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A Philosophical Essay on Probabilities

A Philosophical Essay on Probabilities
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Title: A Philosophical Essay on Probabilities
Release Date: 2019-02-13
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Transcriber's Note:

Apparent typographical errors have been corrected.

The corrections noted in the Errata section have been incorporated in thetext. Three further corrections have also been made:9/10 in place of 1/10, and 9/10 in place of 6/10, on page 110;and "ex voto" in place of "ex veto" on page 173.

A PHILOSOPHICAL ESSAY
ON
PROBABILITIES.

BY
PIERRE SIMON, Marquis de LAPLACE.

TRANSLATED FROM THE SIXTH FRENCH EDITION
BY

FREDERICK WILSON TRUSCOTT, Ph.D. (Harv.),
Professor of Germanic Languages in the West Virginia University,

AND

FREDERICK LINCOLN EMORY, M.E. (Wor. Poly. Inst.),
Professor of Mechanics and Applied Mathematics in the West Virginia
University; Mem. Amer. Soc. Mech. Eng.

FIRST EDITION.
FIRST THOUSAND.

NEW YORK:
JOHN WILEY & SONS.
London: CHAPMAN & HALL, Limited.
1902.

Copyright, 1902,
BY
F. W. TRUSCOTT
AND
F. L. EMORY.

ROBERT DRUMMOND PRINTER, NEW YORK


TABLE OF CONTENTS.

PAGE
PART I.
A PHILOSOPHICAL ESSAY ON PROBABILITIES.
CHAPTER I.
Introduction 1
CHAPTER II.
Concerning Probability 3
CHAPTER III.
General Principles of the Calculus of Probabilities 11
CHAPTER IV.
Concerning Hope 20
CHAPTER V.
Analytical Methods of the Calculus of Probabilities 26
PART II.
APPLICATION OF THE CALCULUS OF PROBABILITIES.
CHAPTER VI.
Games of Chance 53
CHAPTER VII.
Concerning the Unknown Inequalities which may Exist among Chances Supposed to be Equal 56
CHAPTER VIII.
Concerning the Laws of Probability which result from the Indefinite Multiplication of Events 60
CHAPTER IX.
Application of the Calculus of Probabilities to Natural Philosophy 73
CHAPTER X.
Application of the Calculus of Probabilities to the Moral Sciences 107
CHAPTER XI.
Concerning the Probability of Testimonies 109
CHAPTER XII.
Concerning the Selections and Decisions of Assemblies 126
CHAPTER XIII.
Concerning the Probability of Testimonies 132
CHAPTER XIV.
Concerning Tables of Mortality, and the Mean Durations of Life, Marriage, and Some Associations 140
CHAPTER XV.
Concerning the Benefits of Institutions which Depend upon the Probability of Events 149
CHAPTER XVI.
Concerning Illusions in the Estimation of Probabilities 160
CHAPTER XVII.
Concerning the Various Means of Approaching Certainty 176
CHAPTER XVIII.
Historical Notice of the Calculus of Probabilities to 1816 185

ERRATA.

Page 89, line 22, for Pline read Pliny
" 102, lines 14, 16, " minutes " days
" 143, line 25, " sun " soil
" 177, lines 15, 17, 18, 21, 22, 24,  for  primary read prime
" 182, line 5, for conjunctions read being binary

PART I.
A PHILOSOPHICAL ESSAY ON PROBABILITIES.

CHAPTER I.
INTRODUCTION.

This philosophical essay is the development of alecture on probabilities which I delivered in 1795 tothe normal schools whither I had been called, by adecree of the national convention, as professor ofmathematics with Lagrange. I have recently publishedupon the same subject a work entitled The AnalyticalTheory of Probabilities. I present here without theaid of analysis the principles and general results of thistheory, applying them to the most important questionsof life, which are indeed for the most part only problemsof probability. Strictly speaking it may even be saidthat nearly all our knowledge is problematical; and inthe small number of things which we are able to knowwith certainty, even in the mathematical sciencesthemselves, the principal means for ascertaining truth—inductionand analogy—are based on probabilities;{2}so that the entire system of human knowledge is connectedwith the theory set forth in this essay. Doubtlessit will be seen here with interest that in considering,even in the eternal principles of reason, justice, andhumanity, only the favorable chances which are constantlyattached to them, there is a great advantage infollowing these principles and serious inconvenience indeparting from them: their chances, like those favorableto lotteries, always end by prevailing in the midstof the vacillations of hazard. I hope that the reflectionsgiven in this essay may merit the attention ofphilosophers and direct it to a subject so worthy ofengaging their minds.

CHAPTER II.
CONCERNING PROBABILITY.

All events, even those which on account of theirinsignificance do not seem to follow the great laws ofnature, are a result of it just as necessarily as the revolutionsof the sun. In ignorance of the ties which unitesuch events to the entire system of the universe, theyhave been made to depend upon final causes or uponhazard, according as they occur and are repeated withregularity, or appear without regard to order; but theseimaginary causes have gradually receded with thewidening bounds of knowledge and disappear entirelybefore sound philosophy, which sees in them only theexpression of our ignorance of the true causes.

Present events are connected with preceding onesby a tie based upon the evident principle that a thingcannot occur without a cause which produces it. Thisaxiom, known by the name of the principle of sufficientreason, extends even to actions which are consideredindifferent; the freest will is unable without a determinativemotive to give them birth; if we assume twopositions with exactly similar circumstances and findthat the will is active in the one and inactive in the{4}other, we say that its choice is an effect without a cause.It is then, says Leibnitz, the blind chance of theEpicureans. The contrary opinion is an illusion of themind, which, losing sight of the evasive reasons of thechoice of the will in indifferent things, believes thatchoice is determined of itself and without motives.

We ought then to regard the present state of theuniverse as the effect of its anterior state and as thecause of the one which is to follow. Given for oneinstant an intelligence which could comprehend all theforces by which nature is animated and the respectivesituation of the beings who compose it—an intelligencesufficiently vast to submit these data to analysis—itwould embrace in the same formula the movements ofthe greatest bodies of the universe and those of thelightest atom; for it, nothing would be uncertain andthe future, as the past, would be present to its eyes.The human mind offers, in the perfection which it hasbeen able to give to astronomy, a feeble idea of this intelligence.Its discoveries in mechanics and geometry,added to that of universal gravity, have enabled it tocomprehend in the same analytical expressions thepast and future states of the system of the world.Applying the same method to some other objects of itsknowledge, it has succeeded in referring to general lawsobserved phenomena and in foreseeing those whichgiven circumstances ought to produce. All these effortsin the search for truth tend to lead it back continuallyto the vast intelligence which we have just mentioned,but from which it will always remain infinitely removed.This tendency, peculiar to the human race, is thatwhich renders it superior to animals; and their progress{5}in this respect distinguishes nations and ages and constitutestheir true glory.

Let us recall that formerly, and at no remote epoch,an unusual rain or an extreme drought, a comet havingin train a very long tail, the eclipses, the auroraborealis, and in general all the unusual phenomenawere regarded as so many signs of celestial wrath.Heaven was invoked in order to avert their banefulinfluence. No one prayed to have the planets and thesun arrested in their courses: observation had soonmade apparent the futility of such prayers. But asthese phenomena, occurring and disappearing at longintervals, seemed to oppose the order of nature, it wassupposed that Heaven, irritated by the crimes of theearth, had created them to announce its vengeance.Thus the long tail of the comet of 1456 spread terrorthrough Europe, already thrown into consternation bythe rapid successes of the Turks, who had just overthrownthe Lower Empire. This star after four revolutionshas excited among us a very different interest.The knowledge of the laws of the system of the worldacquired in the interval had dissipated the fearsbegotten by the ignorance of the true relationship ofman to the universe; and Halley, having recognizedthe identity of this comet with those of the years 1531,1607, and 1682, announced its next return for the endof the year 1758 or the beginning of the year 1759.The learned world awaited with impatience this returnwhich was to confirm one of the greatest discoveriesthat have been made in the sciences, and fulfil theprediction of Seneca when he said, in speaking of therevolutions of those stars which fall from an enormous{6}height: "The day will come when, by study pursuedthrough several ages, the things now concealed willappear with evidence; and posterity will be astonishedthat truths so clear had escaped us." Clairaut thenundertook to submit to analysis the perturbations whichthe comet had experienced by the action of the twogreat planets, Jupiter and Saturn; after immense calculationshe fixed its next passage at the periheliontoward the beginning of April, 1759, which was actuallyverified by observation. The regularity which astronomyshows us in the movements of the comets doubtlessexists also in all phenomena.

The curve described by a simple molecule of air orvapor is regulated in a manner just as certain as theplanetary orbits; the only difference between them isthat which comes from our ignorance.

Probability is relative, in part to this ignorance, inpart to our knowledge. We know that of three or agreater number of events a single one ought to occur;but nothing induces us to believe that one of them willoccur rather than the others. In this state of indecisionit is impossible for us to announce their occurrence withcertainty. It is, however, probable that one of theseevents, chosen at will, will not occur because we seeseveral cases equally possible which exclude its occurrence,while only a single one favors it.

The theory of chance consists in reducing all theevents of the same kind to a certain number of casesequally possible, that is to say, to such as we may beequally undecided about in regard to their existence,and in determining the number of cases favorable tothe event whose probability is sought. The ratio of{7}this number to that of all the cases possible is themeasure of this probability, which is thus simply afraction whose numerator is the number of favorablecases and whose denominator is the number of all thecases possible.

The preceding notion of probability supposes that,in increasing in the same ratio the number of favorablecases and that of all the cases possible, the probabilityremains the same. In order to convince ourselves letus take two urns, A and B, the first containing fourwhite and two black balls, and the second containingonly two white balls and one black one. We mayimagine the two black balls of the first urn attached bya thread which breaks at the moment when one ofthem is seized in order to be drawn out, and the fourwhite balls thus forming two similar

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