{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_58881/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_58881","id":"gutenberg_58881","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/58881","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Probabilities"],"text":"Produced by Turgut Dincer, Chris Pinfield, and the Online\nDistributed Proofreading Team at http://www.pgdp.net (This\nfile was produced from images generously made available\nby The Internet Archive)\n\n\n\n\n\n\n\n\n\nTranscriber's Note:\n\nApparent typographical errors have been corrected.\n\nItalics are indicated by _underscores_. Small capitals have been\nreplaced by full capitals.\n\nThe corrections noted in the Errata section have been incorporated in\nthe text. Two further corrections have also been made: 9/10 in place of\n1/10, and 9/10 in place of 6/10, on page 110; and \"ex voto\" in place of\n\"ex veto\" on page 173.\n\n\n\n\n A PHILOSOPHICAL ESSAY\n ON\n PROBABILITIES.\n\n BY\n PIERRE SIMON, MARQUIS DE LAPLACE.\n\n\n _TRANSLATED FROM THE SIXTH FRENCH EDITION_\n BY\n\n FREDERICK WILSON TRUSCOTT, PH.D. (HARV.),\n _Professor of Germanic Languages in the West Virginia University_,\n\n AND\n\n FREDERICK LINCOLN EMORY, M.E. (WOR. POLY. INST.),\n _Professor of Mechanics and Applied Mathematics in the West Virginia\n University; Mem. Amer. Soc. Mech. Eng._\n\n\n _FIRST EDITION._\n FIRST THOUSAND.\n\n\n NEW YORK:\n JOHN WILEY & SONS.\n LONDON: CHAPMAN & HALL, LIMITED.\n 1902.\n\n\n Copyright, 1902,\n BY\n F. W. TRUSCOTT\n AND\n F. L. EMORY.\n\n\n ROBERT DRUMMOND PRINTER, NEW YORK\n\n\n\n\nTABLE OF CONTENTS.\n\n\n                                                                    PAGE\n PART I.\n _A PHILOSOPHICAL ESSAY ON PROBABILITIES._\n\n CHAPTER I.\n Introduction                                                          1\n\n CHAPTER II.\n Concerning Probability                                                3\n\n CHAPTER III.\n General Principles of the Calculus of Probabilities                  11\n\n CHAPTER IV.\n Concerning Hope                                                      20\n\n CHAPTER V.\n Analytical Methods of the Calculus of Probabilities                  26\n\n\n PART II.\n _APPLICATION OF THE CALCULUS OF PROBABILITIES._\n\n CHAPTER VI.\n Games of Chance                                                      53\n\n CHAPTER VII.\n Concerning the Unknown Inequalities which may Exist among\n Chances Supposed to be Equal                                         56\n\n CHAPTER VIII.\n Concerning the Laws of Probability which result from the\n Indefinite Multiplication of Events                                  60\n\n CHAPTER IX.\n Application of the Calculus of Probabilities to Natural\n Philosophy                                                           73\n\n CHAPTER X.\n Application of the Calculus of Probabilities to the Moral\n Sciences                                                            107\n\n CHAPTER XI.\n Concerning the Probability of Testimonies                           109\n\n CHAPTER XII.\n Concerning the Selections and Decisions of Assemblies               126\n\n CHAPTER XIII.\n Concerning the Probability of the Judgments of Tribunals            132\n\n CHAPTER XIV.\n Concerning Tables of Mortality, and the Mean Durations\n of Life, Marriage, and Some Associations                            140\n\n CHAPTER XV.\n Concerning the Benefits of Institutions which Depend upon\n the Probability of Events                                           149\n\n CHAPTER XVI.\n Concerning Illusions in the Estimation of Probabilities             160\n\n CHAPTER XVII.\n Concerning the Various Means of Approaching Certainty               176\n\n CHAPTER XVIII.\n Historical Notice of the Calculus of Probabilities to 1816          185\n\n\n\n\nERRATA.\n\n\n Page  89, line 22,      _for_  Pline         _read_  Pliny\n\n   \"  102, lines 14, 16,   \"    minutes         \"     days\n\n   \"  143, line 25,        \"    sun             \"     soil\n\n   \"  177, lines 15, 17, 18, 21, 22, 24, _for_ primary _read_ prime\n\n   \"  182, line 5,       _for_  conjunctions  _read_  being binary\n\n\n\n\nA PHILOSOPHICAL ESSAY ON PROBABILITIES.\n\n\n\n\nCHAPTER I.\n\n_INTRODUCTION._\n\n\nThis philosophical essay is the development of a lecture on\nprobabilities which I delivered in 1795 to the normal schools whither I\nhad been called, by a decree of the national convention, as professor of\nmathematics with Lagrange. I have recently published upon the same\nsubject a work entitled _The Analytical Theory of Probabilities_. I\npresent here without the aid of analysis the principles and general\nresults of this theory, applying them to the most important questions of\nlife, which are indeed for the most part only problems of probability.\nStrictly speaking it may even be said that nearly all our knowledge is\nproblematical; and in the small number of things which we are able to\nknow with certainty, even in the mathematical sciences themselves, the\nprincipal means for ascertaining truth\u2014induction and analogy\u2014are based\non probabilities; so that the entire system of human knowledge is\nconnected with the theory set forth in this essay. Doubtless it will be\nseen here with interest that in considering, even in the eternal\nprinciples of reason, justice, and humanity, only the favorable chances\nwhich are constantly attached to them, there is a great advantage in\nfollowing these principles and serious inconvenience in departing from\nthem: their chances, like those favorable to lotteries, always end by\nprevailing in the midst of the vacillations of hazard. I hope that the\nreflections given in this essay may merit the attention of philosophers\nand direct it to a subject so worthy of engaging their minds.\n\n\n\n\nCHAPTER II.\n\n_CONCERNING PROBABILITY._\n\n\nAll events, even those which on account of their insignificance do not\nseem to follow the great laws of nature, are a result of it just as\nnecessarily as the revolutions of the sun. In ignorance of the ties\nwhich unite such events to the entire system of the universe, they have\nbeen made to depend upon final causes or upon hazard, according as they\noccur and are repeated with regularity, or appear without regard to\norder; but these imaginary causes have gradually receded with the\nwidening bounds of knowledge and disappear entirely before sound\nphilosophy, which sees in them only the expression of our ignorance of\nthe true causes.\n\nPresent events are connected with preceding ones by a tie based upon the\nevident principle that a thing cannot occur without a cause which\nproduces it. This axiom, known by the name of _the principle of\nsufficient reason_, extends even to actions which are considered\nindifferent; the freest will is unable without a determinative motive to\ngive them birth; if we assume two positions with exactly similar\ncircumstances and find that the will is active in the one and inactive\nin the other, we say that its choice is an effect without a cause. It is\nthen, says Leibnitz, the blind chance of the Epicureans. The contrary\nopinion is an illusion of the mind, which, losing sight of the evasive\nreasons of the choice of the will in indifferent things, believes that\nchoice is determined of itself and without motives.\n\nWe ought then to regard the present state of the universe as the effect\nof its anterior state and as the cause of the one which is to follow.\nGiven for one instant an intelligence which could comprehend all the\nforces by which nature is animated and the respective situation of the\nbeings who compose it\u2014an intelligence sufficiently vast to submit these\ndata to analysis\u2014it would embrace in the same formula the movements of\nthe greatest bodies of the universe and those of the lightest atom; for\nit, nothing would be uncertain and the future, as the past, would be\npresent to its eyes. The human mind offers, in the perfection which it\nhas been able to give to astronomy, a feeble idea of this intelligence.\nIts discoveries in mechanics and geometry, added to that of universal\ngravity, have enabled it to comprehend in the same analytical\nexpressions the past and future states of the system of the world.\nApplying the same method to some other objects of its knowledge, it has\nsucceeded in referring to general laws observed phenomena and in\nforeseeing those which given circumstances ought to produce. All these\nefforts in the search for truth tend to lead it back continually to the\nvast intelligence which we have just mentioned, but from which it will\nalways remain infinitely removed. This tendency, peculiar to the human\nrace, is that which renders it superior to animals; and their progress\nin this respect distinguishes nations and ages and constitutes their\ntrue glory.\n\nLet us recall that formerly, and at no remote epoch, an unusual rain or\nan extreme drought, a comet having in train a very long tail, the\neclipses, the aurora borealis, and in general all the unusual phenomena\nwere regarded as so many signs of celestial wrath. Heaven was invoked in\norder to avert their baneful influence. No one prayed to have the\nplanets and the sun arrested in their courses: observation had soon made\napparent the futility of such prayers. But as these phenomena, occurring\nand disappearing at long intervals, seemed to oppose the order of\nnature, it was supposed that Heaven, irritated by the crimes of the\nearth, had created them to announce its vengeance. Thus the long tail of\nthe comet of 1456 spread terror through Europe, already thrown into\nconsternation by the rapid successes of the Turks, who had just\noverthrown the Lower Empire. This star after four revolutions has\nexcited among us a very different interest. The knowledge of the laws of\nthe system of the world acquired in the interval had dissipated the\nfears begotten by the ignorance of the true relationship of man to the\nuniverse; and Halley, having recognized the identity of this comet with\nthose of the years 1531, 1607, and 1682, announced its next return for\nthe end of the year 1758 or the beginning of the year 1759. The learned\nworld awaited with impatience this return which was to confirm one of\nthe greatest discoveries that have been made in the sciences, and fulfil\nthe prediction of Seneca when he said, in speaking of the revolutions of\nthose stars which fall from an enormous height: \"The day will come when,\nby study pursued through several ages, the things now concealed will\nappear with evidence; and posterity will be astonished that truths so\nclear had escaped us.\" Clairaut then undertook to submit to analysis the\nperturbations which the comet had experienced by the action of the two\ngreat planets, Jupiter and Saturn; after immense calculations he fixed\nits next passage at the perihelion toward the beginning of April, 1759,\nwhich was actually verified by observation. The regularity which\nastronomy shows us in the movements of the comets doubtless exists also\nin all phenomena.\n\nThe curve described by a simple molecule of air or vapor is regulated in\na manner just as certain as the planetary orbits; the only difference\nbetween them is that which comes from our ignorance.\n\nProbability is relative, in part to this ignorance, in part to our\nknowledge. We know that of three or a greater number of events a single\none ought to occur; but nothing induces us to believe that one of them\nwill occur rather than the others. In this state of indecision it is\nimpossible for us to announce their occurrence with certainty. It is,\nhowever, probable that one of these events, chosen at will, will not\noccur because we see several cases equally possible which exclude its\noccurrence, while only a single one favors it.\n\nThe theory of chance consists in reducing all the events of the same\nkind to a certain number of cases equally possible, that is to say, to\nsuch as we may be equally undecided about in regard to their existence,\nand in determining the number of cases favorable to the event whose\nprobability is sought. The ratio of this number to that of all the cases\npossible is the measure of this probability, which is thus simply a\nfraction whose numerator is the number of favorable cases and whose\ndenominator is the number of all the cases possible.\n\nThe preceding notion of probability supposes that, in increasing in the\nsame ratio the number of favorable cases and that of all the cases\npossible, the probability remains the same. In order to convince\nourselves let us take two urns, A and B, the first containing four white\nand two black balls, and the second containing only two white balls and\none black one. We may imagine the two black balls of the first urn\nattached by a thread which breaks at the moment when one of them is\nseized in order to be drawn out, and the four white balls thus forming\ntwo similar systems. All the chances which will favor the seizure of one\nof the balls of the black system will lead to a black ball. If we\nconceive now that the threads which unite the balls do not break at all,\nit is clear that the number of possible chances will not change any more\nthan that of the chances favorable to the extraction of the black balls;\nbut two balls will be drawn from the urn at the same time; the\nprobability of drawing a black ball from the urn A will then be the same\nas at first. But then we have obviously the case of urn B with the\nsingle difference that the three balls of this last urn would be\nreplaced by three systems of two balls invariably connected.\n\nWhen all the cases are favorable to an event the probability changes to\ncertainty and its expression becomes equal to unity. Upon this\ncondition, certainty and probability are comparable, although there may\nbe an essential difference between the two states of the mind when a\ntruth is rigorously demonstrated to it, or when it still perceives a\nsmall source of error.\n\nIn things which are only probable the difference of the data, which each\nman has in regard to them, is one of the principal causes of the\ndiversity of opinions which prevail in regard to the same objects. Let\nus suppose, for example, that we have three urns, A, B, C, one of which\ncontains only black balls while the two others contain only white balls;\na ball is to be drawn from the urn C and the probability is demanded\nthat this ball will be black. If we do not know which of the three urns\ncontains black balls only, so that there is no reason to believe that it\nis C rather than B or A, these three hypotheses will appear equally\npossible, and since a black ball can be drawn only in the first\nhypothesis, the probability of drawing it is equal to one third. If it\nis known that the urn A contains white balls only, the indecision then\nextends only to the urns B and C, and the probability that the ball\ndrawn from the urn C will be black is one half. Finally this probability\nchanges to certainty if we are assured that the urns A and B contain\nwhite balls only.\n\nIt is thus that an incident related to a numerous assembly finds various\ndegrees of credence, according to the extent of knowledge of the\nauditors. If the man who reports it is fully convinced of it and if, by\nhis position and character, he inspires great confidence, his statement,\nhowever extraordinary it may be, will have for the auditors who lack\ninformation the same degree of probability as an ordinary statement made\nby the same man, and they will have entire faith in it. But if some one\nof them knows that the same incident is rejected by other equally\ntrustworthy men, he will be in doubt and the incident will be\ndiscredited by the enlightened auditors, who will reject it whether it\nbe in regard to facts well averred or the immutable laws of nature.\n\nIt is to the influence of the opinion of those whom the multitude judges\nbest informed and to whom it has been accustomed to give its confidence\nin regard to the most important matters of life that the propagation of\nthose errors is due which in times of ignorance have covered the face of\nthe earth. Magic and astrology offer us two great examples. These errors\ninculcated in infancy, adopted without examination, and having for a\nbasis only universal credence, have maintained themselves during a very\nlong time; but at last the progress of science has destroyed them in the\nminds of enlightened men, whose opinion consequently has caused them to\ndisappear even among the common people, through the power of imitation\nand habit which had so generally spread them abroad. This power, the\nrichest resource of the moral world, establishes and conserves in a\nwhole nation ideas entirely contrary to those which it upholds elsewhere\nwith the same authority. What indulgence ought we not then to have for\nopinions different from ours, when this difference often depends only\nupon the various points of view where circumstances have placed us! Let\nus enlighten those whom we judge insufficiently instructed; but first\nlet us examine critically our own opinions and weigh with impartiality\ntheir respective probabilities.\n\nThe difference of opinions depends, however, upon the manner in which\nthe influence of known data is determined. The theory of probabilities\nholds to considerations so delicate that it is not surprising that with\nthe same data two persons arrive at different results, especially in\nvery complicated questions. Let us examine now the general principles of\nthis theory.\n\n\n\n\nCHAPTER III.\n\n_THE GENERAL PRINCIPLES OF THE CALCULUS OF PROBABILITIES._\n\n\n_First Principle._\u2014The first of these principles is the definition\nitself of probability, which, as has been seen, is the ratio of the\nnumber of favorable cases to that of all the cases possible.\n\n_Second Principle._\u2014But that supposes the various cases equally\npossible. If they are not so, we will determine first their respective\npossibilities, whose exact appreciation is one of the most delicate\npoints of the theory of chance. Then the probability will be the sum of\nthe possibilities of each favorable case. Let us illustrate this\nprinciple by an example.\n\nLet us suppose that we throw into the air a large and very thin coin\nwhose two large opposite faces, which we will call heads and tails, are\nperfectly similar. Let us find the probability of throwing heads at\nleast one time in two throws. It is clear that four equally possible\ncases may arise, namely, heads at the first and at the second throw;\nheads at the first throw and tails at the second; tails at the first\nthrow and heads at the second; finally, tails at both throws. The first\nthree cases are favorable to the event whose probability is sought;\nconsequently this probability is equal to \u00be; so that it is a bet of\nthree to one that heads will be thrown at least once in two throws.\n\nWe can count at this game only three different cases, namely, heads at\nthe first throw, which dispenses with throwing a second time; tails at\nthe first throw and heads at the second; finally, tails at the first and\nat the second throw. This would reduce the probability to \u2154 if we should\nconsider with d'Alembert these three cases as equally possible. But it\nis apparent that the probability of throwing heads at the first throw is\n\u00bd, while that of the two other cases is \u00bc, the first case being a simple\nevent which corresponds to two events combined: heads at the first and\nat the second throw, and heads at the first throw, tails at the second.\nIf we then, conforming to the second principle, add the possibility \u00bd of\nheads at the first throw to the possibility \u00bc of tails at the first\nthrow and heads at the second, we shall have \u00be for the probability\nsought, which agrees with what is found in the supposition when we play\nthe two throws. This supposition does not change at all the chance of\nthat one who bets on this event; it simply serves to reduce the various\ncases to the cases equally possible.\n\n_Third Principle._\u2014One of the most important points of the theory of\nprobabilities and that which lends the most to illusions is the manner\nin which these probabilities increase or diminish by their mutual\ncombination. If the events are independent of one another, the\nprobability of their combined existence is the product of their\nrespective probabilities. Thus the probability of throwing one ace with\na single die is \u2159; that of throwing two aces in throwing two dice at the\nsame time is 1/36. Each face of the one being able to combine with the\nsix faces of the other, there are in fact thirty-six equally possible\ncases, among which one single case gives two aces. Generally the\nprobability that a simple event in the same circumstances will occur\nconsecutively a given number of times is equal to the probability of\nthis simple event raised to the power indicated by this number. Having\nthus the successive powers of a fraction less than unity diminishing\nwithout ceasing, an event which depends upon a series of very great\nprobabilities may become extremely improbable. Suppose then an incident\nbe transmitted to us by twenty witnesses in such manner that the first\nhas transmitted it to the second, the second to the third, and so on.\nSuppose again that the probability of each testimony be equal to the\nfraction 9/10; that of the incident resulting from the testimonies will\nbe less than \u215b. We cannot better compare this diminution of the\nprobability than with the extinction of the light of objects by the\ninterposition of several pieces of glass. A relatively small number of\npieces suffices to take away the view of an object that a single piece\nallows us to perceive in a distinct manner. The historians do not appear\nto have paid sufficient attention to this degradation of the probability\nof events when seen across a great number of successive generations;\nmany historical events reputed as certain would be at least doubtful if\nthey were submitted to this test.\n\nIn the purely mathematical sciences the most distant consequences\nparticipate in the certainty of the principle from which they are\nderived. In the applications of analysis to physics the results have all\nthe certainty of facts or experiences. But in the moral sciences, where\neach inference is deduced from that which precedes it only in a probable\nmanner, however probable these deductions may be, the chance of error\nincreases with their number and ultimately surpasses the chance of truth\nin the consequences very remote from the principle.\n\n_Fourth Principle._\u2014When two events depend upon each other, the\nprobability of the compound event is the product of the probability of\nthe first event and the probability that, this event having occurred,\nthe second will occur. Thus in the preceding case of the three urns A,\nB, C, of which two contain only white balls and one contains only black\nballs, the probability of drawing a white ball from the urn C is \u2154,\nsince of the three urns only two contain balls of that color. But when a\nwhite ball has been drawn from the urn C, the indecision relative to\nthat one of the urns which contain only black balls extends only to the\nurns A and B; the probability of drawing a white ball from the urn B is\n\u00bd; the product of \u2154 by \u00bd, or \u2153, is then the probability of drawing two\nwhite balls at one time from the urns B and C.\n\nWe see by this example the influence of past events upon the probability\nof future events. For the probability of drawing a white ball from the\nurn B, which primarily is \u2154, becomes \u00bd when a white ball has been drawn\nfrom the urn C; it would change to certainty if a black ball had been\ndrawn from the same urn. We will determine this influence by means of\nthe following principle, which is a corollary of the preceding one.\n\n_Fifth Principle._\u2014If we calculate _\u00e0 priori_ the probability of the\noccurred event and the probability of an event composed of that one and\na second one which is expected, the second probability divided by the\nfirst will be the probability of the event expected, drawn from the\nobserved event.\n\nHere is presented the question raised by some philosophers touching the\ninfluence of the past upon the probability of the future. Let us suppose\nat the play of heads and tails that heads has occurred oftener than\ntails. By this alone we shall be led to believe that in the constitution\nof the coin there is a secret cause which favors it. Thus in the conduct\nof life constant happiness is a proof of competency which should induce\nus to employ preferably happy persons. But if by the unreliability of\ncircumstances we are constantly brought back to a state of absolute\nindecision, if, for example, we change the coin at each throw at the\nplay of heads and tails, the past can shed no light upon the future and\nit would be absurd to take account of it.\n\n_Sixth Principle._\u2014Each of the causes to which an observed event may be\nattributed is indicated with just as much likelihood as there is\nprobability that the event will take place, supposing the event to be\nconstant. The probability of the existence of any one of these causes is\nthen a fraction whose numerator is the probability of the event\nresulting from this cause and whose denominator is the sum of the\nsimilar probabilities relative to all the causes; if these various\ncauses, considered _\u00e0 priori_, are unequally probable, it is necessary,\nin place of the probability of the event resulting from each cause, to\nemploy the product of this probability by the possibility of the cause\nitself. This is the fundamental principle of this branch of the analysis\nof chances which consists in passing from events to causes.\n\nThis principle gives the reason why we attribute regular events to a\nparticular cause. Some philosophers have thought that these events are\nless possible than others and that at the play of heads and tails, for\nexample, the combination in which heads occurs twenty successive times\nis less easy in its nature than those where heads and tails are mixed in\nan irregular manner. But this opinion supposes that past events have an\ninfluence on the possibility of future events, which is not at all\nadmissible. The regular combinations occur more rarely only because they\nare less numerous. If we seek a cause wherever we perceive symmetry, it\nis not that we regard a symmetrical event as less possible than the\nothers, but, since this event ought to be the effect of a regular cause\nor that of chance, the first of these suppositions is more probable than\nthe second. On a table we see letters arranged in this order,\n\n _C o n s t a n t i n o p l e_,\n\nand we judge that this arrangement is not the result of chance, not\nbecause it is less possible than the others, for if this word were not\nemployed in any language we should not suspect it came from any\nparticular cause, but this word being in use among us, it is\nincomparably more probable that some person has thus arranged the\naforesaid letters than that this arrangement is due to chance.\n\nThis is the place to define the word _extraordinary_. We arrange in our\nthought all possible events in various classes; and we regard as\n_extraordinary_ those classes which include a very small number. Thus at\nthe play of heads and tails the occurrence of heads a hundred successive\ntimes appears to us extraordinary because of the almost infinite number\nof combinations which may occur in a hundred throws; and if we divide\nthe combinations into regular series containing an order easy to\ncomprehend, and into irregular series, the latter are incomparably more\nnumerous. The drawing of a white ball from an urn which among a million\nballs contains only one of this color, the others being black, would\nappear to us likewise extraordinary, because we form only two classes of\nevents relative to the two colors. But the drawing of the number 475813,\nfor example, from an urn that contains a million numbers seems to us an\nordinary event; because, comparing individually the numbers with one\nanother without dividing them into classes, we have no reason to believe\nthat one of them will appear sooner than the others.\n\nFrom what precedes, we ought generally to conclude that the more\nextraordinary the event, the greater the need of its being supported by\nstrong proofs. For those who attest it, being able to deceive or to have\nbeen deceived, these two causes are as much more probable as the reality\nof the event is less. We shall see this particularly when we come to\nspeak of the probability of testimony.\n\n_Seventh Principle._\u2014The probability of a future event is the sum of the\nproducts of the probability of each cause, drawn from the event\nobserved, by the probability that, this cause existing, the future event\nwill occur. The following example will illustrate this principle.\n\nLet us imagine an urn which contains only two balls, each of which may\nbe either white or black. One of these balls is drawn and is put back\ninto the urn before proceeding to a new draw. Suppose that in the first\ntwo draws white balls have been drawn; the probability of again drawing\na white ball at the third draw is required.\n\nOnly two hypotheses can be made here: either one of the balls is white\nand the other black, or both are white. In the first hypothesis the\nprobability of the event observed is \u00bc; it is unity or certainty in the\nsecond. Thus in regarding these hypotheses as so many causes, we shall\nhave for the sixth principle \u2155 and \u2158 for their respective probabilities.\nBut if the first hypothesis occurs, the probability of drawing a white\nball at the third draw is \u00bd; it is equal to certainty in the second\nhypothesis; multiplying then the last probabilities by those of the\ncorresponding hypotheses, the sum of the products, or 9/10, will be the\nprobability of drawing a white ball at the third draw.\n\nWhen the probability of a single event is unknown we may suppose it\nequal to any value from zero to unity. The probability of each of these\nhypotheses, drawn from the event observed, is, by the sixth principle, a\nfraction whose numerator is the probability of the event in this\nhypothesis and whose denominator is the sum of the similar probabilities\nrelative to all the hypotheses. Thus the probability that the\npossibility of the event is comprised within given limits is the sum of\nthe fractions comprised within these limits. Now if we multiply each\nfraction by the probability of the future event, determined in the\ncorresponding hypothesis, the sum of the products relative to all the\nhypotheses will be, by the seventh principle, the probability of the\nfuture event drawn from the event observed. Thus we find that an event\nhaving occurred successively any number of times, the probability that\nit will happen again the next time is equal to this number increased by\nunity divided by the same number, increased by two units. Placing the\nmost ancient epoch of history at five thousand years ago, or at 1826213\ndays, and the sun having risen constantly in the interval at each\nrevolution of twenty-four hours, it is a bet of 1826214 to one that it\nwill rise again to-morrow. But this number is incomparably greater for\nhim who, recognizing in the totality of phenomena the principal\nregulator of days and seasons, sees that nothing at the present moment\ncan arrest the course of it.\n\nBuffon in his _Political Arithmetic_ calculates differently the\npreceding probability. He supposes that it differs from unity only by a\nfraction whose numerator is unity and whose denominator is the number 2\nraised to a power equal to the number of days which have elapsed since\nthe epoch. But the true manner of relating past events with the\nprobability of causes and of future events was unknown to this\nillustrious writer.\n\n\n\n\nCHAPTER IV.\n\n_CONCERNING HOPE._\n\n\nThe probability of events serves to determine the hope or the fear of\npersons interested in their existence. The word _hope_ has various\nacceptations; it expresses generally the advantage of that one who\nexpects a certain benefit in suppositions which are only probable. This\nadvantage in the theory of chance is a product of the sum hoped for by\nthe probability of obtaining it; it is the partial sum which ought to\nresult when we do not wish to run the risks of the event in supposing\nthat the division is made proportional to the probabilities. This\ndivision is the only equitable one when all strange circumstances are\neliminated; because an equal degree of probability gives an equal right\nto the sum hoped for. We will call this advantage _mathematical hope_.\n\n_Eighth Principle._\u2014When the advantage depends on several events it is\nobtained by taking the sum of the products of the probability of each\nevent by the benefit attached to its occurrence.\n\nLet us apply this principle to some examples. Let us suppose that at the\nplay of heads and tails Paul receives two francs if he throws heads at\nthe first throw and five francs if he throws it only at the second.\nMultiplying two francs by the probability \u00bd of the first case, and five\nfrancs by the probability \u00bc of the second case, the sum of the products,\nor two and a quarter francs, will be Paul's advantage. It is the sum\nwhich he ought to give in advance to that one who has given him this\nadvantage; for, in order to maintain the equality of the play, the throw\nought to be equal to the advantage which it procures.\n\nIf Paul receives two francs by throwing heads at the first and five\nfrancs by throwing it at the second throw, whether he has thrown it or\nnot at the first, the probability of throwing heads at the second throw\nbeing \u00bd, multiplying two francs and five francs by \u00bd the sum of these\nproducts will give three and one half francs for Paul's advantage and\nconsequently for his stake at the game.\n\n_Ninth Principle._\u2014In a series of probable events of which the ones\nproduce a benefit and the others a loss, we shall have the advantage\nwhich results from it by making a sum of the products of the probability\nof each favorable event by the benefit which it procures, and\nsubtracting from this sum that of the products of the probability of\neach unfavorable event by the loss which is attached to it. If the\nsecond sum is greater than the first, the benefit becomes a loss and\nhope is changed to fear.\n\nConsequently we ought always in the conduct of life to make the product\nof the benefit hoped for, by its probability, at least equal to the\nsimilar product relative to the loss. But it is necessary, in order to\nattain this, to appreciate exactly the advantages, the losses, and their\nrespective probabilities. For this a great accuracy of mind, a delicate\njudgment, and a great experience in affairs is necessary; it is\nnecessary to know how to guard one's self against prejudices, illusions\nof fear or hope, and erroneous ideas, ideas of fortune and happiness,\nwith which the majority of people feed their self-love.\n\nThe application of the preceding principles to the following question\nhas greatly exercised the geometricians. Paul plays at heads and tails\nwith the condition of receiving two francs if he throws heads at the\nfirst throw, four francs if he throws it only at the second throw, eight\nfrancs if he throws it only at the third, and so on. His stake at the\nplay ought to be, according to the eighth principle, equal to the number\nof throws, so that if the game continues to infinity the stake ought to\nbe infinite. However, no reasonable man would wish to risk at this game\neven a small sum, for example five francs. Whence comes this difference\nbetween the result of calculation and the indication of common sense? We\nsoon recognize that it amounts to this: that the moral advantage which a\nbenefit procures for us is not proportional to this benefit and that it\ndepends upon a thousand circumstances, often very difficult to define,\nbut of which the most general and most important is that of fortune.\n\nIndeed it is apparent that one franc has much greater value for him who\npossesses only a hundred than for a millionaire. We ought then to\ndistinguish in the hoped-for benefit its absolute from its relative\nvalue. But the latter is regulated by the motives which make it\ndesirable, whereas the first is independent of them. The general\nprinciple for appreciating this relative value cannot be given, but here\nis one proposed by Daniel Bernoulli which will serve in many cases.\n\n_Tenth Principle._\u2014The relative value of an infinitely small sum is\nequal to its absolute value divided by the total benefit of the person\ninterested. This supposes that every one has a certain benefit whose\nvalue can never be estimated as zero. Indeed even that one who possesses\nnothing always gives to the product of his labor and to his hopes a\nvalue at least equal to that which is absolutely necessary to sustain\nhim.\n\nIf we apply analysis to the principle just propounded, we obtain the\nfollowing rule: Let us designate by unity the part of the fortune of an\nindividual, independent of his expectations. If we determine the\ndifferent values that this fortune may have by virtue of these\nexpectations and their probabilities, the product of these values raised\nrespectively to the powers indicated by their probabilities will be the\nphysical fortune which would procure for the individual the same moral\nadvantage which he receives from the part of his fortune taken as unity\nand from his expectations; by subtracting unity from the product, the\ndifference will be the increase of the physical fortune due to\nexpectations: we will call this increase _moral hope_. It is easy to see\nthat it coincides with mathematical hope when the fortune taken as unity\nbecomes infinite in reference to the variations which it receives from\nthe expectations. But when these variations are an appreciable part of\nthis unity the two hopes may differ very materially among themselves.\n\nThis rule conduces to results conformable to the indications of common\nsense which can by this means be appreciated with some exactitude. Thus\nin the preceding question it is found that if the fortune of Paul is two\nhundred francs, he ought not reasonably to stake more than nine francs.\nThe same rule leads us again to distribute the danger over several parts\nof a benefit expected rather than to expose the entire benefit to this\ndanger. It results similarly that at the fairest game the loss is always\ngreater than the gain. Let us suppose, for example, that a player having\na fortune of one hundred francs risks fifty at the play of heads and\ntails; his fortune after his stake at the play will be reduced to\neighty-seven francs, that is to say, this last sum would procure for the\nplayer the same moral advantage as the state of his fortune after the\nstake. The play is then disadvantageous even in the case where the stake\nis equal to the product of the sum hoped for, by its probability. We can\njudge by this of the immorality of games in which the sum hoped for is\nbelow this product. They subsist only by false reasonings and by the\ncupidity which they excite and which, leading the people to sacrifice\ntheir necessaries to chimerical hopes whose improbability they are not\nin condition to appreciate, are the source of an infinity of evils.\n\nThe disadvantage of games of chance, the advantage of not exposing to\nthe same danger the whole benefit that is expected, and all the similar\nresults indicated by common sense, subsist, whatever may be the function\nof the physical fortune which for each individual expresses his moral\nfortune. It is enough that the proportion of the increase of this\nfunction to the increase of the physical fortune diminishes in the\nmeasure that the latter increases.\n\n\n\n\nCHAPTER V.\n\n_CONCERNING THE ANALYTICAL METHODS OF THE CALCULUS OF PROBABILITIES._\n\n\nThe application of the principle which we have just expounded to the\nvarious questions of probability requires methods whose investigation\nhas given birth to several methods of analysis and especially to the\ntheory of combinations and to the calculus of finite differences.\n\nIf we form the product of the binomials, unity plus the first letter,\nunity plus the second letter, unity plus the third letter, and so on up\nto _n_ letters, and subtract unity from this developed product, the\nresult will be the sum of the combination of all these letters taken one\nby one, two by two, three by three, etc., each combination having unity\nfor a coefficient. In order to have the number of combinations of these\n_n_ letters taken _s_ by _s_ times, we shall observe that if we suppose\nthese letters equal among themselves, the preceding product will become\nthe _n_th power of the binomial one plus the first letter; thus the\nnumber of combinations of _n_ letters taken _s_ by _s_ times will be the\ncoefficient of the _s_th power of the first letter in the development in\nthis binomial; and this number is obtained by means of the known\nbinomial formula.\n\nAttention must be paid to the respective situations of the letters in\neach combination, observing that if a second letter is joined to the\nfirst it may be placed in the first or second position which gives two\ncombinations. If we join to these combinations a third letter, we can\ngive it in each combination the first, the second, and the third rank\nwhich forms three combinations relative to each of the two others, in\nall six combinations. From this it is easy to conclude that the number\nof arrangements of which _s_ letters are susceptible is the product of\nthe numbers from unity to _s_. In order to pay regard to the respective\npositions of the letters it is necessary then to multiply by this\nproduct the number of combinations of _n_ letters _s_ by _s_ times,\nwhich is tantamount to taking away the denominator of the coefficient of\nthe binomial which expresses this number.\n\nLet us imagine a lottery composed of _n_ numbers, of which _r_ are drawn\nat each draw. The probability is demanded of the drawing of _s_ given\nnumbers in one draw. To arrive at this let us form a fraction whose\ndenominator will be the number of all the cases possible or of the\ncombinations of _n_ letters taken _r_ by _r_ times, and whose numerator\nwill be the number of all the combinations which contain the given _s_\nnumbers. This last number is evidently that of the combinations of the\nother numbers taken _n_ less _s_ by _n_ less _s_ times. This fraction\nwill be the required probability, and we shall easily find that it can\nbe reduced to a fraction whose numerator is the number of combinations\nof _r_ numbers taken _s_ by _s_ times, and whose denominator is the\nnumber of combinations of _n_ numbers taken similarly _s_ by _s_ times.\nThus in the lottery of France, formed as is known of 90 numbers of which\nfive are drawn at each draw, the probability of drawing a given\ncombination is 5/90, or 1/18; the lottery ought then for the equality of\nthe play to give eighteen times the stake. The total number of\ncombinations two by two of the 90 numbers is 4005, and that of the\ncombinations two by two of 5 numbers is 10. The probability of the\ndrawing of a given pair is then 1/4005, and the lottery ought to give\nfour hundred and a half times the stake; it ought to give 11748 times\nfor a given tray, 511038 times for a quaternary, and 43949268 times for\na quint. The lottery is far from giving the player these advantages.\n\nSuppose in an urn _a_ white balls, _b_ black balls, and after having\ndrawn a ball it is put back into the urn; the probability is asked that\nin _n_ number of draws _m_ white balls and _n_ - _m_ black balls will be\ndrawn. It is clear that the number of cases that may occur at each\ndrawing is _a_ + _b_. Each case of the second drawing being able to\ncombine with all the cases of the first, the number of possible cases in\ntwo drawings is the square of the binomial _a_ + _b_. In the development\nof this square, the square of a expresses the number of cases in which a\nwhite ball is twice drawn, the double product of _a_ by _b_ expresses\nthe number of cases in which a white ball and a black ball are drawn.\nFinally, the square of _b_ expresses the number of cases in which two\nblack balls are drawn. Continuing thus, we see generally that the _n_th\npower of the binomial _a_ + _b_ expresses the number of all the cases\npossible in _n_ draws; and that in the development of this power the\nterm multiplied by the _m_th power of _a_ expresses the number of cases\nin which _m_ white balls and _n_ - _m_ black balls may be drawn.\nDividing then this term by the entire power of the binomial, we shall\nhave the probability of drawing _m_ white balls and _n_ - _m_ black\nballs. The ratio of the numbers _a_ and _a_ + _b_ being the probability\nof drawing one white ball at one draw; and the ratio of the numbers _b_\nand _a_ + _b_ being the probability of drawing one black ball; if we\ncall these probabilities _p_ and _q_, the probability of drawing _m_\nwhite balls in _n_ draws will be the term multiplied by the _m_th power\nof _p_ in the development of the _n_th power of the binomial _p_ + _q_;\nwe may see that the sum _p_ + _q_ is unity. This remarkable property of\nthe binomial is very useful in the theory of probabilities. But the most\ngeneral and direct method of resolving questions of probability consists\nin making them depend upon equations of differences. Comparing the\nsuccessive conditions of the function which expresses the probability\nwhen we increase the variables by their respective differences, the\nproposed question often furnishes a very simple proportion between the\nconditions. This proportion is what is called _equation of ordinary or\npartial differentials_; _ordinary_ when there is only one variable,\n_partial_ when there are several. Let us consider some examples of this.\n\nThree players of supposed equal ability play together on the following\nconditions: that one of the first two players who beats his adversary\nplays the third, and if he beats him the game is finished. If he is\nbeaten, the victor plays against the second until one of the players has\ndefeated consecutively the two others, which ends the game. The\nprobability is demanded that the game will be finished in a certain\nnumber _n_ of plays. Let us find the probability that it will end\nprecisely at the _n_th play. For that the player who wins ought to enter\nthe game at the play _n_ - 1 and win it thus at the following play. But\nif in place of winning the play _n_ - 1 he should be beaten by his\nadversary who had just beaten the other player, the game would end at\nthis play. Thus the probability that one of the players will enter the\ngame at the play _n_ - 1 and will win it is equal to the probability\nthat the game will end precisely with this play; and as this player\nought to win the following play in order that the game may be finished\nat the _n_th play, the probability of this last case will be only one\nhalf of the preceding one. This probability is evidently a function of\nthe number _n_; this function is then equal to the half of the same\nfunction when _n_ is diminished by unity. This equality forms one of\nthose equations called _ordinary finite differential equations_.\n\nWe may easily determine by its use the probability that the game will\nend precisely at a certain play. It is evident that the play cannot end\nsooner than at the second play; and for this it is necessary that that\none of the first two players who has beaten his adversary should beat at\nthe second play the third player; the probability that the game will end\nat this play is \u00bd. Hence by virtue of the preceding equation we conclude\nthat the successive probabilities of the end of the game are \u00bc for the\nthird play, \u215b for the fourth play, and so on; and in general \u00bd raised to\nthe power _n_ - 1 for the _n_th play. The sum of all these powers of \u00bd\nis unity less the last of these powers; it is the probability that the\ngame will end at the latest in _n_ plays.\n\nLet us consider again the first problem more difficult which may be\nsolved by probabilities and which Pascal proposed to Fermat to solve.\nTwo players, A and B, of equal skill play together on the conditions\nthat the one who first shall beat the other a given number of times\nshall win the game and shall take the sum of the stakes at the game;\nafter some throws the players agree to quit without having finished the\ngame: we ask in what manner the sum ought to be divided between them. It\nis evident that the parts ought to be proportional to the respective\nprobabilities of winning the game. The question is reduced then to the\ndetermination of these probabilities. They depend evidently upon the\nnumber of points which each player lacks of having attained the given\nnumber. Hence the probability of A is a function of the two numbers\nwhich we will call _indices_. If the two players should agree to play\none throw more (an agreement which does not change their condition,\nprovided that after this new throw the division is always made\nproportionally to the new probabilities of winning the game), then\neither A would win this throw and in that case the number of points\nwhich he lacks would be diminished by unity, or the player B would win\nit and in that case the number of points lacking to this last player\nwould be less by unity. But the probability of each of these cases is \u00bd;\nthe function sought is then equal to one half of this function in which\nwe diminish by unity the first index plus the half of the same function\nin which the second variable is diminished by unity. This equality is\none of those equations called _equations of partial differentials_.\n\nWe are able to determine by its use the probabilities of A by dividing\nthe smallest numbers, and by observing that the probability or the\nfunction which expresses it is equal to unity when the player A does not\nlack a single point, or when the first index is zero, and that this\nfunction becomes zero with the second index. Supposing thus that the\nplayer A lacks only one point, we find that his probability is \u00bd, \u00be, \u215e,\netc., according as B lacks one point, two, three, etc. Generally it is\nthen unity less the power of \u00bd, equal to the number of points which B\nlacks. We will suppose then that the player A lacks two points and his\nprobability will be found equal to \u00bc, \u00bd, 11/16, etc., according as B\nlacks one point, two points, three points, etc. We will suppose again\nthat the player A lacks three points, and so on.\n\nThis manner of obtaining the successive values of a quantity by means of\nits equation of differences is long and laborious. The geometricians\nhave sought methods to obtain the general function of indices that\nsatisfies this equation, so that for any particular case we need only to\nsubstitute in this function the corresponding values of the indices. Let\nus consider this subject in a general way. For this purpose let us\nconceive a series of terms arranged along a horizontal line so that each\nof them is derived from the preceding one according to a given law. Let\nus suppose this law expressed by an equation among several consecutive\nterms and their index, or the number which indicates the rank that they\noccupy in the series. This equation I call the _equation of finite\ndifferences by a single index_. The order or the degree of this equation\nis the difference of rank of its two extreme terms. We are able by its\nuse to determine successively the terms of the series and to continue it\nindefinitely; but for that it is necessary to know a number of terms of\nthe series equal to the degree of the equation. These terms are the\narbitrary constants of the expression of the general term of the series\nor of the integral of the equation of differences.\n\nLet us imagine now below the terms of the preceding series a second\nseries of terms arranged horizontally; let us imagine again below the\nterms of the second series a third horizontal series, and so on to\ninfinity; and let us suppose the terms of all these series connected by\na general equation among several consecutive terms, taken as much in the\nhorizontal as in the vertical sense, and the numbers which indicate\ntheir rank in the two senses. This equation is called the _equation of\npartial finite differences by two indices_.\n\nLet us imagine in the same way below the plan of the preceding series a\nsecond plan of similar series, whose terms should be placed respectively\nbelow those of the first plan; let us imagine again below this second\nplan a third plan of similar series, and so on to infinity; let us\nsuppose all the terms of these series connected by an equation among\nseveral consecutive terms taken in the sense of length, width, and\ndepth, and the three numbers which indicate their rank in these three\nsenses. This equation I call the _equation of partial finite differences\nby three indices_.\n\nFinally, considering the matter in an abstract way and independently of\nthe dimensions of space, let us imagine generally a system of\nmagnitudes, which should be functions of a certain number of indices,\nand let us suppose among these magnitudes, their relative differences to\nthese indices and the indices themselves, as many equations as there are\nmagnitudes; these equations will be partial finite differences by a\ncertain number of indices.\n\nWe are able by their use to determine successively these magnitudes. But\nin the same manner as the equation by a single index requires for it\nthat we know a certain number of terms of the series, so the equation by\ntwo indices requires that we know one or several lines of series whose\ngeneral terms should be expressed each by an arbitrary function of one\nof the indices. Similarly the equation by three indices requires that we\nknow one or several plans of series, the general terms of which should\nbe expressed each by an arbitrary function of two indices, and so on. In\nall these cases we shall be able by successive eliminations to determine\na certain term of the series. But all the equations among which we\neliminate being comprised in the same system of equations, all the\nexpressions of the successive terms which we obtain by these\neliminations ought to be comprised in one general expression, a function\nof the indices which determine the rank of the term. This expression is\nthe integral of the proposed equation of differences, and the search for\nit is the object of integral calculus.\n\nTaylor is the first who in his work entitled _Metodus incrementorum_ has\nconsidered linear equations of finite differences. He gives the manner\nof integrating those of the first order with a coefficient and a last\nterm, functions of the index. In truth the relations of the terms of the\narithmetical and geometrical progressions which have always been taken\ninto consideration are the simplest cases of linear equations of\ndifferences; but they had not been considered from this point of view.\nIt was one of those which, attaching themselves to general theories,\nlead to these theories and are consequently veritable discoveries.\n\nAbout the same time Moivre was considering under the name of recurring\nseries the equations of finite differences of a certain order having a\nconstant coefficient. He succeeded in integrating them in a very\ningenious manner. As it is always interesting to follow the progress of\ninventors, I shall expound the method of Moivre by applying it to a\nrecurring series whose relation among three consecutive terms is given.\nFirst he considers the relation among the consecutive terms of a\ngeometrical progression or the equation of two terms which expresses it.\nReferring it to terms less than unity, he multiplies it in this state by\na constant factor and subtracts the product from the first equation.\nThus he obtains an equation among three consecutive terms of the\ngeometrical progression. Moivre considers next a second progression\nwhose ratio of terms is the same factor which he has just used. He\ndiminishes similarly by unity the index of the terms of the equation of\nthis new progression. In this condition he multiplies it by the ratio of\nthe terms of the first progression, and he subtracts the product from\nthe equation of the second progression, which gives him among three\nconsecutive terms of this progression a relation entirely similar to\nthat which he has found for the first progression. Then he observes that\nif one adds term by term the two progressions, the same ratio exists\namong any three of these consecutive terms. He compares the coefficients\nof this ratio to those of the relation of the terms of the proposed\nrecurrent series, and he finds for determining the ratios of the two\ngeometrical progressions an equation of the second degree, whose roots\nare these ratios. Thus Moivre decomposes the recurrent series into two\ngeometrical progressions, each multiplied by an arbitrary constant which\nhe determines by means of the first two terms of the recurrent series.\nThis ingenious process is in fact the one that d'Alembert has since\nemployed for the integration of linear equations of infinitely small\ndifferences with constant coefficients, and Lagrange has transformed\ninto similar equations of finite differences.\n\nFinally, I have considered the linear equations of partial finite\ndifferences, first under the name of _recurro-recurrent_ series and\nafterwards under their own name. The most general and simplest manner of\nintegrating all these equations appears to me that which I have based\nupon the consideration of discriminant functions, the idea of which is\nhere given.\n\nIf we conceive a function _V_ of a variable _t_ developed according to\nthe powers of this variable, the coefficient of any one of these powers\nwill be a function of the exponent or index of this power, which index I\nshall call _x_. _V_ is what I call the discriminant function of this\ncoefficient or of the function of the index.\n\nNow if we multiply the series of the development of _V_ by a function of\nthe same variable, such, for example, as unity plus two times this\nvariable, the product will be a new discriminant function in which the\ncoefficient of the power _x_ of the variable _t_ will be equal to the\ncoefficient of the same power in _V_ plus twice the coefficient of the\npower less unity. Thus the function of the index _x_ in the product will\nbe equal to the function of the index _x_ in _V_ plus twice the same\nfunction in which the index is diminished by unity. This function of the\nindex _x_ is thus a derivative of the function of the same index in the\ndevelopment of _V_, a function which I shall call the _primitive\nfunction_ of the index. Let us designate the derivative function by the\nletter Alembert placed before the primitive function. The derivation\nindicated by this letter will depend upon the multiplier of _V_, which\nwe will call _T_ and which we will suppose developed like _V_ by the\nratio to the powers of the variable _t_. If we multiply anew by _T_ the\nproduct of _V_ by _T_, which is equivalent to multiplying _V_ by _T\u00b2_,\nwe shall form a third discriminant function, in which the coefficient of\nthe _x_th power of _t_ will be a derivative similar to the corresponding\ncoefficient of the preceding product; it may be expressed by the same\ncharacter _\u03b4_ placed before the preceding derivative, and then this\ncharacter will be written twice before the primitive function of _x_.\nBut in place of writing it thus twice we give it 2 for an exponent.\n\nContinuing thus, we see generally that if we multiply _V_ by the _n_th\npower of _T_, we shall have the coefficient of the _x_th power of _t_ in\nthe product of _V_ by the _n_th power of _T_ by placing before the\nprimitive function the character _\u03b4_ with _n_ for an exponent.\n\nLet us suppose, for example, that _T_ be unity divided by _t_; then in\nthe product of _V_ by _T_ the coefficient of the _x_th power of _t_ will\nbe the coefficient of the power greater by unity in _V_; this\ncoefficient in the product of _V_ by the _n_th power of _T_ will then be\nthe primitive function in which _x_ is augmented by _n_ units.\n\nLet us consider now a new function _Z_ of _t_, developed like _V_ and\n_T_ according to the powers of _t_; let us designate by the character\n_\u0394_ placed before the primitive function the coefficient of the _x_th\npower of _t_ in the product of _V_ by _Z_; this coefficient in the\nproduct of _V_ by the _n_th power of _Z_ will be expressed by the\ncharacter _\u0394_ affected by the exponent _n_ and placed before the\nprimitive function of _x_.\n\nIf, for example, _Z_ is equal to unity divided by _t_ less one, the\ncoefficient of the _x_th power of _t_ in the product of _V_ by _Z_ will\nbe the coefficient of the _x_ + 1 power of _t_ in _V_ less the\ncoefficient of the _x_th power. It will be then the finite difference of\nthe primitive function of the index _x_. Then the character _\u0394_\nindicates a finite difference of the primitive function in the case\nwhere the index varies by unity; and the _n_th power of this character\nplaced before the primitive function will indicate the finite _n_th\ndifference of this function. If we suppose that _T_ be unity divided by\n_t_, we shall have _T_ equal to the binomial _Z_ + 1. The product of _V_\nby the _n_th power of _T_ will then be equal to the product of _V_ by\nthe _n_th power of the binomial _Z_ + 1. Developing this power in the\nratio of the powers of _Z_, the product of _V_ by the various terms of\nthis development will be the discriminant functions of these same terms\nin which we substitute in place of the powers of _Z_ the corresponding\nfinite differences of the primitive function of the index.\n\nNow the product of _V_ by the _n_th power of _T_ is the primitive\nfunction in which the index _x_ is augmented by _n_ units; repassing\nfrom the discriminant functions to their coefficients, we shall have\nthis primitive function thus augmented equal to the development of the\n_n_th power of the binomial _Z_ + 1, provided that in this development\nwe substitute in place of the powers of _Z_ the corresponding\ndifferences of the primitive function and that we multiply the\nindependent term of these powers by the primitive function. We shall\nthus obtain the primitive function whose index is augmented by any\nnumber _n_ by means of its differences.\n\nSupposing that _T_ and _Z_ always have the preceding values, we shall\nhave _Z_ equal to the binomial _T_ - 1; the product of _V_ by the _n_th\npower of _Z_ will then be equal to the product of _V_ by the development\nof the _n_th power of the binomial _T_ - 1. Repassing from the\ndiscriminant functions to their coefficients as has just been done, we\nshall have the _n_th difference of the primitive function expressed by\nthe development of the _n_th power of the binomial _T_ - 1, in which we\nsubstitute for the powers of _T_ this same function whose index is\naugmented by the exponent of the power, and for the independent term of\n_t_, which is unity, the primitive function, which gives this difference\nby means of the consecutive terms of this function.\n\nPlacing _\u03b4_ before the primitive function expressing the derivative of\nthis function, which multiplies the _x_ power of _t_ in the product of\n_V_ by _T_, and _\u0394_ expressing the same derivative in the product of _V_\nby _Z_, we are led by that which precedes to this general result:\nwhatever may be the function of the variable _t_ represented by _T_ and\n_Z_, we may, in the development of all the identical equations\nsusceptible of being formed among these functions, substitute the\ncharacters _\u03b4_ and _\u0394_ in place of _T_ and _Z_, provided that we write\nthe primitive function of the index in series with the powers and with\nthe products of the powers of the characters, and that we multiply by\nthis function the independent terms of these characters.\n\nWe are able by means of this general result to transform any certain\npower of a difference of the primitive function of the index _x_, in\nwhich _x_ varies by unity, into a series of differences of the same\nfunction in which _x_ varies by a certain number of units and\nreciprocally. Let us suppose that _T_ be the _i_ power of unity divided\nby _t_ - 1, and that _Z_ be always unity divided by _t_ - 1; then the\ncoefficient of the _x_ power of _t_ in the product of _V_ by _T_ will be\nthe coefficient of the _x_ + _i_ power of _t_ in _V_ less the\ncoefficient of the _x_ power of _t_; it will then be the finite\ndifference of the primitive function of the index _x_ in which we vary\nthis index by the number _i_. It is easy to see that _T_ is equal to the\ndifference between the _i_ power of the binomial _Z_ + 1 and unity. The\n_n_th power of _T_ is equal to the _n_th power of this difference. If in\nthis equality we substitute in place of _T_ and _Z_ the characters _\u03b4_\nand _\u0394_, and after the development we place at the end of each term the\nprimitive function of the index _x_, we shall have the _n_th difference\nof this function in which _x_ varies by _i_ units expressed by a series\nof differences of the same function in which _x_ varies by unity. This\nseries is only a transformation of the difference which it expresses and\nwhich is identical with it; but it is in similar transformations that\nthe power of analysis resides.\n\nThe generality of analysis permits us to suppose in this expression that\n_n_ is negative. Then the negative powers of _\u03b4_ and _\u0394_ indicate the\nintegrals. Indeed the _n_th difference of the primitive function having\nfor a discriminant function the product of _V_ by the _n_th power of the\nbinomial one divided by _t_ less unity, the primitive function which is\nthe _n_th integral of this difference has for a discriminant function\nthat of the same difference multiplied by the _n_th power taken less\nthan the binomial one divided by _t_ minus one, a power to which the\nsame power of the character _\u0394_ corresponds; this power indicates then\nan integral of the same order, the index _x_ varying by unity; and the\nnegative powers of _\u03b4_ indicate equally the integrals _x_ varying by _i_\nunits. We see, thus, in the clearest and simplest manner the rationality\nof the analysis observed among the positive powers and differences, and\namong the negative powers and the integrals.\n\nIf the function indicated by _\u03b4_ placed before the primitive function is\nzero, we shall have an equation of finite differences, and _V_ will be\nthe discriminant function of its integral. In order to obtain this\ndiscriminant function we shall observe that in the product of _V_ by _T_\nall the powers of _t_ ought to disappear except the powers inferior to\nthe order of the equation of differences; _V_ is then equal to a\nfraction whose denominator is _T_ and whose numerator is a polynomial in\nwhich the highest power of _t_ is less by unity than the order of the\nequation of differences. The arbitrary coefficients of the various\npowers of _t_ in this polynomial, including the power zero, will be\ndetermined by as many values of the primitive function of the index when\nwe make successively _x_ equal to zero, to one, to two, etc. When the\nequation of differences is given we determine _T_ by putting all its\nterms in the first member and zero in the second; by substituting in the\nfirst member unity in place of the function which has the largest index;\nthe first power of _t_ in place of the primitive function in which this\nindex is diminished by unity; the second power of _t_ for the primitive\nfunction where this index is diminished by two units, and so on. The\ncoefficient of the _x_th power of _t_ in the development of the\npreceding expression of _V_ will be the primitive function of _x_ or the\nintegral of the equation of finite differences. Analysis furnishes for\nthis development various means, among which we may choose that one which\nis most suitable for the question proposed; this is an advantage of this\nmethod of integration.\n\nLet us conceive now that _V_ be a function of the two variables _t_ and\n_t\u00b4_ developed according to the powers and products of these variables;\nthe coefficient of any product of the powers _x_ and _x\u00b4_ of _t_ and\n_t\u00b4_ will be a function of the exponents or indices _x_ and _x\u00b4_ of\nthese powers; this function I shall call the _primitive function_ of\nwhich _V_ is the discriminant function.\n\nLet us multiply _V_ by a function _T_ of the two variables _t_ and _t\u00b4_\ndeveloped like _V_ in ratio of the powers and the products of these\nvariables; the product will be the discriminant function of a derivative\nof the primitive function; if _T_, for example, is equal to the variable\n_t_ plus the variable _t\u00b4_ minus two, this derivative will be the\nprimitive function of which we diminish by unity the index _x_ plus this\nsame primitive function of which we diminish by unity the index _x\u00b4_\nless two times the primitive function. Designating whatever _T_ may be\nby the character _\u03b4_ placed before the primitive function, this\nderivative, the product of _V_ by the _n_th power of _T_, will be the\ndiscriminant function of the derivative of the primitive function before\nwhich one places the _n_th power of the character _\u03b4_. Hence result the\ntheorems analogous to those which are relative to functions of a single\nvariable.\n\nSuppose the function indicated by the character _\u03b4_ be zero; one will\nhave an equation of partial differences. If, for example, we make as\nbefore _T_ equal to the variable _t_ plus the variable _t\u00b4_ - 2, we have\nzero equal to the primitive function of which we diminish by unity the\nindex _x_ plus the same function of which we diminish by unity the index\n_x\u00b4_ minus two times the primitive function. The discriminant function\n_V_ of the primitive function or of the integral of this equation ought\nthen to be such that its product by _T_ does not include at all the\nproducts of _t_ by _t\u00b4_; but _V_ may include separately the powers of\n_t_ and those of _t\u00b4_, that is to say, an arbitrary function of _t_ and\nan arbitrary function of _t\u00b4_; _V_ is then a fraction whose numerator is\nthe sum of these two arbitrary functions and whose denominator is _T_.\nThe coefficient of the product of the _x_th power of _t_ by the _x\u00b4_\npower of _t\u00b4_ in the development of this fraction will then be the\nintegral of the preceding equation of partial differences. This method\nof integrating this kind of equations seems to me the simplest and the\neasiest by the employment of the various analytical processes for the\ndevelopment of rational fractions.\n\nMore ample details in this matter would be scarcely understood without\nthe aid of calculus.\n\nConsidering equations of infinitely small partial differences as\nequations of finite partial differences in which nothing is neglected,\nwe are able to throw light upon the obscure points of their calculus,\nwhich have been the subject of great discussions among geometricians. It\nis thus that I have demonstrated the possibility of introducing\ndiscontinued functions in their integrals, provided that the\ndiscontinuity takes place only for the differentials of the order of\nthese equations or of a superior order. The transcendent results of\ncalculus are, like all the abstractions of the understanding, general\nsigns whose true meaning may be ascertained only by repassing by\nmetaphysical analysis to the elementary ideas which have led to them;\nthis often presents great difficulties, for the human mind tries still\nless to transport itself into the future than to retire within itself.\nThe comparison of infinitely small differences with finite differences\nis able similarly to shed great light upon the metaphysics of\ninfinitesimal calculus.\n\nIt is easily proven that the finite _n_th difference of a function in\nwhich the increase of the variable is _E_ being divided by the _n_th\npower of _E_, the quotient reduced in series by ratio to the powers of\nthe increase _E_ is formed by a first term independent of _E._ In the\nmeasure that _E_ diminishes, the series approaches more and more this\nfirst term from which it can differ only by quantities less than any\nassignable magnitude. This term is then the limit of the series and\nexpresses in differential calculus the infinitely small _n_th difference\nof the function divided by the _n_th power of the infinitely small\nincrease.\n\nConsidering from this point of view the infinitely small differences, we\nsee that the various operations of differential calculus amount to\ncomparing separately in the development of identical expressions the\nfinite terms or those independent of the increments of the variables\nwhich are regarded as infinitely small; this is rigorously exact, these\nincrements being indeterminant. Thus differential calculus has all the\nexactitude of other algebraic operations.\n\nThe same exactitude is found in the applications of differential\ncalculus to geometry and mechanics. If we imagine a curve cut by a\nsecant at two adjacent points, naming _E_ the interval of the ordinates\nof these two points, _E_ will be the increment of the abscissa from the\nfirst to the second ordinate. It is easy to see that the corresponding\nincrement of the ordinate will be the product of _E_ by the first\nordinate divided by its subsecant; augmenting then in this equation of\nthe curve the first ordinate by this increment, we shall have the\nequation relative to the second ordinate. The difference of these two\nequations will be a third equation which, developed by the ratio of the\npowers of _E_ and divided by _E_, will have its first term independent\nof _E_, which will be the limit of this development. This term, equal to\nzero, will give then the limit of the subsecants, a limit which is\nevidently the subtangent.\n\nThis singularly happy method of obtaining the subtangent is due to\nFermat, who has extended it to transcendent curves. This great\ngeometrician expresses by the character _E_ the increment of the\nabscissa; and considering only the first power of this increment, he\ndetermines exactly as we do by differential calculus the subtangents of\nthe curves, their points of inflection, the _maxima_ and _minima_ of\ntheir ordinates, and in general those of rational functions. We see\nlikewise by his beautiful solution of the problem of the refraction of\nlight inserted in the _Collection of the Letters of Descartes_ that he\nknows how to extend his methods to irrational functions in freeing them\nfrom irrationalities by the elevation of the roots to powers. Fermat\nshould be regarded, then, as the true discoverer of Differential\nCalculus. Newton has since rendered this calculus more analytical in his\n_Method of Fluxions_, and simplified and generalized the processes by\nhis beautiful theorem of the binomial. Finally, about the same time\nLeibnitz has enriched differential calculus by a notation which, by\nindicating the passage from the finite to the infinitely small, adds to\nthe advantage of expressing the general results of calculus that of\ngiving the first approximate values of the differences and of the sums\nof the quantities; this notation is adapted of itself to the calculus of\npartial differentials.\n\nWe are often led to expressions which contain so many terms and factors\nthat the numerical substitutions are impracticable. This takes place in\nquestions of probability when we consider a great number of events.\nMeanwhile it is necessary to have the numerical value of the formul\u00e6 in\norder to know with what probability the results are indicated, which the\nevents develop by multiplication. It is necessary especially to have the\nlaw according to which this probability continually approaches\ncertainty, which it will finally attain if the number of events were\ninfinite. In order to obtain this law I considered that the definite\nintegrals of differentials multiplied by the factors raised to great\npowers would give by integration the formul\u00e6 composed of a great number\nof terms and factors. This remark brought me to the idea of transforming\ninto similar integrals the complicated expressions of analysis and the\nintegrals of the equation of differences. I fulfilled this condition by\na method which gives at the same time the function comprised under the\nintegral sign and the limits of the integration. It offers this\nremarkable thing, that the function is the same discriminant function of\nthe expressions and the proposed equations; this attaches this method to\nthe theory of discriminant functions of which it is thus the complement.\nFurther, it would only be a question of reducing the definite integral\nto a converging series. This I have obtained by a process which makes\nthe series converge with as much more rapidity as the formula which it\nrepresents is more complicated, so that it is more exact as it becomes\nmore necessary. Frequently the series has for a factor the square root\nof the ratio of the circumference to the diameter; sometimes it depends\nupon other transcendents whose number is infinite.\n\nAn important remark which pertains to great generality of analysis, and\nwhich permits us to extend this method to formul\u00e6 and to equations of\ndifference which the theory of probability presents most frequently, is\nthat the series to which one comes by supposing the limits of the\ndefinite integrals to be real and positive take place equally in the\ncase where the equation which determines these limits has only negative\nor imaginary roots. These passages from the positive to the negative and\nfrom the real to the imaginary, of which I first have made use, have led\nme further to the values of many singular definite integrals, which I\nhave accordingly demonstrated directly. We may then consider these\npassages as a means of discovery parallel to induction and analogy long\nemployed by geometricians, at first with an extreme reserve, afterwards\nwith entire confidence, since a great number of examples has justified\nits use. In the mean time it is always necessary to confirm by direct\ndemonstrations the results obtained by these divers means.\n\nI have named the ensemble of the preceding methods the _Calculus of\nDiscriminant Functions_; this calculus serves as a basis for the work\nwhich I have published under the title of the _Analytical Theory of\nProbabilities_. It is connected with the simple idea of indicating the\nrepeated multiplications of a quantity by itself or its entire and\npositive powers by writing toward the top of the letter which expresses\nit the numbers which mark the degrees of these powers.\n\nThis notation, employed by Descartes in his _Geometry_ and generally\nadopted since the publication of this important work, is a little thing,\nespecially when compared with the theory of curves and variable\nfunctions by which this great geometrician has established the\nfoundations of modern calculus. But the language of analysis, most\nperfect of all, being in itself a powerful instrument of discoveries,\nits notations, especially when they are necessary and happily conceived,\nare so many germs of new calculi. This is rendered appreciable by this\nexample.\n\nWallis, who in his work entitled _Arithmetica Infinitorum_, one of those\nwhich have most contributed to the progress of analysis, has interested\nhimself especially in following the thread of induction and analogy,\nconsidered that if one divides the exponent of a letter by two, three,\netc., the quotient will be accordingly the Cartesian notation, and when\ndivision is possible the exponent of the square, cube, etc., root of the\nquantity which represents the letter raised to the dividend exponent.\nExtending by analogy this result to the case where division is\nimpossible, he considered a quantity raised to a fractional exponent as\nthe root of the degree indicated by the denominator of this\nfraction\u2014namely, of the quantity raised to a power indicated by the\nnumerator. He observed then that, according to the Cartesian notation,\nthe multiplication of two powers of the same letter amounts to adding\ntheir exponents, and that their division amounts to subtracting the\nexponents of the power of the divisor from that of the power of the\ndividend, when the second of these exponents is greater than the first.\nWallis extended this result to the case where the first exponent is\nequal to or greater than the second, which makes the difference zero or\nnegative. He supposed then that a negative exponent indicates unity\ndivided by the quantity raised to the same exponent taken positively.\nThese remarks led him to integrate generally the monomial differentials,\nwhence he inferred the definite integrals of a particular kind of\nbinomial differentials whose exponent is a positive integral number. The\nobservation then of the law of the numbers which express these\nintegrals, a series of interpolations and happy inductions where one\nperceives the germ of the calculus of definite integrals which has so\nmuch exercised geometricians and which is one of the fundaments of my\nnew _Theory of Probabilities_, gave him the ratio of the area of the\ncircle to the square of its diameter expressed by an infinite product,\nwhich, when one stops it, confines this ratio to limits more and more\nconverging; this is one of the most singular results in analysis. But it\nis remarkable that Wallis, who had so well considered the fractional\nexponents of radical powers, should have continued to note these powers\nas had been done before him. Newton in his _Letters to Oldembourg_, if I\nam not mistaken, was the first to employ the notation of these powers by\nfractional exponents. Comparing by the way of induction, of which Wallis\nhad made such a beautiful use, the exponents of the powers of the\nbinomial with the coefficients of the terms of its development in the\ncase where this exponent is integral and positive, he determined the law\nof these coefficients and extended it by analogy to fractional and\nnegative powers. These various results, based upon the notation of\nDescartes, show his influence on the progress of analysis. It has still\nthe advantage of giving the simplest and fairest idea of logarithms,\nwhich are indeed only the exponents of a magnitude whose successive\npowers, increasing by infinitely small degrees, can represent all\nnumbers.\n\nBut the most important extension that this notation has received is that\nof variable exponents, which constitutes exponential calculus, one of\nthe most fruitful branches of modern analysis. Leibnitz was the first to\nindicate the transcendents by variable exponents, and thereby he has\ncompleted the system of elements of which a finite function can be\ncomposed; for every finite explicit function of a variable may be\nreduced in the last analysis to simple magnitudes, combined by the\nmethod of addition, subtraction, multiplication, and division and raised\nto constant or variable powers. The roots of the equations formed from\nthese elements are the implicit functions of the variable. It is thus\nthat a variable has for a logarithm the exponent of the power which is\nequal to it in the series of the powers of the number whose hyperbolic\nlogarithm is unity, and the logarithm of a variable of it is an implicit\nfunction.\n\nLeibnitz thought to give to his differential character the same\nexponents as to magnitudes; but then in place of indicating the repeated\nmultiplications of the same magnitude these exponents indicate the\nrepeated differentiations of the same function. This new extension of\nthe Cartesian notation led Leibnitz to the analogy of positive powers\nwith the differentials, and the negative powers with the integrals.\nLagrange has followed this singular analogy in all its developments; and\nby series of inductions which may be regarded as one of the most\nbeautiful applications which have ever been made of the method of\ninduction he has arrived at general formul\u00e6 which are as curious as\nuseful on the transformations of differences and of integrals the ones\ninto the others when the variables have divers finite increments and\nwhen these increments are infinitely small. But he has not given the\ndemonstrations of it which appear to him difficult. The theory of\ndiscriminant functions extends the Cartesian notations to some of its\ncharacters; it shows with proof the analogy of the powers and operations\nindicated by these characters; so that it may still be regarded as the\nexponential calculus of characters. All that concerns the series and the\nintegration of equations of differences springs from it with an extreme\nfacility.\n\n\n\n\nPART II.\n\nAPPLICATIONS OF THE CALCULUS OF PROBABILITIES.\n\n\n\n\nCHAPTER VI.\n\n_GAMES OF CHANCE._\n\n\nThe combinations which games present were the object of the first\ninvestigations of probabilities. In an infinite variety of these\ncombinations many of them lend themselves readily to calculus; others\nrequire more difficult calculi; and the difficulties increasing in the\nmeasure that the combinations become more complicated, the desire to\nsurmount them and curiosity have excited geometricians to perfect more\nand more this kind of analysis. It has been seen already that the\nbenefits of a lottery are easily determined by the theory of\ncombinations. But it is more difficult to know in how many draws one can\nbet one against one, for example that all the numbers will be drawn, _n_\nbeing the number of numbers, _r_ that of the numbers drawn at each draw,\nand _i_ the unknown number of draws. The expression of the probability\nof drawing all the numbers depends upon the _n_th finite difference of\nthe _i_ power of a product of _r_ consecutive numbers. When the number\n_n_ is considerable the search for the value of _i_ which renders this\nprobability equal to \u00bd becomes impossible at least unless this\ndifference is converted into a very converging series. This is easily\ndone by the method here below indicated by the approximations of\nfunctions of very large numbers. It is found thus since the lottery is\ncomposed of ten thousand numbers, one of which is drawn at each draw,\nthat there is a disadvantage in betting one against one that all the\nnumbers will be drawn in 95767 draws and an advantage in making the same\nbet for 95768 draws. In the lottery of France this bet is\ndisadvantageous for 85 draws and advantageous for 86 draws.\n\nLet us consider again two players, A and B, playing together at heads\nand tails in such a manner that at each throw if heads turns up A gives\none counter to B, who gives him one if tails turns up; the number of\ncounters of B is limited, while that of A is unlimited, and the game is\nto end only when B shall have no more counters. We ask in how many\nthrows one should bet one to one that the game will end. The expression\nof the probability that the game will end in an _i_ number of throws is\ngiven by a series which comprises a great number of terms and factors if\nthe number of counters of B is considerable; the search for the value of\nthe unknown _i_ which renders this series \u00bd would then be impossible if\nwe did not reduce the same to a very convergent series. In applying to\nit the method of which we have just spoken, we find a very simple\nexpression for the unknown from which it results that if, for example, B\nhas a hundred counters, it is a bet of a little less than one against\none that the game will end in 23780 throws, and a bet of a little more\nthan one against one that it will end in 23781 throws.\n\nThese two examples added to those we have already given are sufficient\nto shows how the problems of games have contributed to the perfection of\nanalysis.\n\n\n\n\nCHAPTER VII.\n\n_CONCERNING THE UNKNOWN INEQUALITIES WHICH MAY EXIST AMONG CHANCES WHICH\nARE SUPPOSED EQUAL._\n\n\nInequalities of this kind have upon the results of the calculation of\nprobabilities a sensible influence which deserves particular attention.\nLet us take the game of heads and tails, and let us suppose that it is\nequally easy to throw the one or the other side of the coin. Then the\nprobability of throwing heads at the first throw is \u00bd and that of\nthrowing it twice in succession is \u00bc. But if there exist in the coin an\ninequality which causes one of the faces to appear rather than the other\nwithout knowing which side is favored by this inequality, the\nprobability of throwing heads at the first throw will always \u00bd; because\nof our ignorance of which face is favored by the inequality the\nprobability of the simple event is increased if this inequality is\nfavorable to it, just so much is it diminished if the inequality is\ncontrary to it. But in this same ignorance the probability of throwing\nheads twice in succession is increased. Indeed this probability is that\nof throwing heads at the first throw multiplied by the probability that\nhaving thrown it at the first throw it will be thrown at the second; but\nits happening at the first throw is a reason for belief that the\ninequality of the coin favors it; the unknown inequality increases,\nthen, the probability of throwing heads at the second throw; it\nconsequently increases the product of these two probabilities. In order\nto submit this matter to calculus let us suppose that this inequality\nincreases by a twentieth the probability of the simple event which it\nfavors. If this event is heads, its probability will be \u00bd plus 1/20, or\n11/20, and the probability of throwing it twice in succession will be\nthe square of 11/20, or 121/400. If the favored event is tails, the\nprobability of heads, will be \u00bd minus 1/20, or 9/20, and the probability\nof throwing it twice in succession will be 81/400. Since we have at\nfirst no reason for believing that the inequality favors one of these\nevents rather than the other, it is clear that in order to have the\nprobability of the compound event heads heads it is necessary to add the\ntwo preceding probabilities and take the half of their sum, which gives\n101/400 for this probability, which exceeds \u00bc by 1/400 or by the square\nof the favor 1/20 that the inequality adds to the possibilities of the\nevent which it favors. The probability of throwing tails tails is\nsimilarly 101/400, but the probability of throwing heads tails or tails\nheads is each 99/400; for the sum of these four probabilities ought to\nequal certainty or unity. We find thus generally that the constant and\nunknown causes which favor simple events which are judged equally\npossible always increase the probability of the repetition of the same\nsimple event.\n\nIn an even number of throws heads and tails ought both to happen either\nan even number of times or odd number of times. The probability of each\nof these cases is \u00bd if the possibilities of the two faces are equal; but\nif there is between them an unknown inequality, this inequality is\nalways favorable to the first case.\n\nTwo players whose skill is supposed to be equal play on the conditions\nthat at each throw that one who loses gives a counter to his adversary,\nand that the game continues until one of the players has no more\ncounters. The calculation of the probabilities shows us that for the\nequality of the play the throws of the players ought to be an inverse\nratio to their counters. But if there is between the players a small\nunknown inequality, it favors that one of the players who has the\nsmallest number of counters. His probability of winning the game\nincreases if the players agree to double or triple their counters; and\nit will be \u00bd or the same as the probability of the other player in the\ncase where the number of their counters should become infinite,\npreserving always the same ratio.\n\nOne may correct the influence of these unknown inequalities by\nsubmitting them themselves to the chances of hazard. Thus at the play of\nheads and tails, if one has a second coin which is thrown each time with\nthe first and one agrees to name constantly heads the face turned up by\nthe second coin, the probability of throwing heads twice in succession\nwith the first coin will approach much nearer \u00bc than in the case of a\nsingle coin. In this last case the difference is the square of the small\nincrement of possibility that the unknown inequality gives to the face\nof the first coin which it favors; in the other case this difference is\nthe quadruple product of this square by the corresponding square\nrelative to the second coin.\n\nLet there be thrown into an urn a hundred numbers from 1 to 100 in the\norder of numeration, and after having shaken the urn in order to mix the\nnumbers one is drawn; it is clear that if the mixing has been well done\nthe probabilities of the drawing of the numbers will be the same. But if\nwe fear that there is among them small differences dependent upon the\norder according to which the numbers have been thrown into the urn, we\nshall diminish considerably these differences by throwing into a second\nurn the numbers according to the order of their drawing from the first\nurn, and by shaking then this second urn in order to mix the numbers. A\nthird urn, a fourth urn, etc., would diminish more and more these\ndifferences already inappreciable in the second urn.\n\n\n\n\nCHAPTER VIII.\n\n_CONCERNING THE LAWS OF PROBABILITY WHICH RESULT FROM THE INDEFINITE\nMULTIPLICATION OF EVENTS._\n\n\nAmid the variable and unknown causes which we comprehend under the name\nof _chance_, and which render uncertain and irregular the march of\nevents, we see appearing, in the measure that they multiply, a striking\nregularity which seems to hold to a design and which has been considered\nas a proof of Providence. But in reflecting upon this we soon recognize\nthat this regularity is only the development of the respective\npossibilities of simple events which ought to present themselves more\noften when they are more probable. Let us imagine, for example, an urn\nwhich contains white balls and black balls; and let us suppose that each\ntime a ball is drawn it is put back into the urn before proceeding to a\nnew draw. The ratio of the number of the white balls drawn to the number\nof black balls drawn will be most often very irregular in the first\ndrawings; but the variable causes of this irregularity produce effects\nalternately favorable and unfavorable to the regular march of events\nwhich destroy each other mutually in the totality of a great number of\ndraws, allowing us to perceive more and more the ratio of white balls to\nthe black balls contained in the urn, or the respective possibilities of\ndrawing a white ball or black ball at each draw. From this results the\nfollowing theorem.\n\nThe probability that the ratio of the number of white balls drawn to the\ntotal number of balls drawn does not deviate beyond a given interval\nfrom the ratio of the number of white balls to the total number of balls\ncontained in the urn, approaches indefinitely to certainty by the\nindefinite multiplication of events, however small this interval.\n\nThis theorem indicated by common sense was difficult to demonstrate by\nanalysis. Accordingly the illustrious geometrician Jacques Bernoulli,\nwho first has occupied himself with it, attaches great importance to the\ndemonstrations he has given. The calculus of discriminant functions\napplied to this matter not only demonstrates with facility this theorem,\nbut still more it gives the probability that the ratio of the events\nobserved deviates only in certain limits from the true ratio of their\nrespective possibilities.\n\nOne may draw from the preceding theorem this consequence which ought to\nbe regarded as a general law, namely, that the ratios of the acts of\nnature are very nearly constant when these acts are considered in great\nnumber. Thus in spite of the variety of years the sum of the productions\nduring a considerable number of years is sensibly the same; so that man\nby useful foresight is able to provide against the irregularity of the\nseasons by spreading out equally over all the seasons the goods which\nnature distributes in an unequal manner. I do not except from the above\nlaw results due to moral causes. The ratio of annual births to the\npopulation, and that of marriages to births, show only small variations;\nat Paris the number of annual births is almost the same, and I have\nheard it said at the post-office in ordinary seasons the number of\nletters thrown aside on account of defective addresses changes little\neach year; this has likewise been observed at London.\n\nIt follows again from this theorem that in a series of events\nindefinitely prolonged the action of regular and constant causes ought\nto prevail in the long run over that of irregular causes. It is this\nwhich renders the gains of the lotteries just as certain as the products\nof agriculture; the chances which they reserve assure them a benefit in\nthe totality of a great number of throws. Thus favorable and numerous\nchances being constantly attached to the observation of the eternal\nprinciples of reason, of justice, and of humanity which establish and\nmaintain societies, there is a great advantage in conforming to these\nprinciples and of grave inconvenience in departing from them. If one\nconsult histories and his own experience, one will see all the facts\ncome to the aid of this result of calculus. Consider the happy effects\nof institutions founded upon reason and the natural rights of man among\nthe peoples who have known how to establish and preserve them. Consider\nagain the advantages which good faith has procured for the governments\nwho have made it the basis of their conduct and how they have been\nindemnified for the sacrifices which a scrupulous exactitude in keeping\ntheir engagements has cost them. What immense credit at home! What\npreponderance abroad! On the contrary, look into what an abyss of\nmisfortunes nations have often been precipitated by the ambition and the\nperfidy of their chiefs. Every time that a great power intoxicated by\nthe love of conquest aspires to universal domination the sentiment of\nindependence produces among the menaced nations a coalition of which it\nbecomes almost always the victim. Similarly in the midst of the variable\ncauses which extend or restrain the divers states, the natural limits\nacting as constant causes ought to end by prevailing. It is important\nthen to the stability as well as to the happiness of empires not to\nextend them beyond those limits into which they are led again without\ncessation by the action of the causes; just as the waters of the seas\nraised by violent tempests fall again into their basins by the force of\ngravity. It is again a result of the calculus of probabilities confirmed\nby numerous and melancholy experiences. History treated from the point\nof view of the influence of constant causes would unite to the interest\nof curiosity that of offering to man most useful lessons. Sometimes we\nattribute the inevitable results of these causes to the accidental\ncircumstances which have produced their action. It is, for example,\nagainst the nature of things that one people should ever be governed by\nanother when a vast sea or a great distance separates them. It may be\naffirmed that in the long run this constant cause, joining itself\nwithout ceasing to the variable causes which act in the same way and\nwhich the course of time develops, will end by finding them sufficiently\nstrong to give to a subjugated people its natural independence or to\nunite it to a powerful state which may be contiguous.\n\nIn a great number of cases, and these are the most important of the\nanalysis of hazards, the possibilities of simple events are unknown and\nwe are forced to search in past events for the indices which can guide\nus in our conjectures about the causes upon which they depend. In\napplying the analysis of discriminant functions to the principle\nelucidated above on the probability of the causes drawn from the events\nobserved, we are led to the following theorem.\n\nWhen a simple event or one composed of several simple events, as, for\ninstance, in a game, has been repeated a great number of times the\npossibilities of the simple events which render most probable that which\nhas been observed are those that observation indicates with the greatest\nprobability; in the measure that the observed event is repeated this\nprobability increases and would end by amounting to certainty if the\nnumbers of repetitions should become infinite.\n\nThere are two kinds of approximations: the one is relative to the limits\ntaken on all sides of the possibilities which give to the past the\ngreatest probability; the other approximation is related to the\nprobability that these possibilities fall within these limits. The\nrepetition of the compound event increases more and more this\nprobability, the limits remaining the same; it reduces more and more the\ninterval of these limits, the probability remaining the same; in\ninfinity this interval becomes zero and the probability changes to\ncertainty.\n\nIf we apply this theorem to the ratio of the births of boys to that of\ngirls observed in the different countries of Europe, we find that this\nratio, which is everywhere about equal to that of 22 to 21, indicates\nwith an extreme probability a greater facility in the birth of boys.\nConsidering further that it is the same at Naples and at St. Petersburg,\nwe shall see that in this regard the influence of climate is without\neffect. We might then suspect, contrary to the common belief, that this\npredominance of masculine births exists even in the Orient. I have\nconsequently invited the French scholars sent to Egypt to occupy\nthemselves with this interesting question; but the difficulty in\nobtaining exact information about the births has not permitted them to\nsolve it. Happily, M. de Humboldt has not neglected this matter among\nthe innumerable new things which he has observed and collected in\nAmerica with so much sagacity, constancy, and courage. He has found in\nthe tropics the same ratio of the births as we observe in Paris; this\nought to make us regard the greater number of masculine births as a\ngeneral law of the human race. The laws which the different kinds of\nanimals follow in this regard seem to me worthy of the attention of\nnaturalists.\n\nThe fact that the ratio of births of boys to that of girls differs very\nlittle from unity even in the great number of the births observed in a\nplace would offer in this regard a result contrary to the general law,\nwithout which we should be right in concluding that this law did not\nexist. In order to arrive at this result it is necessary to employ great\nnumbers and to be sure that it is indicated by great probability. Buffon\ncites, for example, in his _Political Arithmetic_ several communities of\nBourgogne where the births of girls have surpassed those of boys. Among\nthese communities that of Carcelle-le-Grignon presents in 2009 births\nduring five years 1026 girls and 983 boys. Although these numbers are\nconsiderable, they indicate, however, only a greater possibility in the\nbirths of girls with a probability of 9/10, and this probability,\nsmaller than that of not throwing heads four times in succession in the\ngame of heads and tails, is not sufficient to investigate the cause for\nthis anomaly, which, according to all probability, would disappear if\none should follow during a century the births in this community.\n\nThe registers of births, which are kept with care in order to assure the\ncondition of the citizens, may serve in determining the population of a\ngreat empire without recurring to the enumeration of its inhabitants\u2014a\nlaborious operation and one difficult to make with exactitude. But for\nthis it is necessary to know the ratio of the population to the annual\nbirths. The most precise means of obtaining it consists, first, in\nchoosing in the empire districts distributed in an almost equal manner\nover its whole surface, so as to render the general result independent\nof local circumstances; second, in enumerating with care for a given\nepoch the inhabitants of several communities in each of these districts;\nthird, by determining from the statement of the births during several\nyears which precede and follow this epoch the mean number corresponding\nto the annual births. This number, divided by that of the inhabitants,\nwill give the ratio of the annual births to the population in a manner\nmore and more accurate as the enumeration becomes more considerable. The\ngovernment, convinced of the utility of a similar enumeration, has\ndecided at my request to order its execution. In thirty districts spread\nout equally over the whole of France, communities have been chosen which\nwould be able to furnish the most exact information. Their enumerations\nhave given 2037615 individuals as the total number of their inhabitants\non the 23d of September, 1802. The statement of the births in these\ncommunities during the years 1800, 1801, and 1802 have given:\n\n   Births.      Marriages.        Deaths.\n 110312 boys      46037         103659 men\n 105287 girls                    99443 women\n\nThe ratio of the population to annual births is then 28 + (352845/1000000);\nit is greater than had been estimated up to this time. Multiplying the\nnumber of annual births in France by this ratio, we shall have the\npopulation of this kingdom. But what is the probability that the\npopulation thus determined will not deviate from the true population\nbeyond a given limit? Resolving this problem and applying to its\nsolution the preceding data, I have found that, the number of annual\nbirths in France being supposed to be 1000000, which brings the\npopulation to 28352845 inhabitants, it is a bet of almost 300000 against\n1 that the error of this result is not half a million.\n\nThe ratio of the births of boys to that of girls which the preceding\nstatement offers is that of 22 to 21; and the marriages are to the\nbirths as 3 is to 4.\n\nAt Paris the baptisms of children of both sexes vary a little from the\nratio of 22 to 21. Since 1745, the epoch in which one has commenced to\ndistinguish the sexes upon the birth-registers, up to the end of 1784,\nthere have been baptized in this capital 393386 boys and 377555 girls.\nThe ratio of the two numbers is almost that of 25 to 24; it appears then\nat Paris that a particular cause approximates an equality of baptisms of\nthe two sexes. If we apply to this matter the calculus of probabilities,\nwe find that it is a bet of 238 to 1 in favor of the existence of this\ncause, which is sufficient to authorize the investigation. Upon\nreflection it has appeared to me that the difference observed holds to\nthis, that the parents in the country and the provinces, finding some\nadvantage in keeping the boys at home, have sent to the Hospital for\nFoundlings in Paris fewer of them relative to the number of girls\naccording to the ratio of births of the two sexes. This is proved by the\nstatement of the registers of this hospital. From the beginning of 1745\nto the end of 1809 there were entered 163499 boys and 159405 girls. The\nfirst of these numbers exceeds only by 1/38 the second, which it ought\nto have surpassed at least by 1/24. This confirms the existence of the\nassigned cause, namely, that the ratio of births of boys to those of\ngirls is at Paris that of 22 to 21, no attention having been paid to\nfoundlings.\n\nThe preceding results suppose that we may compare the births to the\ndrawings of balls from an urn which contains an infinite number of white\nballs and black balls so mixed that at each draw the chances of drawing\nought to be the same for each ball; but it is possible that the\nvariations of the same seasons in different years may have some\ninfluence upon the annual ratio of the births of boys to those of girls.\nThe Bureau of Longitudes of France publishes each year in its annual the\ntables of the annual movement of the population of the kingdom. The\ntables already published commence in 1817; in that year and in the five\nfollowing years there were born 2962361 boys and 2781997 girls, which\ngives about 16/15 for the ratio of the births of boys to that of girls.\nThe ratios of each year vary little from this mean result; the smallest\nratio is that of 1822, where it was only 17/16; the greatest is of the\nyear 1817, when it was 15/14. These ratios vary appreciably from the\nratio of 22/21 found above. Applying to this deviation the analysis of\nprobabilities in the hypothesis of the comparison of births to the\ndrawings of balls from an urn, we find that it would be scarcely\nprobable. It appears, then, to indicate that this hypothesis, although\nclosely approximated, is not rigorously exact. In the number of births\nwhich we have just stated there are of natural children 200494 boys and\n190698 girls. The ratio of masculine and feminine births was then in\nthis regard 20/19, smaller than the mean ratio of 16/15. This result is\nin the same sense as that of the births of foundlings; and it seems to\nprove that in the class of natural children the births of the two sexes\napproach more nearly equality than in the class of legitimate children.\nThe difference of the climates from the north to the south of France\ndoes not appear to influence appreciably the ratio of the births of boys\nand girls. The thirty most southern districts have given 16/15 for this\nratio, the same as that of entire France.\n\nThe constancy of the superiority of the births of boys over girls at\nParis and at London since they have been observed has appeared to some\nscholars to be a proof of Providence, without which they have thought\nthat the irregular causes which disturb without ceasing the course of\nevents ought several times to have rendered the annual births of girls\nsuperior to those of boys.\n\nBut this proof is a new example of the abuse which has been so often\nmade of final causes which always disappear on a searching examination\nof the questions when we have the necessary data to solve them. The\nconstancy in question is a result of regular causes which give the\nsuperiority to the births of boys and which extend it to the anomalies\ndue to hazard when the number of annual births is considerable. The\ninvestigation of the probability that this constancy will maintain\nitself for a long time belongs to that branch of the analysis of hazards\nwhich passes from past events to the probability of future events; and\ntaking as a basis the births observed from 1745 to 1784, it is a bet of\nalmost 4 against 1 that at Paris the annual births of boys will\nconstantly surpass for a century the births of girls; there is then no\nreason to be astonished that this has taken place for a half-century.\n\nLet us take another example of the development of constant ratios which\nevents present in the measure that they are multiplied. Let us imagine a\nseries of urns arranged circularly, and each containing a very great\nnumber of white balls and black balls; the ratio of white balls to the\nblack in the urns being originally very different and such, for example,\nthat one of these urns contains only white balls, while another contains\nonly black balls. If one draws a ball from the first urn in order to put\nit into the second, and, after having shaken the second urn in order to\nmix well the new ball with the others, one draws a ball to put it into\nthe third urn, and so on to the last urn, from which is drawn a ball to\nput into the first, and if this series is recommenced continually, the\nanalysis of probability shows us that the ratios of the white balls to\nthe black in these urns will end by being the same and equal to the\nratio of the sum of all the white balls to the sum of all the black\nballs contained in the urns. Thus by this regular mode of change the\nprimitive irregularity of these ratios disappears eventually in order to\nmake room for the most simple order. Now if among these urns one\nintercalate new ones in which the ratio of the sum of the white balls to\nthe sum of the black balls which they contain differs from the\npreceding, continuing indefinitely in the totality of the urns the\ndrawings which we have just indicated, the simple order established in\nthe old urns will be at first disturbed, and the ratios of the white\nballs to the black balls will become irregular; but little by little\nthis irregularity will disappear in order to make room for a new order,\nwhich will finally be that of the equality of the ratios of the white\nballs to the black balls contained in the urns. We may apply these\nresults to all the combinations of nature in which the constant forces\nby which their elements are animated establish regular modes of action,\nsuited to bring about in the very heart of chaos systems governed by\nadmirable laws.\n\nThe phenomena which seem the most dependent upon hazard present, then,\nwhen multiplied a tendency to approach without ceasing fixed ratios, in\nsuch a manner that if we conceive on all sides of each of these ratios\nan interval as small as desired, the probability that the mean result of\nthe observations falls within this interval will end by differing from\ncertainty only by a quantity greater than an assignable magnitude. Thus\nby the calculations of probabilities applied to a great number of\nobservations we may recognize the existence of these ratios. But before\nseeking the causes it is necessary, in order not to be led into vain\nspeculations, to assure ourselves that they are indicated by a\nprobability which does not permit us to regard them as anomalies due to\nhazard. The theory of discriminant functions gives a very simple\nexpression for this probability, which is obtained by integrating the\nproduct of the differential of the quantity of which the result deduced\nfrom a great number of observations varies from the truth by a constant\nless than unity, dependent upon the nature of the problem, and raised to\na power whose exponent is the ratio of the square of this variation to\nthe number of observations. The integral taken between the limits given\nand divided by the same integral, applied to a positive and negative\ninfinity, will express the probability that the variation from the truth\nis comprised between these limits. Such is the general law of the\nprobability of results indicated by a great number of observations.\n\n\n\n\nCHAPTER IX.\n\n_THE APPLICATION OF THE CALCULUS OF PROBABILITIES TO NATURAL PHILOSOPHY._\n\n\nThe phenomena of nature are most often enveloped by so many strange\ncircumstances, and so great a number of disturbing causes mix their\ninfluence, that it is very difficult to recognize them. We may arrive at\nthem only by multiplying the observations or the experiences, so that\nthe strange effects finally destroy reciprocally each other, the mean\nresults putting in evidence those phenomena and their divers elements.\nThe more numerous the number of observations and the less they vary\namong themselves the more their results approach the truth. We fulfil\nthis last condition by the choice of the methods of observations, by the\nprecision of the instruments, and by the care which we take to observe\nclosely; then we determine by the theory of probabilities the most\nadvantageous mean results or those which give the least value of the\nerror. But that is not sufficient; it is further necessary to appreciate\nthe probability that the errors of these results are comprised in the\ngiven limits; and without this we have only an imperfect knowledge of\nthe degree of exactitude obtained. Formul\u00e6 suitable to these matters are\nthen true improvements of the method of sciences, and it is indeed\nimportant to add them to this method. The analysis which they require is\nthe most delicate and the most difficult of the theory of probabilities;\nit is one of the principal objects of the work which I have published\nupon this theory, and in which I have arrived at formul\u00e6 of this kind\nwhich have the remarkable advantage of being independent of the law of\nthe probability of errors and of including only the quantities given by\nthe observations themselves and their expressions.\n\nEach observation has for an analytic expression a function of the\nelements which we wish to determine; and if these elements are nearly\nknown, this function becomes a linear function of their corrections. In\nequating it to the observation itself there is formed _an equation of\ncondition._ If we have a great number of similar equations, we combine\nthem in such a manner as to obtain as many final equations as there are\nelements whose corrections we determine then by resolving these\nequations. But what is the most advantageous manner of combining\nequations of condition in order to obtain final equations? What is the\nlaw of the probabilities of errors of which the elements are still\nsusceptible that we draw from them? This is made clear to us by the\ntheory of probabilities. The formation of a final equation by means of\nthe equation of condition amounts to multiplying each one of these by an\nindeterminate factor and by uniting the products; it is necessary to\nchoose the system of factors which gives the smallest opportunity for\nerror. But it is apparent that if we multiply the possible errors of an\nelement by their respective probabilities, the most advantageous system\nwill be that in which the sum of these products all, taken, positively\nis a _minimum_; for a positive or a negative error ought to be\nconsidered as a loss. Forming, then, this sum of products, the condition\nof the _minimum_ will determine the system of factors which it is\nexpedient to adopt, or the most advantageous system. We find thus that\nthis system is that of the coefficients of the elements in each equation\nof condition; so that we form a first final equation by multiplying\nrespectively each equation of condition by its coefficient of the first\nelement and by uniting all these equations thus multiplied. We form a\nsecond final equation by employing in the same manner the coefficients\nof the second element, and so on. In this manner the elements and the\nlaws of the phenomena obtained in the collection of a great number of\nobservations are developed with the most evidence.\n\nThe probability of the errors which each element still leaves to be\nfeared is proportional to the number whose hyperbolic logarithm is unity\nraised to a power equal to the square of the error taken as a minus\nquantity and multiplied by a constant coefficient which may be\nconsidered as the modulus of the probability of the errors; because, the\nerror remaining the same, its probability decreases with rapidity when\nthe former increases; so that the element obtained weighs, if I may thus\nspeak toward the truth, as much more as this modulus is greater. I would\ncall for this reason this modulus the _weight_ of the element or of the\nresult. This weight is the greatest possible in the system of\nfactors\u2014the most advantageous; it is this which gives to this system\nsuperiority over others. By a remarkable analogy of this weight with\nthose of bodies compared at their common centre of gravity it results\nthat if the same element is given by divers systems, composed each of a\ngreat number of observations, the most advantageous, the mean result of\ntheir totality is the sum of the products of each partial result by its\nweight. Moreover, the total weight of the results of the divers systems\nis the sum of their partial weights; so that the probability of the\nerrors of the mean result of their totality is proportional to the\nnumber which has unity for an hyperbolic logarithm raised to a power\nequal to the square of the error taken as minus and multiplied by the\nsum of the weights. Each weight depends in truth upon the law of the\nprobability of error of each system, and almost always this law is\nunknown; but happily I have been able to eliminate the factor which\ncontains it by means of the sum of the squares of the variations of the\nobservations in this system from their mean result. It would then be\ndesirable in order to complete our knowledge of the results obtained by\nthe totality of a great number of observations that we write by the side\nof each result the weight which corresponds to it; analysis furnishes\nfor this object both general and simple methods. When we have thus\nobtained the exponential which represents the law of the probability of\nerrors, we shall have the probability that the error of the result is\nincluded within given limits by taking within the limits the integral of\nthe product of this exponential by the differential of the error and\nmultiplying it by the square root of the weight of the result divided by\nthe circumference whose diameter is unity. Hence it follows that for the\nsame probability the errors of the results are reciprocal to the square\nroots of their weights, which serves to compare their respective\nprecision.\n\nIn order to apply this method with success it is necessary to vary the\ncircumstances of the observations or the experiences in such a manner as\nto avoid the constant causes of error. It is necessary that the\nobservations should be numerous, and that they should be so much the\nmore so as there are more elements to determine; for the weight of the\nmean result increases as the number of observations divided by the\nnumber of the elements. It is still necessary that the elements follow\nin these observations a different course; for if the course of the two\nelements were exactly the same, which would render their coefficients\nproportional in equation of conditions, these elements would form only a\nsingle unknown quantity and it would be impossible to distinguish them\nby these observations. Finally it is necessary that the observations\nshould be precise; this condition, the first of all, increases greatly\nthe weight of the result the expression of which has for a divisor the\nsum of the squares of the deviations of the observations from this\nresult. With these precautions we shall be able to make use of the\npreceding method and measure the degree of confidence which the results\ndeduced from a great number of observations merit.\n\nThe rule which we have just given to conclude equations of condition,\nfinal equations, amount to rendering a minimum the sum of the squares of\nthe errors of observations; for each equation of condition becomes exact\nby substituting in it the observation plus its error; and if we draw\nfrom it the expression of this error, it is easy to see that the\ncondition of the _minimum_ of the sum of the squares of these\nexpressions gives the rule in question. This rule is the more precise as\nthe observations are more numerous; but even in the case where their\nnumber is small it appears natural to employ the same rule which in all\ncases offers a simple means of obtaining without groping the corrections\nwhich we seek to determine. It serves further to compare the precision\nof the divers astronomical tables of the same star. These tables may\nalways be supposed as reduced to the same form, and then they differ\nonly by the epochs, the mean movements and the coefficients of the\narguments; for if one of them contains a coefficient which is not found\nin the others, it is clear that this amounts to supposing zero in them\nas the coefficient of this argument. If now we rectify these tables by\nthe totality of the good observations, they would satisfy the condition\nthat the sum of the squares of the errors should be a minimum; the\ntables which, compared to a considerable number of observations,\napproach nearest this condition merit then the preference.\n\nIt is principally in astronomy that the method explained above may be\nemployed with advantage. The astronomical tables owe the truly\nastonishing exactitude which they have attained to the precision of\nobservations and of theories, and to the use of equations of conditions\nwhich cause to concur a great number of excellent observations in the\ncorrection of the same element. But it remains to determine the\nprobability of the errors that this correction leaves still to be\nfeared; and the method which I have just explained enables us to\nrecognize the probability of these errors. In order to give some\ninteresting applications of it I have profited by the immense work which\nM. Bouvard has just finished on the movements of Jupiter and Saturn, of\nwhich he has formed very precise tables. He has discussed with the\ngreatest care the oppositions and quadratures of these two planets\nobserved by Bradley and by the astronomers who have followed him down to\nthe last years; he has concluded the corrections of the elements of\ntheir movement and their masses compared to that of the sun taken as\nunity. His calculations give him the mass of Saturn equal to the 3512th\npart of that of the sun. Applying to them my formul\u00e6 of probability, I\nfind that it is a bet of 11,000 against one that the error of this\nresult is not 1/100 of its value, or that which amounts to almost the\nsame\u2014that after a century of new observations added to the preceding\nones, and examined in the same manner, the new result will not differ by\n1/100 from that of M. Bouvard. This wise astronomer finds again the mass\nof Jupiter equal to the 1071th part of the sun; and my method of\nprobability gives a bet of 1,000,000 to one that this result is not\n1/100 in error.\n\nThis method may be employed again with success in geodetic operations.\nWe determine the length of the great arc on the surface of the earth by\ntriangulation, which depends upon a base measured with exactitude. But\nwhatever precision may be brought to the measure of the angles, the\ninevitable errors can, by accumulating, cause the value of the arc\nconcluded from a great number of triangles to deviate appreciably from\nthe truth. We recognize this value, then, only imperfectly unless the\nprobability that its error is comprised within given limits can be\nassigned. The error of a geodetic result is a function of the errors of\nthe angles of each triangle. I have given in the work cited general\nformul\u00e6 in order to obtain the probability of the values of one or of\nseveral linear functions of a great number of partial errors of which we\nknow the law of probability; we may then by means of these formul\u00e6\ndetermine the probability that the error of a geodetic result is\ncontained within the assigned limits, whatever may be the law of the\nprobability of partial errors. It is moreover more necessary to render\nourselves independent of the law, since the most simple laws themselves\nare always infinitely less probable, seeing the infinite number of those\nwhich may exist in nature. But the unknown law of partial errors\nintroduces into the formul\u00e6 an indeterminant which does not permit of\nreducing them to numbers unless we are able to eliminate it. We have\nseen that in astronomical questions, where each observation furnishes an\nequation of condition for obtaining the elements, we eliminate this\ndeterminant by means of the sum of the squares of the remainders when\nthe most probable values of the elements have been substituted in each\nequation. Geodetic questions not offering similar equations, it is\nnecessary to seek another means of elimination. The quantity by which\nthe sum of the angles of each observed triangle surpasses two right\nangles plus the spherical excess furnishes this means. Thus we replace\nby the sum of the squares of these quantities the sum of the squares of\nthe remainders of the equations of condition; and we may assign in\nnumbers the probability that the error of the final result of a series\nof geodetic operations will not exceed a given quantity. But what is the\nmost advantageous manner of dividing among the three angles of each\ntriangle the observed sum of their errors? The analysis of probabilities\nrenders it apparent that each angle ought to be diminished by a third of\nthis sum, provided that the weight of a geodetic result be the greatest\npossible, which renders the same error less probable. There is then a\ngreat advantage in observing the three angles of each triangle and of\ncorrecting them as we have just said. Simple common sense indicates this\nadvantage; but the calculation of probabilities alone is able to\nappreciate it and to render apparent that by this correction it becomes\nthe greatest possible.\n\nIn order to assure oneself of the exactitude of the value of a great arc\nwhich rests upon a base measured at one of its extremities one measures\na second base toward the other extremity; and one concludes from one of\nthese bases the length of the other. If this length varies very little\nfrom the observation, there is all reason to believe that the chain of\ntriangles which unites these bases is very nearly exact and likewise the\nvalue of the large arc which results from it. One corrects, then, this\nvalue by modifying the angles of the triangles in such a manner that the\nbase is calculated according to the bases measured. But this may be done\nin an infinity of ways, among which is preferred that of which the\ngeodetic result has the greatest weight, inasmuch as the same error\nbecomes less probable. The analysis of probabilities gives formul\u00e6 for\nobtaining directly the most advantageous correction which results from\nthe measurements of the several bases and the laws of probability which\nthe multiplicity of the bases makes\u2014laws which become very rapidly\ndecreasing by this multiplicity.\n\nGenerally the errors of the results deduced from a great number of\nobservations are the linear functions of the partial errors of each\nobservation. The coefficients of these functions depend upon the nature\nof the problem and upon the process followed in order to obtain the\nresults. The most advantageous process is evidently that in which the\nsame error in the results is less probable than according to any other\nprocess. The application of the calculus of probabilities to natural\nphilosophy consists, then, in determining analytically the probability\nof the values of these functions and in choosing their indeterminant\ncoefficients in such a manner that the law of this probability should be\nmost rapidly descending. Eliminating, then, from the formul\u00e6 by the data\nof the question the factor which is introduced by the almost always\nunknown law of the probability of partial errors, we may be able to\nevaluate numerically the probability that the errors of the results do\nnot exceed a given quantity. We shall thus have all that may be desired\ntouching the results deduced from a great number of observations.\n\nVery approximate results may be obtained by other considerations.\nSuppose, for example, that one has a thousand and one observations of\nthe same quantity; the arithmetical mean of all these observations is\nthe result given by the most advantageous method. But one would be able\nto choose the result according to the condition that the sum of the\nvariations from each partial value all taken positively should be a\n_minimum_. It appears indeed natural to regard as very approximate the\nresult which satisfies this condition. It is easy to see that if one\ndisposes the values given by the observations according to the order of\nmagnitude, the value which will occupy the mean will fulfil the\npreceding condition, and calculus renders it apparent that in the case\nof an infinite number of observations it would coincide with the truth;\nbut the result given by the most advantageous method is still preferable.\n\nWe see by that which precedes that the theory of probabilities leaves\nnothing arbitrary in the manner of distributing the errors of the\nobservations; it gives for this distribution the most advantageous\nformul\u00e6 which diminishes as much as possible the errors to be feared in\nthe results.\n\nThe consideration of probabilities can serve to distinguish the small\nirregularities of the celestial movements enveloped in the errors of\nobservations, and to repass to the cause of the anomalies observed in\nthese movements.\n\nIn comparing all the observations it was Ticho-Brah\u00e9 who recognized the\nnecessity of applying to the moon an equation of time different from\nthat which had been applied to the sun and to the planets. It was\nsimilarly the totality of a great number of observations which made\nMayer recognize that the coefficient of the inequality of the precession\nought to be diminished a little for the moon. But since this diminution,\nalthough confirmed and even augmented by Mason, did not appear to result\nfrom universal gravitation, the majority of astronomers neglect it in\ntheir calculations. Having submitted to the calculation of probabilities\na considerable number of lunar observations chosen for this purpose and\nwhich M. Bouvard consented to examine at my request, it appeared to me\nto be indicated with so strong a probability that I believed the cause\nof it ought to be investigated. I soon saw that it would be only the\nellipticity of the terrestrial spheroid, neglected up to that time in\nthe theory of the lunar movement as being able to produce only\nimperceptible terms. I concluded that these terms became perceptible by\nthe successive integrations of differential equations. I determined then\nthose terms by a particular analysis, and I discovered first the\ninequality of the lunar movement in latitude which is proportional to\nthe sine of the longitude of the moon, which no astronomer before had\nsuspected. I recognized then by means of this inequality that another\nexists in the lunar movement in longitude which produces the diminution\nobserved by Mayer in the equation of the precession applicable to the\nmoon. The quantity of this diminution and the coefficient of the\npreceding inequality in latitude are very appropriate to fix the\noblateness of the earth. Having communicated my researches to M. Burg,\nwho was occupied at that time in perfecting the tables of the moon by\nthe comparison of all the good observations, I requested him to\ndetermine with a particular care these two quantities. By a very\nremarkable agreement the values which he has found give to the earth the\nsame oblateness, 1/305, which differs little from the mean derived from\nthe measurements of the degrees of the meridian and the pendulum; but\nthose regarded from the point of view of the influence of the errors of\nthe observations and of the perturbing causes in these measurements, did\nnot appear to me exactly determined by these lunar inequalities.\n\nIt was again by the consideration of probabilities that I recognized the\ncause of the secular equation of the moon. The modern observations of\nthis star compared to the ancient eclipses had indicated to astronomers\nan acceleration in the lunar movement; but the geometricians, and\nparticularly Lagrange, having vainly sought in the perturbations which\nthis movement experienced the terms upon which this acceleration\ndepends, reject it. An attentive examination of the ancient and modern\nobservations and of the intermediary eclipses observed by the Arabians\nconvinced me that it was indicated with a great probability. I took up\nagain then from this point of view the lunar theory, and I recognized\nthat the secular equation of the moon is due to the action of the sun\nupon this satellite, combined with the secular variation of the\neccentricity of the terrestrial orb; this brought me to the discovery of\nthe secular equations of the movements of the nodes and of the perigees\nof the lunar orbit, which equations had not been even suspected by\nastronomers. The very remarkable agreement of this theory with all the\nancient and modern observations has brought it to a very high degree of\nevidence.\n\nThe calculus of probabilities has led me similarly to the cause of the\ngreat irregularities of Jupiter and Saturn. Comparing modern\nobservations with ancient, Halley found an acceleration in the movement\nof Jupiter and a retardation in that of Saturn. In order to conciliate\nthe observations he reduced the movements to two secular equations of\ncontrary signs and increasing as the squares of the times passed since\n1700. Euler and Lagrange submitted to analysis the alterations which the\nmutual attraction of these two planets ought to produce in these\nmovements. They found in doing this the secular equations; but their\nresults were so different that one of the two at least ought to be\nerroneous. I determined then to take up again this important problem of\n_celestial mechanics_, and I recognized the invariability of the mean\nplanetary movements, which nullified the secular equations introduced by\nHalley in the tables of Jupiter and Saturn. Thus there remain, in order\nto explain the great irregularity of these planets, only the attractions\nof the comets to which many astronomers had effective recourse, or the\nexistence of an irregularity over a long period produced in the\nmovements of the two planets by their reciprocal action and affected by\ncontrary signs for each of them. A theorem which I found in regard to\nthe inequalities of this kind rendered this inequality very probable.\nAccording to this theorem, if the movement of Jupiter is accelerated,\nthat of Saturn is retarded, which has already conformed to what Halley\nhad noticed; moreover, the acceleration of Jupiter resulting from the\nsame theorem is to the retardation of Saturn very nearly in the ratio of\nthe secular equations proposed by Halley. Considering the mean movements\nof Jupiter and Saturn I was enabled easily to recognize that two times\nthat of Jupiter differed only by a very small quantity from five times\nthat of Saturn. The period of an irregularity which would have for an\nargument this difference would be about nine centuries. Indeed its\ncoefficient would be of the order of the cubes of the eccentricities of\nthe orbits; but I knew that by virtue of successive integrations it\nacquired for divisor the square of the very small multiplier of the time\nin the argument of this inequality which is able to give it a great\nvalue; the existence of this inequality appeared to me then very\nprobable. The following observation increased then its probability.\nSupposing its argument zero toward the epoch of the observations of\nTicho-Brah\u00e9, I saw that Halley ought to have found by the comparison of\nmodern with ancient observations the alterations which he had indicated;\nwhile the comparison of the modern observations among themselves ought\nto offer contrary alterations similar to those which Lambert had\nconcluded from this comparison. I did not then hesitate at all to\nundertake this long and tedious calculation necessary to assure myself\nof this inequality. It was entirely confirmed by the result of this\ncalculation, which moreover made me recognize a great number of other\ninequalities of which the totality has inclined the tables of Jupiter\nand Saturn to the precision of the same observations.\n\nIt was again by means of the calculus of probabilities that I recognized\nthe remarkable law of the mean movements of the three first satellites\nof Jupiter, according to which the mean longitude of the first minus\nthree times that of the second plus two times that of the third is\nrigorously equal to the half-circumference. The approximation with which\nthe mean movements of these stars satisfy this law since their discovery\nindicates its existence with an extreme probability. I sought then the\ncause of it in their mutual action. The searching examination of this\naction convinced me that it was sufficient if in the beginning the\nratios of their mean movements had approached this law within certain\nlimits, because their mutual action had established and maintained it\nrigorously. Thus these three bodies will balance one another eternally\nin space according to the preceding law unless strange causes, such as\ncomets, should change suddenly their movements about Jupiter.\n\nAccordingly it is seen how necessary it is to be attentive to the\nindications of nature when they are the result of a great number of\nobservations, although in other respects they may be inexplicable by\nknown means. The extreme difficulty of problems relative to the system\nof the world has forced geometricians to recur to the approximation\nwhich always leaves room for the fear that the quantities neglected may\nhave an appreciable influence. When they have been warned of this\ninfluence by the observations, they have recurred to their analysis; in\nrectifying it they have always found the cause of the anomalies\nobserved; they have determined the laws and often they have anticipated\nthe observations in discovering the inequalities which it had not yet\nindicated. Thus one may say that nature itself has concurred in the\nanalytical perfection of the theories based upon the principle of\nuniversal gravity; and this is to my mind one of the strongest proofs of\nthe truth of this admirable principle.\n\nIn the cases which I have just considered the analytical solution of the\nquestion has changed the probability of the causes into certainty. But\nmost often this solution is impossible and it remains only to augment\nmore and more this probability. In the midst of numerous and\nincalculable modifications which the action of the causes receives then\nfrom strange circumstances these causes conserve always with the effects\nobserved the proper ratios to make them recognizable and to verify their\nexistence. Determining these ratios and comparing them with a great\nnumber of observations if one finds that they constantly satisfy it, the\nprobability of the causes may increase to the point of equalling that of\nfacts in regard to which there is no doubt. The investigation of these\nratios of causes to their effects is not less useful in natural\nphilosophy than the direct solution of problems whether it be to verify\nthe reality of these causes or to determine the laws from their effects;\nsince it may be employed in a great number of questions whose direct\nsolution is not possible, it replaces it in the most advantageous\nmanner. I shall discuss here the application which I have made of it to\none of the most interesting phenomena of nature, the flow and the ebb of\nthe sea.\n\nPliny has given of this phenomenon a description remarkable for its\nexactitude, and in it one sees that the ancients had observed that the\ntides of each month are greatest toward the syzygies and smallest toward\nthe quadratures; that they are higher in the perigees than in the\napogees of the moon, and higher in the equinoxes than in the solstices.\nThey concluded from this that this phenomenon is due to the action of\nthe sun and moon upon the sea. In the preface of his work _De Stella\nMartis_ Kepler admits a tendency of the waters of the sea toward the\nmoon; but, ignorant of the law of this tendency, he was able to give on\nthis subject only a probable idea. Newton converted into certainty the\nprobability of this idea by attaching it to his great principle of\nuniversal gravity. He gave the exact expression of the attractive forces\nwhich produced the flood and the ebb of the sea; and in order to\ndetermine the effects he supposed that the sea takes at each instant the\nposition of equilibrium which is agreeable to these forces. He explained\nin this manner the principal phenomena of the tides; but it followed\nfrom this theory that in our ports the two tides of the same day would\nbe very unequal if the sun and the moon should have a great declination.\nAt Brest, for example, the evening tide would be in the syzygies of the\nsolstices about eight times greater than the morning tide, which is\ncertainly contrary to the observations which prove that these two tides\nare very nearly equal. This result from the Newtonian theory might hold\nto the supposition that the sea is agreeable at each instant to a\nposition of equilibrium, a supposition which is not at all admissible.\nBut the investigation of the true figure of the sea presents great\ndifficulties. Aided by the discoveries which the geometricians had just\nmade in the theory of the movement of fluids and in the calculus of\npartial differences, I undertook this investigation, and I gave the\ndifferential equations of the movement of the sea by supposing that it\ncovers the entire earth. In drawing thus near to nature I had the\nsatisfaction of seeing that my results approached the observations,\nespecially in regard to the little difference which exists in our ports\nbetween the two tides of the solstitial syzygies of the same day. I\nfound that they would be equal if the sea had everywhere the same depth;\nI found further that in giving to this depth convenient values one was\nable to augment the height of the tides in a port conformably to the\nobservations. But these investigations, in spite of their generality,\ndid not satisfy at all the great differences which even adjacent ports\npresent in this regard and which prove the influence of local\ncircumstances. The impossibility of knowing these circumstances and the\nirregularity of the basin of the seas and that of integrating the\nequations of partial differences which are relative has compelled me to\nmake up the deficiency by the method I have indicated above. I then\nendeavored to determine the greatest ratios possible among the forces\nwhich affect all the molecules of the sea, and their effects observable\nin our ports. For this I made use of the following principle, which may\nbe applied to many other phenomena.\n\n\"The state of the system of a body in which the primitive conditions of\nthe movement have disappeared by the resistances which this movement\nmeets is periodic as the forces which animate it.\"\n\nCombining this principle with that of the coexistence of very small\noscillations, I have found an expression of the height of the tides\nwhose arbitraries contain the effect of local circumstances of each port\nand are reduced to the smallest number possible; it is only necessary to\ncompare it to a great number of observations.\n\nUpon the invitation of the Academy of Sciences, observations were made\nat the beginning of the last century at Brest upon the tides, which were\ncontinued during six consecutive years. The situation of this port is\nvery favorable to this sort of observations; it communicates with the\nsea by a canal which empties into a vast roadstead at the far end of\nwhich the port has been constructed. The irregularities of the sea\nextend thus only to a small degree into the port, just as the\noscillations which the irregular movement of a vessel produces in a\nbarometer are diminished by a throttling made in the tube of this\ninstrument. Moreover, the tides being considerable at Brest, the\naccidental variations caused by the winds are only feeble; likewise we\nnotice in the observations of these tides, however little we multiply\nthem, a great regularity which induced me to propose to the government\nto order in this port a new series of observations of the tides,\ncontinued during a period of the movement of the nodes of the lunar\norbit. This has been done. The observations began June 1, 1806; and\nsince this time they have been made every day without interruption. I am\nindebted to the indefatigable zeal of M. Bouvard, for all that interests\nastronomy, the immense calculations which the comparison of my analysis\nwith the observations has demanded. There have been used about six\nthousand observations, made during the year 1807 and the fifteen years\nfollowing. It results from this comparison that my formul\u00e6 represent\nwith a remarkable precision all the varieties of the tides relative to\nthe digression of the moon, from the sun, to the declination of these\nstars, to their distances from the earth, and to the laws of variation\nat the _maximum_ and _minimum_ of each of these elements. There results\nfrom this accord a probability that the flow and the ebb of the sea is\ndue to the attraction of the sun and moon, so approaching certainty that\nit ought to leave room for no reasonable doubt. It changes into\ncertainty when we consider that this attraction is derived from the law\nof universal gravity demonstrated by all the celestial phenomena.\n\nThe action of the moon upon the sea is more than double that of the sun.\nNewton and his successors in the development of this action have paid\nattention only to the terms divided by the cube of the distance from the\nmoon to the earth, judging that the effects due to the following terms\nought to be inappreciable. But the calculation of probabilities makes it\nclear to us that the smallest effects of regular causes may manifest\nthemselves in the results of a great number of observations arranged in\nthe order most suitable to indicate them. This calculation again\ndetermines their probability and up to what point it is necessary to\nmultiply the observations to make it very great. Applying it to the\nnumerous observations discussed by M. Bouvard I recognized that at Brest\nthe action of the moon upon the sea is greater in the full moons than in\nthe new moons, and greater when the moon is austral than when it is\nboreal\u2014phenomena which can result only from the terms of the lunar\naction divided by the fourth power of the distance from the moon to the\nearth.\n\nTo arrive at the ocean the action of the sun and the moon traverses the\natmosphere, which ought consequently to feel its influence and to be\nsubjected to movements similar to those of the sea.\n\nThese movements produce in the barometer periodic oscillations. Analysis\nhas made it clear to me that they are inappreciable in our climates. But\nas local circumstances increase considerably the tides in our ports, I\nhave inquired again if similar circumstances have made appreciable these\noscillations of the barometer. For this I have made use of the\nmeteorological observations which have been made every day for many\nyears at the royal observatory. The heights of the barometer and of the\nthermometer are observed there at nine o'clock in the morning, at noon,\nat three o'clock in the afternoon, and at eleven o'clock in the evening.\nM. Bouvard has indeed wished to take up the consideration of\nobservations of the eight years elapsed from October 1, 1815, to October\n1, 1823, on the registers. In disposing the observations in the manner\nmost suitable to indicate the lunar atmospheric flood at Paris, I find\nonly one eighteenth of a millimeter for the extent of the corresponding\noscillation of the barometer. It is this especially which has made us\nfeel the necessity of a method for determining the probability of a\nresult, and without this method one is forced to present as the laws of\nnature the results of irregular causes which has often happened in\nmeteorology. This method applied to the preceding result shows the\nuncertainty of it in spite of the great number of observations employed,\nwhich it would be necessary to increase tenfold in order to obtain a\nresult sufficiently probable.\n\nThe principle which serves as a basis for my theory of the tides may be\nextended to all the effects of hazard to which variable causes are\njoined according to regular laws. The action of these causes produces in\nthe mean results of a great number of effects varieties which follow the\nsame laws and which one may recognize by the analysis of probabilities.\nIn the measure which these effects are multiplied those varieties are\nmanifested with an ever-increasing probability, which would approach\ncertainty if the number of the effects of the results should become\ninfinite. This theorem is analogous to that which I have already\ndeveloped upon the action of constant causes. Every time, then, that a\ncause whose progress is regular can have influence upon a kind of\nevents, we may seek to discover its influence by multiplying the\nobservations and arranging them in the most suitable order to indicate\nit. When this influence appears to manifest itself the analysis of\nprobabilities determines the probability of its existence and that of\nits intensity; thus the variation of the temperature from day to night\nmodifying the pressure of the atmosphere and consequently the height of\nthe barometer, it is natural to think that the multiplied observations\nof these heights ought to show the influence of the solar heat. Indeed\nthere has long been recognized at the equator, where this influence\nappears to be greatest, a small diurnal variation in the height of the\nbarometer of which the _maximum_ occurs about nine o'clock in the\nmorning and the _minimum_ about three o'clock in the afternoon. A second\n_maximum_ occurs about eleven o'clock in the evening and a second\n_minimum_ about four o'clock in the morning. The oscillations of the\nnight are less than those of the day, the extent of which is about two\nmillimeters. The inconstancy of our climate has not taken this variation\nfrom our observers, although it may be less appreciable than in the\ntropics. M. Ramond has recognized and determined it at Clermont, the\nchief place of the district of Puy-de-D\u00f4me, by a series of precise\nobservations made during several years; he has even found that it is\nsmaller in the months of winter than in other months. The numerous\nobservations which I have discussed in order to estimate the influence\nof attractions of the sun and the moon upon the barometric heights at\nParis have served me in determining their diurnal variation. Comparing\nthe heights at nine o'clock in the morning with those of the same days\nat three o'clock in the afternoon, this variation is manifested with so\nmuch evidence that its mean value each month has been constantly\npositive for each of the seventy-two months from January 1, 1817, to\nJanuary 1, 1823; its mean value in these seventy-two months has been\nalmost .8 of a millimeter, a little less than at Clermont and much less\nthan at the equator. I have recognized that the mean result of the\ndiurnal variations of the barometer from 9 o'clock A.M. to 3 P.M. has\nbeen only .5428 millimeter in the three months of November, December,\nJanuary, and that it has risen to 1.0563 millimeters in the three\nfollowing months, which coincides with the observations of M. Ramond.\nThe other months offer nothing similar.\n\nIn order to apply to these phenomena the calculation of these\nprobabilities, I commenced by determining the law of the probability of\nthe anomalies of the diurnal variation due to hazard. Applying it then\nto the observations of this phenomenon, I found that it was a bet of\nmore than 300,000 against one that a regular cause produced it. I do not\nseek to determine this cause; I content myself with stating its\nexistence. The period of the diurnal variation regulated by the solar\nday indicates evidently that this variation is due to the action of the\nsun. The extreme smallness of the attractive action of the sun upon the\natmosphere is proved by the smallness of the effects due to the united\nattractions of the sun and the moon. It is then by the action of its\nheat that the sun produces the diurnal variation of the barometer; but\nit is impossible to subject to calculus the effects of its action on the\nheight of the barometer and upon the winds. The diurnal variation of the\nmagnetic needle is certainly a result of the action of the sun. But does\nthis star act here as in the diurnal variation of the barometer by its\nheat or by its influence upon electricity and upon magnetism, or finally\nby the union of these influences? A long series of observations made in\ndifferent countries will enable us to apprehend this.\n\nOne of the most remarkable phenomena of the system of the world is that\nof all the movements of rotation and of revolution of the planets and\nthe satellites in the sense of the rotation of the sun and about in the\nsame plane of its equator. A phenomenon so remarkable is not the effect\nof hazard: it indicates a general cause which has determined all its\nmovements. In order to obtain the probability with which this cause is\nindicated we shall observe that the planetary system, such as we know it\nto-day, is composed of eleven planets and of eighteen satellites at\nleast, if we attribute with Herschel six satellites to the planet\nUranus. The movements of the rotation of the sun, of six planets, of the\nmoon, of the satellites of Jupiter, of the ring of Saturn, and of one of\nits satellites have been recognized. These movements form with those of\nrevolution a totality of forty-three movements directed in the same\nsense; but one finds by the analysis of probabilities that it is a bet\nof more than 4000000000000 against one that this disposition is not the\nresult of hazard; this forms a probability indeed superior to that of\nhistorical events in regard to which no doubt exists. We ought then to\nbelieve at least with equal confidence that a primitive cause has\ndirected the planetary movements, especially if we consider that the\ninclination of the greatest number of these movements at the solar\nequator is very small.\n\nAnother equally remarkable phenomenon of the solar system is the small\ndegree of the eccentricity of the orbs of the planets and the\nsatellites, while those of the comets are very elongated, the orbs of\nthe system not offering any intermediate shades between a great and a\nsmall eccentricity. We are again forced to recognize here the effect of\na regular cause; chance has certainly not given an almost circular form\nto the orbits of all the planets and their satellites; it is then that\nthe cause which has determined the movements of these bodies has\nrendered them almost circular. It is necessary, again, that the great\neccentricities of the orbits of the comets should result from the\nexistence of this cause without its having influenced the direction of\ntheir movements; for it is found that there are almost as many\nretrograde comets as direct comets, and that the mean inclination of all\ntheir orbits to the ecliptic approaches very nearly half a right angle,\nas it ought to be if the bodies had been thrown at hazard.\n\nWhatever may be the nature of the cause in question, since it has\nproduced or directed the movement of the planets, it is necessary that\nit should have embraced all the bodies and considered all the distances\nwhich separate them, it can have been only a fluid of an immense\nextension. Therefore in order to have given them in the same sense an\nalmost circular movement about the sun it is necessary that this fluid\nshould have surrounded this star as an atmosphere. The consideration of\nthe planetary movements leads us then to think that by virtue of an\nexcessive heat the atmosphere of the sun was originally extended beyond\nthe orbits of all the planets, and that it has contracted gradually to\nits present limits.\n\nIn the primitive state where we imagine the sun it resembled the nebul\u00e6\nthat the telescope shows us composed of a nucleus more or less brilliant\nsurrounded by a nebula which, condensing at the surface, ought to\ntransform it some day into a star. If one conceives by analogy all the\nstars formed in this manner, one can imagine their anterior state of\nnebulosity itself preceded by other stars in which the nebulous matter\nwas more and more diffuse, the nucleus being less and less luminous and\ndense. Going back, then, as far as possible, one would arrive at a\nnebulosity so diffuse that one would be able scarcely to suspect its\nexistence.\n\nSuch is indeed the first state of the nebul\u00e6 which Herschel observed\nwith particular care by means of his powerful telescopes, and in which\nhe has followed the progress of condensation, not in a single one, these\nstages not becoming appreciable to us except after centuries, but in\ntheir totality, just about as one can in a vast forest follow the\nincrease of the trees by the individuals of the divers ages which the\nforest contains. He has observed from the beginning nebulous matter\nspread out in divers masses in the different parts of the heavens, of\nwhich it occupies a great extent. He has seen in some of these masses\nthis matter slightly condensed about one or several faintly luminous\nnebul\u00e6. In the other nebul\u00e6 these nuclei shine, moreover, in proportion\nto the nebulosity which surrounds them. The atmospheres of each nucleus\nbecoming separated by an ulterior condensation, there result the\nmultifold nebul\u00e6 formed of brilliant nuclei very adjacent and surrounded\neach by an atmosphere; sometimes the nebulous matter, by condensing in a\nuniform manner, has produced the nebul\u00e6 which are called _planetary_.\nFinally a greater degree of condensation transforms all these nebul\u00e6\ninto stars. The nebul\u00e6 classed according to this philosophic view\nindicate with an extreme probability their future transformation into\nstars and the anterior state of nebulosity of existing stars. The\nfollowing considerations come to the aid of proofs drawn from these\nanalogies.\n\nFor a long time the particular disposition of certain stars visible to\nthe naked eye has struck the attention of philosophical observers.\nMitchel has already remarked how improbable it is that the stars of the\nPleiades, for example, should have been confined in the narrow space\nwhich contain them by the chances of hazard alone, and he has concluded\nfrom this that this group of stars and the similar groups that the\nheaven presents us are the results of a primitive cause or of a general\nlaw of nature. These groups are a necessary result of the condensation\nof the nebul\u00e6 at several nuclei; it is apparent that the nebulous matter\nbeing attracted continuously by the divers nuclei, they ought to form in\ntime a group of stars equal to that of the Pleiades. The condensation of\nthe nebul\u00e6 at two nuclei forms similarly very adjacent stars, revolving\nthe one about the other, equal to those whose respective movements\nHerschel has already considered. Such are, further, the 61st of the Swan\nand its following one in which Bessel has just recognized particular\nmovements so considerable and so little different that the proximity of\nthese stars to one another and their movement about the common centre of\ngravity ought to leave no doubt. Thus one descends by degrees from the\ncondensation of nebulous matter to the consideration of the sun\nsurrounded formerly by a vast atmosphere, a consideration to which one\nrepasses, as has been seen, by the examination of the phenomena of the\nsolar system. A case so remarkable gives to the existence of this\nanterior state of the sun a probability strongly approaching certainty.\n\nBut how has the solar atmosphere determined the movements of rotation\nand revolution of the planets and the satellites? If these bodies had\npenetrated deeply the atmosphere its resistance would have caused them\nto fall upon the sun; one is then led to believe with much probability\nthat the planets have been formed at the successive limits of the solar\natmosphere which, contracting by the cold, ought to have abandoned in\nthe plane of its equator zones of vapors which the mutual attraction of\ntheir molecules has changed into divers spheroids. The satellites have\nbeen similarly formed by the atmospheres of their respective planets.\n\nI have developed at length in my _Exposition of the System of the World_\nthis hypothesis, which appears to me to satisfy all the phenomena which\nthis system presents us. I shall content myself here with considering\nthat the angular velocity of rotation of the sun and the planets being\naccelerated by the successive condensation of their atmospheres at their\nsurfaces, it ought to surpass the angular velocity of revolution of the\nnearest bodies which revolve about them. Observation has indeed\nconfirmed this with regard to the planets and satellites, and even in\nratio to the ring of Saturn, the duration of whose revolution is .438\ndays, while the duration of the rotation of Saturn is .427 days.\n\nIn this hypothesis the comets are strangers to the planetary system. In\nattaching their formation to that of the nebul\u00e6 they may be regarded as\nsmall nebul\u00e6 at the nuclei, wandering from systems to solar systems, and\nformed by the condensation of the nebulous matter spread out in such\ngreat profusion in the universe. The comets would be thus, in relation\nto our system, as the aerolites are relatively to the Earth, to which\nthey would appear strangers. When these stars become visible to us they\noffer so perfect resemblance to the nebul\u00e6 that they are often\nconfounded with them; and it is only by their movement, or by the\nknowledge of all the nebul\u00e6 confined to that part of the heavens where\nthey appear, that we succeed in distinguishing them. This supposition\nexplains in a happy manner the great extension which the heads and tails\nof comets take in the measure that they approach the sun, and the\nextreme rarity of these tails which, in spite of their immense depth, do\nnot weaken at all appreciably the light of the stars which we look\nacross.\n\nWhen the little nebul\u00e6 come into that part of space where the attraction\nof the sun is predominant, and which we shall call the _sphere of\nactivity_ of this star, it forces them to describe elliptic or\nhyperbolic orbits. But their speed being equally possible in all\ndirections they ought to move indifferently in all the senses and under\nall inclinations of the elliptic, which is conformable to that which has\nbeen observed.\n\nThe great eccentricity of the cometary orbits results again from the\npreceding hypothesis. Indeed if these orbits are elliptical they are\nvery elongated, since their great axes are at least equal to the radius\nof the sphere of activity of the sun. But these orbits may be\nhyperbolic; and if the axes of these hyperbol\u00e6 are not very large in\nproportion to the mean distance from the sun to the earth, the movement\nof the comets which describe them will appear sensibly hyperbolic.\nHowever, of the hundred comets of which we already have the elements,\nnot one has appeared certainly to move in an hyperbola; it is necessary,\nthen, that the chances which give an appreciable hyperbola should be\nextremely rare in proportion to the contrary chances.\n\nThe comets are so small that, in order to become visible, their\nperihelion distance ought to be inconsiderable. Up to the present this\ndistance has surpassed only twice the diameter of the terrestrial orbit,\nand most often it has been below the radius of this orbit. It is\nconceived that, in order to approach so near the sun, their speed at the\nmoment of their entrance into its sphere of activity ought to have a\nmagnitude and a direction confined within narrow limits. In determining\nby the analysis of probabilities the ratio of the chances which, in\nthese limits, give an appreciable hyperbola, to the chances which give\nan orbit which may be confounded with a parabola, I have found that it\nis a bet of at least 6000 against one that a nebula which penetrates\ninto the activity of the sun in such a manner as to be observed will\ndescribe either a very elongated ellipse or an hyperbola. By the\nmagnitude of its axis, the latter will be appreciably confounded with a\nparabola in the part which is observed; it is then not surprising that,\nup to this time, hyperbolic movements have not been recognized.\n\nThe attraction of the planets, and, perhaps further, the resistance of\nthe ethereal centres, ought to have changed many cometary orbits in the\nellipses whose great axis is less than the radius of the sphere of\nactivity of the sun, which augments the chances of the elliptical\norbits. We may believe that this change has taken place with the comet\nof 1759, and with the comet whose duration is only twelve hundred days,\nand which will reappear without ceasing in this short interval, unless\nthe evaporation which it meets at each of its returns to the perihelion\nends by rendering it invisible.\n\nWe are able further, by the analysis of probabilities, to verify the\nexistence or the influence of certain causes whose action is believed to\nexist upon organized beings. Of all the instruments that we are able to\nemploy in order to recognize the imperceptible agents of nature the most\nsensitive are the nerves, especially when particular causes increase\ntheir sensibility. It is by their aid that the feeble electricity which\nthe contact of two heterogeneous metals develops has been discovered;\nthis has opened a vast field to the researches of physicists and\nchemists. The singular phenomena which results from extreme sensibility\nof the nerves in some individuals have given birth to divers opinions\nabout the existence of a new agent which has been named _animal\nmagnetism_, about the action on ordinary magnetism, and about the\ninfluence of the sun and moon in some nervous affections, and finally,\nabout the impressions which the proximity of metals or of running water\nmakes felt. It is natural to think that the action of these causes is\nvery feeble, and that it may be easily disturbed by accidental\ncircumstances; thus because in some cases it is not manifested at all\nits existence ought not to be denied. We are so far from recognizing all\nthe agents of nature and their divers modes of action that it would be\nunphilosophical to deny the phenomena solely because they are\ninexplicable in the present state of our knowledge. But we ought to\nexamine them with an attention as much the more scrupulous as it appears\nthe more difficult to admit them; and it is here that the calculation of\nprobabilities becomes indispensable in determining to just what point it\nis necessary to multiply the observations or the experiences in order to\nobtain in favor of the agents which they indicate, a probability\nsuperior to the reasons which can be obtained elsewhere for not\nadmitting them.\n\nThe calculation of probabilities can make appreciable the advantages and\nthe inconveniences of the methods employed in the speculative sciences.\nThus in order to recognize the best of the treatments in use in the\nhealing of a malady, it is sufficient to test each of them on an equal\nnumber of patients, making all the conditions exactly similar; the\nsuperiority of the most advantageous treatment will manifest itself more\nand more in the measure that the number is increased; and the\ncalculation will make apparent the corresponding probability of its\nadvantage and the ratio according to which it is superior to the others.\n\n\n\n\nCHAPTER X.\n\n_APPLICATION OF THE CALCULUS OF PROBABILITIES TO THE MORAL SCIENCES._\n\n\nWe have just seen the advantages of the analysis of probabilities in the\ninvestigation of the laws of natural phenomena whose causes are unknown\nor so complicated that their results cannot be submitted to calculus.\nThis is the case of nearly all subjects of the moral sciences. So many\nunforeseen causes, either hidden or inappreciable, influence human\ninstitutions that it is impossible to judge _\u00e0 priori_ the results. The\nseries of events which time brings about develops these results and\nindicates the means of remedying those that are harmful. Wise laws have\noften been made in this regard; but because we had neglected to conserve\nthe motives many have been abrogated as useless, and the fact that\nvexatious experiences have made the need felt anew ought to have\nre\u00ebstablished them.\n\nIt is very important to keep in each branch of the public administration\nan exact register of the results which the various means used have\nproduced, and which are so many experiences made on a large scale by\ngovernments. Let us apply to the political and moral sciences the method\nfounded upon observation and upon calculus, the method which has served\nus so well in the natural sciences. Let us not offer in the least a\nuseless and often dangerous resistance to the inevitable effects of the\nprogress of knowledge; but let us change only with an extreme\ncircumspection our institutions and the usages to which we have already\nso long conformed. We should know well by the experience of the past the\ndifficulties which they present; but we are ignorant of the extent of\nthe evils which their change can produce. In this ignorance the theory\nof probability directs us to avoid all change; especially is it\nnecessary to avoid the sudden changes which in the moral world as well\nas in the physical world never operate without a great loss of vital\nforce.\n\nAlready the calculus of probabilities has been applied with success to\nseveral subjects of the moral sciences. I shall present here the\nprincipal results.\n\n\n\n\nCHAPTER XI.\n\n_CONCERNING THE PROBABILITIES OF TESTIMONIES._\n\n\nThe majority of our opinions being founded on the probability of proofs\nit is indeed important to submit it to calculus. Things it is true often\nbecome impossible by the difficulty of appreciating the veracity of\nwitnesses and by the great number of circumstances which accompany the\ndeeds they attest; but one is able in several cases to resolve the\nproblems which have much analogy with the questions which are proposed\nand whose solutions may be regarded as suitable approximations to guide\nand to defend us against the errors and the dangers of false reasoning\nto which we are exposed. An approximation of this kind, when it is well\nmade, is always preferable to the most specious reasonings. Let us try\nthen to give some general rules for obtaining it.\n\nA single number has been drawn from an urn which contains a thousand of\nthem. A witness to this drawing announces that number 79 is drawn; one\nasks the probability of drawing this number. Let us suppose that\nexperience has made known that this witness deceives one time in ten, so\nthat the probability of his testimony is 1/10. Here the event observed\nis the witness attesting that number 79 is drawn. This event may result\nfrom the two following hypotheses, namely: that the witness utters the\ntruth or that he deceives. Following the principle that has been\nexpounded on the probability of causes drawn from events observed it is\nnecessary first to determine _\u00e0 priori_ the probability of the event in\neach hypothesis. In the first, the probability that the witness will\nannounce number 79 is the probability itself of the drawing of this\nnumber, that is to say, 1/1000. It is necessary to multiply it by the\nprobability 6/10 of the veracity of the witness; one will have then\n9/10000 for the probability of the event observed in this hypothesis. If\nthe witness deceives, number 79 is not drawn, and the probability of\nthis case is 999/1000. But to announce the drawing of this number the\nwitness has to choose it among the 999 numbers not drawn; and as he is\nsupposed to have no motive of preference for the ones rather than the\nothers, the probability that he will choose number 79 is 1/999;\nmultiplying, then, this probability by the preceding one, we shall have\n1/1000 for the probability that the witness will announce number 79 in\nthe second hypothesis. It is necessary again to multiply this\nprobability by 1/10 of the hypothesis itself, which gives 1/10000 for\nthe probability of the event relative to this hypothesis. Now if we form\na fraction whose numerator is the probability relative to the first\nhypothesis, and whose denominator is the sum of the probabilities\nrelative to the two hypotheses, we shall have, by the sixth principle,\nthe probability of the first hypothesis, and this probability will be\n9/10; that is to say, the veracity itself of the witness. This is\nlikewise the probability of the drawing of number 79. The probability of\nthe falsehood of the witness and of the failure of drawing this number\nis 1/10.\n\nIf the witness, wishing to deceive, has some interest in choosing number\n79 among the numbers not drawn,\u2014if he judges, for example, that having\nplaced upon this number a considerable stake, the announcement of its\ndrawing will increase his credit, the probability that he will choose\nthis number will no longer be as at first, 1/999, it will then be \u00bd, \u2153,\netc., according to the interest that he will have in announcing its\ndrawing. Supposing it to be 1/9, it will be necessary to multiply by\nthis fraction the probability 999/1000 in order to get in the hypothesis\nof the falsehood the probability of the event observed, which it is\nnecessary still to multiply by 1/10, which gives 111/10000 for the\nprobability of the event in the second hypothesis. Then the probability\nof the first hypothesis, or of the drawing of number 79, is reduced by\nthe preceding rule to 9/120. It is then very much decreased by the\nconsideration of the interest which the witness may have in announcing\nthe drawing of number 79. In truth this same interest increases the\nprobability 9/10 that the witness will speak the truth if number 79 is\ndrawn. But this probability cannot exceed unity or 10/10; thus the\nprobability of the drawing of number 79 will not surpass 10/121. Common\nsense tells us that this interest ought to inspire distrust, but\ncalculus appreciates the influence of it.\n\nThe probability _\u00e0 priori_ of the number announced by the witness is\nunity divided by the number of the numbers in the urn; it is changed by\nvirtue of the proof into the veracity itself of the witness; it may then\nbe decreased by the proof. If, for example, the urn contains only two\nnumbers, which gives \u00bd for the probability _\u00e0 priori_ of the drawing of\nnumber 1, and if the veracity of a witness who announces it is 4/10,\nthis drawing becomes less probable. Indeed it is apparent, since the\nwitness has then more inclination towards a falsehood than towards the\ntruth, that his testimony ought to decrease the probability of the fact\nattested every time that this probability equals or surpasses \u00bd. But if\nthere are three numbers in the urn the probability _\u00e0 priori_ of the\ndrawing of number 1 is increased by the affirmation of a witness whose\nveracity surpasses \u2153.\n\nSuppose now that the urn contains 999 black balls and one white ball,\nand that one ball having been drawn a witness of the drawing announces\nthat this ball is white. The probability of the event observed,\ndetermined _\u00e0 priori_ in the first hypothesis, will be here, as in the\npreceding question, equal to 9/10000. But in the hypothesis where the\nwitness deceives, the white ball is not drawn and the probability of\nthis case is 999/1000. It is necessary to multiply it by the probability\n1/10 of the falsehood, which gives 999/10000 for the probability of the\nevent observed relative to the second hypothesis. This probability was\nonly 1/10000 in the preceding question; this great difference results\nfrom this\u2014that a black ball having been drawn the witness who wishes to\ndeceive has no choice at all to make among the 999 balls not drawn in\norder to announce the drawing of a white ball. Now if one forms two\nfractions whose numerators are the probabilities relative to each\nhypothesis, and whose common denominator is the sum of these\nprobabilities, one will have 9/1008 for the probability of the first\nhypothesis and of the drawing of a white ball, and 999/1008 for the\nprobability of the second hypothesis and of the drawing of a black ball.\nThis last probability strongly approaches certainty; it would approach\nit much nearer and would become 999999/1000008 if the urn contained a\nmillion balls of which one was white, the drawing of a white ball\nbecoming then much more extraordinary. We see thus how the probability\nof the falsehood increases in the measure that the deed becomes more\nextraordinary.\n\nWe have supposed up to this time that the witness was not mistaken at\nall; but if one admits, however, the chance of his error the\nextraordinary incident becomes more improbable. Then in place of the two\nhypotheses one will have the four following ones, namely: that of the\nwitness not deceiving and not being mistaken at all; that of the witness\nnot deceiving at all and being mistaken; the hypothesis of the witness\ndeceiving and not being mistaken at all; finally, that of the witness\ndeceiving and being mistaken. Determining _\u00e0 priori_ in each of these\nhypotheses the probability of the event observed, we find by the sixth\nprinciple the probability that the fact attested is false equal to a\nfraction whose numerator is the number of black balls in the urn\nmultiplied by the sum of the probabilities that the witness does not\ndeceive at all and is mistaken, or that he deceives and is not mistaken,\nand whose denominator is this numerator augmented by the sum of the\nprobabilities that the witness does not deceive at all and is not\nmistaken at all, or that he deceives and is mistaken at the same time.\nWe see by this that if the number of black balls in the urn is very\ngreat, which renders the drawing of the white ball extraordinary, the\nprobability that the fact attested is not true approaches most nearly to\ncertainty.\n\nApplying this conclusion to all extraordinary deeds it results from it\nthat the probability of the error or of the falsehood of the witness\nbecomes as much greater as the fact attested is more extraordinary. Some\nauthors have advanced the contrary on this basis that the view of an\nextraordinary fact being perfectly similar to that of an ordinary fact\nthe same motives ought to lead us to give the witness the same credence\nwhen he affirms the one or the other of these facts. Simple common sense\nrejects such a strange assertion; but the calculus of probabilities,\nwhile confirming the findings of common sense, appreciates the greatest\nimprobability of testimonies in regard to extraordinary facts.\n\nThese authors insist and suppose two witnesses equally worthy of belief,\nof whom the first attests that he saw an individual dead fifteen days\nago whom the second witness affirms to have seen yesterday full of life.\nThe one or the other of these facts offers no improbability. The\nreservation of the individual is a result of their combination; but the\ntestimonies do not bring us at all directly to this result, although the\ncredence which is due these testimonies ought not to be decreased by the\nfact that the result of their combination is extraordinary.\n\nBut if the conclusion which results from the combination of the\ntestimonies was impossible one of them would be necessarily false; but\nan impossible conclusion is the limit of extraordinary conclusions, as\nerror is the limit of improbable conclusions; the value of the\ntestimonies which becomes zero in the case of an impossible conclusion\nought then to be very much decreased in that of an extraordinary\nconclusion. This is indeed confirmed by the calculus of probabilities.\n\nIn order to make it plain let us consider two urns, A and B, of which\nthe first contains a million white balls and the second a million black\nballs. One draws from one of these urns a ball, which he puts back into\nthe other urn, from which one then draws a ball. Two witnesses, the one\nof the first drawing, the other of the second, attest that the ball\nwhich they have seen drawn is white without indicating the urn from\nwhich it has been drawn. Each testimony taken alone is not improbable;\nand it is easy to see that the probability of the fact attested is the\nveracity itself of the witness. But it follows from the combination of\nthe testimonies that a white ball has been extracted from the urn A at\nthe first draw, and that then placed in the urn B it has reappeared at\nthe second draw, which is very extraordinary; for this second urn,\ncontaining then one white ball among a million black balls, the\nprobability of drawing the white ball is 1/1000001. In order to\ndetermine the diminution which results in the probability of the thing\nannounced by the two witnesses we shall notice that the event observed\nis here the affirmation by each of them that the ball which he has seen\nextracted is white. Let us represent by 9/10 the probability that he\nannounces the truth, which can occur in the present case when the\nwitness does not deceive and is not mistaken at all, and when he\ndeceives and is mistaken at the same time. One may form the four\nfollowing hypotheses:\n\n1st. The first and second witness speak the truth. Then a white ball has\nat first been drawn from the urn A, and the probability of this event is\n\u00bd, since the ball drawn at the first draw may have been drawn either\nfrom the one or the other urn. Consequently the ball drawn, placed in\nthe urn B, has reappeared at the second draw; the probability of this\nevent is 1/1000001, the probability of the fact announced is then\n1/2000002. Multiplying it by the product of the probabilities 9/10 and\n9/10 that the witnesses speak the truth one will have 81/200000200 for\nthe probability of the event observed in this first hypothesis.\n\n2d. The first witness speaks the truth and the second does not, whether\nhe deceives and is not mistaken or he does not deceive and is mistaken.\nThen a white ball has been drawn from the urn A at the first draw, and\nthe probability of this event is \u00bd. Then this ball having been placed in\nthe urn B a black ball has been drawn from it: the probability of such\ndrawing is 1000000/1000001; one has then 1000000/2000002 for the\nprobability of the compound event. Multiplying it by the product of the\ntwo probabilities 9/10 and 1/10 that the first witness speaks the truth\nand that the second does not, one will have 9000000/200000200 for the\nprobability for the event observed in the second hypothesis.\n\n3d. The first witness does not speak the truth and the second announces\nit. Then a black ball has been drawn from the urn B at the first\ndrawing, and after having been placed in the urn A a white ball has been\ndrawn from this urn. The probability of the first of these events is \u00bd\nand that of the second is 1000000/1000001; the probability of the\ncompound event is then 1000000/2000002. Multiplying it by the product of\nthe probabilities 1/10 and 9/10 that the first witness does not speak\nthe truth and that the second announces it, one will have\n9000000/200000200 for the probability of the event observed relative to\nthis hypothesis.\n\n4th. Finally, neither of the witnesses speaks the truth. Then a black\nball has been drawn from the urn B at the first draw; then having been\nplaced in the urn A it has reappeared at the second drawing: the\nprobability of this compound event is 1/2000002. Multiplying it by the\nproduct of the probabilities 1/10 and 1/10 that each witness does not\nspeak the truth one will have 1/200000200 for the probability of the\nevent observed in this hypothesis.\n\nNow in order to obtain the probability of the thing announced by the two\nwitnesses, namely, that a white ball has been drawn at each draw, it is\nnecessary to divide the probability corresponding to the first\nhypothesis by the sum of the probabilities relative to the four\nhypotheses; and then one has for this probability 81/18000082, an\nextremely small fraction.\n\nIf the two witnesses affirm the first, that a white ball has been drawn\nfrom one of the two urns A and B; the second that a white ball has been\nlikewise drawn from one of the two urns A\u00b4 and B\u00b4, quite similar to the\nfirst ones, the probability of the thing announced by the two witnesses\nwill be the product of the probabilities of their testimonies, or\n81/100; it will then be at least a hundred and eighty thousand times\ngreater than the preceding one. One sees by this how much, in the first\ncase, the reappearance at the second draw of the white ball drawn at the\nfirst draw, the extraordinary conclusion of the two testimonies\ndecreases the value of it.\n\nWe would give no credence to the testimony of a man who should attest to\nus that in throwing a hundred dice into the air they had all fallen on\nthe same face. If we had ourselves been spectators of this event we\nshould believe our own eyes only after having carefully examined all the\ncircumstances, and after having brought in the testimonies of other eyes\nin order to be quite sure that there had been neither hallucination nor\ndeception. But after this examination we should not hesitate to admit it\nin spite of its extreme improbability; and no one would be tempted, in\norder to explain it, to recur to a denial of the laws of vision. We\nought to conclude from it that the probability of the constancy of the\nlaws of nature is for us greater than this, that the event in question\nhas not taken place at all\u2014a probability greater than that of the\nmajority of historical facts which we regard as incontestable. One may\njudge by this the immense weight of testimonies necessary to admit a\nsuspension of natural laws, and how improper it would be to apply to\nthis case the ordinary rules of criticism. All those who without\noffering this immensity of testimonies support this when making recitals\nof events contrary to those laws, decrease rather than augment the\nbelief which they wish to inspire; for then those recitals render very\nprobable the error or the falsehood of their authors. But that which\ndiminishes the belief of educated men increases often that of the\nuneducated, always greedy for the wonderful.\n\nThere are things so extraordinary that nothing can balance their\nimprobability. But this, by the effect of a dominant opinion, can be\nweakened to the point of appearing inferior to the probability of the\ntestimonies; and when this opinion changes an absurd statement admitted\nunanimously in the century which has given it birth offers to the\nfollowing centuries only a new proof of the extreme influence of the\ngeneral opinion upon the more enlightened minds. Two great men of the\ncentury of Louis XIV.\u2014Racine and Pascal\u2014are striking examples of this.\nIt is painful to see with what complaisance Racine, this admirable\npainter of the human heart and the most perfect poet that has ever\nlived, reports as miraculous the recovery of Mlle. Perrier, a niece of\nPascal and a day pupil at the monastery of Port-Royal; it is painful to\nread the reasons by which Pascal seeks to prove that this miracle should\nbe necessary to religion in order to justify the doctrine of the monks\nof this abbey, at that time persecuted by the Jesuits. The young Perrier\nhad been afflicted for three years and a half by a lachrymal fistula;\nshe touched her afflicted eye with a relic which was pretended to be one\nof the thorns of the crown of the Saviour and she had faith in instant\nrecovery. Some days afterward the physicians and the surgeons attest the\nrecovery, and they declare that nature and the remedies have had no part\nin it. This event, which took place in 1656, made a great sensation, and\n\"all Paris rushed,\" says Racine, \"to Port-Royal. The crowd increased\nfrom day to day, and God himself seemed to take pleasure in authorizing\nthe devotion of the people by the number of miracles which were\nperformed in this church.\" At this time miracles and sorcery did not yet\nappear improbable, and one did not hesitate at all to attribute to them\nthe singularities of nature which could not be explained otherwise.\n\nThis manner of viewing extraordinary results is found in the most\nremarkable works of the century of Louis XIV.; even in the Essay on the\nHuman Understanding by the philosopher Locke, who says, in speaking of\nthe degree of assent: \"Though the common experience and the ordinary\ncourse of things have justly a mighty influence on the minds of men, to\nmake them give or refuse credit to anything proposed to their belief;\nyet there is one case, wherein the strangeness of the fact lessens not\nthe assent to a fair testimony of it. For where such supernatural events\nare suitable to ends aimed at by him who has the power to change the\ncourse of nature, there, under such circumstances, they may be the\nfitter to procure belief, by how much the more they are beyond or\ncontrary to ordinary observation.\" The true principles of the\nprobability of testimonies having been thus misunderstood by\nphilosophers to whom reason is principally indebted for its progress, I\nhave thought it necessary to present at length the results of calculus\nupon this important subject.\n\nThere comes up naturally at this point the discussion of a famous\nargument of Pascal, that Craig, an English mathematician, has produced\nunder a geometric form. Witnesses declare that they have it from\nDivinity that in conforming to a certain thing one will enjoy not one or\ntwo but an infinity of happy lives. However feeble the probability of\nthe proofs may be, provided that it be not infinitely small, it is clear\nthat the advantage of those who conform to the prescribed thing is\ninfinite since it is the product of this probability and an infinite\ngood; one ought not to hesitate then to procure for oneself this\nadvantage.\n\nThis argument is based upon the infinite number of happy lives promised\nin the name of the Divinity by the witnesses; it is necessary then to\nprescribe them, precisely because they exaggerate their promises beyond\nall limits, a consequence which is repugnant to good sense. Also\ncalculus teaches us that this exaggeration itself enfeebles the\nprobability of their testimony to the point of rendering it infinitely\nsmall or zero. Indeed this case is similar to that of a witness who\nshould announce the drawing of the highest number from an urn filled\nwith a great number of numbers, one of which has been drawn and who\nwould have a great interest in announcing the drawing of this number.\nOne has already seen how much this interest enfeebles his testimony. In\nevaluating only at \u00bd the probability that if the witness deceives he\nwill choose the largest number, calculus gives the probability of his\nannouncement as smaller than a fraction whose numerator is unity and\nwhose denominator is unity plus the half of the product of the number of\nthe numbers by the probability of falsehood considered _\u00e0 priori_ or\nindependently of the announcement. In order to compare this case to that\nof the argument of Pascal it is sufficient to represent by the numbers\nin the urn all the possible numbers of happy lives which the number of\nthese numbers renders infinite; and to observe that if the witnesses\ndeceive they have the greatest interest, in order to accredit their\nfalsehood, in promising an eternity of happiness. The expression of the\nprobability of their testimony becomes then infinitely small.\nMultiplying it by the infinite number of happy lives promised, infinity\nwould disappear from the product which expresses the advantage resultant\nfrom this promise which destroys the argument of Pascal.\n\nLet us consider now the probability of the totality of several\ntestimonies upon an established fact. In order to fix our ideas let us\nsuppose that the fact be the drawing of a number from an urn which\ncontains a hundred of them, and of which one single number has been\ndrawn. Two witnesses of this drawing announce that number 2 has been\ndrawn, and one asks for the resultant probability of the totality of\nthese testimonies. One may form these two hypotheses: the witnesses\nspeak the truth; the witnesses deceive. In the first hypothesis the\nnumber 2 is drawn and the probability of this event is 1/100. It is\nnecessary to multiply it by the product of the veracities of the\nwitnesses, veracities which we will suppose to be 9/10 and 7/10: one\nwill have then 63/10000 for the probability of the event observed in\nthis hypothesis. In the second, the number 2 is not drawn and the\nprobability of this event is 99/100. But the agreement of the witnesses\nrequires then that in seeking to deceive they both choose the number 2\nfrom the 99 numbers not drawn: the probability of this choice if the\nwitnesses do not have a secret agreement is the product of the fraction\n1/99 by itself; it becomes necessary then to multiply these two\nprobabilities together, and by the product of the probabilities 1/10 and\n3/10 that the witnesses deceive; one will have thus 1/330000 for the\nprobability of the event observed in the second hypothesis. Now one will\nhave the probability of the fact attested or of the drawing of number 2\nin dividing the probability relative to the first hypothesis by the sum\nof the probabilities relative to the two hypotheses; this probability\nwill be then 2079/2080, and the probability of the failure to draw this\nnumber and of the falsehood of the witnesses will be 1/2080.\n\nIf the urn should contain only the numbers 1 and 2 one would find in the\nsame manner 21/22 for the probability of the drawing of number 2, and\nconsequently 1/22 for the probability of the falsehood of the witnesses,\na probability at least ninety-four times larger than the preceding one.\nOne sees by this how much the probability of the falsehood of the\nwitnesses diminishes when the fact which they attest is less probable in\nitself. Indeed one conceives that then the accord of the witnesses, when\nthey deceive, becomes more difficult, at least when they do not have a\nsecret agreement, which we do not suppose here at all.\n\nIn the preceding case where the urn contained only two numbers the _\u00e0\npriori_ probability of the fact attested is \u00bd, the resultant probability\nof the testimonies is the product of the veracities of the witnesses\ndivided by this product added to that of the respective probabilities of\ntheir falsehood.\n\nIt now remains for us to consider the influence of time upon the\nprobability of facts transmitted by a traditional chain of witnesses. It\nis clear that this probability ought to diminish in proportion as the\nchain is prolonged. If the fact has no probability itself, such as the\ndrawing of a number from an urn which contains an infinity of them, that\nwhich it acquires by the testimonies decreases according to the\ncontinued product of the veracity of the witnesses. If the fact has a\nprobability in itself; if, for example, this fact is the drawing of the\nnumber 2 from an urn which contains an infinity of them, and of which it\nis certain that one has drawn a single number; that which the\ntraditional chain adds to this probability decreases, following a\ncontinued product of which the first factor is the ratio of the number\nof numbers in the urn less one to the same number, and of which each\nother factor is the veracity of each witness diminished by the ratio of\nthe probability of his falsehood to the number of the numbers in the urn\nless one; so that the limit of the probability of the fact is that of\nthis fact considered _\u00e0 priori_, or independently of the testimonies, a\nprobability equal to unity divided by the number of the numbers in the\nurn.\n\nThe action of time enfeebles then, without ceasing, the probability of\nhistorical facts just as it changes the most durable monuments. One can\nindeed diminish it by multiplying and conserving the testimonies and the\nmonuments which support them. Printing offers for this purpose a great\nmeans, unfortunately unknown to the ancients. In spite of the infinite\nadvantages which it procures the physical and moral revolutions by which\nthe surface of this globe will always be agitated will end, in\nconjunction with the inevitable effect of time, by rendering doubtful\nafter thousands of years the historical facts regarded to-day as the\nmost certain.\n\nCraig has tried to submit to calculus the gradual enfeebling of the\nproofs of the Christian religion; supposing that the world ought to end\nat the epoch when it will cease to be probable, he finds that this ought\nto take place 1454 years after the time when he writes. But his analysis\nis as faulty as his hypothesis upon the duration of the moon is bizarre.\n\n\n\n\nCHAPTER XII.\n\n_CONCERNING THE SELECTIONS AND THE DECISIONS OF ASSEMBLIES._\n\n\nThe probability of the decisions of an assembly depends upon the\nplurality of votes, the intelligence and the impartiality of the members\nwho compose it. So many passions and particular interests so often add\ntheir influence that it is impossible to submit this probability to\ncalculus. There are, however, some general results dictated by simple\ncommon sense and confirmed by calculus. If, for example, the assembly is\npoorly informed about the subject submitted to its decision, if this\nsubject requires delicate considerations, or if the truth on this point\nis contrary to established prejudices, so that it would be a bet of more\nthan one against one that each voter will err; then the decision of the\nmajority will be probably wrong, and the fear of it will be the better\nbased as the assembly is more numerous. It is important then, in public\naffairs, that assemblies should have to pass upon subjects within reach\nof the greatest number; it is important for them that information be\ngenerally diffused and that good works founded upon reason and\nexperience should enlighten those who are called to decide the lot of\ntheir fellows or to govern them, and should forewarn them against false\nideas and the prejudices of ignorance. Scholars have had frequent\noccasion to remark that first conceptions often deceive and that the\ntruth is not always probable.\n\nIt is difficult to understand and to define the desire of an assembly in\nthe midst of a variety of opinions of its members. Let us attempt to\ngive some rules in regard to this matter by considering the two most\nordinary cases: the election among several candidates, and that among\nseveral propositions relative to the same subject.\n\nWhen an assembly has to choose among several candidates who present\nthemselves for one or for several places of the same kind, that which\nappears simplest is to have each voter write upon a ticket the names of\nall the candidates according to the order of merit that he attributes to\nthem. Supposing that he classifies them in good faith, the inspection of\nthese tickets will give the results of the elections in such a manner\nthat the candidates may be compared among themselves; so that new\nelections can give nothing more in this regard. It is a question now to\nconclude the order of preference which the tickets establish among the\ncandidates. Let us imagine that one gives to each voter an urn which\ncontains an infinity of balls by means of which he is able to shade all\nthe degrees of merit of the candidates; let us conceive again that he\ndraws from his urn a number of balls proportional to the merit of each\ncandidate, and let us suppose this number written upon a ticket at the\nside of the name of the candidate. It is clear that by making a sum of\nall the numbers relative to each candidate upon each ticket, that one of\nall the candidates who shall have the largest sum will be the candidate\nwhom the assembly prefers; and that in general the order of preference\nof the candidates will be that of the sums relative to each of them. But\nthe tickets do not mark at all the number of balls which each voter\ngives to the candidates; they indicate solely that the first has more of\nthem than the second, the second more than the third, and so on. In\nsupposing then at first upon a given ticket a certain number of balls\nall the combinations of the inferior numbers which fulfil the preceding\nconditions are equally admissible; and one will have the number of balls\nrelative to each candidate by making a sum of all the numbers which each\ncombination gives him and dividing it by the entire number of\ncombinations. A very simple analysis shows that the numbers which must\nbe written upon each ticket at the side of the last name, of the one\nbefore the last, etc., are proportional to the terms of the arithmetical\nprogression 1, 2, 3, etc. Writing then thus upon each ticket the terms\nof this progression, and adding the terms relative to each candidate\nupon these tickets, the divers sums will indicate by their magnitude the\norder of their preference which ought to be established among the\ncandidates. Such is the mode of election which The Theory of\nProbabilities indicates. Without doubt it would be better if each voter\nshould write upon his ticket the names of the candidates in the order of\nmerit which he attributes to them. But particular interests and many\nstrange considerations of merit would affect this order and place\nsometimes in the last rank the candidate most formidable to that one\nwhom one prefers, which gives too great an advantage to the candidates\nof mediocre merit. Likewise experience has caused the abandonment of\nthis mode of election in the societies which had adopted it.\n\nThe election by the absolute majority of the suffrages unites to the\ncertainty of not admitting any one of the candidates whom this majority\nrejects, the advantage of expressing most often the desire of the\nassembly. It always coincides with the preceding mode when there are\nonly two candidates. Indeed it exposes an assembly to the inconvenience\nof rendering elections interminable. But experience has shown that this\ninconvenience is nil, and that the general desire to put an end to\nelections soon unites the majority of the suffrages upon one of the\ncandidates.\n\nThe choice among several propositions relative to the same object ought\nto be subjected, seemingly, to the same rules as the election among\nseveral candidates. But there exists between the two cases this\ndifference, namely, that the merit of a candidate does not exclude that\nof his competitors; but if it is necessary to choose among propositions\nwhich are contrary, the truth of the one excludes the truth of the\nothers. Let us see how one ought then to view this question.\n\nLet us give to each voter an urn which contains an infinite number of\nballs, and let us suppose that he distributes them upon the divers\npropositions according to the respective probabilities which he\nattributes to them. It is clear that the total number of balls\nexpressing certainty, and the voter being by the hypothesis assured that\none of the propositions ought to be true, he will distribute this number\nat length upon the propositions. The problem is reduced then to this,\nnamely, to determine the combinations in which the balls will be\ndistributed in such a manner that there may be more of them upon the\nfirst proposition of the ticket than upon the second, more upon the\nsecond than upon the third, etc.; to make the sums of all the numbers of\nballs relative to each proposition in the divers combinations, and to\ndivide this sum by the number of combinations; the quotients will be the\nnumbers of balls that one ought to attribute to the propositions upon a\ncertain ticket. One finds by analysis that in going from the last\nproposition these quotients are among themselves as the following\nquantities: first, unity divided by the number of propositions; second,\nthe preceding quantity, augmented by unity, divided by the number of\npropositions less one; third, this second quantity, augmented by unity,\ndivided by the number of propositions less two, and so on for the\nothers. One will write then upon each ticket these quantities at the\nside of the corresponding propositions, and adding the relative\nquantities to each proposition upon the divers tickets the sums will\nindicate by their magnitude the order of preference which the assembly\ngives to these propositions.\n\nLet us speak a word about the manner of renewing assemblies which should\nchange in totality in a definite number of years. Ought the renewal to\nbe made at one time, or is it advantageous to divide it among these\nyears? According to the last method the assembly would be formed under\nthe influence of the divers opinions dominant during the time of its\nrenewal; the opinion which obtained then would be probably the mean of\nall these opinions. The assembly would receive thus at the time the same\nadvantage that is given to it by the extension of the elections of its\nmembers to all parts of the territory which it represents. Now if one\nconsiders what experience has only too clearly taught, namely, that\nelections are always directed in the greatest degree by dominant\nopinions, one will feel how useful it is to temper these opinions, the\nones by the others, by means of a partial renewal.\n\n\n\n\nCHAPTER XIII.\n\n_CONCERNING THE PROBABILITY OF THE JUDGMENTS OF TRIBUNALS._\n\n\nAnalysis confirms what simple common sense teaches us, namely, the\ncorrectness of judgments is as much more probable as the judges are more\nnumerous and more enlightened. It is important then that tribunals of\nappeal should fulfil these two conditions. The tribunals of the first\ninstance standing in closer relation to those amenable offer to the\nhigher tribunal the advantage of a first judgment already probable, and\nwith which the latter often agree, be it in compromising or in desisting\nfrom their claims. But if the uncertainty of the matter in litigation\nand its importance determine a litigant to have recourse to the tribunal\nof appeals, he ought to find in a greater probability of obtaining an\nequitable judgment greater security for his fortune and the compensation\nfor the trouble and expense which a new procedure entails. It is this\nwhich had no place in the institution of the reciprocal appeal of the\ntribunals of the district, an institution thereby very prejudicial to\nthe interest of the citizens. It would be perhaps proper and conformable\nto the calculus of probabilities to demand a majority of at least two\nvotes in a tribunal of appeal in order to invalidate the sentence of the\nlower tribunal. One would obtain this result if the tribunal of appeal\nbeing composed of an even number of judges the sentence should stand in\nthe case of the equality of votes.\n\nI shall consider particularly the judgments in criminal matters.\n\nIn order to condemn an accused it is necessary without doubt that the\njudges should have the strongest proofs of his offence. But a moral\nproof is never more than a probability; and experience has only too\nclearly shown the errors of which criminal judgments, even those which\nappear to be the most just, are still susceptible. The impossibility of\namending these errors is the strongest argument of the philosophers who\nhave wished to proscribe the penalty of death. We should then be obliged\nto abstain from judging if it were necessary for us to await\nmathematical evidence. But the judgment is required by the danger which\nwould result from the impunity of the crime. This judgment reduces\nitself, if I am not mistaken, to the solution of the following question:\nHas the proof of the offence of the accused the high degree of\nprobability necessary so that the citizens would have less reason to\ndoubt the errors of the tribunals, if he is innocent and condemned, than\nthey would have to fear his new crimes and those of the unfortunate ones\nwho would be emboldened by the example of his impunity if he were guilty\nand acquitted? The solution of this question depends upon several\nelements very difficult to ascertain. Such is the eminence of danger\nwhich would threaten society if the criminal accused should remain\nunpunished. Sometimes this danger is so great that the magistrate sees\nhimself constrained to waive forms wisely established for the protection\nof innocence. But that which renders almost always this question\ninsoluble is the impossibility of appreciating exactly the probability\nof the offence and of fixing that which is necessary for the\ncondemnation of the accused. Each judge in this respect is forced to\nrely upon his own judgment. He forms his opinion by comparing the divers\ntestimonies and the circumstances by which the offence is accompanied,\nto the results of his reflections and his experiences, and in this\nrespect a long habitude of interrogating and judging accused persons\ngives great advantage in ascertaining the truth in the midst of indices\noften contradictory.\n\nThe preceding question depends again upon the care taken in the\ninvestigation of the offence; for one demands naturally much stronger\nproofs for imposing the death penalty than for inflicting a detention of\nsome months. It is a reason for proportioning the care to the offence,\ngreat care taken with an unimportant case inevitably clearing many\nguilty ones. A law which gives to the judges power of moderating the\ncare in the case of attenuating circumstances is then conformable at the\nsame time to principles of humanity towards the culprit, and to the\ninterest of society. The product of the probability of the offence by\nits gravity being the measure of the danger to which the acquittal of\nthe accused can expose society, one would think that the care taken\nought to depend upon this probability. This is done indirectly in the\ntribunals where one retains for some time the accused against whom there\nare very strong proofs, but insufficient to condemn him; in the hope of\nacquiring new light one does not place him immediately in the midst of\nhis fellow citizens, who would not see him again without great alarm.\nBut the arbitrariness of this measure and the abuse which one can make\nof it have caused its rejection in the countries where one attaches the\ngreatest price to individual liberty.\n\nNow what is the probability that the decision of a tribunal which can\ncondemn only by a given majority will be just, that is to say, conform\nto the true solution of the question proposed above? This important\nproblem well solved will give the means of comparing among themselves\nthe different tribunals. The majority of a single vote in a numerous\ntribunal indicates that the affair in question is very doubtful; the\ncondemnation of the accused would be then contrary to the principles of\nhumanity, protectors of innocence. The unanimity of the judges would\ngive very strong probability of a just decision; but in abstaining from\nit too many guilty ones would be acquitted. It is necessary, then,\neither to limit the number of judges, if one wishes that they should be\nunanimous, or increase the majority necessary for a condemnation, when\nthe tribunal becomes more numerous. I shall attempt to apply calculus to\nthis subject, being persuaded that it is always the best guide when one\nbases it upon the data which common sense suggests to us.\n\nThe probability that the opinion of each judge is just enters as the\nprincipal element into this calculation. If in a tribunal of a thousand\nand one judges, five hundred and one are of one opinion, and five\nhundred are of the contrary opinion, it is apparent that the probability\nof the opinion of each judge surpasses very little \u00bd; for supposing it\nobviously very large a single vote of difference would be an improbable\nevent. But if the judges are unanimous, this indicates in the proofs\nthat degree of strength which entails conviction; the probability of the\nopinion of each judge is then very near unity or certainty, provided\nthat the passions or the ordinary prejudices do not affect at the same\ntime all the judges. Outside of these cases the ratio of the votes for\nor against the accused ought alone to determine this probability. I\nsuppose thus that it can vary from \u00bd to unity, but that it cannot be\nbelow \u00bd. If that were not the case the decision of the tribunal would be\nas insignificant as chance; it has value only in so far as the opinion\nof the judge has a greater tendency to truth than to error. It is thus\nby the ratio of the numbers of votes favorable, and contrary to the\naccused, that I determine the probability of this opinion.\n\nThese data suffice to ascertain the general expression of the\nprobability that the decision of a tribunal judging by a known majority\nis just. In the tribunals where of eight judges five votes would be\nnecessary for the condemnation of an accused, the probability of the\nerror to be feared in the justice of the decision would surpass \u00bc. If\nthe tribunal should be reduced to six members who are able to condemn\nonly by a plurality of four votes, the probability of the error to be\nfeared would be below \u00bc. There would be then for the accused an\nadvantage in this reduction of the tribunal. In both cases the majority\nrequired is the same and is equal to two. Thus the majority remaining\nconstant, the probability of error increases with the number of judges;\nthis is general whatever may be the majority required, provided that it\nremains the same. Taking, then, for the rule the arithmetical ratio, the\naccused finds himself in a position less and less advantageous in the\nmeasure that the tribunal becomes more numerous. One might believe that\nin a tribunal where one might demand a majority of twelve votes,\nwhatever the number of the judges was, the votes of the minority,\nneutralizing an equal number of votes of the majority, the twelve\nremaining votes would represent the unanimity of a jury of twelve\nmembers, required in England for the condemnation of an accused; but one\nwould be greatly mistaken. Common sense shows that there is a difference\nbetween the decision of a tribunal of two hundred and twelve judges, of\nwhich one hundred and twelve condemn the accused, while one hundred\nacquit him, and that of a tribunal of twelve judges unanimous for\ncondemnation. In the first case the hundred votes favorable to the\naccused warrant in thinking that the proofs are far from attaining the\ndegree of strength which entails conviction; in the second case, the\nunanimity of the judges leads to the belief that they have attained this\ndegree. But simple common sense does not suffice at all to appreciate\nthe extreme difference of the probability of error in the two cases. It\nis necessary then to recur to calculus, and one finds nearly one fifth\nfor the probability of error in the first case, and only 1/8192 for this\nprobability in the second case, a probability which is not one\nthousandth of the first. It is a confirmation of the principle that the\narithmetical ratio is unfavorable to the accused when the number of\njudges increases. On the contrary, if one takes for a rule the\ngeometrical ratio, the probability of the error of the decision\ndiminishes when the number of judges increases. For example, in the\ntribunals which can condemn only by a plurality of two thirds of the\nvotes, the probability of the error to be feared is nearly one fourth if\nthe number of the judges is six; it is below 1/7 if this number is\nincreased to twelve. Thus one ought to be governed neither by the\narithmetical ratio nor by the geometrical ratio if one wishes that the\nprobability of error should never be above nor below a given fraction.\n\nBut what fraction ought to be determined upon? It is here that the\narbitrariness begins and the tribunals offer in this regard the greatest\nvariety. In the special tribunals where five of the eight votes suffice\nfor the condemnation of the accused, the probability of the error to be\nfeared in regard to justice of the judgment is 65/256, or more than \u00bc.\nThe magnitude of this fraction is dreadful; but that which ought to\nreassure us a little is the consideration that most frequently the judge\nwho acquits an accused does not regard him as innocent; he pronounces\nsolely that it is not attained by proofs sufficient for condemnation.\nOne is especially reassured by the pity which nature has placed in the\nheart of man and which disposes the mind to see only with reluctance a\nculprit in the accused submitted to his judgment. This sentiment, more\nactive in those who have not the habitude of criminal judgments,\ncompensates for the inconveniences attached to the inexperience of the\njurors. In a jury of twelve members, if the plurality demanded for the\ncondemnation is eight of twelve votes, the probability of the error to\nbe feared 1093/8192, or a little more than one eighth, it is almost 1/22\nif this plurality consists of nine votes. In the case of unanimity the\nprobability of the error to be feared is 1/8192, that is to say, more\nthan a thousand times less than in our juries. This supposes that the\nunanimity results only from proofs favorable or contrary to the accused;\nbut motives that are entirely strange, ought oftentimes to concur in\nproducing it, when it is imposed upon the jury as a necessary condition\nof its judgment. Then its decisions depending upon the temperament, the\ncharacter, the habits of the jurors, and the circumstances in which they\nare placed, they are sometimes contrary to the decisions which the\nmajority of the jury would have made if they had listened only to the\nproofs; this seems to me to be a great fault of this manner of judging.\n\nThe probability of the decision is too feeble in our juries, and I think\nthat in order to give a sufficient guarantee to innocence, one ought to\ndemand at least a plurality of nine votes in twelve.\n\n\n\n\nCHAPTER XIV.\n\n_CONCERNING TABLES OF MORTALITY, AND OF MEAN DURATIONS OF LIFE, OF\nMARRIAGES, AND OF ASSOCIATIONS._\n\n\nThe manner of preparing tables of mortality is very simple. One takes in\nthe civil registers a great number of individuals whose birth and death\nare indicated. One determines how many of these individuals have died in\nthe first year of their age, how many in the second year, and so on. It\nis concluded from these the number of individuals living at the\ncommencement of each year, and this number is written in the table at\nthe side of that which indicates the year. Thus one writes at the side\nof zero the number of births; at the side of the year 1 the number of\ninfants who have attained one year; at the side of the year 2 the number\nof infants who have attained two years, and so on for the rest. But\nsince in the first two years of life the mortality is very great, it is\nnecessary for the sake of greater exactitude to indicate in this first\nage the number of survivors at the end of each half year.\n\nIf we divide the sum of the years of the life of all the individuals\ninscribed in a table of mortality by the number of these individuals we\nshall have the mean duration of life which corresponds to this table.\nFor this, we will multiply by a half year the number of deaths in the\nfirst year, a number equal to the difference of the numbers of\nindividuals inscribed at the side of the years 0 and 1. Their mortality\nbeing distributed over the entire year the mean duration of their life\nis only a half year. We will multiply by a year and a half the number of\ndeaths in the second year; by two years and a half the number of deaths\nin the third year; and so on. The sum of these products divided by the\nnumber of births will be the mean duration of life. It is easy to\nconclude from this that we will obtain this duration, by making the sum\nof the numbers inscribed in the table at the side of each year, dividing\nit by the number of births and subtracting one half from the quotient,\nthe year being taken as unity. The mean duration of life that remains,\nstarting from any age, is determined in the same manner, working upon\nthe number of individuals who have arrived at this age, as has just been\ndone with the number of births. But it is not at the moment of birth\nthat the mean duration of life is the greatest; it is when one has\nescaped the dangers of infancy and it is then about forty-three years.\nThe probability of arriving at a certain age, starting from a given age\nis equal to the ratio of the two numbers of individuals indicated in the\ntable at these two ages.\n\nThe precision of these results demands that for the formation of tables\nwe should employ a very great number of births. Analysis gives then very\nsimple formul\u00e6 for appreciating the probability that the numbers\nindicated in these tables will vary from the truth only within narrow\nlimits. We see by these formul\u00e6 that the interval of the limits\ndiminishes and that the probability increases in proportion as we take\ninto consideration more births; so that the tables would represent\nexactly the true law of mortality if the number of births employed were\ninfinite.\n\nA table of mortality is then a table of the probability of human life.\nThe ratio of the individuals inscribed at the side of each year to the\nnumber of births is the probability that a new birth will attain this\nyear. As we estimate the value of hope by making a sum of the products\nof each benefit hoped for, by the probability of obtaining it, so we can\nequally evaluate the mean duration of life by adding the products of\neach year by half the sum of the probabilities of attaining the\ncommencement and the end of it, which leads to the result found above.\nBut this manner of viewing the mean duration of life has the advantage\nof showing that in a stationary population, that is to say, such that\nthe number of births equals that of deaths, the mean duration of life is\nthe ratio itself of the population to the annual births; for the\npopulation being supposed stationary, the number of individuals of an\nage comprised between two consecutive years of the table is equal to the\nnumber of annual births, multiplied by half the sum of the probabilities\nof attaining these years; the sum of all these products will be then the\nentire population. Now it is easy to see that this sum, divided by the\nnumber of annual births, coincides with the mean duration of life as we\nhave just defined it.\n\nIt is easy by means of a table of mortality to form the corresponding\ntable of the population supposed to be stationary. For this we take the\narithmetical means of the numbers of the table of mortality\ncorresponding to the ages zero and one year, one and two years, two and\nthree years, etc. The sum of all these means is the entire population;\nit is written at the side of the age zero. There is subtracted from this\nsum the first mean and the remainder is the number of individuals of one\nyear and upwards; it is written at the side of the year 1. There is\nsubtracted from this first remainder the second mean; this second\nremainder is the number of individuals of two years and upwards; it is\nwritten at the side of the year 2, and so on.\n\nSo many variable causes influence mortality that the tables which\nrepresent it ought to be changed according to place and time. The divers\nstates of life offer in this regard appreciable differences relative to\nthe fatigues and the dangers inseparable from each state and of which it\nis indispensable to keep account in the calculations founded upon the\nduration of life. But these differences have not been sufficiently\nobserved. Some day they will be and then will be known what sacrifice of\nlife each profession demands and one will profit by this knowledge to\ndiminish the dangers.\n\nThe greater or less salubrity of the soil, its elevation, its\ntemperature, the customs of the inhabitants, and the operations of\ngovernments have a considerable influence upon mortality. But it is\nalways necessary to precede the investigation of the cause of the\ndifferences observed by that of the probability with which this cause is\nindicated. Thus the ratio of the population to annual births, which one\nhas seen raised in France to twenty-eight and one third, is not equal to\ntwenty-five in the ancient duchy of Milan. These ratios, both\nestablished upon a great number of births, do not permit of calling into\nquestion the existence among the Milanese of a special cause of\nmortality, which it is of moment for the government of our country to\ninvestigate and remove.\n\nThe ratio of the population to the births would increase again if we\ncould diminish and remove certain dangerous and widely spread maladies.\nThis has happily been done for the smallpox, at first by the inoculation\nof this disease, then in a manner much more advantageous, by the\ninoculation of vaccine, the inestimable discovery of Jenner, who has\nthereby become one of the greatest benefactors of humanity.\n\nThe smallpox has this in particular, namely, that the same individual is\nnot twice affected by it, or at least such cases are so rare that they\nmay be abstracted from the calculation. This malady, from which few\nescaped before the discovery of vaccine, is often fatal and causes the\ndeath of one seventh of those whom it attacks. Sometimes it is mild, and\nexperience has taught that it can be given this latter character by\ninoculating it upon healthy persons, prepared for it by a proper diet\nand in a favorable season. Then the ratio of the individuals who die to\nthe inoculated ones is not one three hundredth. This great advantage of\ninoculation, joined to those of not altering the appearance and of\npreserving from the grievous consequences which the natural smallpox\noften brings, caused it to be adopted by a great number of persons. The\npractice was strongly recommended, but it was strongly combated, as is\nnearly always the case in things subject to inconvenience. In the midst\nof this dispute Daniel Bernoulli proposed to submit to the calculus of\nprobabilities the influence of inoculation upon the mean duration of\nlife. Since precise data of the mortality produced by the smallpox at\nthe various ages of life were lacking, he supposed that the danger of\nhaving this malady and that of dying of it are the same at every age. By\nmeans of these suppositions he succeeded by a delicate analysis in\nconverting an ordinary table of mortality into that which would be used\nif smallpox did not exist, or if it caused the death of only a very\nsmall number of those affected, and he concludes from it that\ninoculation would augment by three years at least the mean duration of\nlife, which appeared to him beyond doubt the advantage of this\noperation. D'Alembert attacked the analysis of Bernoulli: at first in\nregard to the uncertainty of his two hypotheses, then in regard to its\ninsufficiency in this, that no comparison was made of the immediate\ndanger, although very small, of dying of inoculation, to the very great\nbut very remote danger of succumbing to natural smallpox. This\nconsideration, which disappears when one considers a great number of\nindividuals, is for this reason immaterial for governments and the\nadvantages of inoculation for them still remain; but it is of great\nweight for the father of a family who must fear, in having his children\ninoculated, to see that one perish whom he holds most dear and to be the\ncause of it. Many parents were restrained by this fear, which the\ndiscovery of vaccine has happily dissipated. By one of those mysteries\nwhich nature offers to us so frequently, vaccine is a preventive of\nsmallpox just as certain as variolar virus, and there is no danger at\nall; it does not expose to any malady and demands only very little care.\nTherefore the practice of it has spread quickly; and to render it\nuniversal it remains only to overcome the natural inertia of the people,\nagainst which it is necessary to strive continually, even when it is a\nquestion of their dearest interests.\n\nThe simplest means of calculating the advantage which the extinction of\na malady would produce consists in determining by observation the number\nof individuals of a given age who die of it each year and subtracting\nthis number from the number of deaths at the same age. The ratio of the\ndifference to the total number of individuals of the given age would be\nthe probability of dying in the year at this age if the malady did not\nexist. Making, then, a sum of these probabilities from birth up to any\ngiven age, and subtracting this sum from unity, the remainder will be\nthe probability of living to that age corresponding to the extinction of\nthe malady. The series of these probabilities will be the table of\nmortality relative to this hypothesis, and we may conclude from it, by\nwhat precedes, the mean duration of life. It is thus that Duvilard has\nfound that the increase of the mean duration of life, due to inoculation\nwith vaccine, is three years at the least. An increase so considerable\nwould produce a very great increase in the population if the latter, for\nother reasons, were not restrained by the relative diminution of\nsubsistences.\n\nIt is principally by the lack of subsistences that the progressive march\nof the population is arrested. In all kinds of animals and vegetables,\nnature tends without ceasing to augment the number of individuals until\nthey are on a level of the means of subsistence. In the human race moral\ncauses have a great influence upon the population. If easy clearings of\nthe forest can furnish an abundant nourishment for new generations, the\ncertainty of being able to support a numerous family encourages\nmarriages and renders them more productive. Upon the same soil the\npopulation and the births ought to increase at the same time\nsimultaneously in geometric progression. But when clearings become more\ndifficult and more rare then the increase of population diminishes; it\napproaches continually the variable state of subsistences, making\noscillations about it just as a pendulum whose periodicity is retarded\nby changing the point of suspension, oscillates about this point by\nvirtue of its own weight. It is difficult to evaluate the _maximum_\nincrease of the population; it appears after observations that in\nfavorable circumstances the population of the human race would be\ndoubled every fifteen years. We estimate that in North America the\nperiod of this doubling is twenty-two years. In this state of things,\nthe population, births, marriages, mortality, all increase according to\nthe same geometric progression of which we have the constant ratio of\nconsecutive terms by the observation of annual births at two epochs.\n\nBy means of a table of mortality representing the probabilities of human\nlife, we may determine the duration of marriages. Supposing in order to\nsimplify the matter that the mortality is the same for the two sexes, we\nshall obtain the probability that the marriage will subsist one year, or\ntwo, or three, etc., by forming a series of fractions whose common\ndenominator is the product of the two numbers of the table corresponding\nto the ages of the consorts, and whose numerators are the successive\nproducts of the numbers corresponding to these ages augmented by one, by\ntwo, by three, etc., years. The sum of these fractions augmented by one\nhalf will be the mean duration of marriage, the year being taken as\nunity. It is easy to extend the same rule to the mean duration of an\nassociation formed of three or of a greater number of individuals.\n\n\n\n\nCHAPTER XV.\n\n_CONCERNING THE BENEFITS OF INSTITUTIONS WHICH DEPEND UPON THE\nPROBABILITY OF EVENTS._\n\n\nLet us recall here what has been said in speaking of hope. It has been\nseen that in order to obtain the advantage which results from several\nsimple events, of which the ones produce a benefit and the others a\nloss, it is necessary to add the products of the probability of each\nfavorable event by the benefit which it procures, and subtract from\ntheir sum that of the products of the probability of each unfavorable\nevent by the loss which is attached to it. But whatever may be the\nadvantage expressed by the difference of these sums, a single event\ncomposed of these simple events does not guarantee against the fear of\nexperiencing a loss. One imagines that this fear ought to decrease when\none multiplies the compound event. The analysis of probabilities leads\nto this general theorem.\n\nBy the repetition of an advantageous event, simple or compound, the real\nbenefit becomes more and more probable and increases without ceasing; it\nbecomes certain in the hypothesis of an infinite number of repetitions;\nand dividing it by this number the quotient or the mean benefit of each\nevent is the mathematical hope itself or the advantage relative to the\nevent. It is the same with a loss which becomes certain in the long run,\nhowever small the disadvantage of the event may be.\n\nThis theorem upon benefits and losses is analogous to those which we\nhave already given upon the ratios which are indicated by the indefinite\nrepetition of events simple or compound; and, like them, it proves that\nregularity ends by establishing itself even in the things which are most\nsubordinated to that which we name _hazard_.\n\nWhen the events are in great number, analysis gives another very simple\nexpression of the probability that the benefit will be comprised within\ndetermined limits. This is the expression which enters again into the\ngeneral law of probability given above in speaking of the probabilities\nwhich result from the indefinite multiplication of events.\n\nThe stability of institutions which are based upon probabilities depends\nupon the truth of the preceding theorem. But in order that it may be\napplied to them it is necessary that those institutions should multiply\nthese advantageous events for the sake of numerous things.\n\nThere have been based upon the probabilities of human life divers\ninstitutions, such as life annuities and tontines. The most general and\nthe most simple method of calculating the benefits and the expenses of\nthese institutions consists in reducing these to actual amounts. The\nannual interest of unity is that which is called _the rate of interest_.\nAt the end of each year an amount acquires for a factor unity plus the\nrate of interest; it increases then according to a geometrical\nprogression of which this factor is the ratio. Thus in the course of\ntime it becomes immense. If, for example, the rate of interest is 1/20\nor five per cent, the capital doubles very nearly in fourteen years,\nquadruples in twenty-nine years, and in less than three centuries it\nbecomes two million times larger.\n\nAn increase so prodigious has given birth to the idea of making use of\nit in order to pay off the public debt. One forms for this purpose a\nsinking fund to which is devoted an annual fund employed for the\nredemption of public bills and without ceasing increased by the interest\nof the bills redeemed. It is clear that in the long run this fund will\nabsorb a great part of the national debt. If, when the needs of the\nState make a loan necessary, a part of this loan is devoted to the\nincreasing of the annual sinking fund, the variation of public bills\nwill be less; the confidence of the lenders and the probability of\nretiring without loss of capital loaned when one desires will be\naugmented and will render the conditions of the loan less onerous.\nFavorable experiences have fully confirmed these advantages. But the\nfidelity in engagements and the stability, so necessary to the success\nof such institutions, can be guaranteed only by a government in which\nthe legislative power is divided among several independent powers. The\nconfidence which the necessary co\u00f6peration of these powers inspires,\ndoubles the strength of the State, and the sovereign himself gains then\nin legal power more than he loses in arbitrary power.\n\nIt results from that which precedes that the actual capital equivalent\nto a sum which is to be paid only after a certain number of years is\nequal to this sum multiplied by the probability that it will be paid at\nthat time and divided by unity augmented by the rate of interest and\nraised to a power expressed by the number of these years.\n\nIt is easy to apply this principle to life annuities upon one or several\npersons, and to savings banks, and to assurance societies of any nature.\nSuppose that one proposes to form a table of life annuities according to\na given table of mortality. A life annuity payable at the end of five\nyears, for example, and reduced to an actual amount is, by this\nprinciple, equal to the product of the two following quantities, namely,\nthe annuity divided by the fifth power of unity augmented by the rate of\ninterest and the probability of paying it. This probability is the\ninverse ratio of the number of individuals inscribed in the table\nopposite to the age of that one who settles the annuity to the number\ninscribed opposite to this age augmented by five years. Forming, then, a\nseries of fractions whose denominators are the products of the number of\npersons indicated in the table of mortality as living at the age of that\none who settles the annuity, by the successive powers of unity augmented\nby the rate of interest, and whose numerators are the products of the\nannuity by the number of persons living at the same age augmented\nsuccessively by one year, by two years, etc., the sum of these fractions\nwill be the amount required for the life annuity at that age.\n\nLet us suppose that a person wishes by means of a life annuity to assure\nto his heirs an amount payable at the end of the year of his death. In\norder to determine the value of this annuity, one may imagine that the\nperson borrows in life at a bank this capital and that he places it at\nperpetual interest in the same bank. It is clear that this same capital\nwill be due by the bank to his heirs at the end of the year of his\ndeath; but he will have paid each year only the excess of the life\ninterest over the perpetual interest. The table of life annuities will\nthen show that which the person ought to pay annually to the bank in\norder to assure this capital after his death.\n\nMaritime assurance, that against fire and storms, and generally all the\ninstitutions of this kind, are computed on the same principles. A\nmerchant having vessels at sea wishes to assure their value and that of\ntheir cargoes against the dangers that they may run; in order to do\nthis, he gives a sum to a company which becomes responsible to him for\nthe estimated value of his cargoes and his vessels. The ratio of this\nvalue to the sum which ought to be given for the price of the assurance\ndepends upon the dangers to which the vessels are exposed and can be\nappreciated only by numerous observations upon the fate of vessels which\nhave sailed from port for the same destination.\n\nIf the persons assured should give to the assurance company only the sum\nindicated by the calculus of probabilities, this company would not be\nable to provide for the expenses of its institution; it is necessary\nthen that they should pay a sum much greater than the cost of such\ninsurance. What then is their advantage? It is here that the\nconsideration of the moral disadvantage attached to an uncertainty\nbecomes necessary. One conceives that the fairest game becomes, as has\nalready been seen, disadvantageous, because the player exchanges a\ncertain stake for an uncertain benefit; assurance by which one exchanges\nthe uncertain for the certain ought to be advantageous. It is indeed\nthis which results from the rule which we have given above for\ndetermining moral hope and by which one sees moreover how far the\nsacrifice may extend which ought to be made to the assurance company by\nreserving always a moral advantage. This company can then in procuring\nthis advantage itself make a great benefit, if the number of the assured\npersons is very large, a condition necessary to its continued existence.\nThen its benefits become certain and the mathematical and moral hopes\ncoincide; for analysis leads to this general theorem, namely, that if\nthe expectations are very numerous the two hopes approach each other\nwithout ceasing and end by coinciding in the case of an infinite number.\n\nWe have said in speaking of mathematical and moral hopes that there is a\nmoral advantage in distributing the risks of a benefit which one expects\nover several of its parts. Thus in order to send a sum of money to a\ndistant part it is much better to send it on several vessels than to\nexpose it on one. This one does by means of mutual assurances. If two\npersons, each having the same sum upon two different vessels which have\nsailed from the same port to the same destination, agree to divide\nequally all the money which may arrive, it is clear that by this\nagreement each of them divides equally between the two vessels the sum\nwhich he expects. Indeed this kind of assurance always leaves\nuncertainty as to the loss which one may fear. But this uncertainty\ndiminishes in proportion as the number of policy-holders increases; the\nmoral advantage increases more and more and ends by coinciding with the\nmathematical advantage, its natural limit. This renders the association\nof mutual assurances when it is very numerous more advantageous to the\nassured ones than the companies of assurance which, in proportion to the\nbenefit that they give, give a moral advantage always inferior to the\nmathematical advantage. But the surveillance of their administration can\nbalance the advantage of the mutual assurances. All these results are,\nas has already been seen, independent of the law which expresses the\nmoral advantage.\n\nOne may look upon a free people as a great association whose members\nsecure mutually their properties by supporting proportionally the\ncharges of this guaranty. The confederation of several peoples would\ngive to them advantages analogous to those which each individual enjoys\nin the society. A congress of their representatives would discuss\nobjects of a utility common to all and without doubt the system of\nweights, measures, and moneys proposed by the French scientists would be\nadopted in this congress as one of the things most useful to commercial\nrelations.\n\nAmong the institutions founded upon the probabilities of human life the\nbetter ones are those in which, by means of a light sacrifice of his\nrevenue, one assures his existence and that of his family for a time\nwhen one ought to fear to be unable to satisfy their needs. As far as\ngames are immoral, so far these institutions are advantageous to customs\nby favoring the strongest bents of our nature. The government ought then\nto encourage them and respect them in the vicissitudes of public\nfortune; since the hopes which they present look toward a distant\nfuture, they are able to prosper only when sheltered from all inquietude\nduring their existence. It is an advantage that the institution of a\nrepresentative government assures them.\n\nLet us say a word about loans. It is clear that in order to borrow\nperpetually it is necessary to pay each year the product of the capital\nby the rate of interest. But one may wish to discharge this principal in\nequal payments made during a definite number of years, payments which\nare called _annuities_ and whose value is obtained in this manner. Each\nannuity in order to be reduced at the actual moment ought to be divided\nby a power of unity augmented by the rate of interest equal to the\nnumber of years after which this annuity ought to be paid. Forming then\na geometric progression whose first term is the annuity divided by unity\naugmented by the rate of interest, and whose last term is this annuity\ndivided by the same quantity raised to a power equal to the number of\nyears during which the payment should have been made, the sum of this\nprogression will be equivalent to the capital borrowed, which will\ndetermine the value of the annuity. A sinking fund is at bottom only a\nmeans of converting into annuities a perpetual rent with the sole\ndifference that in the case of a loan by annuities the interest is\nsupposed constant, while the interest of funds acquired by the sinking\nfund is variable. If it were the same in both cases, the annuity\ncorresponding to the funds acquired would be formed by these funds and\nfrom this annuity the State contributes annually to the sinking fund.\n\nIf one wishes to make a life loan it will be observed that the tables of\nlife annuities give the capital required to constitute a life annuity at\nany age, a simple proportion will give the rent which one ought to pay\nto the individual from whom the capital is borrowed. From these\nprinciples all the possible kinds of loans may be calculated.\n\nThe principles which we have just expounded concerning the benefits and\nthe losses of institutions may serve to determine the mean result of any\nnumber of observations already made, when one wishes to regard the\ndeviations of the results corresponding to divers observations. Let us\ndesignate by _x_ the correction of the least result and by _x_ augmented\nsuccessively by _q_, _q\u00b4_, _q\u00b4\u00b4_, etc., the corrections of the following\nresults. Let us name _e_, _e\u00b4_, _e\u00b4\u00b4_, etc., the errors of the\nobservations whose law of probability we will suppose known. Each\nobservation being a function of the result, it is easy to see that by\nsupposing the correction _x_ of this result to be very small, the error\n_e_ of the first observation will be equal to the product of _x_ by a\ndetermined coefficient. Likewise the error _e\u00b4_ of the second\nobservation will be the product of the sum _q_ plus _x_, by a determined\ncoefficient, and so on. The probability of the error _e_ being given by\na known function, it will be expressed by the same function of the first\nof the preceding products. The probability of _e\u00b4_ will be expressed by\nthe same function of the second of these products, and so on of the\nothers. The probability of the simultaneous existence of the errors _e_,\n_e\u00b4_, _e\u00b4\u00b4_, etc., will be then proportional to the product of these\ndivers functions, a product which will be a function of _x_. This being\ngranted, if one conceives a curve whose abscissa is _x_, and whose\ncorresponding ordinate is this product, this curve will represent the\nprobability of the divers values of _x_, whose limits will be determined\nby the limits of the errors _e_, _e\u00b4_, _e\u00b4\u00b4_, etc. Now let us designate\nby _X_ the abscissa which it is necessary to choose; _X_ diminished by\n_x_ will be the error which would be committed if the abscissa _x_ were\nthe true correction. This error, multiplied by the probability of _x_ or\nby the corresponding ordinate of the curve, will be the product of the\nloss by its probability, regarding, as one should, this error as a loss\nattached to the choice _X_. Multiplying this product by the differential\nof _x_ the integral taken from the first extremity of the curve to _X_\nwill be the disadvantage of _X_ resulting from the values of _x_\ninferior to _X_. For the values of _x_ superior to _X_, _x_ less _X_\nwould be the error of _X_ if _x_ were the true correction; the integral\nof the product of _x_ by the corresponding ordinate of the curve and by\nthe differential of _x_ will be then the disadvantage of _X_ resulting\nfrom the values _x_ superior to _x_, this integral being taken from _x_\nequal to _X_ up to the last extremity of the curve. Adding this\ndisadvantage to the preceding one, the sum will be the disadvantage\nattached to the choice of _X_. This choice ought to be determined by the\ncondition that this disadvantage be a _minimum_; and a very simple\ncalculation shows that for this, _X_ ought to be the abscissa whose\nordinate divides the curve into two equal parts, so that it is thus\nprobable that the true value of _x_ falls on neither the one side nor\nthe other of _X_.\n\nCelebrated geometricians have chosen for _X_ the most probable value of\n_x_ and consequently that which corresponds to the largest ordinate of\nthe curve; but the preceding value appears to me evidently that which\nthe theory of probability indicates.\n\n\n\n\nCHAPTER XVI.\n\n_CONCERNING ILLUSIONS IN THE ESTIMATION OF PROBABILITIES._\n\n\nThe mind has its illusions as the sense of sight; and in the same manner\nthat the sense of feeling corrects the latter, reflection and\ncalculation correct the former. Probability based upon a daily\nexperience, or exaggerated by fear and by hope, strikes us more than a\nsuperior probability but it is only a simple result of calculus. Thus we\ndo not fear in return for small advantages to expose our life to dangers\nmuch less improbable than the drawing of a quint in the lottery of\nFrance; and yet no one would wish to procure for himself the same\nadvantages with the certainty of losing his life if this quint should be\ndrawn.\n\nOur passions, our prejudices, and dominating opinions, by exaggerating\nthe probabilities which are favorable to them and by attenuating the\ncontrary probabilities, are the abundant sources of dangerous illusions.\n\nPresent evils and the cause which produced them effect us much more than\nthe remembrance of evils produced by the contrary cause; they prevent us\nfrom appreciating with justice the inconveniences of the ones and the\nothers, and the probability of the proper means to guard ourselves\nagainst them. It is this which leads alternately to despotism and to\nanarchy the people who are driven from the state of repose to which they\nnever return except after long and cruel agitations.\n\nThis vivid impression which we receive from the presence of events, and\nwhich allows us scarcely to remark the contrary events observed by\nothers, is a principal cause of error against which one cannot\nsufficiently guard himself.\n\nIt is principally at games of chance that a multitude of illusions\nsupport hope and sustain it against unfavorable chances. The majority of\nthose who play at lotteries do not know how many chances are to their\nadvantage, how many are contrary to them. They see only the possibility\nby a small stake of gaining a considerable sum, and the projects which\ntheir imagination brings forth, exaggerate to their eyes the probability\nof obtaining it; the poor man especially, excited by the desire of a\nbetter fate, risks at play his necessities by clinging to the most\nunfavorable combinations which promise him a great benefit. All would be\nwithout doubt surprised by the immense number of stakes lost if they\ncould know of them; but one takes care on the contrary to give to the\nwinnings a great publicity, which becomes a new cause of excitement for\nthis funereal play.\n\nWhen a number in the lottery of France has not been drawn for a long\ntime the crowd is eager to cover it with stakes. They judge since the\nnumber has not been drawn for a long time that it ought at the next\ndrawing to be drawn in preference to others. So common an error appears\nto me to rest upon an illusion by which one is carried back\ninvoluntarily to the origin of events. It is, for example, very\nimprobable that at the play of heads and tails one will throw heads ten\ntimes in succession. This improbability which strikes us indeed when it\nhas happened nine times, leads us to believe that at the tenth throw\ntails will be thrown. But the past indicating in the coin a greater\npropensity for heads than for tails renders the first of the events more\nprobable than the second; it increases as one has seen the probability\nof throwing heads at the following throw. A similar illusion persuades\nmany people that one can certainly win in a lottery by placing each time\nupon the same number, until it is drawn, a stake whose product surpasses\nthe sum of all the stakes. But even when similar speculations would not\noften be stopped by the impossibility of sustaining them they would not\ndiminish the mathematical disadvantage of speculators and they would\nincrease their moral disadvantage, since at each drawing they would risk\na very large part of their fortune.\n\nI have seen men, ardently desirous of having a son, who could learn only\nwith anxiety of the births of boys in the month when they expected to\nbecome fathers. Imagining that the ratio of these births to those of\ngirls ought to be the same at the end of each month, they judged that\nthe boys already born would render more probable the births next of\ngirls. Thus the extraction of a white ball from an urn which contains a\nlimited number of white balls and of black balls increases the\nprobability of extracting a black ball at the following drawing. But\nthis ceases to take place when the number of balls in the urn is\nunlimited, as one must suppose in order to compare this case with that\nof births. If, in the course of a month, there were born many more boys\nthan girls, one might suspect that toward the time of their conception a\ngeneral cause had favored masculine conception, which would render more\nprobable the birth next of a boy. The irregular events of nature are not\nexactly comparable to the drawing of the numbers of a lottery in which\nall the numbers are mixed at each drawing in such a manner as to render\nthe chances of their drawing perfectly equal. The frequency of one of\nthese events seems to indicate a cause slightly favoring it, which\nincreases the probability of its next return, and its repetition\nprolonged for a long time, such as a long series of rainy days, may\ndevelop unknown causes for its change; so that at each expected event we\nare not, as at each drawing of a lottery, led back to the same state of\nindecision in regard to what ought to happen. But in proportion as the\nobservation of these events is multiplied, the comparison of their\nresults with those of lotteries becomes more exact.\n\nBy an illusion contrary to the preceding ones one seeks in the past\ndrawings of the lottery of France the numbers most often drawn, in order\nto form combinations upon which one thinks to place the stake to\nadvantage. But when the manner in which the mixing of the numbers in\nthis lottery is considered, the past ought to have no influence upon the\nfuture. The very frequent drawings of a number are only the anomalies of\nchance; I have submitted several of them to calculation and have\nconstantly found that they are included within the limits which the\nsupposition of an equal possibility of the drawing of all the numbers\nallows us to admit without improbability.\n\nIn a long series of events of the same kind the single chances of hazard\nought sometimes to offer the singular veins of good luck or bad luck\nwhich the majority of players do not fail to attribute to a kind of\nfatality. It happens often in games which depend at the same time upon\nhazard and upon the competency of the players, that that one who loses,\ntroubled by his loss, seeks to repair it by hazardous throws which he\nwould shun in another situation; thus he aggravates his own ill luck and\nprolongs its duration. It is then that prudence becomes necessary and\nthat it is of importance to convince oneself that the moral disadvantage\nattached to unfavorable chances is increased by the ill luck itself.\n\nThe opinion that man has long been placed in the centre of the universe,\nconsidering himself the special object of the cares of nature, leads\neach individual to make himself the centre of a more or less extended\nsphere and to believe that hazard has preference for him. Sustained by\nthis belief, players often risk considerable sums at games when they\nknow that the chances are unfavorable. In the conduct of life a similar\nopinion may sometimes have advantages; but most often it leads to\ndisastrous enterprises. Here as everywhere illusions are dangerous and\ntruth alone is generally useful.\n\nOne of the great advantages of the calculus of probabilities is to teach\nus to distrust first opinions. As we recognize that they often deceive\nwhen they may be submitted to calculus, we ought to conclude that in\nother matters confidence should be given only after extreme\ncircumspection. Let us prove this by example.\n\nAn urn contains four balls, black and white, but which are not all of\nthe same color. One of these balls has been drawn whose color is white\nand which has been put back in the urn in order to proceed again to\nsimilar drawings. One demands the probability of extracting only black\nballs in the four following drawings.\n\nIf the white and black were in equal number this probability would be\nthe fourth power of the probability \u00bd of extracting a black ball at each\ndrawing; it would be then 1/16. But the extraction of a white ball at\nthe first drawing indicates a superiority in the number of white balls\nin the urn; for if one supposes in the urn three white balls and one\nblack the probability of extracting a white ball is \u00be; it is 2/4 if one\nsupposes two white balls and two black; finally it is reduced to \u00bc if\none supposes three black balls and one white. Following the principle of\nthe probability of causes drawn from events the probabilities of these\nthree suppositions are among themselves as the quantities \u00be, 2/4, \u00bc;\nthey are consequently equal to 3/6, 2/6, \u2159. It is thus a bet of 5\nagainst 1 that the number of black balls is inferior, or at the most\nequal, to that of the white. It seems then that after the extraction of\na white ball at the first drawing, the probability of extracting\nsuccessively four black balls ought to be less than in the case of the\nequality of the colors or smaller than one sixteenth. However, it is\nnot, and it is found by a very simple calculation that this probability\nis greater than one fourteenth. Indeed it would be the fourth power of\n\u00bc, of 2/4, and of \u00be in the first, the second, and the third of the\npreceding suppositions concerning the colors of the balls in the urn.\nMultiplying respectively each power by the probability of the\ncorresponding supposition, or by 3/6, 2/6, and \u2159, the sum of the\nproducts will be the probability of extracting successively four black\nballs. One has thus for this probability 29/384, a fraction greater than\n1/14. This paradox is explained by considering that the indication of\nthe superiority of white balls over the black ones at the first drawing\ndoes not exclude at all the superiority of the black balls over the\nwhite ones, a superiority which excludes the supposition of the equality\nof the colors. But this superiority, though but slightly probable, ought\nto render the probability of drawing successively a given number of\nblack balls greater than in this supposition if the number is\nconsiderable; and one has just seen that this commences when the given\nnumber is equal to four. Let us consider again an urn which contains\nseveral white and black balls. Let us suppose at first that there is\nonly one white ball and one black. It is then an even bet that a white\nball will be extracted in one drawing. But it seems for the equality of\nthe bet that one who bets on extracting the white ball ought to have two\ndrawings if the urn contains two black and one white, three drawings if\nit contains three black and one white, and so on; it is supposed that\nafter each drawing the extracted ball is placed again in the urn.\n\nWe are convinced easily that this first idea is erroneous. Indeed in the\ncase of two black and one white ball, the probability of extracting two\nblack in two drawings is the second power of \u2154 or 4/9; but this\nprobability added to that of drawing a white ball in two drawings is\ncertainty or unity, since it is certain that two black balls or at least\none white ball ought to be drawn; the probability in this last case is\nthen 5/9, a fraction greater than \u00bd. There would still be a greater\nadvantage in the bet of drawing one white ball in five draws when the\nurn contains five black and one white ball; this bet is even\nadvantageous in four drawings; it returns then to that of throwing six\nin four throws with a single die.\n\nThe Chevalier de Mer\u00e9, who caused the invention of the calculus of\nprobabilities by encouraging his friend Pascal, the great geometrician,\nto occupy himself with it, said to him \"that he had found error in the\nnumbers by this ratio. If we undertake to make six with one die there is\nan advantage in undertaking it in four throws, as 671 to 625. If we\nundertake to make two sixes with two dice, there is a disadvantage in\nundertaking in 24 throws. At least 24 is to 36, the number of the faces\nof the two dice, as 4 is to 6, the number of faces of one die.\" \"This\nwas,\" wrote Pascal to Fermat, \"his great scandal which caused him to say\nboldly that the propositions were not constant and that arithmetic was\ndemented.... He has a very good mind, but he is not a geometrician,\nwhich is, as you know, a great fault.\" The Chevalier de Mer\u00e9, deceived\nby a false analogy, thought that in the case of the equality of bets the\nnumber of throws ought to increase in proportion to the number of all\nthe chances possible, which is not exact, but which approaches exactness\nas this number becomes larger.\n\nOne has endeavored to explain the superiority of the births of boys over\nthose of girls by the general desire of fathers to have a son who would\nperpetuate the name. Thus by imagining an urn filled with an infinity of\nwhite and black balls in equal number, and supposing a great number of\npersons each of whom draws a ball from this urn and continues with the\nintention of stopping when he shall have extracted a white ball, one has\nbelieved that this intention ought to render the number of white balls\nextracted superior to that of the black ones. Indeed this intention\ngives necessarily after all the drawings a number of white balls equal\nto that of persons, and it is possible that these drawings would never\nlead a black ball. But it is easy to see that this first notion is only\nan illusion; for if one conceives that in the first drawing all the\npersons draw at once a ball from the urn, it is evident that their\nintention can have no influence upon the color of the balls which ought\nto appear at this drawing. Its unique effect will be to exclude from the\nsecond drawing the persons who shall have drawn a white one at the\nfirst. It is likewise apparent that the intention of the persons who\nshall take part in the new drawing will have no influence upon the color\nof the balls which shall be drawn, and that it will be the same at the\nfollowing drawings. This intention will have no influence then upon the\ncolor of the balls extracted in the totality of drawings; it will,\nhowever, cause more or fewer to participate at each drawing. The ratio\nof the white balls extracted to the black ones will differ thus very\nlittle from unity. It follows that the number of persons being supposed\nvery large, if observation gives between the colors extracted a ratio\nwhich differs sensibly from unity, it is very probable that the same\ndifference is found between unity and the ratio of the white balls to\nthe black contained in the urn.\n\nI count again among illusions the application which Liebnitz and Daniel\nBernoulli have made of the calculus of probabilities to the summation of\nseries. If one reduces the fraction whose numerator is unity and whose\ndenominator is unity plus a variable, in a series prescribed by the\nratio to the powers of this variable, it is easy to see that in\nsupposing the variable equal to unity the fraction becomes \u00bd, and the\nseries becomes plus one, minus one, plus one, minus one, etc. In adding\nthe first two terms, the second two, and so on, the series is\ntransformed into another of which each term is zero. Grandi, an Italian\nJesuit, concluded from this the possibility of the creation; because the\nseries being always \u00bd, he saw this fraction spring from an infinity of\nzeros or from nothing. It was thus that Liebnitz believed he saw the\nimage of creation in his binary arithmetic where he employed only the\ntwo characters, unity and zero. He imagined, since God can be\nrepresented by unity and nothing by zero, that the Supreme Being had\ndrawn from nothing all beings, as unity with zero expresses all the\nnumbers in this system of arithmetic. This idea was so pleasing to\nLiebnitz that he communicated it to the Jesuit Grimaldi, president of\nthe tribunal of mathematics in China, in the hope that this emblem of\ncreation would convert to Christianity the emperor there who\nparticularly loved the sciences. I report this incident only to show to\nwhat extent the prejudices of infancy can mislead the greatest men.\n\nLiebnitz, always led by a singular and very loose metaphysics,\nconsidered that the series plus one, minus one, plus one, etc., becomes\nunity or zero according as one stops at a number of terms odd or even;\nand as in infinity there is no reason to prefer the even number to the\nodd, one ought following the rules of probability, to take the half of\nthe results relative to these two kinds of numbers, and which are zero\nand unity, which gives \u00bd for the value of the series. Daniel Bernoulli\nhas since extended this reasoning to the summation of series formed from\nperiodic terms. But all these series have no values properly speaking;\nthey get them only in the case where their terms are multiplied by the\nsuccessive powers of a variable less than unity. Then these series are\nalways convergent, however small one supposes the difference of the\nvariable from unity; and it is easy to demonstrate that the values\nassigned by Bernoulli, by virtue of the rule of probabilities, are the\nsame values of the generative fraction of the series, when one supposes\nin these fractions the variable equal to unity. These values are again\nthe limits which the series approach more and more, in proportion as the\nvariable approaches unity. But when the variable is exactly equal to\nunity the series cease to be convergent; they have values only as far as\none arrests them. The remarkable ratio of this application of the\ncalculus of probabilities with the limits of the values of periodic\nseries supposes that the terms of these series are multiplied by all the\nconsecutive powers of the variable. But this series may result from the\ndevelopment of an infinity of different fractions in which this did not\noccur. Thus the series plus one, minus one, plus one, etc., may spring\nfrom the development of a fraction whose numerator is unity plus the\nvariable, and whose denominator is this numerator augmented by the\nsquare of the variable. Supposing the variable equal to unity, this\ndevelopment changes, in the series proposed, and the generative fraction\nbecomes equal to \u2154; the rules of probabilities would give then a false\nresult, which proves how dangerous it would be to employ similar\nreasoning, especially in the mathematical sciences, which ought to be\nespecially distinguished by the rigor of their operations.\n\nWe are led naturally to believe that the order according to which we see\nthings renewed upon the earth has existed from all times and will\ncontinue always. Indeed if the present state of the universe were\nexactly similar to the anterior state which has produced it, it would\ngive birth in its turn to a similar state; the succession of these\nstates would then be eternal. I have found by the application of\nanalysis to the law of universal gravity that the movement of rotation\nand of revolution of the planets and satellites, and the position of the\norbits and of their equators are subjected only to periodic\ninequalities. In comparing with ancient eclipses the theory of the\nsecular equation of the moon I have found that since Hipparchus the\nduration of the day has not varied by the hundredth of a second, and\nthat the mean temperature of the earth has not diminished the\none-hundredth of a degree. Thus the stability of actual order appears\nestablished at the same time by theory and by observations. But this\norder is effected by divers causes which an attentive examination\nreveals, and which it is impossible to submit to calculus.\n\nThe actions of the ocean, of the atmosphere, and of meteors, of\nearthquakes, and the eruptions of volcanoes, agitate continually the\nsurface of the earth and ought to effect in the long run great changes.\nThe temperature of climates, the volume of the atmosphere, and the\nproportion of the gases which constitute it, may vary in an\ninappreciable manner. The instruments and the means suitable to\ndetermine these variations being new, observation has been unable up to\nthis time to teach us anything in this regard. But it is hardly probable\nthat the causes which absorb and renew the gases constituting the air\nmaintain exactly their respective proportions. A long series of\ncenturies will show the alterations which are experienced by all these\nelements so essential to the conservation of organized beings. Although\nhistorical monuments do not go back to a very great antiquity they offer\nus nevertheless sufficiently great changes which have come about by the\nslow and continued action of natural agents. Searching in the bowels of\nthe earth one discovers numerous d\u00e9bris of former nature, entirely\ndifferent from the present. Moreover, if the entire earth was in the\nbeginning fluid, as everything appears to indicate, one imagines that in\npassing from that state to the one which it has now, its surface ought\nto have experienced prodigious changes. The heavens itself in spite of\nthe order of its movements, is not unchangeable. The resistance of light\nand of other ethereal fluids, and the attraction of the stars ought,\nafter a great number of centuries, to alter considerably the planetary\nmovements. The variations already observed in the stars and in the form\nof the nebul\u00e6 give us a presentiment of those which time will develop in\nthe system of these great bodies. One may represent the successive\nstates of the universe by a curve, of which time would be the abscissa\nand of which the ordinates are the divers states. Scarcely knowing an\nelement of this curve we are far from being able to go back to its\norigin; and if in order to satisfy the imagination, always restless from\nour ignorance of the cause of the phenomena which interest it, one\nventures some conjectures it is wise to present them only with extreme\nreserve.\n\nThere exists in the estimation of probabilities a kind of illusions,\nwhich depending especially upon the laws of the intellectual\norganization demands, in order to secure oneself against them, a\nprofound examination of these laws. The desire to penetrate into the\nfuture and the ratios of some remarkable events, to the predictions of\nastrologers, of diviners and soothsayers, to presentiments and dreams,\nto the numbers and the days reputed lucky or unlucky, have given birth\nto a multitude of prejudices still very widespread. One does not reflect\nupon the great number of non-coincidences which have made no impression\nor which are unknown. However, it is necessary to be acquainted with\nthem in order to appreciate the probability of the causes to which the\ncoincidences are attributed. This knowledge would confirm without doubt\nthat which reason tells us in regard to these prejudices. Thus the\nphilosopher of antiquity to whom is shown in a temple, in order to exalt\nthe power of the god who is adored there, the _ex veto_ of all those who\nafter having invoked it were saved from shipwreck, presents an incident\nconsonant with the calculus of probabilities, observing that he does not\nsee inscribed the names of those who, in spite of this invocation, have\nperished. Cicero has refuted all these prejudices with much reason and\neloquence in his _Treatise on Divination_, which he ends by a passage\nwhich I shall cite; for one loves to find again among the ancients the\nthunderbolts of reason, which, after having dissipated all the\nprejudices by its light, shall become the sole foundation of human\ninstitutions.\n\n\"It is necessary,\" says the Roman orator, \"to reject divination by\ndreams and all similar prejudices. Widespread superstition has\nsubjugated the majority of minds and has taken possession of the\nfeebleness of men. It is this we have expounded in our books upon the\nnature of the gods and especially in this work, persuaded that we shall\nrender a service to others and to ourselves if we succeed in destroying\nsuperstition. However (and I desire especially in this regard my thought\nbe well comprehended), in destroying superstition I am far from wishing\nto disturb religion. Wisdom enjoins us to maintain the institutions and\nthe ceremonies of our ancestors, touching the cult of the gods.\nMoreover, the beauty of the universe and the order of celestial things\nforce us to recognize some superior nature which ought to be remarked\nand admired by the human race. But as far as it is proper to propagate\nreligion, which is joined to the knowledge of nature, so far it is\nnecessary to work toward the extirpation of superstition, for it\ntorments one, importunes one, and pursues one continually and in all\nplaces. If one consult a diviner or a soothsayer, if one immolates a\nvictim, if one regards the flight of a bird, if one encounters a\nChaldean or an aruspex, if it lightens, if it thunders, if the\nthunderbolt strikes, finally, if there is born or is manifested a kind\nof prodigy, things one of which ought often to happen, then superstition\ndominates and leaves no repose. Sleep itself, this refuge of mortals in\ntheir troubles and their labors, becomes by it a new source of\ninquietude and fear.\"\n\nAll these prejudices and the terrors which they inspire are connected\nwith physiological causes which continue sometimes to operate strongly\nafter reason has disabused us of them. But the repetition of acts\ncontrary to these prejudices can always destroy them.\n\n\n\n\nCHAPTER XVII.\n\n_CONCERNING THE VARIOUS MEANS OF APPROACHING CERTAINTY._\n\n\nInduction, analogy, hypotheses founded upon facts and rectified\ncontinually by new observations, a happy tact given by nature and\nstrengthened by numerous comparisons of its indications with experience,\nsuch are the principal means for arriving at truth.\n\nIf one considers a series of objects of the same nature one perceives\namong them and in their changes ratios which manifest themselves more\nand more in proportion as the series is prolonged, and which, extending\nand generalizing continually, lead finally to the principle from which\nthey were derived. But these ratios are enveloped by so many strange\ncircumstances that it requires great sagacity to disentangle them and to\nrecur to this principle: it is in this that the true genius of sciences\nconsists. Analysis and natural philosophy owe their most important\ndiscoveries to this fruitful means, which is called _induction_. Newton\nwas indebted to it for his theorem of the binomial and the principle of\nuniversal gravity. It is difficult to appreciate the probability of the\nresults of induction, which is based upon this that the simplest ratios\nare the most common; this is verified in the formul\u00e6 of analysis and is\nfound again in natural phenomena, in crystallization, and in chemical\ncombinations. This simplicity of ratios will not appear astonishing if\nwe consider that all the effects of nature are only mathematical results\nof a small number of immutable laws.\n\nYet induction, in leading to the discovery of the general principles of\nthe sciences, does not suffice to establish them absolutely. It is\nalways necessary to confirm them by demonstrations or by decisive\nexperiences; for the history of the sciences shows us that induction has\nsometimes led to inexact results. I shall cite, for example, a theorem\nof Fermat in regard to prime numbers. This great geometrician, who had\nmeditated profoundly upon this theorem, sought a formula which,\ncontaining only prime numbers, gave directly a prime number greater than\nany other number assignable. Induction led him to think that two, raised\nto a power which was itself a power of two, formed with unity a prime\nnumber. Thus, two raised to the square plus one, forms the prime number\nfive; two raised to the second power of two, or sixteen, forms with one\nthe prime number seventeen. He found that this was still true for the\neighth and the sixteenth power of two augmented by unity; and this\ninduction, based upon several arithmetical considerations, caused him to\nregard this result as general. However, he avowed that he had not\ndemonstrated it. Indeed, Euler recognized that this does not hold for\nthe thirty-second power of two, which, augmented by unity, gives\n4,294,967,297, a number divisible by 641.\n\nWe judge by induction that if various events, movements, for example,\nappear constantly and have been long connected by a simple ratio, they\nwill continue to be subjected to it; and we conclude from this, by the\ntheory of probabilities, that this ratio is due, not to hazard, but to a\nregular cause. Thus the equality of the movements of the rotation and\nthe revolution of the moon; that of the movements of the nodes of the\norbit and of the lunar equator, and the coincidence of these nodes; the\nsingular ratio of the movements of the first three satellites of\nJupiter, according to which the mean longitude of the first satellite,\nless three times that of the second, plus two times that of the third,\nis equal to two right angles; the equality of the interval of the tides\nto that of the passage of the moon to the meridian; the return of the\ngreatest tides with the syzygies, and of the smallest with the\nquadratures; all these things, which have been maintained since they\nwere first observed, indicate with an extreme probability, the existence\nof constant causes which geometricians have happily succeeded in\nattaching to the law of universal gravity, and the knowledge of which\nrenders certain the perpetuity of these ratios.\n\nThe chancellor Bacon, the eloquent promoter of the true philosophical\nmethod, has made a very strange misuse of induction in order to prove\nthe immobility of the earth. He reasons thus in the _Novum Organum_, his\nfinest work: \"The movement of the stars from the orient to the occident\nincreases in swiftness, in proportion to their distance from the earth.\nThis movement is swiftest with the stars; it slackens a little with\nSaturn, a little more with Jupiter, and so on to the moon and the\nhighest comets. It is still perceptible in the atmosphere, especially\nbetween the tropics, on account of the great circles which the molecules\nof the air describe there; finally, it is almost inappreciable with the\nocean; it is then nil for the earth.\" But this induction proves only\nthat Saturn, and the stars which are inferior to it, have their own\nmovements, contrary to the real or apparent movement which sweeps the\nwhole celestial sphere from the orient to the occident, and that these\nmovements appear slower with the more remote stars, which is conformable\nto the laws of optics. Bacon ought to have been struck by the\ninconceivable swiftness which the stars require in order to accomplish\ntheir diurnal revolution, if the earth is immovable, and by the extreme\nsimplicity with which its rotation explains how bodies so distant, the\nones from the others, as the stars, the sun, the planets, and the moon,\nall seem subjected to this revolution. As to the ocean and to the\natmosphere, he ought not to compare their movement with that of the\nstars which are detached from the earth; but since the air and the sea\nmake part of the terrestrial globe, they ought to participate in its\nmovement or in its repose. It is singular that Bacon, carried to great\nprospects by his genius, was not won over by the majestic idea which the\nCopernican system of the universe offers. He was able, however, to find\nin favor of that system, strong analogies in the discoveries of Galileo,\nwhich were continued by him. He has given for the search after truth the\nprecept, but not the example. But by insisting, with all the force of\nreason and of eloquence, upon the necessity of abandoning the\ninsignificant subtleties of the school, in order to apply oneself to\nobservations and to experiences, and by indicating the true method of\nascending to the general causes of phenomena, this great philosopher\ncontributed to the immense strides which the human mind made in the\ngrand century in which he terminated his career.\n\nAnalogy is based upon the probability, that similar things have causes\nof the same kind and produce the same effects. This probability increase\nas the similitude becomes more perfect. Thus we judge without doubt that\nbeings provided with the same organs, doing the same things, experience\nthe same sensations, and are moved by the same desires. The probability\nthat the animals which resemble us have sensations analogous to ours,\nalthough a little inferior to that which is relative to individuals of\nour species, is still exceedingly great; and it has required all the\ninfluence of religious prejudices to make us think with some\nphilosophers that animals are mere automatons. The probability of the\nexistence of feeling decreases in the same proportion as the similitude\nof the organs with ours diminishes, but it is always very great, even\nwith insects. In seeing those of the same species execute very\ncomplicated things exactly in the same manner from generation to\ngeneration, and without having learned them, one is led to believe that\nthey act by a kind of affinity analogous to that which brings together\nthe molecules of crystals, but which, together with the sensation\nattached to all animal organization, produces, with the regularity of\nchemical combinations, combinations that are much more singular; one\nmight, perhaps, name this mingling of elective affinities and sensations\n_animal affinity_. Although there exists a great analogy between the\norganization of plants and that of animals, it does not seem to me\nsufficient to extend to vegetables the sense of feeling; but nothing\nauthorizes us in denying it to them.\n\nSince the sun brings forth, by the beneficent action of its light and of\nits heat, the animals and plants which cover the earth, we judge by\nanalogy that it produces similar effects upon the other planets; for it\nis not natural to think that the cause whose activity we see developed\nin so many ways should be sterile upon so great a planet as Jupiter,\nwhich, like the terrestrial globe, has its days, its nights, and its\nyears, and upon which observations indicate changes which suppose very\nactive forces. Yet this would be giving too great an extension to\nanalogy to conclude from it the similitude of the inhabitants of the\nplanets and of the earth. Man, made for the temperature which he enjoys,\nand for the element which he breathes, would not be able, according to\nall appearance, to live upon the other planets. But ought there not to\nbe an infinity of organization relative to the various constitutions of\nthe globes of this universe? If the single difference of the elements\nand of the climates make so much variety in terrestrial productions, how\nmuch greater the difference ought to be among those of the various\nplanets and of their satellites! The most active imagination can form no\nidea of it; but their existence is very probable.\n\nWe are led by a strong analogy to regard the stars as so many suns\nendowed, like ours, with an attractive power proportional to the mass\nand reciprocal to the square of the distances; for this power being\ndemonstrated for all the bodies of the solar system, and for their\nsmallest molecules, it appears to appertain to all matter. Already the\nmovements of the small stars, which have been called _double_, on\naccount of their being binary, appear to indicate it; a century at most\nof precise observations, by verifying their movements of revolution, the\nones about the others, will place beyond doubt their reciprocal\nattractions.\n\nThe analogy which leads us to make each star the centre of a planetary\nsystem is far less strong than the preceding one; but it acquires\nprobability by the hypothesis which has been proposed in regard to the\nformation of the stars and of the sun; for in this hypothesis each star,\nhaving been like the sun, primitively environed by a vast atmosphere, it\nis natural to attribute to this atmosphere the same effects as to the\nsolar atmosphere, and to suppose that it has produced, in condensing,\nplanets and satellites.\n\nA great number of discoveries in the sciences is due to analogy. I shall\ncite as one of the most remarkable, the discovery of atmospheric\nelectricity, to which one has been led by the analogy of electric\nphenomena with the effects of thunder.\n\nThe surest method which can guide us in the search for truth, consists\nin rising by induction from phenomena to laws and from laws to forces.\nLaws are the ratios which connect particular phenomena together: when\nthey have shown the general principle of the forces from which they are\nderived, one verifies it either by direct experiences, when this is\npossible, or by examination if it agrees with known phenomena; and if by\na rigorous analysis we see them proceed from this principle, even in\ntheir small details, and if, moreover, they are quite varied and very\nnumerous, then science acquires the highest degree of certainty and of\nperfection that it is able to attain. Such, astronomy has become by the\ndiscovery of universal gravity. The history of the sciences shows that\nthe slow and laborious path of induction has not always been that of\ninventors. The imagination, impatient to arrive at the causes, takes\npleasure in creating hypotheses, and often it changes the facts in order\nto adapt them to its work; then the hypotheses are dangerous. But when\none regards them only as the means of connecting the phenomena in order\nto discover the laws; when, by refusing to attribute them to a reality,\none rectifies them continually by new observations, they are able to\nlead to the veritable causes, or at least put us in a position to\nconclude from the phenomena observed those which given circumstances\nought to produce.\n\nIf we should try all the hypotheses which can be formed in regard to the\ncause of phenomena we should arrive, by a process of exclusion, at the\ntrue one. This means has been employed with success; sometimes we have\narrived at several hypotheses which explain equally well all the facts\nknown, and among which scholars are divided, until decisive observations\nhave made known the true one. Then it is interesting, for the history of\nthe human mind, to return to these hypotheses, to see how they succeed\nin explaining a great number of facts, and to investigate the changes\nwhich they ought to undergo in order to agree with the history of\nnature. It is thus that the system of Ptolemy, which is only the\nrealization of celestial appearances, is transformed into the hypothesis\nof the movement of the planets about the sun, by rendering equal and\nparallel to the solar orbit the circles and the epicycles which he\ncauses to be described annually, and the magnitude of which he leaves\nundetermined. It suffices, then, in order to change this hypothesis into\nthe true system of the world, to transport the apparent movement of the\nsun in a sense contrary to the earth.\n\nIt is almost always impossible to submit to calculus the probability of\nthe results obtained by these various means; this is true likewise for\nhistorical facts. But the totality of the phenomena explained, or of the\ntestimonies, is sometimes such that without being able to appreciate the\nprobability we cannot reasonably permit ourselves any doubt in regard to\nthem. In the other cases it is prudent to admit them only with great\nreserve.\n\n\n\n\nCHAPTER XVIII.\n\n_HISTORICAL NOTICE CONCERNING THE CALCULUS OF PROBABILITIES._\n\n\nLong ago were determined, in the simplest games, the ratios of the\nchances which are favorable or unfavorable to the players; the stakes\nand the bets were regulated according to these ratios. But no one before\nPascal and Fermat had given the principles and the methods for\nsubmitting this subject to calculus, and no one had solved the rather\ncomplicated questions of this kind. It is, then, to these two great\ngeometricians that we must refer the first elements of the science of\nprobabilities, the discovery of which can be ranked among the remarkable\nthings which have rendered illustrious the seventeenth century\u2014the\ncentury which has done the greatest honor to the human mind. The\nprincipal problem which they solved by different methods, consists, as\nwe have seen, in distributing equitably the stake among the players, who\nare supposed to be equally skilful and who agree to stop the game before\nit is finished, the condition of play being that, in order to win the\ngame, one must gain a given number of points different for each of the\nplayers. It is clear that the distribution should be made proportionally\nto the respective probabilities of the players of winning this game, the\nprobabilities depending upon the numbers of points which are still\nlacking. The method of Pascal is very ingenious, and is at bottom only\nthe equation of partial differences of this problem applied in\ndetermining the successive probabilities of the players, by going from\nthe smallest numbers to the following ones. This method is limited to\nthe case of two players; that of Fermat, based upon combinations,\napplies to any number of players. Pascal believed at first that it was,\nlike his own, restricted to two players; this brought about between them\na discussion, at the conclusion of which Pascal recognized the\ngenerality of the method of Fermat.\n\nHuygens united the divers problems which had already been solved and\nadded new ones in a little treatise, the first that has appeared on this\nsubject and which has the title _De Ratiociniis in ludo ale\u00e6_. Several\ngeometricians have occupied themselves with the subject since: Hudde,\nthe great pensionary, Witt in Holland, and Halley in England, applied\ncalculus to the probabilities of human life, and Halley published in\nthis field the first table of mortality. About the same time Jacques\nBernoulli proposed to geometricians various problems of probability, of\nwhich he afterwards gave solutions. Finally he composed his beautiful\nwork entitled _Ars conjectandi_, which appeared seven years after his\ndeath, which occurred in 1706. The science of probabilities is more\nprofoundly investigated in this work than in that of Huygens. The author\ngives a general theory of combinations and series, and applies it to\nseveral difficult questions concerning hazards. This work is still\nremarkable on account of the justice and the cleverness of view, the\nemployment of the formula of the binomial in this kind of questions, and\nby the demonstration of this theorem, namely, that in multiplying\nindefinitely the observations and the experiences, the ratio of the\nevents of different natures approaches that of their respective\nprobabilities in the limits whose interval becomes more and more narrow\nin proportion as they are multiplied, and become less than any\nassignable quantity. This theorem is very useful for obtaining by\nobservations the laws and the causes of phenomena. Bernoulli attaches,\nwith reason, a great importance to his demonstration, upon which he has\nsaid to have meditated for twenty years.\n\nIn the interval, from the death of Jacques Bernoulli to the publication\nof his work, Montmort and Moivre produced two treatises upon the\ncalculus of probabilities. That of Montmort has the title _Essai sur les\nJeux de hasard_; it contains numerous applications of this calculus to\nvarious games. The author has added in the second edition some letters\nin which Nicolas Bernoulli gives the ingenious solutions of several\ndifficult problems. The treatise of Moivre, later than that of Montmort,\nappeared at first in the _Transactions philosophiques_ of the year 1711.\nThen the author published it separately, and he has improved it\nsuccessively in three editions. This work is principally based upon the\nformula of the binomial and the problems which it contains have, like\ntheir solutions, a grand generality. But its distinguishing feature is\nthe theory of recurrent series and their use in this subject. This\ntheory is the integration of linear equations of finite differences with\nconstant coefficients, which Moivre made in a very happy manner.\n\nIn his work, Moivre has taken up again the theory of Jacques Bernoulli\nin regard to the probability of results determined by a great number of\nobservations. He does not content himself with showing, as Bernoulli\ndoes, that the ratio of the events which ought to occur approaches\nwithout ceasing that of their respective probabilities; but he gives\nbesides an elegant and simple expression of the probability that the\ndifference of these two ratios is contained within the given limits. For\nthis purpose he determines the ratio of the greatest term of the\ndevelopment of a very high power of the binomial to the sum of all its\nterms, and the hyperbolic logarithm of the excess of this term above the\nterms adjacent to it.\n\nThe greatest term being then the product of a considerable number of\nfactors, his numerical calculus becomes impracticable. In order to\nobtain it by a convergent approximation, Moivre makes use of a theorem\nof Stirling in regard to the mean term of the binomial raised to a high\npower, a remarkable theorem, especially in this, that it introduces the\nsquare root of the ratio of the circumference to the radius in an\nexpression which seemingly ought to be irrelevant to this transcendent.\nMoreover, Moivre was greatly struck by this result, which Stirling had\ndeduced from the expression of the circumference in infinite products;\nWallis had arrived at this expression by a singular analysis which\ncontains the germ of the very curious and useful theory of definite\nintegrals.\n\nMany scholars, among whom one ought to name Deparcieux, Kersseboom,\nWargentin, Dupr\u00e9 de Saint-Maure, Simpson, Sussmilch, Mess\u00e8ne, Moheau,\nPrice, Bailey, and Duvillard, have collected a great amount of precise\ndata in regard to population, births, marriages, and mortality. They\nhave given formul\u00e6 and tables relative to life annuities, tontines,\nassurances, etc. But in this short notice I can only indicate these\nuseful works in order to adhere to original ideas. Of this number\nspecial mention is due to the mathematical and moral hopes and to the\ningenious principle which Daniel Bernoulli has given for submitting the\nlatter to analysis. Such is again the happy application which he has\nmade of the calculus of probabilities to inoculation. One ought\nespecially to include, in the number of these original ideas, direct\nconsideration of the possibility of events drawn from events observed.\nJacques Bernoulli and Moivre supposed these possibilities known, and\nthey sought the probability that the result of future experiences will\nmore and more nearly represent them. Bayes, in the _Transactions\nphilosophiques_ of the year 1763, sought directly the probability that\nthe possibilities indicated by past experiences are comprised within\ngiven limits; and he has arrived at this in a refined and very ingenious\nmanner, although a little perplexing. This subject is connected with the\ntheory of the probability of causes and future events, concluded from\nevents observed. Some years later I expounded the principles of this\ntheory with a remark as to the influence of the inequalities which may\nexist among the chances which are supposed to be equal. Although it is\nnot known which of the simple events these inequalities favor,\nnevertheless this ignorance itself often increases the probability of\ncompound events.\n\nIn generalizing analysis and the problems concerning probabilities, I\nwas led to the calculus of partial finite differences, which Lagrange\nhas since treated by a very simple method, elegant applications of which\nhe has used in this kind of problems. The theory of generative functions\nwhich I published about the same time includes these subjects among\nthose it embraces, and is adapted of itself and with the greatest\ngenerality to the most difficult questions of probability. It determines\nagain, by very convergent approximations, the values of the functions\ncomposed of a great number of terms and factors; and in showing that the\nsquare root of the ratio of the circumference to the radius enters most\nfrequently into these values, it shows that an infinity of other\ntranscendents may be introduced.\n\nTestimonies, votes, and the decisions of electoral and deliberative\nassemblies, and the judgments of tribunals, have been submitted likewise\nto the calculus of probabilities. So many passions, divers interests,\nand circumstances complicate the questions relative to the subjects,\nthat they are almost always insoluble. But the solution of very simple\nproblems which have a great analogy with them, may often shed upon\ndifficult and important questions great light, which the surety of\ncalculus renders always preferable to the most specious reasonings.\n\nOne of the most interesting applications of the calculus of\nprobabilities concerns the mean values which must be chosen among the\nresults of observations. Many geometricians have studied the subject,\nand Lagrange has published in the _M\u00e9moires de Turin_ a beautiful method\nfor determining these mean values when the law of the errors of the\nobservations is known. I have given for the same purpose a method based\nupon a singular contrivance which may be employed with advantage in\nother questions of analysis; and this, by permitting indefinite\nextension in the whole course of a long calculation of the functions\nwhich ought to be limited by the nature of the problem, indicates the\nmodifications which each term of the final result ought to receive by\nvirtue of these limitations. It has already been seen that each\nobservation furnishes an equation of condition of the first degree,\nwhich may always be disposed of in such a manner that all its terms be\nin the first member, the second being zero. The use of these equations\nis one of the principal causes of the great precision of our\nastronomical tables, because an immense number of excellent observations\nhas thus been made to concur in determining their elements. When there\nis only one element to be determined C\u00f4tes prescribed that the equations\nof condition should be prepared in such a manner that the coefficient of\nthe unknown element be positive in each of them; and that all these\nequations should be added in order to form a final equation, whence is\nderived the value of this element. The rule of C\u00f4tes was followed by all\ncalculators, but since he failed to determine several elements, there\nwas no fixed rule for combining the equations of condition in such a\nmanner as to obtain the necessary final equations; but one chose for\neach element the observations most suitable to determine it. It was in\norder to obviate these gropings that Legendre and Gauss concluded to add\nthe squares of the first members of the equations of condition, and to\nrender the sum a minimum, by varying each unknown element; by this means\nis obtained directly as many final equations as there are elements. But\ndo the values determined by these equations merit the preference over\nall those which may be obtained by other means? This question, the\ncalculus of probabilities alone was able to answer. I applied it, then,\nto this subject, and obtained by a delicate analysis a rule which\nincludes the preceding method, and which adds to the advantage of\ngiving, by a regular process, the desired elements that of obtaining\nthem with the greatest show of evidence from the totality of\nobservations, and of determining the values which leave only the\nsmallest possible errors to be feared.\n\nHowever, we have only an imperfect knowledge of the results obtained, as\nlong as the law of the errors of which they are susceptible is unknown;\nwe must be able to assign the probability that these errors are\ncontained within given limits, which amounts to determining that which I\nhave called the _weight_ of a result. Analysis leads to general and\nsimple formul\u00e6 for this purpose. I have applied this analysis to the\nresults of geodetic observations. The general problem consists in\ndetermining the probabilities that the values of one or of several\nlinear functions of the errors of a very great number of observations\nare contained within any limits.\n\nThe law of the possibility of the errors of observations introduces into\nthe expressions of these probabilities a constant, whose value seems to\nrequire the knowledge of this law, which is almost always unknown.\nHappily this constant can be determined from the observations.\n\nIn the investigation of astronomical elements it is given by the sum of\nthe squares of the differences between each observation and the\ncalculated one. The errors equally probable being proportional to the\nsquare root of this sum, one can, by the comparison of these squares,\nappreciate the relative exactitude of the different tables of the same\nstar. In geodetic operations these squares are replaced by the squares\nof the errors of the sums observed of the three angles of each triangle.\nThe comparison of the squares of these errors will enable us to judge of\nthe relative precision of the instruments with which the angles have\nbeen measured. By this comparison is seen the advantage of the repeating\ncircle over the instruments which it has replaced in geodesy.\n\nThere often exists in the observations many sources of errors: thus the\npositions of the stars being determined by means of the meridian\ntelescope and of the circle, both susceptible of errors whose law of\nprobability ought not to be supposed the same, the elements that are\ndeduced from these positions are affected by these errors. The equations\nof condition, which are made to obtain these elements, contain the\nerrors of each instrument and they have various coefficients. The most\nadvantageous system of factors by which these equations ought to be\nmultiplied respectively, in order to obtain, by the union of the\nproducts, as many final equations as there are elements to be\ndetermined, is no longer that of the coefficients of the elements in\neach equation of condition. The analysis which I have used leads easily,\nwhatever the number of the sources of error may be, to the system of\nfactors which gives the most advantageous results, or those in which the\nsame error is less probable than in any other system. The same analysis\ndetermines the laws of probability of the errors of these results. These\nformul\u00e6 contain as many unknown constants as there are sources of error,\nand they depend upon the laws of probability of these errors. It has\nbeen seen that, in the case of a single source, this constant can be\ndetermined by forming the sum of the squares of the residuals of each\nequation of condition, when the values found for these elements have\nbeen substituted. A similar process generally gives values of these\nconstants, whatever their number may be, which completes the application\nof the calculus of probabilities to the results of observations.\n\nI ought to make here an important remark. The small uncertainty that the\nobservations, when they are not numerous, leave in regard to the values\nof the constants of which I have just spoken, renders a little uncertain\nthe probabilities determined by analysis. But it almost always suffices\nto know if the probability, that the errors of the results obtained are\ncomprised within narrow limits, approaches closely to unity; and when it\nis not, it suffices to know up to what point the observations should be\nmultiplied, in order to obtain a probability such that no reasonable\ndoubt remains in regard to the correctness of the results. The analytic\nformul\u00e6 of probabilities satisfy perfectly this requirement; and in this\nconnection they may be viewed as the necessary complement of the\nsciences, based upon a totality of observations susceptible of error.\nThey are likewise indispensable in solving a great number of problems in\nthe natural and moral sciences. The regular causes of phenomena are most\nfrequently either unknown, or too complicated to be submitted to\ncalculus; again, their action is often disturbed by accidental and\nirregular causes; but its impression always remains in the events\nproduced by all these causes, and it leads to modifications which only a\nlong series of observations can determine. The analysis of probabilities\ndevelops these modifications; it assigns the probability of their causes\nand it indicates the means of continually increasing this probability.\nThus in the midst of the irregular causes which disturb the atmosphere,\nthe periodic changes of solar heat, from day to night, and from winter\nto summer, produce in the pressure of this great fluid mass and in the\ncorresponding height of the barometer, the diurnal and annual\noscillations; and numerous barometric observations have revealed the\nformer with a probability at least equal to that of the facts which we\nregard as certain. Thus it is again that the series of historical events\nshows us the constant action of the great principles of ethics in the\nmidst of the passions and the various interests which disturb societies\nin every way. It is remarkable that a science, which commenced with the\nconsideration of games of chance, should be elevated to the rank of the\nmost important subjects of human knowledge.\n\nI have collected all these methods in my _Th\u00e9orie analytique des\nProbabilit\u00e9s_, in which I have proposed to expound in the most general\nmanner the principles and the analysis of the calculus of probabilities,\nlikewise the solutions of the most interesting and most difficult\nproblems which calculus presents.\n\nIt is seen in this essay that the theory of probabilities is at bottom\nonly common sense reduced to calculus; it makes us appreciate with\nexactitude that which exact minds feel by a sort of instinct without\nbeing able ofttimes to give a reason for it. It leaves no arbitrariness\nin the choice of opinions and sides to be taken; and by its use can\nalways be determined the most advantageous choice. Thereby it\nsupplements most happily the ignorance and the weakness of the human\nmind. If we consider the analytical methods to which this theory has\ngiven birth; the truth of the principles which serve as a basis; the\nfine and delicate logic which their employment in the solution of\nproblems requires; the establishments of public utility which rest upon\nit; the extension which it has received and which it can still receive\nby its application to the most important questions of natural philosophy\nand the moral science; if we consider again that, even in the things\nwhich cannot be submitted to calculus, it gives the surest hints which\ncan guide us in our judgments, and that it teaches us to avoid the\nillusions which ofttimes confuse us, then we shall see that there is no\nscience more worthy of our meditations, and that no more useful one\ncould be incorporated in the system of public instruction.","title":"A Philosophical Essay on Probabilities","tradition":"philosophy-ethics","translated_text":null,"word_count":52789}