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Jean Trembley was born at Geneva in 1749 and he died 18 September 1811. Trembley wrote on a wide range of topics including the calculus, differential equations, finite differences, probability and various applied problems. His papers were published in the Memoirs of the Academies of Göttingen, Berlin, Turin and St. Petersburg. Although a prolific author, he was so overshadowed by his contemporaries such as Laplace and Lagrange that he has been all but forgotten.
With respect to probability and related matters, Trembley contributed eight minor papers. The first two were published in volumes XII and XIII of the Commentationes Societatis Regiae Scientiarum Gottingensis in the years 1796 and 1799. The other six appeared in the Mémoires de l'Académie Royale des Sciences et Belles-Lettres between the years 1799 and 1804. These papers share a common theme: they attempt to "simplify"or "explain" work done by more capable mathematicians.
Todhunter does not speak well of Trembley. On the other hand, he does devote an entire chapter to an explication of these papers.
Elementaris circa Calculum Probabilium." Commentationes
Societatis Regiae Scientiarum Gottingensis, Vol. XII, 1793/4, pp.
99-136, published in 1796.
This paper examines 9 problems previously solved by Moivre, Lagrange and Daniel Bernoulli. Trembley references the second edition (1738) of the Doctrine of Chances by Moivre, the paper "Recherches sur le suites recurrentes..." by Lagrange which was published in the Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin in 1775 and, for the 9th Problem, the paper "Disquisitiones analyticae de novo problemate coniecturali" by Daniel Bernoulli which was published in the Novi Commentarii Acad. Petrop. Vol. XIV for 1769.
A concordance of the problems of Trembley with those of Lagrange and the third edition of the Doctrine of Chances is given in the following table.
|To find the probability an event happen
exactly b times in a trials.
|Introduction||Corollary to Problem I|
|To find the probability an event happen
at least b times in a trials.
|Problems III, IV and V||Problem I|
|The Problem of Points for two players.||Introduction|
|The Problem of Points for three players.||Problem VI|
|The Problem of Points for four players.||Problem VI|
|Duration of Play||Problem LXV|
|Duration of Play - To bring forth an event b
more than it is not or c times fewer than it is not.
|Problems LXIII, LXIV,
LXVI & LXVII
|Duration of Play - To bring forth an event at least
b times, another at least c times, in a trials.
|To find the distribution of balls in urns||Problem VII|
Causarum ab effectibus oriunda," Commentationes Societatis
Regiae Scientiarum Gottingensis, Vol. XIII, 1795/8, pp. 64-119,
published in 1799. Page 64 is apparently a misprint for 84 since it is
followed by page 85. This paper references the following memoirs:
Laplace, "Mémoire sur la probabilité des causes par les événemens," Savants étranges 6, 1774, p. 621-656.
Laplace, "Mémoire sur les probabilités," Mém. Acad. R. Sci. Paris, 1778 (1781), p. 227-332.
Laplace, "Suite du mémoire sur les approximations des Formules qui sont fonctions de très-grands nombres," Mém. Acad. R. Sci. Paris 1783 (1786), p. 423-467.
Lagrange, "Recherches sur les suites récurrentes," Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, 1775.1 (1777) pp. 183-272.
The paper is somewhat loosely structured in that Trembley is not particularly clear in announcing where he is headed. Briefly, the paper deals with urn problems. The contents, following the outline of Todhunter, are these:
|To find the probability that m white and n
black balls will be
extracted from an urn from which previously p white and q
black had been extracted.
|To find the probability the ratio of white to black lies
0 and a given fraction given that previously p white and q black
had been extracted.
|An application to the births observed at Vitteaux in Bourgogne.||Laplace, 1783|
|To find the probability that white shall not exceed black
2a more drawings are made given that p white and q black had
been extracted previously.
|Division of stakes for 2 players with unknown skills.||Laplace, 1774|
|To approximate a probability arising from the observation
the ratio of births of boys to births of girls is greater in London
than in Paris.
In addition, one may consult Prevost & Lhulier, "Sur les Probabilités," Mémoires de l'Académie des sciences et belles-lettres...Berlin, 1796, pp. 117-142.
une question relative au calcul des probabilités." Mémoires
de l'Académie des sciences et belles-lettres...Berlin,
1794/5, pp. 69-108, published in 1799. Trembley considers problems
which arise from the Genoise Lottery, for example, what is the
probability that after a sequence of independent lotteries all of the
numbers will have been brought forth at least once?
Regarding this paper one may refer first to 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. 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.
An approximation formula obtained by Trembley was previously produced by Laplace. See "Suite du mémoire sur les approximations des Formules qui sont fonctions de très-grands nombres," Mém. Acad. R. Sci. Paris 1783 (1786), p. 423-467.
Finally, in E600, "Solutio quarundam quaestionum difficiliorum in calculo probabilis." Opuscula Analytica Vol. II, 1785, p. 331-346, Euler investigated the probability that all numbers or some fewer numbers be drawn in a sequence of lotteries.
"Recherches sur la
mortalité de la petite vérole." Mémoires
de l'Académie des sciences et belles-lettres...Berlin, 1796,
pp. 17-38, published in 1799. Corresponding to this paper are some
corrections which appeared in the volume for 1804 and published in
1807. It is "Eclairissement relatif au
Mémoire sur la mortalité..." pp. 80-82.
This paper is very closely related to that of Daniel Bernoulli, "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. Here are the tables of Bernoulli referenced by Trembley.
"Essai sur la
manière de trouver le terme général des
séries récurrentes." Mémoires de
l'Académie des sciences et belles-lettres...Berlin, 1797,
pp. 84-105, published in 1800.
The aim of this paper is to show how to solve recurrence relations without the need to find the zeros of the denominator of the generating fraction. Trembley illustrates his results with several series employed by Euler in his Introductio Analysin Infinitorum (1748). The paper on the whole is extremely tedious. The last portion takes up Problem XII of Laplace solved in "Recherches, sur l'integration des équations differentielles aux différences finies, & sur leur usage dans la théorie des hasards." Savants étranges, 1773 (1776) p. 37-162.
les calculs relatifs à la durée des mariages et au nombre
des époux subsistans." Mémoires de
l'Académie des sciences et belles-lettres...Berlin,
1799/1800, pp. 110-130. It was published in 1803.
The original idea for this paper appears to be in a problem discussed by Jean Bernoulli III in "Mémoire sur un probleme de la Doctrine du Hazard," Histoire de l'Academie des sciences et belles lettres de Berlin for 1768, (1770), pp. 384-408. It is this:
Any number of persons of one same
age, half men, half women, are married together the same year,
Trembley refers also to the papers of Daniel Bernoulli, "De usu algorithmi
infinitesimalis in arte coniectandi specimen," Novi Commentarii
Acad. Petrop. Vol. XII, 1766-7 (1768), pp. 89-98 and the one
which immediately follows, "De duratione matrimoniorum
media pro quacunque coniugum aetate, aliisque quaestionibus affinibus,"
Novi Commentarii Acad. Petrop. Vol. XII, 1766/7 (1768), pp.
Three other individuals were mentioned by Trembley - Wenceslaus Johann Gustav Karstens, author of Theorie von Wittwencassen (Theory of Widows' Insurance), Johann Nicolas Tetens and Johann Andrea Christian Michelsen. Johann Tetens (1736 - 1807) wrote Einleitung zur Berechnung der Leibrenten und Anwartschaften Vol 1 (1785) and Vol. 2 (1786). Johann Michelsen (1749 - 1797) was a Professor of Mathematics and Physics.
The author apparently never followed through with a threat to publish further investigations on this topic.
la méthode de prendre les milieux entre les observations." Mémoires
de l'Académie des sciences et belles-lettres...Berlin, 1801,
pp. 29-58. It was published in 1804.
This paper concerns the method of taking the mean among observations or rather, the theory of errors. Trembley cites Daniel Bernoulli, J.H. Lambert, P.S. Laplace and J.L. Lagrange as eminent mathematicians who have devoted themselves to its study.
Daniel Bernoulli, of course, had treated this topic in his "Diiudicatio maxime probabilis plurium observationem discrepantium atque verisimillima inductio inde formanda." Acta Acad. Sci. Imp. Petrop., 1777 (1778), 1, 3-23. This paper 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.
Now Trembley is known to have been familiar with Lambert's Beyträge zum Gebrauche der Mathematik und deren Anwendung since he had referred to it in an earlier paper. In this may be found "Anmerkungen und Zusätz zur practischen Geometria" and "Theorie der Zuverläßigkeit der Beobachtungen und Versuche." Both are contained in Part I (1765) of the Beyträge. Lambert also studies the problem of errors in the Photometria (1760).
With Laplace we may first refer to the "Mémoire sur l'inclination moyenne des orbites des comètes, sur la figure de la terre, et sur les fonctions" Savants étranges 7, 1773 (1776), p. 503-540. In this Laplace asked if it is possible to determine the probability that the mean fall within certain limits seeking to apply his solution to the mean inclination of the comets. It is possible Trembley had examined his "Mémoire sur les probabilités," Mém. Acad. R. Sci. Paris, 1778 (1781), p. 227-332 in which Laplace derived his logarithmic error law, computed the area under the "normal" curve of errors and used his logarithmic error law to give a rule to correct instrument error.
Of course, the relevant paper of Joseph Louis Lagrange is "Memoir on the utility of taking the mean among the results of several observations in which one examines the advantage of this method by the calculus of probabilities, and where one solves different problems related to this material," Miscellanea Taurinensia, t. V, 1770-1773. Refer here particularly to Problems VII and VIII, sections 25-29.
le calcul d'un Jeu de hasard." Mémoires de
l'Académie des sciences et belles-lettres...Berlin, 1802,
pp. 86-102. Publication date is 1804.
This paper is concerned with a problem posed by Montmort on the game of Her. The problem was discussed during the years 1711 and 1713 by Montmort, Waldegrave, the Abbé of Monsoury, and Nicolas Bernoulli and their conclusions are preserved in the letters exchanged by Montmort and Bernoulli which were printed in the second edition of Montmort's Essay d'analyse sur les jeux de hasard. In the two player version of the game of Her, optimal play requires that the players employ a mixed strategy. This mixed strategy was discovered accidentally by Waldegrave but a theory of mixed stategies was not developed until the 20th century. Trembley, after discussing at length the two player game, claims to solve the problem originally posed by Montmort. His work is flawed.
Ronald A. Fisher discovered independently Waldegrave's solution to the card game of Her. It appears in "Randomisation, and an Old Enigma of Card Play" published in the Mathematical Gazette 18, 1934 pp. 294-297.