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Probability and Statistics

In Croix ou Pile d'Alembert introduced his famous error that the probability that at least one head should appear in two consecutive tosses of a fair coin is 2/3 rather than 3/4. In addition, he discussed the Petersburg problem. The article Gageure is noteworthy in that it contains the text of a letter by the Swiss mathematician Necker responding to the error of the earlier article.

In Absent d'Alembert
called attention to the
dissertation of Nicholas Bernoulli, *De usu artis conjectandi in iuri*.
See Chapter 3 of the dissertation entitled *Concerning the
absent
person considered as dead*.

In 1733, M. le
Clerc, Comte de Buffon
introduced to the world the mathematical analysis of the game of
*franc-carreau* (*open-tile* or *free-tile*) as well as
his
famous **Needle Problem**. A summary
was printed in the *Histoire de l'Academie...Paris* for that
year. The
article Carreau discussed both *franc-carreau*
and the needle problem.

These eight volumes, published between 1761 and 1780, contain short works on various mathematical topics. It is useful to partition them into several groups.

*Firstly*, there are five memoirs which
continued the discussion of
**Croix ou Pile** and the **Petersburg Problem**.

The Petersburg problem was introduced by
Nicholas Bernoulli in
correspondence
to Montmort which occurred
in
September 1713. The two exchanged ideas until December 1716. Some of
the
correspondence was printed in the third edition of *Essay d'analyse
sur
les jeux de hazard* dated in 1713. There is also interesting
correspondence
between Nicholas Bernoulli and others on the problem which began with a
letter
from Cramer in 1728.

Daniel Bernoulli
wrote an important
paper, "Specimen theoriae novae de mensura sortis" on the Petersburg
problem.
This was printed in Book V of *Commentarii Academiae Scientiarum
Imperialis
Petropolitanae*, p. 175-192, for the years 1730-31, but published in
1738.
An English translation may be found in *Econometrica*, XXII:
23-36,
January 1954 prepared by Louise Sommer. It is to this paper that
d'Alembert
refers.

The mathematician Alexis Fontaine de Bertins (1704-1771) offered a solution to the Petersburg problem in the paper "Solution d'un Problème sur les Jeux de Hasard." This work appeared in a special collection of his memoirs published by the Paris Academy in 1764. An explication of his reasoning is available.

Volume 2 | 1761 | Memoir 10 |
Réflexions sur le Calcul de Probabilités | pp. 1-25 |

Volume 4 | 1768 | Memoir 23, V |
Sur le Calcul des Probabilités | pp.73-79 |

Memoir 23, VI |
Sur l'analyse des Jeux | pp. 79-92 | ||

Memoir 27, I |
Extraits de letters sur le calcul des Probabilités, | pp. 283-310 | ||

Volume 7 | 1780 | Memoir 52 |
Sur le Calcul des Probabilités | pp. 39-60 |

*Secondly*, Daniel Bernoulli had studied the question of whether
or
not to inoculate against the smallpox. In an article written for the
general
reader published in June of 1760 and in a paper presented to the
Academy
of Sciences of Paris that same year, he stated that the two chief
motives
for inoculation are humanity and the interest of the state. This paper
was
published as "Essai d'une nouvelle analyse de la
mortalité
causée par la petite vérole, et des avantages de
l'inoculation
pour la prévenir"; (Essay on a new analysis of the
mortality
caused by the smallpox and on the advantages of inoculation for its
prevention.)
appearing in the *Histoires et Mémoires de l'Académie
Royale
des Sciences de Paris*, p. 1-45 in the year 1766. There is an
English
translation by L. Bradley, in the small book with title Smallpox
Inoculation:
An Eighteenth Century Mathematical Controversy published by the Adult
Education
Dept. University of Nottingham, 1971.

D'Alembert took exception to Bernoulli's treatment
of the problem. His responses
were recorded in four memoirs. The celebrated geometer cited in **Memoir
11** is Alexis Fontaine des Bertins. The book by Bradley mentioned
above
includes a translation of **Memoir 11** and **Memoir 23, VII **below.
Associated with **Memoir 11** are Notes which occupy pages 47-95 of
the
same volume.

Volume 2 | 1761 | Memoir 11 |
Sur l'application du Calcul des Probabilités à l'inoculation de la petite Vérole |
pp. 26-46 |

Volume 4 | 1768 | Memoir 23, VII |
Sur un Mémoire de M. Bernoulli concernant l'Inoculation |
pp. 98-105 |

Memoir 27 |
Sur les Calculs relatifs à l'Inoculation | pp. 310-341 | ||

Volume 5 | 1768 | Memoir 44 |
Sur les Calculs relatifs à l'Inoculation; addition au Vingt-septième Mémoire |
pp. 508-510 |

Denis Diderot felt compelled to
respond
to **Memoirs 10** and **11** in "Sur deux mémoires de
d'Alembert
l'un concernant le calcul des probabilités l'autre
l'inoculation."
This work is divided into three sections: Sur
les probabilités, Quelques observations sur ce mémoire,
and De l'inoculation. Only the first two sections are rendered here.

These pages were destined to the *Correspondence
of Grimm*, but were
not published there. Diderot, in his letters to his mistress,
Madamoiselle
Voland, returns three times to this subject and the last time says: "The morsel on the probabilities is a grimoire, which will
not
amuse you." (25 October 1761.) It certainly is not a polished
piece
of prose. Rather, it has the marks of being hastily written. The work
is
rambling, repetitive, and in places obscure. Nonetheless, Diderot does
a
fair assessment of d'Alembert and it bears reading.

Lastly, there are two memoirs on the duration of life. These are

Volume 4 | 1768 | Memoir 23, VI |
Sur la durée de la vie | pp. 92-98 |

Volume 5 | 1768 | Memoir 36, III |
Sur les tables de mortalité | pp. 228-231 |

In Memoir 36, III, d'Alembert
makes
reference to the mortality tables of the Curé of *** and of
Sweden
published by Deparcieux.

In early 1753, two volumes of essays previously presented before the
Académie Français were published under the title,
*Mélanges de littérature et de philosophie*. Two
more
volumes were printed in 1759. The fifth and last was published in 1767.
In
this fifth volume appeared the memoirs "Doutes
et questions
sur le calcul des probabilités." and "Réflexions sur
l'Inoculation." These were apparently derivatives of **Memoirs 10**
and
**11 **written** **for the general public. The basis for the
present
version is taken from the Oeuvres de
D'Alembert, Tome Premier, Part I, published at Paris in 1821. It
appears
on pages 451-462. The "Réflexions sur l'Inoculation."
follows
immediately on pages 463-514.

The first memoir, "Doutes et questions sur le calcul des probabilités" references two papers of Daniel Bernoulli.

The first paper of Bernoulli concerns the Petersburg
Problem, "Specimen theoriae
novae de mensura sortis. *Comment. Acad. Imp. Petrop.* **5**
175-192.
(1730-1731). This may also be found in his *Werke* **2**, pp.
223-234.
It has been translated into English as "Exposition of a new theory on
the
measurement of risk," *Econometrica ***22**, 23-36 (1954).

An interesting work published anonymously in 1801 under the title Réfutation de Quelques Erreurs Singulieres de Mr. D'Alembert sur les Principes du Calcul des Probabilités et Solution d'un Problème connu sous le nom de Problème de Pétersbourg sur le Jeu de Croix et Pile que Personne n'avoit résolu jusqu'à présent et que Mr. D'Alembert a jugé insoluble was the response by Josef Niklas Windisch-Grätz.

The second paper concerns the orbit of the planets and comets about the sun. The known planets each orbit the sun in a plane very nearly that of the solar equator. The Paris Academy had offered many years previously a prize for a model to explain why the planets all fall into roughly the same plane of orbit. Daniel Bernoulli earned a share of the prize in 1734 with his paper, "Physical and astronomical researches on the problem proposed for the second time by the Academie Royale des Sciences de Paris." Here he hypothesized that the solar atmosphere was the cause. But he also concluded from the fact that the planes of the planets deviate so slightly from that of the solar equator, that the orbital planes could not be determined due to chance alone. On the other hand, he noted that the comets appear to have no liaison with the solar equator.

This encyclopedia for the most part reprinted articles from the Diderot
*Encyclopedia*. The first volume (Aba-Ext) appeared in 1784, the
second
(Fac-Rud) in 1785, and the third (Sag-Zon) in 1789. It does contain
several
items of note. The article **Absent **possesses an addition by
both
Diderot and
Condorcet. There are two articles
**Probabilité**: the first (pages 640-649) is that attributed
to
Diderot from his *Encyclopedia*,
the second (pages 649-663) is by
Condorcet. The article **Milieu**
was contributed by Jean Bernoulli.
D'Alembert may himself have added the new article
Cartes, although it was published after his
death.
The mathematics of this article is defective and has been commented on
by
Binet.

Last updated 17 July 2009.