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Pierre Simon Laplace
on
Probability and Statistics

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.

1771 - 1786

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

Recurrent series

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.

Inverse Probability, the theory of errors and tests of significance.

The study of inverse probability had its inception with the work of Thomas Bayes. Laplace apparently was unaware of Bayes' paper.

Generating functions and asymptotic expansions

1795

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.

1809-1820

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.

Generating functions and the Central Limit Theorem.

Théorie Analytique des Probabilités.

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.

Mention should be made also of the "Notice sur les probabilités," which appeared in the Annuaire présenté à S.M. l'Empereur et Roi par le Bureau des Longitudes pour l'an 1811 (1810), pp. 98-128.  This is an early version of the Essai.  See Gillispies' paper: "Mémoires inédits ou anonymes de Laplace sur la théorie des erreurs, les polynômes de Legendre et la philosophie des probabilités," Revue d'histoire des sciences (1979), Tome 32, No. 3, pp. 223-279.

The texts available below are based upon the Oeuvres Complète de Laplace, Volume 7, published in 1886. If we compare it to the 1820 edition, the last published during the lifetime of Laplace, we find that some editing was done.

Table of Contents of TAP

  1. INTRODUCTION

  2. BOOK I. Calculus of generating functions. [Full Table of Contents of Book I]

    FIRST PART. General considerations on the elements of magnitudes.
    Chapter I. On generating functions of one variable.
    Chapter II. On generating functions of two variables.
    SECOND PART. Theory of the approximations of formulas which are functions of very great numbers.
    Chapter I. On the integration by approximation of the differentials which contain factors raised to great powers.
    Chapter II. On the integration by approximation of the linear equations in finite and infinitely small differences.
    Chapter III. Application of the preceding methods to the approximation of diverse functions of very great numbers.
  3. BOOK II. General theory of probabilities.
    Chapter I. General principles of this theory. (Sections 1 and 2, pages 181-190.)
    Chapter II. On the probability of the events composed of simple events of which the respective probabilities are given. (Sections 3-15, pages 191-279.)
    Chapter III. On the laws of probability which result from the indefinite multiplication of events. (Sections 16-17, pages 280-308.) 
    Chapter IV. On the probabilility of errors of the mean results of a great number of observations and of the most advantageous mean results. (Sections 18-24, pages 309-354.)
    Chapter V. Application of the calculus of probabilities to the research of phenomena and of their causes. (Section 25, pages 355-369.)
    Chapter VI. On the probabilities of causes and of future events, drawn from observed events. (Sections 26-33, pages 370-409.)
    Chapter VII. On the influence of the unknown inequalities which can exist among some chances which we suppose perfectly equal. (Section 34, pages 410-415.)
    Chapter VIII. On the mean duration of life, of marriages and of any associations. (Sections 35-37, pages 416-427.)
    Chapter IX. On the benefits depending on the probability of future events. (Sections 38-40, pages 428-440.)
    Chapter X. On moral expectation. (Sections 41-43, pages 441-454.)
    Chapter XI. On the probability of testimonies. (Sections 44-50, pages 455-470.)
  4. ADDITIONS
    I. We deduce the expression of the ratio of the circumference to the radius, given by Wallis, by infinite product in No. 34 of Book I. (Pages 471-479.)
    II. Direct demonstration of the expression in No. 40 of Book I. (Pages 480-485.)
    III. Demonstration of a formula in No. 42 of Book I. (Pages 485-493.)

  5. SUPPLEMENTS

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.

Post TAP and Supplements

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.

NEW Since creating the previous materials, a complete translation of the 1820 edition of TAP with the supplements has been completed. Book I and Book II.

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."