Bernoulli identifies series solutions. These are reduced to closed form solutions by varying the initial conditions to the solution of a related differential equation. (Sections 7 - 14)

Next Bernoulli takes up the differential equations which describe the mixing process. From these he computes the same solution as obtained previously. (Section 15).

Finally, Bernoulli notes that the expected number of white tickets in  the urns converges to a uniform distribution (n/3 in each urn) but that in so doing the number varies according to a damped oscillatory pattern. The last sections (16 - 20) examine the rate of convergence and the magnitude of the maximal deviations from n/3.

We have here some notes to assist in reading the paper.

Laplace produces a formula in Chapter III of Book II for any number of urns in his Théorie analytique des Probabilités.

4 & 5. "Mensura sortis ad fortuitam successionem rerum naturaliter contigentium applicata," Novi Commentarii Acad. Petrop. Vol. XIV, 1769, pars prior (1770), pp. 26-45. This is followed by "Continuato argumenti de mensura sortis ad fortuitam successionem rerum naturaliter contingentium applicata," Novi Commentarii Acad. Petrop. Vol. XV 1770 (1771) pp. 3-28.

Bernoulli was apparently unaware that Moivre had obtained as of November 12, 1733 what is today called the normal approximation to the binomial distribution. There was printed at that time, but not widely circulated, the pamphlet Approximatio ad Summam Terminorum Binomii (a + b)n in Seriem expansi. This pamphlet was reprinted with little change in the second (1738) and third (1756, pp. 243-250) editions of his Doctrine of Chances.

Bernoulli studies the distribution of male births and, in particular, the observed deviations from their expected value under two hypotheses. Since the observed frequency of births can be modeled by a binomial distribution, he is concerned expecially with the estimation of binomial probabilities. However, as birth records typically contain a large number of observations, he resorts, whenever possible, to the methods of the infinitesimal calculus.

In the first of these two papers, Bernoulli takes as fiduciary distribution the case of 20000 observations and the hypothesis that the probability of success is 1/2, computing for this distribution the location of the first and third quartiles. He states without proof how binomial probabilities and these quartiles can be estimated quite easily for distributions possessing a different number of observations. Finally, the normal approximation to the binomial is given, again without proof. See here notes to the first paper. Results from this paper are used in "Specimen philosophicum de compensationibus horologicis, et veriori mensura temporis." Acta Acad...Petropol. 1777-2 (1780), pp. 109-128.

The second paper investigates the ratio of male to female births as presented in commonly available tables. Bernoulli's purpose is a statistical analysis. But he must also develop mathematical tools which permit the analysis. To this end he develops simple methods to estimate binomial probabilities when the probability of success is not 1/2 as well as derive the normal approximation to the binomial. In the case of two competing hypotheses, one for which the ratio of males to females is 1:1 and another for which the ratio is as 1055:2055, he asks which is more probable given the data. Moreover, he attempts to assess the suitability of his model by examining annual deviations from the expected value over a series of years. See here notes on this second paper.

6. "Diiudicatio maxime probabilis plurium observationum discrepantium atque verisimillima inductio inde formanda," Acta Acad...Petropol. 1777, pars prior. (1778) pp. 3-23. This paper has been translated by Maurice Kendall and published in Biometrika Vol. 48 (1961), pp. 3-13. To this same paper, Euler has replied in "Observationes in praecedentem dissertationem illustris Bernoulli." Acta. Acad. Sci. Imp. Petrop. 1, 24–33 (1777). This likewise has been translated into English by C.C. Allen as “Observations on the foregoing dissertation of Bernoulli.” Biometrika 48, 13–18, (1961)..