Leonhard Euler was born on 15 April 170 at Basel, Switzerland. He attended the university there where he made the acquaintance of Johann Bernoulli. He was invited to St. Petersburg in 1727 upon the death of Niklaus II Bernoulli where he worked with Daniel Bernoulli. Euler became professor of physics in 1731, and professor of mathematics in 1733, when he replaced Daniel who had departed for Basel that year. Euler was invited to Berlin by Frederick the Great of Prussia where he worked for the 25 year period from 1741. In 1766 Euler returned to St. Petersburg where he remained until his death on 18 September 1783.

Euler's works related to the calculus of probabilities and to statistics are all to be found in

*Opera Omnia *Series I, Volume 7, 1923

Edited. Louis Gustave du Pasquier

Available here are translations into English of the
majority of these works
together with references to various related documents. The papers are
referenced
by their Enestrom numbers. Of the few which have been previously
translated,
I cite the journals in which they appear. I thank A. Berra for the
translation
of E811 and Chapter VII of the *De Usu *of Nicolas Bernoulli; I
am solely
responsible for the remainder as well as for all errors.

These documents may be freely copied for use by educators and educational institutions as long as proper credit is given and they remain unaltered. This site may neither be mirrored nor its files reposted. Comments and corrections are welcome. As time permits I will expand the materials available here.

Euler solves Huygen's Fifth Problem and investigates when the players have equal expectations in the Adversaria mathematica H4. This can only be dated approximately to the period 1740 to around 1750.

The game of *Treize* has sometimes been called *Rencontre*.
Todhunter
mentioned that the game or a generalization had been discussed by

- Montmort,
*Essai d'analyse sur les jeux de hazard,*pages 130-143, - De Moivre,
*Doctrine of Chances*, pages 109-117, - Euler,
*Hist. de l'Acad... Berlin*, for 1751, - Lambert,
*Nouveaux Mémoires de l'Acad... Berlin*, for 1771, - Laplace,
*Théorie Analytic des Probabilities,*1812, Book II, Chapter II, Section 9, pages 219-225, - Michaelis,
*Mémoire sur la probabilité du jeu de rencontre*, Berlin, 1846.

Although Todhunter observed the later paper of Michaelis, he failed to
note
Catalan's paper, "D'un
Problème de
Probabilité, relatif au Jeu de rencontre," in *Journal
de Mathématiques Pures et Appliquées*, 1837.

In the *Essai d'analyse sur les jeux de hazard*,
2nd edition (1713),
Montmort discussed the game of *Treize* on pages 130 to 143. In
the
1st edition of 1708 Montmort did not give a demonstration of any of his
results.
However, in the 2nd edition he gave two demonstrations which he had
received
from Nikolaus Bernoulli (pages 301 and 302). This includes a series
with
value **e**. Translation
of Montmort.

In the 1st edition of the *Doctrine of Chances*
(1717), the game is
mentioned in the *Preface *and in *Problems 35 & 36*.
These
problems are contained in the 2nd edition of 1738 and the 3rd edition
of
1756. The
third
edition contains the series for the constant **e** for the
first time.

Euler's paper on the game of *Rencontre*,
published in 1753, is E201,
"Calcul de la probabilité dans le jeu de
rencontre," *Mémoires de l'académie de Berlin *[7]
(1751), 1753, p. 255-270. Regarding this work, the editor says that a
memoir
entitled "Calcul des probabilités dans les jeux de hasard" was
presented
to the Academy of Berlin 8 March 1758. He asserts that it is probably
memoir
201: "Calcul de la probabilité dans le jeu de rencontre." An
analysis
of it appeared in the *Nova Acta eruditorum*, Leipzig 1754, p,
179.

Lambert produced a rather interesting paper entitled
"Examen d'une espece de Superstition
ramenée au calcul des probabilités" published in *Nouveaux
Mémoires de l'Académie Royale des Sciences et Belles
Lettres*
for 1771. In this paper he cites Euler and gives a general solution for
the
number of encounters. However, the purpose of the paper is not to study
this
game but rather to give an argument to support of the "accuracy" of
weather
almanacs.

In Book II
Chapter II Section 9
of the *Théorie Analytic des Probabilities*, Laplace
considers
the following urn problem: In an urn are *r* balls labeled
1,
*r* balls labeled 2, and so on to *r* balls labeled *n*.
If
the balls are drawn successively, what is the probability that at least
one
of the balls exits at the position indicated by its label? At least
two?
This same section treats a related problem: There are *i* urns
each
containing *n* balls, all of different colors. All the balls are
drawn
from each urn in turn. He asks, What is the probability that one or
more
balls will be drawn at the same rank from all the urns?

In *Mémoire
sur la
probabilité du jeu de rencontre*, G. Michaelis
generalizes
the two problems of Laplace in this way. He allows that the balls of
each
label vary in number.

Although not printed in volume 7 of his collected works, there is a paper on the number of derangements of a set of numbers which has a place here because of its relationship to E201. It is

E738. "Solution quaestionis
curiosae ex doctrina
combinationum" which was presented to the assembly 18 October 1779
and
published in the *Mémoires de l'académie des sciences
de
St. Pétersbourg ***3** (1809/10), 1811, p.
57-64.

The game of *Pharaon* was discussed by

- Montmort,
*Essai d'analyse sur les jeux de hazard*, pages 77-104, - de Moivre,
*Doctrine of Chances*, pages IX, 77-82, 105-07, - Daniel Bernoulli, "Exercitationes quaedam mathematicae," 1724
- Giovanni Rizzetti, "Ludorum scientia, sive Artis coniectandi elementa ad alias applicata," Acta Eruditorum Supp. IX, 1729. pp. 215 - 229, 294 - 307.
- Euler,
*Mémoires*...*Berlin*for 1753, pages 144-164.

The problem is to determine the advantage to the banker in the game. According to Todhunter, the problem was solved by Montmort and N. Bernoulli. However, the treatment of the problem by Montmort changed with edition. The second edition (1713) is more compact than the first (1708). In the second edition there is also a contribution by Jean III Bernoulli. Columns 2 and 4 of his tables are incorrect. Translation of Montmort.

The article from the
Encyclopedia of Diderot
on
*Pharaon* may be compared to Montmort's treatment of the game. By
the
way, a punter is a gambler, the one who wagers against the banker.

Moivre asserts in his introduction that the chief question concerning Pharaon (and Bassette) is this: What percent does the banker receive of all the money wagered at these Games? He shows that for Pharaon it is nearly 3%, and 0.75%. for Bassette. Moivre on Pharaon.

In 1755, Euler wrote E313, "Sur
l'avantage du banquier
au jeu de Pharaon," *Mémoires de l'académie de
sciences
de Berlin *[20] (1764), 1753, p. 144-164. According to the editor,
the
memoir was presented 27 February 1755 to the Academy of Science of
Berlin,
and a second memoir bearing the same title was presented there 20 July
1758.
It is not clear if these were the same work or two different works. The
publication took place in 1766.

The Genoise lottery was the first number lottery. It and its variants were discussed by many mathematicians because such lotteries were perceived to be unfair and because they gave rise to many interesting problems. Usually it took the form of choosing 5 from 100 with various payoffs depending upon the wager made. It was treated by

- Juan Caramuel,
*De concertationibus Cosmopolitanis*in*Mathesis Biceps. Vetus, et Nova. Syntagma 6*, 1670; - Guiseppe Maria Stampa,
*Ludus Serio Expensus*, 1700; - Nicolas Bernoulli,
*De Use Arte conjectandi in iure, VII*, 1709; - Frenicle de Bessy, "Abregé des combinations," 1729;
- Euler in correspondence with Frederick the Great and in the papers E338, E412, E600, E812, and E813.
- Jean III Bernoulli,
*Mémoires...Berlin*for 1767. - P.S. Laplace
*Mém. Acad. R. Sci. Paris*, 1774 and*Théorie analytique des Probabilités*, 1812, Book II, Chapter II, Section 4, pp. 194-203. - Jean Trembley,
*Mémoires de l'Académie royale des sciences et belles-lettres*. Berlin 1794/5, pp. 69-108. - Giofrancesco Malfatti, Lotto in Nuova Enciclopedia Italiana (1779) and Giuoco del lotto. Antologia Romana, 11.(1785)

Juan Caramuel reprints the *De ratiociniis in ludo aleae* of
Huygens in its totality. As for
the
Genoise lottery, he states that the payoffs are

Correct picks | 1 | 2 | 3 | 4 | 6 |

Payoff | 1 | 10 | 300 | 1500 | 10000 |

Caramuel argues that the corresponding correct payoffs are

8 | 10.7 | 334.1 | 31703.9 | 15090208 |

and hence the lottery is an unfair contract.

Nicolas Bernoulli, nephew of Jakob Bernoulli,
submitted as his
dissertation *De Usu Arte conjectandi in jure,* in partial
fulfillment
for the degree of Doctor of Laws.
In* Chapter VII*
he
took up the subject of lotteries. There is strong evidence that Nicolas
Bernoulli
made use of a notebook in which Jakob himself considered the Genoise
lottery.
He assumes the prizes should be inversely proportional to the
probability
of winning. In this way he obtains

0.95 | 10.9 | 337.3 | 31700 | 15057504 |

Because of his error, he treats Caramuel somewhat shabbily. Bernoulli's position is that this is an unfair contract as well.

Frenicle de Bessy died in 1675. The Paris Academy published his
works in
1729. Among them is this short paper
"*Abregé des combinaisons*."
He analyzes the game for these payoffs

0 | 4 | 300 | 5000 | 20000 |

and claims the payoffs should be

0 | 0 | 500 or 600 | 5000 or 6000 | 20000 |

Euler's interest in lotteries began at the latest in 1749 when he was
commissioned by Frederick the Great to render an opinion on a proposed
lottery.
The first of two letters began
15
September 1749. A second series began on
17
August 1763.

Euler himself wrote several papers prompted by investigations of lotteries.

E812. Read before the Academy of Berlin 10 March
1763 but only published
posthumously in 1862. "Reflexions sur une espese
singulier
de loterie nommée loterie genoise." *Opera postuma I*,
1862,
p. 319-335. The paper determined the probability that a
particular
number be drawn.

E338. "Sur la probabilité
des sequences dans la
loterie genoise." *Mémoires de l'Académie royale
des
sciences et belles-lettres de Berlin* [21] (1765), 1767, p. 191-230.
As
the name implies, Euler asks for the probability that various sequences
of
numbers be drawn.

The volume of the journal which contains E338, contains as well a paper by Jean III Bernoulli, "Sur les suites ou séquences dans la loterie de Genes," pp. 234-253 and two papers by Beguelin, "Sur les suites ou séquences dans la lotterie de Gene: First memoir and second memoir, " pp. 231-280.

E412. Read 29 November 1770. "Solution
d'une questione
tres difficile dans le calcul des probabilités." *Mémoires
de l'Académie royale des sciences et belles-lettres de Berlin*
[25] (1769), 1771, p. 255-302. This is an analysis of a lottery for
which
there are several classes and a guaranteed payment.

E600. "Solutio quarundam
quaestionum difficiliorum
in calculo probabilis." *Opuscula Analytica Vol. II,* 1785,
p. 331-346.
Here Euler investigated the probability that all numbers or some fewer
numbers
be drawn in a sequence of lotteries.

Regarding this latter paper, see De Moivre, 1711, *De
Mensura Sortis*
Problem 18 or its nearly identical counterpart in the *Doctrine of
Chances
*Problem
39. In these places, de Moivre determined the expectation of one
who
would cast a die some number of times so as to produce all faces. P.S.
Laplace
asked for the probability that all tickets will have been
withdrawn
after a prescribed number of drawings. This
problem was solved in
"Mémoire sur les
suites
récurro-récurrentes et sur leurs usages dans la
théorie
des hasards," *Mém. Acad. R. Sci. Paris* (*Savants
étrangers*) 6, 1774, pages 353-371. Here Laplace refers to
the
Genoise Lottery as the **Lottery of the Military School**. Years
later,
in the *Théorie analytique des Probabilités* he
asked
for the number of drawings for which the probability that all tickets
will
have come forth is one-half. This is found in
Book II, Chapter II, No. 4.
The
Genoise Lottery is now called the **Lottery of France**. Jean
Trembley,
citing the papers of both Euler and Laplace, generalized the solution
to
the problem in "Recherches sur
une
question relative au calcul des probabilités," *Mémoires
de l'Académie royale des sciences et belles-lettres*, Berlin
1794/5,
pp. 69-108.

E813 "Analyse d'un probleme du
calcul des probabilites,"
*Opera Postuma I*, 1862, p. 336-341. In this paper, Euler
determined
the probability that a ticket will be drawn 0, 1, 2, ... times in *n
*successive drawings of *r *tickets from an urn.

The so-called St. Petersburg Problem began with a letter from Nicolas Bernoulli to Montmort. Eventually Daniel Bernoulli and Cramer became involved. The salient portions of there correspondence are here: Correspondence of Montmort, Daniel Bernoulli, Cramer and Nicolas Bernoulli.

Daniel Bernoulli
is justly famous
for his paper on risk in which he proposed a solution to the problem.
This
was published in 1738 as "Specimen theoriae novae de mensura
sortis," *Commentarii Acad. Sci. Imp. Petrop.*,
1730-31,
**5**, 175-192. It has been translated into English as "Exposition of a new theory on the measurement of
risk"
*Econometrica*, 1954, **22**, 23-36.

Euler wrote a short paper quite similar in content to that of Daniel Bernoulli. Its time of composition is unknown.

E811. "Vera estimatio sortis in
ludis." *Opera postuma
I*, 1862, p. 315-318.

More documents related to this problem are located in pages devoted to D'Alembert.

Many mathematicians were concerned with the problem of how to combine
discordant
observations. One common method was to compute the arithmetic mean. An
early
paper on this subject was that written by Thomas Simpson. In the work
*Miscellaneous Tracts on some curious, and very interesting Subjects
in
Mechanics, Physical-Astronomy, and Speculative Mathematics...* is a
section
*An Attempt to shew the Advantage arising by Taking the Mean of a
Number
of Observations, in Practical Astronomy*. This item may be found at
the
University of York.
But other methods could be justified.
Daniel Bernoulli wrote on this
problem
in "Diiudicatio maxime probabilis plurium observationem discrepantium
atque
verisimillima inductio inde formanda." *Acta Acad. Sci. Imp. Petrop.*,
1777 (1778), 1, 3-23. It has been translated into English by
C.G.
Allen as "The most probable choice between several discrepant
observations and the formation therefrom of the most likely
induction," *Biometrika*, 1961, **48**,
1-18.

Euler wrote a commentary to Bernoulli's paper which was published in the same issue of the journal. It is

E448. (1778) "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).

Joseph Louis
Lagrange wrote
"Mémoire sur
l'utilité
de la méthode de prendre milieu entre les résultats de
plusiers
observations," *Miscellanea Taurinensia*, t. V,
1770-1773.
Euler responded to this paper with

E628. "Eclaircissements sur le
mémoire de Mr. De
la Grange insere dans le Ve volume des melanges de turior concernant la
methode
de prendre le milieu entre les resultats de plusiere observations etc.,"
*Nova acta academiae scientiarum Petropolitanae* **3** (1785),
1788,
p. 289-297. Summary p. 196-197. The paper was
presented
to the Academy 27 Nov. 1777.

Euler concerned himself with mortality and life expectancy
in E334.
"Recherches générales sur la
mortalité
et la multiplication du genre humain." *Mémoires de
l'Académie
royale des sciences et belles-lettres de Berlin* [16] (1760), 1797,
p.144-164. But see also the section on **Life Assurance**
for
the companion piece E335, "Sur les rentes viageres."
and the mortality table of Kersseboom.

There are two related items included in his collected works. These are

"Von der geschwindigkeit der vermehrung und von der
Zeit der verdoppelung."
which was printed in Süssmilch's *Die göttliche Ordnung*
and
"On the multiplication of the human race." being a fragment from one of
his
notebooks.

Immediately following the memoir on mortality (E334) is the memoir

E335. "Sur les rentes viageres,"
*Mémoires
de l'Académie royale des sciences et belles-lettres de Berlin*[16]
(1760), 1797, p.165-175. Euler viewed this paper as a
continuation
of E334. Here he derived a formula to facilitate the computation of a
life
annuity. Euler observed that there is no advantage to the state to sell
annuities
where the return is greater than the rate of interest earned by the
state.
Therefore he proposed the creation of foreborne annuities, purchased,
for
example, on a child at birth, but due at age 20. This would then permit
the
accumulation of funds as well as allow the annuity to be offered at a
much
lower rate.

The collected works include as well

E403 Des Herrn Leonard Eulers nöthig berechnung zur Einrichtung Einer

E473 D'un Etablisement publie pour payer des pensions a des veuves fonde sur les principles les plus solides de la probabilité. pp. 183-245. This memoir includes a description of a tontine.

E599 Solutio quaestiones ad calculam probabilitatis
pertinentis quantiam
disconceges persolvere debeant ut suis haeradibus post utriusque mortem
certa
argenti summa persolvatur. *Opuscula Analytica II*, 1785, p.
315-330.

Fragment. Sur le calcul des rentes Tontinieres.

This page was last updated on 08/20/11.