Sources in the History of Mathematics and Physical Sciences

13

Editor G.J. Toomer

Advisory Board J.Z. Buchwald P.J. Davis T. Hawkins A.E. Shapiro D. Whiteside Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo Andrew I. Dale

Pierre-Simon Laplace

Philosophical Essay on Probabilities

Translated from the fifth French edition of 1825 With Notes by the Translator

Springer Andrew I. Dale Department of Mathematical Statistics University of Natal King George V Avenue Durban, Natal 4001 Republic of South Africa

Library of Congress Cataloging-in-Publication Data Laplace, Pierre Simon, marquis de. 1749-1827. [Essai philosophique sur les probabilites. English] Philosophical essay on probabilities / Pierre Simon Laplace: translated from the fifth French edition of 1825 by Andrew I. Dale, with notes by the translator. p. cm. - (Sources in the history of mathematics and physical sciences: vol. 13) Includes bibliographical references.

I. Probabilities. I. Dale, Andrew I. II. Title. III. Series: Sources in the history of mathematics and physical sciences ; 13. QA273.18.L3713 1994 519.2--dc20 94-25497

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ISBN-13: 978-1-4612-8689-9 e-ISBN-13: 978-1-4612-4184-3 DOl: 10.1007/978-1-4612-4184-3 To F. J. H. Sources in the History of Mathematics and Physical Sciences

Vol. 1: G.J. Toomer (Ed.) Diodes on Burning Mirrors: The Arabic Translation of the Lost Greek Original, Edited, with English Translation and Commentary by GJ. Toomer

Vol. 2: A. Hermann, K. Y. Meyenn, Y.F. Weisskopf (Eds.) Wolfgang Pauli: Scientific Correspondence I: 1919-1929

Vol. 3: I. Sesiano Books IV to VII of Diophantus' Arithmetica: In the Arabic Translation Attributed to Qusta ibn Liiqil

Vol. 4: P.I. Federico Descartes on Polyhedra: A Study of the De Solidorum Elementis

Vol. 5: O. Neugebauer Astronomical Cuneiform Texts

Vol. 6: K. von Meyenn, A. Hermann, Y.F. Weisskopf (Eds.) Wolfgang Pauli: Scientific Correspondence II: 1930-1939

Vol. 7: J.P. Hogendijk Ibn AI-Haytham's Completion ojthe Conics

Vol. 8: A. Jones Pappus of Alexandria Book 7 of the Collection

Vol. 9: GJ. Toomer (Ed.) Appollonius Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banii Miisil, Edited, with English Translation and Commentary by GJ. Toomer

Vol. 10: K. Andersen Brook Taylor's Role in the History of Linear Perspective

Vol. 11: K. von Meyenn (Ed.) Wolfgang Pauli: Scientific Correspondence III: 1940-1949

Vol. 12: FJ. Ragep Na~ir ai-Din al-Tiisi's Memoir on Astronomy (al-Tadhkira tT

Vol. 13: A.I. Dale Pierre-Simon Laplace. Philosophical Essay on Probabilities, Translated from the fifth French edition of 1825, With Notes by the Translator TRANSLATOR'S PREFACE

Pierre-Simon Laplace (1749-1827) is remembered today among students of probability for his many memoirs on that subject and, more particularly, for his Theorie Analytique des Probabilites, first published in 1812. The second edition of November 1814 of this work had, as an introduction, an Essai Philosophique sur les Probabilitis, the first edition of which had a~ peared in February of that same year. Here, without the aid of symbolic mathematics, Laplace provided a popular exposition of his Theorie. This Essai was based on a lecture on probability given by Laplace at the Bcoles Normales in 1795. It underwent sweeping changes, almost doubling in size, in the three editions of the Theorie that were published during Laplace's lifetime. Part of this increase was effected by the incorporation of extracts from the first edition of the Theorie Analytique des Probabilites into the Essai. Thus a major part of the first article of Book II of the Theorie appears in this translation in the articles headed "On probability" and "General principles of the probability calculus" , while part of the first paragraph of the second article of the same book appears at the start of the article here entitled "On expectation" . Translations of various editions in different languages have appeared over the years. In 1902 an English translation (stated on the title page of the 1952 reprint as being of the sixth French edition) was published. Ade• quate as this might have been at the time, it reads awkwardly today, and several mathematical phrases are obscured in translation - for example, who would recognize "generating functions" in the phrase "discriminant functions"? It was the presence of such obscurities that made me take up the task of providing a translation of the Essai. The fifth edition of the Essai has recently been re-issued with new notes and with a preface by Rene Thom and a postscript by Bernard Bru. Other works that proved useful were the German translation of the Essai published by Richard von Mises in 1932 and Karl Pearson's History of Statistics in the 17th & 18th Centuries.

vii NOTE ON THE TRANSLATION

'Imnslation is always a more or less crim• inal act against both author and readers. W.H. Donahue.

H it is to read well, a translation must not be too literal, and if it is accu• rately to reflect the original work, it must not be too free. What is aimed at here, then, is in fact, in some degree, a paraphrase of Laplace's Essai Philosophique sur lea Probabilites. To reproduce Laplace's style in transla.• tion is impossible, and I have occasionally allowed myself the liberty of a rude rendering to preserve the sense of the original. In his Higher English Rahtz suggests that On the one hand, the paraphrase should contain as much as possible of the meaning of the original, even to the smallest details; and, on the other hand, the result should be a readable bit of prose, developing in logical order and proportion the ideas rather than the words of the original. [1921, p. 327] The production of the best results, he further avers, depends on the taste of the writer and the nature of the subject. Whether I have achieved such production is for the reader to decide, though I would like to be able to say, with Gibbon,

The curious reader ... will perhaps accuse me of giving a bold and licentious paraphrase; but, if he considers it with atten• tion, he will acknowledge that my interpretation is probable and consistent. [1896, vol. 1, p. 429]

The difficulties of a translator are well summed up by Lin Yutang in The Wisdom of Confucius as follows: In the actual act of translation, the translator is faced with two jobs after he has grasped the meaning of the sentence. First he is faced with the choice of one of a number of synonyms, and failure to get at the exact word would completely fail to render the meaning of the remark clear to the reader ... In the second

ix x Note on the translation

place, the translator cannot avoid putting the thought in the more precise concepts of a modern language. [1943, pp. 45-46]

Being also assured of the truth of these words, I have not tried to be com• pletely consistent in my translating. Thus while vraisemblance, hasard and probabilite are given as "likelihood", "chance" and "probability" respec• tively, Laplace's espemnce is translated sometimes by "hope" (when it is contrasted with "fear") and sometimes by "expectation" (when it is used in its modern probabilistic sense). Ordinary footnotes are labelled with the usual footnote symbols ., t, &c.; these footnotes are usually my comments on the text. Additional words of explanation, my additions, are given in braces { ... }. Numbers in brackets [... ] (some of which may be in the footnotes) refer to the Notes following the translation: all of these notes, unless otherwise indicated, are mine. I have indicated differences between the first and the fifth editions of the Essai by using italic Greek and Roman superscripts. Thus, for exam• ple, on page 1, the text passage" aof mathematics with Lagrange by decree of the National Conventiona " has a footnote "a-a and which appeared in the Journal de l'Ecole Polytechnique." This means that the text passage between the superscripts appeared in the first edition in the form shown in the footnote. Similarly, the labelling on page 19 of the footnote as "p-p20" indicates that the passage (from the first edition) in the footnote occurs in the main text between the superscripts p and P, the second superscript P appearing on page 20. The appearance of main text between superscripts for which there is no footnote means that the distinguished passage did not appear in the first edition, but is in the fifth. Finally, Greek letters in the main text, appearing as centred superscripts, indicate that something that appeared in the first edition is missing in the fifth: such omissions occur in the footnotes. Within each article Roman and Greek superscripts run alphabetically, with re-runs if necessary. This system, although perhaps clumsy at first sight, allows an easy com• parison of the first and fifth editions, showing how Laplace's views changed over the years, and providing in essence translations of both the first and last versions of the Essai that were published during Laplace's lifetime. Indications are frequently given here of textual passages from the first edition of the Theone Analytique des Probabilites that were incorporated into the Essai: I have seldom, however, pointed out where the verbal pas• sages from the latter are symbolized and elaborated in the former. It is fairly safe to assume that explanations and mathematical details may be found in the third edition of the Theone. Laplace used few symbols in the Essai. Believing, however, that the ubiquitous man in the street is seldom advantaged by the omission of sym• bols, and that a thought expressed symbolically rather than literally is often more easily absorbed, I have introduced some more symbolism into the text, and formulae will often be found in the Notes. Note on the translation xi

Writing in The London Magazine in November 1781, Boswell said

But surely good Learning is a valuable acquisition, and ancient Learning, as it has come down to us through such a long succes• sion of ages, must make us wiser, and better, and happier. The dross has been left behind, and only what is pure and precious has been preserved till now. It is a favourite mode with many in this age, to separate Knowledge from Learning, and to hold, that we may have all the substance of ancient attainments by means of translation, without understanding the languages in which they are contained. But although I am not able clearly to explain it, I am fully convinced, and every man who has a relish of Greek or Latin, or of one of them, will agree with me, that the science must be very dry indeed which can be equally well communicated to the mind through the medium of transla• tion, as by the direct expression of its author. This is true, even as to literary compositions, in modern languages, but has much greater force when applied to those writings which were finished with the utmost care and nicety, in times when Language was in a far higher state of perfection.

Making suitable changes, one must concur with the sentiment expressed here. No translation is ever as good as the original, and the serious reader will find that the trouble spent in studying the original Essai will be amply rewarded. A Glossary, in which definitions of some of the terms appearing in the translation, as well as some short biographical notes on persons mentioned in the text, may be found, is provided at the end of the book. Let this Note be concluded, as it was begun, with a quotation from Donahue's Johannes Kepler, New Astronomy:

It is my fervent hope that the light my efforts shed upon this dif• ficult work will outweigh any errors in translation that remain. [1992, p. 16] ACKNOWLEDGMENTS

As is often stated in the payment of such recognition (and sometimes, as indeed on this occasion, even truthfully so), it is a pleasure to acknowledge the help of those who, however indirectly it might have been, have had a hand in the preparation of this work. Some financial support during the translating was provided by the Foun• dation for Research Development. Part of this support was used during a period of leave, and part was devoted to the obtaining of photocopies of documents that are rare or (at least in this country) inaccessible; I am grateful to the librarians who made such obtainment possible. Many of the references in the Notes were identified and collected while I was on sabbatical leave in the Department of Statistics of the University of Chicago in 1988. The hospitality of that department in providing me with an office and access to the University's excellent libraries has contributed greatly to whatever merit there may be in the Notes. I am grateful to the following for granting permission for quotation from the works mentioned: Academic Press, from R. A. Epstein's The Theory of Gambling and Statistical Logic of 1967; Ayer Company Publishers, Inc., from their 1981 reprint of the 1838 edition of A. de Morgan's An Essay on Probabilities; the Biometrika Trustees, from K. Pearson's article pub• lished in Biometrika vol. 21 in 1929; Cambridge University Press, from (i) Johannes Kepler New Astronomy, translated by William H. Donahue, @Cambridge University Press 1992, (ii) The Taming of Chance, by I. Hack• ing, @Cambridge University Press 1990, (iii) The Correspondence of Isaac Newton, vol. II, edited by H. W. Thrnbull, Notes & Translation @The Royal Society of London 1960 & published by the Syndics of the Cam• bridge University Press, and (iv) Anecdotes of Samuel Johnson, by H. L. Piozzi (ed. S. C. Roberts), @Cambridge University Press 1932: these ex• tracts are reprinted with the permission of Cambridge University Press; Chelsea Publishing Co., from C. Jordan's Calculus of Finite Differences (3rd edition), 1965; J .M. Dent & Sons Ltd, from F. Galton's Inquiries into Human Faculty and its Development, published by Everyman's Library in 1907; Dover Publications, Inc., from (i) C.B. Boyer's The History of the Calculus and its Conceptual Development of 1959, and (ii) W. Burnside's Theory of Probability of 1959; Edward Arnold, from K. Pearson's The His• tory of Statistics in the 17th and 18th Centuries of 1978; Encyclopredia xiii xiv Acknowledgments

Britannica, Inc., for a passage reprinted with permission from "French Law and Institutions" in the Encyclopt£dia Britannica, 14th edition, @1939 by Encyclopredia Britannica, Inc.; Harmony Books, Crown Publishers Inc., from C. Mackay's Extraordinary Popular Delusions and the Madness of Crowds of 1980, copyright @by Andrew Tobias; extracts from (i) Seneca, X Naturales Quaestiones II, translated by T.H. Corcoran (1972), (ii) Ci• cero, De Senectute, De Amicitia, De Divinatione Book II, translated by W.A. Falconer (1959), and (iii) Pliny, Naturalis Historiae I, translated by H. Rackham (1958), these extracts being reprinted by permission of the publishers and the Loeb Classical Library, Cambridge, Mass.: Harvard University Press; Hodder & Stoughton Limited, from A.W.F. Edwards's Pascal's Arithmetical Triangle of 1987; an extract from W. Feller's An In• troduction to Probability Theory and Its Applications vol. 1 (third edition), copyright @(1968) by John Wiley & Sons, Inc. and reprinted by permis• sion of John Wiley & Sons, Inc.; Johns Hopkins University Press, from E. Halley's Degrees of Mortality of Mankind, (ed. L. J. Reed) 1942; Macmillan Publishing Company, from (i) O.B. Sheynin: "Kepler, Johannes," from the International Encyclopedia of Statistics, edited by William H. Kruskal and Judith M. Tanur, vol. 1, p. 847, copyright @1978 by The Free Press, a Division of Macmillan, Inc., (ii) J.M. Keynes's Essays in Biography @The Royal Econometric Society 1972, (iii) J.M.Keynes's A Treatise on Proba• bility @The Royal Econometric Society 1973, and (iv) F.R. Moulton's An Introduction to Astronomy @1906 and 1916, by the Macmillan Company; McGraw-Hill, from M. H. DeGroot's Optimal Statistical Decisions, 1970; Methuen & Co., from (i) F.J. Rahtz's Higher English (14th edition) of 1921, and (ii) E. Gibbon (ed. J.B. Burg) The History of the Decline and Fall of the Roman Empire, vol. 1, of 1896; Open Court Publishing Company, from (i) F. Cajori's A History of Mathematical Notations, vol. 2 of 1929, and D. E. Smith's edition of A. de Morgan's A Budget of Paradoxes, (2nd edition) of 1915; Penguin Books Limited, from pp. 150-152 of B. Pascal's Pensees, translated with an introduction by A.J. Krailsheimer (Penguin Classics, 1966) copyright @A.J. Krailsheimer, 1966; Random House, Inc., from Lin Yutang's edition of The Wisdom of Confucius, 1943; Reed Book Services Ltd, from P. Moore, G. Hunt, I. Nicolson & P. Cattermole's The Atlas of the Solar System, published by Mitchell Beazley Ltd in 1983; D. Reidel, from G.J. Szekely's Paradoxes in Probability Theory and Mathematical Statistics, 1986, reprinted by permission of Kluwer Academic Publishers; Routledge, from Francis Bacon's Novum Organum of 1905; Springer-Verlag, Heidel• berg and (i) C.A. Wilson, from the latter's article printed in the Archive for History of Exact Sciences vol. 33 in 1985, and (ii) O.B. Sheynin, from the latter's article in the Archive for History of Exact Sciences vol. 39 in 1989; Springer-Verlag New York, Inc., from E.B. McBride's Obtaining Gen• erating Functions of 1971; The Econometric Society, from the translation of D. Bernoulli's "Specimen theoriae novae de mensura sortis" published in Econometrica vol. 22 by L. Sommer in 1954; The MIT Press, from D. Acknowledgments xv

C. Dennett's The Intentional Stance, 1987; The Science Press, from H. Poincare's The Foundations of Science of 1946; the University of Califor• nia Press, from Florian Cajori's edition of Isaac Newton's Mathematical Principles of Natural Philosophy and his System of the World, as edited & translated by Andrew Motte, copyright @1934, renewed 1962 Regents of the University of California; the University of Toronto Press, from W.W. Rouse Ball & H.S.M. Coxeter's Mathematical Recreations and Essays, 12th edition, 1974; A.P. Watt Ltd, on behalf of The Trustees of the Robert Graves Copyright Trust, from Suetonius: The Twelve Caesars, 1957, trans• lated by Robert Graves and published by Penguin Books Ltd. I am indebted to my colleague Steven B. Jackson for the translations from the Latin works of Fermat, Kepler and Tycho Brahe that appear in the Notes. Ivor Grattan-Guinness and Sandy Zabell have read parts of the manuscript and offered helpful suggestions: the presence of a Glossary is at their urging, though any inadequacies in it, or in the translation, should be laid at my doorstep. This tribute would be incomplete without mention of my indebtedness to Jackie de Gaye and Linda Hauptfleisch for their typing of early versions of the translation, and to the editorial staff of Springer-Verlag, New York, for their assistance.

A. I. DALE Durban, Natal April, 1994 CONTENTS

TRANSLATOR'S PREFACE ... vii NOTE ON THE TRANSLATION ix ACKNOWLEDGMENTS ...... xiii

Philosophical Essay on Probabilities

On probability ...... 2 General principles of the probability calculus . 6 On expectation ...... 11 On analytical methods in the probability calculus. . 14

Applications of the probability calculus . 33 On games of chance ...... 33 On unknown inequalities that may exist between supposedly equal chances ...... 34 On laws of probability resulting from the indefinite repetition of events ...... 36 Application of the probability calculus to natural philosophy ...... 43 Application of the probability calculus to the moral sciences ...... 61 On the means of the results of a large number of observations ...... 62 On the probability of testimony . . . . . 65 On elections and decisions of assemblies 74 On the probability of judicial decisions ...... 77 On tables of mortality and the mean duration of life, marriages and associations in general . . . . 81 On the benefits of institutions that depend on the probability of events ...... 86 On illusions in the estimation of probabilities 91 xvii xviii Contents

On various approaches to certainty . 112 Historical note on the probability calculus 117

NOTES ...... · 125 APPENDIX: EDITIONS OF THE ESSAI . · 225 BIBLIOGRAPHY · 227 GLOSSARY ... · 253