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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.
"Disquisitio
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 9^{th}
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.
Trembley |
Problem |
Moivre |
Lagrange |
1 |
To find the probability an event happen exactly b times in a trials. |
Introduction | Corollary to Problem I |
2 |
To find the probability an event happen at least b times in a trials. |
Problems III, IV and V | Problem I |
3 |
The Problem of Points for two players. | Introduction | |
4 |
The Problem of Points for three players. | Problem VI | |
5 |
The Problem of Points for four players. | Problem VI | |
6 |
Duration of Play | Problem LXV | |
7 |
Duration of Play - To bring forth an event b
times more than it is not or c times fewer than it is not. |
Problems LXIII, LXIV, LXVI & LXVII |
Problem VI |
8 |
Duration of Play - To bring forth an event at least b times, another at least c times, in a trials. |
Problem II | |
9 |
To find the distribution of balls in urns | Problem VII |
"De Probabilitate
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:
Sections | Problem |
Reference |
1-2 |
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. |
Laplace, 1774 |
3-7 |
To find the probability the ratio of white to black lies
between 0 and a given fraction given that previously p white and q black had been extracted. |
Laplace, 1778 |
8-9 |
An application to the births observed at Vitteaux in Bourgogne. | Laplace, 1783 |
10-11 |
To find the probability that white shall not exceed black
that if 2a more drawings are made given that p white and q black had been extracted previously. |
Laplace, 1778 |
12-13 |
Division of stakes for 2 players with unknown skills. | Laplace, 1774 |
14 |
To approximate a probability arising from the observation
that the ratio of births of boys to births of girls is greater in London than in Paris. |
Laplace, 1778 |
In addition, one may consult Prevost
& Lhulier, "Sur les
Probabilités," Mémoires de l'Académie des
sciences et belles-lettres...Berlin, 1796, pp. 117-142.
"Recherches sur
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.
"Observations sur
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.
99-126.
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.
"Observations sur
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.
"Observations sur
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 20^{th} 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.