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Nearly all surviving works of Laplace have been collected in the 14 volume edition of Oeuvres complètes de Laplace. This was published under the auspices of the Academy of Sciences of Paris by Gauthier-Villars between 1878 and 1912. Page images from these volumes are available through Gallica. Their contents are
|T. 1-5||Traité de mécanique céleste|
|T. 6||Exposition du système du monde|
|T. 7||Théorie analytique des probabilités|
|T. 8-12||Mémoires extraits des recueils de
l'Académie des sciences de Paris et de la classe
des sciences mathématiques et physiques de l'Institute de France
|T. 13-14||Mémoires divers|
With regard to the life and work of Laplace, several sources may be consulted with profit. These are
The Source Book Entwicklung der Wahrscheinlichkeitsrechnung von den Anfängen bis 1933. Einführung und Texte, Wissenschaftliche Buchgesellschaft 1988 contains several selections from Laplace's Analytic Theory of Probabilities. These selections may be found here.
Laplace began submitting papers to the Paris Academy on 28 March 1770. By the time he was admitted to membership as adjoint on 24 April 1773 he had presented 13 papers.
His memoirs relating to or concerning probability and statistics were written during two distinct periods. The first extended from 1771 to 1786, the second from 1809 to 1820. The monumental Théorie analytique des probabilités appeared in 1812 and consolidated much of his earlier work. Its two subsequent editions enlarged upon the first and incorporated several other recent publications as appendices. References below are to the third edition as published in the Oeuvres Complète, Volume 7. I will use the customary abbreviation TAP to refer to the work.
I am placing here the bulk of his writings on probability and statistics together with related works by other authors. In addition, I am including worked examples of solutions to many of Laplace's problems using, when feasible, Laplace's own solution.
Several further comments are in order. The French revolution, begun in 1789, resulted in a breakdown of academic life. As a result, the Paris Academy ceased operations 8 August 1793 when it was suppressed by the National Convention. The Bureau of Longitudes was founded with the law of 25 June 1795. Its journal was the Connaissance des temps in which Laplace published often.
The École Normale, a teachers' college at which Laplace lectured, opened on 21 January 1795 but closed in three months. It did not reopen until 1812. Classes were first held at what became known as the École Polytechnique on 21 December 1794. Here Laplace was an examiner.
The Institute de France, the successor to the Académie des Sciences, began to hold regular meetings on 27 December 1795.
The Convention of 21 September 1792 created a Revolutionary Calendar for which year I began 22 September 1792 and ended 21 September 1793. The months were named according to the seasons; the first month was named Vendémaire. This calendar went into effect 24 November 1793 and its use ceased at the end of 1805.
The first four papers of Laplace concerning probability and statistics were published in the journal bearing the full title Mémoires de mathématique et de physique présentés à l'Académie royale des sciences, pars divers savans, & lûs dans ses assemblies. This journal is usually referenced either as Mémoires ... pars divers savans or as Savants Étranges. All of these papers were submitted prior to his admission to the Academy, that is, when Laplace was a "stranger" to it. His subsequent papers during this period, submitted after joining the Academy, were published in Mémoires de l'Académie royale des Sciences de Paris.
The memoirs of Laplace can be subdivided into several somewhat crude natural groupings. These are
Two problems tended to dominate the early literature in probability. These are the Problem of Points and the Problem of Duration of Play. Both problems are treated by Laplace in his early work and solved using the theory of recurrent series, what we know today as the theory of finite differences.
Abraham de Moivre was the first to present a theory of recurrent series. He gave a treatment of the integration of linear equations in finite differences in the Doctrine of Chances, pages 220-229, 3rd edition, and in his Miscellanea Analytica de Seriebus et Quadraturis, published in 1730.
Apparently some later research by Lagrange on recurrent series caught the interest of Laplace. The relevant paper of Lagrange is "Sur l'integration d'une Équation Différentielle à différence finies qui contient la théorie des suites récurrentes," Miscellanea Taurensia, I, 1759. Lagrange, promising to follow this paper with one illustrating applications to the theory of chances, did so much later in "Recherches sur le suites recurrentes...", Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, 1775.1.
Laplace's first memoir on
recurrent series was "Recherches
sur le calcul intégral aux différences infiniment
petites, & aux différences finies" which
was published in Mélanges
de philosophie et de
mathématiques de la Société royale de
Turin, pour les années 1766-1769 (Miscellanea
273-345, 1771. An early
draft was read to the Paris Academy on 18 July 1770 as "Sur quelques
usages du calcul intégral appliqué aux différences
finies," and a revised
version 17 May 1771 recorded as "Sur le calcul intégral
appliqué aux différences finies à plusieurs
variables." This work was omitted from the Oeuvres
Complètes. Pearson says that Laplace wrote this
paper while at Caen prior to the age of 20.
The paper offers a new proof of the theorem of Lagrange on the solvability of differential equations. Laplace then shows that the same theorem holds for difference equations. There is no indication here of applications to probability.
suites récurro-récurrentes et sur leur usages
dans la théorie des hasards." Savants
étranges 6, 1774, p. 353-371. Oeuvres 8,
This paper was read to the academy on 5 February 1772. A recurrent series is what we today call a recurrence relation in one variable. A récurro-récurrentes series is one in two variables. In this memoir, Laplace investigates three probability problems by means of recurrence relations. The Duration of Play problem is often called Gambler's Ruin.
|Problem III||Duration of play||Lagrange's
|see also||De Moivre, Doctrine of Chances, LXIII
Lagrange, "Recherches sur le suites recurrentes..."
|Problem IV||Lottery||Solution||see also||Euler, "Solutio
quarundam quaestionum difficiliorum in calculo probabilis."
Trembley, "Recherches sur une question relative au calcul des probabilités."
|Problem V||Even-odd||Solution||see also||Mairan, "Sur le jeu de pair ou non."|
The last section of the memoir, section 8, concerns the solution of a differential equation and is unrelated to the theory of chances. For this reason, I have omitted it.
On 10 March and 17 March 1773, as
reported in the Procès-Verbaux of the
Paris Academy, Laplace read the paper "Recherches sur l'integration des
differentielles aux différences finies et sur leur
application à l'analyse des hasards." This was subsequently
published as "Recherches, sur
l'integration des équations differentielles aux
différences finies, & sur leur usage dans la
théorie des hasards." Savants
étranges, 1773 (1776) p. 37-163. Oeuvres 8,
p. 69-197. However, a note in the margin of
the journal states that the paper was read 10 February 1773.
It should be noted that on 17 May 1771, Laplace read to the academy a memoir entitled "Sur le calcul intégral aux differences finies à plusiers variables." As this was not published separately, it is possible that some of it was incorporated into this present paper. The memoir is quite lengthy. I have divided the work into two sections. The first portion is devoted to exposition of the theory and consists of sections I through XXIV. The portion which offers applications to probability occupies pages 113 to 163 in the original and includes sections XXV to XXXV. Be warned that because this paper uses very awkward notation I have been unable to duplicate it exactly. Errors undoubtedly occurred in the printing. I have adhered to the text as printed in most cases, correcting only the obvious.
This paper was published together with a second part, reprinted as a separate memoir in the Oeuvres Complètes, "2, sur le principe de la gravitation universelle, et sur les inégalités séculaires des planètes qui en dépendent." This is omitted.
In section XXV of this memoir, Laplace discusses probability and expectation. In so doing, he makes reference to previous work by himself and others. These references are to
The remaining sections each contain a problem illustrating the method. These are solved by means of recurrence relations and in many respects, the memoir is similar to memoir .
|XXVI||X||Even-odd||Solution||see also||Mairan, "Sur le jeu de pair ou non."|
|XXVII||XI||Annuity or compound interest||Solution|
|XXVIII||XII||Formation of sequences||Solution||see also||De Moivre, Problem LXXIV.
Trembley, "Essai sur la manière de trouver le terme général des séries récurrentes."
|XIX||XIII||Variation of Waldegrave's Problem||Solution||Montmort & Nicolas Bernoulli
Correspondence on Waldegrave's Problem.
|XXX||XIV||Problem of points
|XXXI||XV||Problem of points
|Solution||see also||Montmort, Propositions XL and XLI|
|XXXII||XVI||Extension of the problem of points||Solution|
|XXXIII||XVII||Duration of play
unequal skill, equal capital
|see also|| "Mémoire sur les suites récurro-récurrentes..."|
|XXXIV||XVIII||Duration of play
unequal skill, unequal capital
|see also||"Mémoire sur les suites récurro-récurrentes..."|
|XXXV||XIX||Duration of play
potential of no winner on a trial
The study of inverse probability had its inception with the work of Thomas Bayes. Laplace apparently was unaware of Bayes' paper.
la probabilité des causes par les
étranges 6, 1774, p. 621-656. Oeuvres 8,
p. 27-65. Based upon internal evidence, the
date of composition can be placed between March and December of 1773.
this memoir, Laplace mentions the publication of the memoir "Mémoire sur les
suites récurro-récurrentes et sur leur usages
dans la théorie des hasards" and having read "Recherches sur l'integration des
équations differentielles aux différences finies,
& sur leur usage dans la théorie des hasards"
before the Academy.
This memoir is one of the most significant in the development of mathematical statistics. Laplace develops in it his theory of inverse probability.
It has been translated by Steven Stigler. See "Laplace's 1774 Memoir on Inverse Probability," Statistical Science, Vol. 1, Issue 3 (Aug. 1986) 359-363 and "Memoir on the Probability of the Cause of Events," in the same issue, pp. 364-378.
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. Oeuvres 8,
Based on internal evidence, this work must have been written no earlier than 1775 although it appears in the record for 1773. Note, however, the publication date is 1776. Only the first portion of the paper is available here.
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.
Laplace now asks if it is possible to determine the probability that the mean fall within certain limits. He seeks to apply his solution to the mean inclination of the comets.
The same question regarding the mean is investigated by Joseph Louis Lagrange in "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. Trembley has also written on the mean in "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. In this paper he cites Daniel Bernoulli, Lagrange and Lambert as well as Laplace.
probabilités," Mém. Acad. R.
Sci. Paris, 1778 (1781), p. 227-332. Oeuvres 9, p.
383-485. Laplace read the memoir
"Mémoire sur le calcul aux suites appliqué aux
probabilités" on 31 May 1780 and submitted it for
publication 19 July of the same year. The review appearing in the Histoire
section of the same issue is interesting in that it gives historical
background to the method of inverse probability. According to Bernard
Bru and Pierre Crépel in "Condorcet: Arithmétique politique, textes
rares ou unédits (1767-1789), pp. 159-161, (Paris, 1994) the article
was composed by Condorcet.
This memoir again treats the subject of inverse probability. In particular, Laplace wants to compute (1) the probability of events which are composed of many simple events of unknown probability and (2) the probability of future events given past events.
Large sample tests of significance are developed in the investigation as to whether the excess of male births over female births may be due to chance or to cause, and if the probability of a male birth differs between two cities. For this, Laplace has need of asymptotic expansions.
Lastly, Laplace derives his logarithmic error law and computes the area under the "normal" curve of errors. Using his logarithmic error law, Laplace gives a rule to correct instrument error.
les naissances, les mariages et les morts à Paris, depuis
1771 jusqu'en 1784, et dans toute l'étendue de la France,
pendant les années 1781 et 1782,"
Mém. Acad. R. Sci. Paris, 1783 (1786), p. 693-702. Oeuvres
11 p. 35-46. Read to
the Academy 30 November 1785.
This memoir shares content with the "Mémoire sur la probabilités" and with the "Suite du mémoire sur les approximations..." in that Laplace studies the problem of estimating the probability of a male birth.
Generating functions and asymptotic expansions
sur les suites," Mém. Acad. R. Sci. Paris,
1779 (1782), p. 207-309. Oeuvres 10,
This paper develops the theory of generating functions. There is no explicit application to the theory of chances. However, generating functions ultimately become the cornerstone of his work in this area.
sur les approximations des formules qui sont fonctions de
très-grands nombres," Mém.
Acad. R. Sci. Paris, 1782 (1785), p. 1-88. Oeuvres 10,
Once again, Laplace investigates asympotic expansions. What is new is the introduction of the characteristic function and the inversion formula. Here, in his study of the distribution of a discrete random variable, he comes very close to discovering the central limit theorem. The result is generalized in TAP, II.18.
sur les approximations des Formules qui sont
fonctions de très-grands nombres (suite)," Mém.
Acad. R. Sci. Paris 1783 (1786), p. 423-467. Oeuvres 10,
p.295-338. The draft was read to
the Academy on 25 and 28 June 1785.
Here Laplace applies the results obtained in the previous memoir to the theory of chances.
A school for teacher training, the École normale, was created in 1795. This particular school closed after but a few months. Nonetheless, Laplace presented an elementary course in mathematics there. The lessons of this course were published as
"Leçons de Mathématiques professées à l'École normale en 1795," Séances de l'École normale (an VIII [1799-1800]) I, pp. 16-32, 268-80, 381-93; II. pp. 3-23, 130-34, 116-929, 302-18; III, pp. 24-39; IV, pp. 32-70, 223-63; V, pp. 201-19; VI, pp. 32-73. These were reprinted in Journal de l'Ecole Polytechnique 2 7e et 8e cahiers. (June 1812) pp. 1-172. and Oeuvres Complètes 14, pp. 10-177.
The lessons were elementary.
|First session||On numeration and arithmetic operations.|
|Second session||On functions; powers and extraction of roots; proportions; progressions and logarithms.|
|Third Session||On algebra; of the first operations of algebra; of powers and exponents.|
|Fourth session||On the theory of equations.|
|Fifth session||On the resolution of equations. Theorem on the form of their imaginary roots.|
|Sixth session||On the elimination of unknowns in equations. Resolution of equations by approximation.|
|Seventh session||On elementary geometry; notions on the limit;
principles of rectilinear
trigonometry and of spherical geometry.
|Eighth session||On the application of algebra to geometry. On the
division of angles. Theorem of Cotes.
Use of trigonometric tables for the resolution of equations. Applications of algebra to
the theory of lines and of surfaces.
|Ninth session||On the new system of weights and measures|
|Tenth session||On probabilities|
The tenth and final session (Dixième Séance) is entitled "Sur les probabilités," Oeuvres Complètes 14, pp. 146-177. The contents of this lesson were reordered and expanded into the introduction in TAP. This latter version has subsequently been known as the Philosophical Essay on Probabilities. The link here is to the fifth edition. Actually, there existed two related earlier publications. These are Séance 57 in the Séances des Écoles Normales 6 (1795-1796), printed in 1800, pages 32-73 and the "Notice sur les probabilités," Annuaire du Bureau des Longitudes (1811, published in 1810) , pages 98- 128.
Two translations into English of this essay have been published.
A Philosophical Essay on Probabilities, translated from the sixth edition by F.W. Truscott and F.L. Emory, John Wiley and Sons, New York, 1902. Reprinted by Dover Publications with an introductory note by E.T. Bell,, New York, 1995. The text is taken from the Oeuvres Complètes T. 7 published by Gauthier-Villars.
Pierre-Simon Laplace. Philosophical Essay on Probabilities, translated by A.I. Dale, Springer-Verlag, New York, 1995. The text is taken from the fifth French edition of 1825.
After a period of twenty-five years, Laplace returned to the study of probability. By this time, the Institute de France had become established. Between year VI and 1818, the fourteen volumes of its journal were published under the title Mémoires de l'Institut National des Sciences et Arts; sciences mathématiques et physiques. This was replaced by the Mémoires de l'Academie Royale des Sciences de l'Institut de France. Its first volume, for the year 1816, appeared in 1818.
sur divers points d'analyse." Journal de l'Ecole
Polytechnique 8 15e
cahier (December 1809) pp. 229-65. Oeuvres 14.178-214.
This paper, as the name suggests, holds a miscellany of results. It is composed of five sections:
On the calculus of generating functions,
On the definite integrals of equations in partial differences,
On the reciprocal passage from real results to imaginary results,
On the integration of non-linear equations in finite differences, and
On the reduction of functions in tables.
I have chosen to present the entire paper for several reasons. The first section contains several new theorems in the calculus of generating functions. This calculus was originally introduced in the Mémoire sur les suites. In this earlier memoir, Laplace discussed the solution of partial differential equations which is likewise taken up in the second section. The third section presents some important definite integrals which are used in the paper which follows. The remaining sections, while having no bearing on probability, are nonetheless interesting.
sur les approximations des formules qui sont fonctions de
très-grands nombres, et sur leur application aux
probabilités" and "Supplement
au mémoire sur les
approximations des formules qui sont fonctions de
très-grands nombres." Mém.
l'Institut 1809 (1810), 353-415, 559-565. Oeuvres 12
p.301-345, p.349-353. The first paper was read on 9 April 1810.
The first paper here begins with a new analysis of the inclination of the orbits of comets for which the inclination is assumed to be distributed uniformly (Sections I-IV). Laplace obtains a limiting distribution for the mean inclination in this case. The remainder of the paper (Sections V-VIII) generalizes the proof and he obtains a central limit theorem.
The second paper was written after Laplace had obtained the Theoria motus corporum coelestium in sectionibus conicis solem ambientium (1809). In this work (Sections 172-189) Gauss based a development of the method of least squares on the assumption that errors follow a normal distribution. Laplace realized that he could give a probabilistic justification to the method of least squares without assuming a normal distribution of errors through his central limit theorem. With a large sample, it is a valid method for any error distribution.
"Notice sur les probabilités," Annuaire publié par le Bureau des Longitudes 1811 (1810), pp. 98-125. Not in Oeuvres Complètes. This anonymous publication is derivative of the lesson on probability given in 1795 at the École Normale. Nearly all was incorporated in the first edition of the Essai philosophique sur les probabilités in 1814.
sur les Intégrales Définies, et leur application
aux Probabilités, et spécialment à la
recherche du milieu qu'il faut choisir entre les résultats
des observations," Mém. l'Institut 1810,
(1811) 279-347. Oeuvres Complètes 12. 357-412.
Read to the Academy 29 April 1811.
This memoir contains a discussion of three problems to illustrate the theory. The first is Gambler's Ruin. The remaining two are urn problems.
qu'il faut choisir
entre les résultats d'un grand nombre d'observations," Connaissance
des Temps for the year 1813 (July 1811), pp. 213-223. Oeuvres
This memoir is the nearly literal reproduction of Section VIII (pp. 401-412) of the previous memoir, "Mémoire sur les Intégrales Définies, et leur application aux Probabilités, et spécialment à la recherche du milieu qu'il faut choisir entre les résultats des observations." This section concerns the use of the method of least squares to correct values only known approximately.
This work was published in 1812 with the following dedication to Napoleon.
To Napoleon-the-Grand. Sire, The benevolence with which Your Majesty has deigned to greet the homage of my Traité de Mécanique Céleste, has inspired me to des ire to dedicate to You this Work on the Calculus of the Probabilities. This delicate calculus extends to the most important questions of life, which are in fact, for the most part, only some problems of probability. It must, under this relationship, interest Your Majesty of whom the genius knows so well to appreciate and so deservedly to encourage all that which can contribute to the progress of knowledge and to the public prosperity. I dare to emplore Her to accept this new homage dictated by the most lively appreciation, and by the profound sentiments of admiration and of respect, with which I am, Sire, of Your Majesty, The very humble and very obedient servant and faithful subject, Laplace.
A review of this work was published in 1812 in the Connaissance des Tems pour l'anné 1815.
The second edition is dated 1814. This new edition added an introduction of 106 pages, which includes the Essai philosophique sur les probabilités. The text expanded from 445 to 481 in length as a result of additions. The dedication to Napolean was replaced with the following Forward by Laplace.
This Work has appeared in the course of 1812, namely, the first Part around the beginning of the year, and the second Part some months after the first. Since this time, the Author has occupied himself especially to perfect it, either by correcting slight faults which were slipped into it, or by some useful additions. The principal is a quite extended Introduction, in which the principles of the Theory of Probabilities and their most interesting applications are exposed without the help of the calculus. This Introduction, which serves as preface to the Work, appears yet separately under this title: Essai philosophique sur les Probabilités. The theory of the probability of testimonies, omitted in the first edition, is here presented with the development which its importance requires. Many analytic theorems, to which the Author had arrived by some indirect paths, are demonstrated directly in the Additions, which contain, moreover, a short extract of the Arithmetic of the infinity of Wallis, one of the Works which have most contributed to the progress of Analysis and where we find the germ of the theory of the definite integrals, one of the foundations of this new Calculus of Probabilities. The Author desires that his Work, increasing by one third at least by these diverse Additions, merits the attention of the geometers, and excites them to cultivate a branch so curious and so important to human knowledge.
The third edition is dated 1820. Here the introduction expands to 142 pages but the text remains the same as in the second. Three supplements were added. These were published in 1816, 1818 and 1820 respectively. The fourth had been added in 1825 by Laplace to the copies of this third edition which were still at his disposal. A second forward by Laplace is added to this edition. It is as follows.
This third Edition differs from the preceding: 1 by a new Introduction which has appeared last year, under this title: Essai philosophique sur les Probabilités, fourth Edition; 2 by three Supplements which are related to the application of the Calculus of Probabilities in the natural sciences and to the geodesic operations. The first two have already been published separately; the third, relative to the operations of surveying, is terminated by the exposition of a general method of the Calculus of Probabilities, whatever be the sources of error.
Here are links to each of the editions: 1812, 2nd edition of 1814, 3rd edition of 1820, the Oeuvres de Laplace Volume 7 of 1847 and the Oeuvres complètes de Laplace Volume 7 published in 1886.
The Essai Philosophique sur les probabilités passed through 7 editions - 1814 (with the 2nd edition of TAP), 1814, 1816, 1819 and 1825 respectively.
See the original edition of 1814, the second edition of 1814, or the third edition of 1816 included as Volume 7 of the Oeuvres complètes. Also we find the fourth edition of 1819, the fifth edition of 1825, the sixth edition of 1840, and a seventh edition of 1840 published in Belgium.An English translation as A Philosophical Essay on Probabilities was made by Truscott and Emory in 1902 from the sixth French edition.
Table of Contents of TAP
BOOK I. Calculus of generating functions. [Full Table of Contents of Book I]
Book I is essentially a reprinting with some revisions of the "Mémoire sur les suites" (1782), the "Mémoire sur les approximations des formules qui sont fonctions de très-grands nombres" (1785), and the "Mémoire sur les approximations des Formules qui sont fonctions de très-grands nombres (suite)" (1786). Much of Book II is new although it does contain a reworking of problems and results derived in earlier papers.
After publication of the first edition of Théorie Analytique des Probabilités, Laplace continued to make application of his work. Note that the publication date of Connaissance des Temps is earlier than the year to which it applies. The reason for this is that it contained astronomical tables and therefore was published prior to the year in which the tables would be useful.
l'application du calcul des probabilités à la
philosophie naturelle," Connaissance des
Temps for the year 1818 (1815), 361-377. Oeuvres
Complètes 13.98-116 and "Sur
le calcul des probabilités appliqué à
la philosophie naturelle," Connaissance des Temps
for the year 1818, (1815), 378-381. Oeuvres
Complètes 13.117-120, supplement to the previous.
The first memoir was read to the first class of the Institute on 18 September 1815. In this memoir, Laplace discusses the masses of Jupiter, Saturn and Uranus expressed as a fraction of the mass of the Sun using values obtained from Bouvard and also the length of the pendulum expressed in tenths of a second using the value of Mathieu. For the sake of comparison, we present the present masses of these planets as given by NASA.
Fraction of the mass of the Sun
The articles are reproduced with some changes in the first supplement to TAP. The first article is reproduced as the introduction and sections 2 through 5. The second article is inserted as section 1 of this supplement. Be warned that these papers require that Chapter IV of TAP be available for consultation.
FIRST SUPPLEMENT. On the application of the Calculus of Probabilities to natural Philosophy. (1816) As mentioned above, this supplement reprints, with changes, the two articles from the Connaissance des Tems for the year 1818. Appended to it is a discussion on optimizing jury sizes.
du calcul des
probabilités aux opérations
géodésiques de la méridienne." Connaissance
des Temps for the year 1822 (1820), pp. 346-348. Oeuvres
Complètes 7, pp.
581-85. See also Oeuvres Complètes 13, p. 188 for correction of an
error. It was read on 20 December 1819.
This memoir is reproduced in Section I of the third supplement to TAP, pages 1-7: "Application des formules géodésiques de probabilité à la Méridienne de France." Oeuvres Complètes 7.581-616.
l'application calcul des probabilités aux observations et
spéciallement aux opérations du nivellement."
de Chimie et de Physique, t. XII; 1819 on pages 337- 341. This memoir concerns
surveying and is an abstract of the third supplement. The paper was
read on 20 December 1819. It is reproduced in Oeuvres
THIRD SUPPLEMENT. Application of geodesic formulas of probability to the meridian of France. (Continuation of Second Supplement, 1820) This appeared in the Annales de Chemie et Physique Vol 12 (1819), pp. 37-41 as "Application du calcul des probabilités aux opérations géodésiques de la méridienne de France." This is translated into English in Tilloch's Philosophical Magazine 58 (1821), pp. 133-136.
FOURTH SUPPLEMENT. (1825) Supplement 4 revisits the urn problem solved by means of partial difference equations. See also sections 8 and 10 of Book II of TAP.
A criticism of Laplace by Paolo Ruffini was reprinted in Nuovo giornale de'letterati, Vol. I (1822), pp. 201--212. The title is "Riflessioni critiche sopra il saggio filosofico intorno alle probabilità del Sig. C. Laplace fatte dal Dott. Paolo ruffini Professore di Clinica Medica, di Medicina pratica, e di Matematica applicaaa nella R. Università di Modena, Rettore della medisima, ec. ed. Modena 18211 in 8. p. 188."