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Jean Le Rond d'Alembert
Probability and Statistics

D'Alembert's earliest writings which bear upon probability and statistics may be found in the great Encyclopedia of Diderot for which he was the scientific editor. Of his contributions to this work, the most important articles are these two: Croix ou Pile (Heads or Tails) and Gageure (Wager) which appeared in 1754 and 1757 respectively. Several other articles written by d'Alembert are Absent, Avantage, Bassette, Carreau, , Loterie and Pari. These articles themselves reference others in the encyclopedia. These include Alternation, Baraicus, Combinaison, Jeu, Pharaon, Probabilité, Rafle, and Trictrac. It is worthwhile to note that the article Jeu does not concern itself with games of chance at all, but rather refers the reader to the article Jouer. This latter article is unsigned and therefore not written by d'Alembert. The article on Trictrac, the game of Backgammon, merely explains the method of play and thus is of no interest. The article Probabilité is also unsigned. It is therefore likely that both were the work of Diderot

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.

The Opuscules Mathématiques.

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
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.

The Mélanges de Philosophie

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.

The Encyclopédie Méthodique - Mathématiques

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.