,transcriber's ,note3 ,apparent typographical errors have been corrected4 ,italics are indicated by underscores4 ,small capitals have been replaced by full capitals4 ,the corrections noted in the ,errata section have been incorporated in the text4 ,two further corrections have also been made3 #i_/aj in place of #a_/aj1 and #i_/aj in place of #f_/aj1 on page #aaj2 and ex voto in place of ex veto on page #agc4 ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,by ,,pierre ,,simon1 ,,marquis ,,de ,,laplace4 ,,translated ,,from ,,the ,,sixth ,,french ,,edition ,,by ,,frederick ,,wilson ,,truscott1 ,,ph4,d4 7,,harv471 ,professor of ,germanic ,languages in the ,west ,virginia ,university1 ,,and ,,frederick ,,lincoln ,,emory1 ,m4,e4 7,,wor4 ,,poly4 ,,inst471 ,professor of ,mechanics and ,applied ,mathematics in the ,west ,virginia ,university2 ,mem4 ,amer4 ,soc4 ,mech4 ,eng4 ,,first ,,edition4 ,,first ,,thousand4 ,,new ,,york3 ,,john ,,wiley ,,sons4 ,,london3 ,,chapman ,,hall1 ,,limited4 #aijb4 ,copyright1 #aijb1 ,,by ,f4 ,w4 ,,truscott ,,and ,f4 ,l4 ,,emory4 ,,robert ,,drummond ,,printer1 ,,new ,,york ,,table ,,of ,,contents4 ,,page ,,part ,i4 ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,chapter ,i4 ,introduction #a ,,chapter ,,ii4 ,concerning ,probability #c ,,chapter ,,iii4 ,general ,principles of the ,calculus of ,probabilities #aa ,,chapter ,,iv4 ,concerning ,hope #bj ,,chapter ,v4 ,analytical ,methods of the ,calculus of ,probabilities #bf ,,part ,,ii4 ,,application ,,of ,,the ,,calculus ,,of ,,probabilities4 ,,chapter ,,vi4 ,games of ,chance #ec ,,chapter ,,vii4 ,concerning the ,unknown ,inequalities which may ,exist among ,chances ,supposed to be ,equal #ef ,,chapter ,,viii4 ,concerning the ,laws of ,probability which result from the ,indefinite ,multiplication of ,events #fj ,,chapter ,,ix4 ,application of the ,calculus of ,probabilities to ,natural ,philosophy #gc ,,chapter ,x4 ,application of the ,calculus of ,probabilities to the ,moral ,sciences #ajg ,,chapter ,,xi4 ,concerning the ,probability of ,testimonies #aji ,,chapter ,,xii4 ,concerning the ,selections and ,decisions of ,assemblies #abf ,,chapter ,,xiii4 ,concerning the ,probability of the ,judgments of ,tribunals #acb ,,chapter ,,xiv4 ,concerning ,tables of ,mortality1 and the ,mean ,durations of ,life1 ,marriage1 and ,some ,associations #adj ,,chapter ,,xv4 ,concerning the ,benefits of ,institutions which ,depend upon the ,probability of ,events #adi ,,chapter ,,xvi4 ,concerning ,illusions in the ,estimation of ,probabilities #afj ,,chapter ,,xvii4 ,concerning the ,various ,means of ,approaching ,certainty #agf ,,chapter ,,xviii4 ,historical ,notice of the ,calculus of ,probabilities to #ahaf #ahe ,,errata4 ,page #hi1 line #bb1 for ,pline read ,pliny #ajb1 lines #ad1 #af1 minutes days #adc1 line #be1 sun soil #agg1 lines #ae1 #ag1 #ah1 #ba1 #bb1 #bd1 for primary read prime #ahb1 line #e1 for conjunctions read being binary ,a ,,philosophical ,,essay ,,on ,,probabilities4 ,,chapter ,i4 ,,introduction4 ,this philosophical essay is the development of a lecture on probabilities which ,i delivered in #agie to the normal schools whither ,i had been called1 by a decree of the national convention1 as professor of mathematics with ,lagrange4 ,i have recently published upon the same subject a work entitled ,the ,analytical ,theory of ,probabilities4 ,i present here without the aid of analysis the principles and general results of this theory1 applying them to the most important questions of life1 which are indeed for the most part only problems of probability4 ,strictly speaking it may even be said that nearly all our knowledge is problematical2 and in the small number of things which we are able to know with certainty1 even in the mathematical sciences themselves1 the principal means for ascertaining truth--induction and analogy--are based on probabilities2 so that the entire system of human knowledge is connected with the theory set forth in this essay4 ,doubtless it will be seen here with interest that in considering1 even in the eternal principles of reason1 justice1 and humanity1 only the favorable chances which are constantly attached to them1 there is a great advantage in following these principles and serious inconvenience in departing from them3 their chances1 like those favorable to lotteries1 always end by prevailing in the midst of the vacillations of hazard4 ,i hope that the reflections given in this essay may merit the attention of philosophers and direct it to a subject so worthy of engaging their minds4 ,,chapter ,,ii4 ,,concerning ,,probability4 ,all events1 even those which on account of their insignificance do not seem to follow the great laws of nature1 are a result of it just as necessarily as the revolutions of the sun4 ,in ignorance of the ties which unite such events to the entire system of the universe1 they have been made to depend upon final causes or upon hazard1 according as they occur and are repeated with regularity1 or appear without regard to order2 but these imaginary causes have gradually receded with the widening bounds of knowledge and disappear entirely before sound philosophy1 which sees in them only the expression of our ignorance of the true causes4 ,present events are connected with preceding ones by a tie based upon the evident principle that a thing cannot occur without a cause which produces it4 ,this axiom1 known by the name of the principle of sufficient reason1 extends even to actions which are considered indifferent2 the freest will is unable without a determinative motive to give them birth2 if we assume two positions with exactly similar circumstances and find that the will is active in the one and inactive in the other1 we say that its choice is an effect without a cause4 ,it is then1 says ,leibnitz1 the blind chance of the ,epicureans4 ,the contrary opinion is an illusion of the mind1 which1 losing sight of the evasive reasons of the choice of the will in indifferent things1 believes that choice is determined of itself and without motives4 ,we ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow4 ,given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it--an intelligence sufficiently vast to submit these data to analysis--it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom2 for it1 nothing would be uncertain and the future1 as the past1 would be present to its eyes4 ,the human mind offers1 in the perfection which it has been able to give to astronomy1 a feeble idea of this intelligence4 ,its discoveries in mechanics and geometry1 added to that of universal gravity1 have enabled it to comprehend in the same analytical expressions the past and future states of the system of the world4 ,applying the same method to some other objects of its knowledge1 it has succeeded in referring to general laws observed phenomena and in foreseeing those which given circumstances ought to produce4 ,all these efforts in the search for truth tend to lead it back continually to the vast intelligence which we have just mentioned1 but from which it will always remain infinitely removed4 ,this tendency1 peculiar to the human race1 is that which renders it superior to animals2 and their progress in this respect distinguishes nations and ages and constitutes their true glory4 ,let us recall that formerly1 and at no remote epoch1 an unusual rain or an extreme drought1 a comet having in train a very long tail1 the eclipses1 the aurora borealis1 and in general all the unusual phenomena were regarded as so many signs of celestial wrath4 ,heaven was invoked in order to avert their baneful influence4 ,no one prayed to have the planets and the sun arrested in their courses3 observation had soon made apparent the futility of such prayers4 ,but as these phenomena1 occurring and disappearing at long intervals1 seemed to oppose the order of nature1 it was supposed that ,heaven1 irritated by the crimes of the earth1 had created them to announce its vengeance4 ,thus the long tail of the comet of #adef spread terror through ,europe1 already thrown into consternation by the rapid successes of the ,turks1 who had just overthrown the ,lower ,empire4 ,this star after four revolutions has excited among us a very different interest4 ,the knowledge of the laws of the system of the world acquired in the interval had dissipated the fears begotten by the ignorance of the true relationship of man to the universe2 and ,halley1 having recognized the identity of this comet with those of the years #aeca1 #afjg1 and #afhb1 announced its next return for the end of the year #ageh or the beginning of the year #agei4 ,the learned world awaited with impatience this return which was to confirm one of the greatest discoveries that have been made in the sciences1 and fulfil the prediction of ,seneca when he said1 in speaking of the revolutions of those stars which fall from an enormous height3 ,the day will come when1 by study pursued through several ages1 the things now concealed will appear with evidence2 and posterity will be astonished that truths so clear had escaped us4 ,clairaut then undertook to submit to analysis the perturbations which the comet had experienced by the action of the two great planets1 ,jupiter and ,saturn2 after immense calculations he fixed its next passage at the perihelion toward the beginning of ,april1 #agei1 which was actually verified by observation4 ,the regularity which astronomy shows us in the movements of the comets doubtless exists also in all phenomena4 ,the curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits2 the only difference between them is that which comes from our ignorance4 ,probability is relative1 in part to this ignorance1 in part to our knowledge4 ,we know that of three or a greater number of events a single one ought to occur2 but nothing induces us to believe that one of them will occur rather than the others4 ,in this state of indecision it is impossible for us to announce their occurrence with certainty4 ,it is1 however1 probable that one of these events1 chosen at will1 will not occur because we see several cases equally possible which exclude its occurrence1 while only a single one favors it4 ,the theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible1 that is to say1 to such as we may be equally undecided about in regard to their existence1 and in determining the number of cases favorable to the event whose probability is sought4 ,the ratio of this number to that of all the cases possible is the measure of this probability1 which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible4 ,the preceding notion of probability supposes that1 in increasing in the same ratio the number of favorable cases and that of all the cases possible1 the probability remains the same4 ,in order to convince ourselves let us take two urns1 ,a and ,b1 the first containing four white and two black balls1 and the second containing only two white balls and one black one4 ,we may imagine the two black balls of the first urn attached by a thread which breaks at the moment when one of them is seized in order to be drawn out1 and the four white balls thus forming two similar systems4 ,all the chances which will favor the seizure of one of the balls of the black system will lead to a black ball4 ,if we conceive now that the threads which unite the balls do not break at all1 it is clear that the number of possible chances will not change any more than that of the chances favorable to the extraction of the black balls2 but two balls will be drawn from the urn at the same time2 the probability of drawing a black ball from the urn ,a will then be the same as at first4 ,but then we have obviously the case of urn ,b with the single difference that the three balls of this last urn would be replaced by three systems of two balls invariably connected4 ,when all the cases are favorable to an event the probability changes to certainty and its expression becomes equal to unity4 ,upon this condition1 certainty and probability are comparable1 although there may be an essential difference between the two states of the mind when a truth is rigorously demonstrated to it1 or when it still perceives a small source of error4 ,in things which are only probable the difference of the data1 which each man has in regard to them1 is one of the principal causes of the diversity of opinions which prevail in regard to the same objects4 ,let us suppose1 for example1 that we have three urns1 ,a1 ,b1 ,c1 one of which contains only black balls while the two others contain only white balls2 a ball is to be drawn from the urn ,c and the probability is demanded that this ball will be black4 ,if we do not know which of the three urns contains black balls only1 so that there is no reason to believe that it is ,c rather than ,b or ,a1 these three hypotheses will appear equally possible1 and since a black ball can be drawn only in the first hypothesis1 the probability of drawing it is equal to one third4 ,if it is known that the urn ,a contains white balls only1 the indecision then extends only to the urns ,b and ,c1 and the probability that the ball drawn from the urn ,c will be black is one half4 ,finally this probability changes to certainty if we are assured that the urns ,a and ,b contain white balls only4 ,it is thus that an incident related to a numerous assembly finds various degrees of credence1 according to the extent of knowledge of the auditors4 ,if the man who reports it is fully convinced of it and if1 by his position and character1 he inspires great confidence1 his statement1 however extraordinary it may be1 will have for the auditors who lack information the same degree of probability as an ordinary statement made by the same man1 and they will have entire faith in it4 ,but if some one of them knows that the same incident is rejected by other equally trustworthy men1 he will be in doubt and the incident will be discredited by the enlightened auditors1 who will reject it whether it be in regard to facts well averred or the immutable laws of nature4 ,it is to the influence of the opinion of those whom the multitude judges best informed and to whom it has been accustomed to give its confidence in regard to the most important matters of life that the propagation of those errors is due which in times of ignorance have covered the face of the earth4 ,magic and astrology offer us two great examples4 ,these errors inculcated in infancy1 adopted without examination1 and having for a basis only universal credence1 have maintained themselves during a very long time2 but at last the progress of science has destroyed them in the minds of enlightened men1 whose opinion consequently has caused them to disappear even among the common people1 through the power of imitation and habit which had so generally spread them abroad4 ,this power1 the richest resource of the moral world1 establishes and conserves in a whole nation ideas entirely contrary to those which it upholds elsewhere with the same authority4 ,what indulgence ought we not then to have for opinions different from ours1 when this difference often depends only upon the various points of view where circumstances have placed us6 ,let us enlighten those whom we judge insufficiently instructed2 but first let us examine critically our own opinions and weigh with impartiality their respective probabilities4 ,the difference of opinions depends1 however1 upon the manner in which the influence of known data is determined4 ,the theory of probabilities holds to considerations so delicate that it is not surprising that with the same data two persons arrive at different results1 especially in very complicated questions4 ,let us examine now the general principles of this theory4 ,,chapter ,,iii4 ,,the ,,general ,,principles ,,of ,,the ,,calculus ,,of ,,probabilities4 ,first ,principle4--,the first of these principles is the definition itself of probability1 which1 as has been seen1 is the ratio of the number of favorable cases to that of all the cases possible4 ,second ,principle4--,but that supposes the various cases equally possible4 ,if they are not so1 we will determine first their respective possibilities1 whose exact appreciation is one of the most delicate points of the theory of chance4 ,then the probability will be the sum of the possibilities of each favorable case4 ,let us illustrate this principle by an example4 ,let us suppose that we throw into the air a large and very thin coin whose two large opposite faces1 which we will call heads and tails1 are perfectly similar4 ,let us find the probability of throwing heads at least one time in two throws4 ,it is clear that four equally possible cases may arise1 namely1 heads at the first and at the second throw2 heads at the first throw and tails at the second2 tails at the first throw and heads at the second2 finally1 tails at both throws4 ,the first three cases are favorable to the event whose probability is sought2 consequently this probability is equal to 2 so that it is a bet of three to one that heads will be thrown at least once in two throws4 ,we can count at this game only three different cases1 namely1 heads at the first throw1 which dispenses with throwing a second time2 tails at the first throw and heads at the second2 finally1 tails at the first and at the second throw4 ,this would reduce the probability to if we should consider with d',alembert these three cases as equally possible4 ,but it is apparent that the probability of throwing heads at the first throw is 1 while that of the two other cases is 1 the first case being a simple event which corresponds to two events combined3 heads at the first and at the second throw1 and heads at the first throw1 tails at the second4 ,if we then1 conforming to the second principle1 add the possibility of heads at the first throw to the possibility of tails at the first throw and heads at the second1 we shall have for the probability sought1 which agrees with what is found in the supposition when we play the two throws4 ,this supposition does not change at all the chance of that one who bets on this event2 it simply serves to reduce the various cases to the cases equally possible4 ,third ,principle4--,one of the most important points of the theory of probabilities and that which lends the most to illusions is the manner in which these probabilities increase or diminish by their mutual combination4 ,if the events are independent of one another1 the probability of their combined existence is the product of their respective probabilities4 ,thus the probability of throwing one ace with a single die is 2 that of throwing two aces in throwing two dice at the same time is #a_/cf4 ,each face of the one being able to combine with the six faces of the other1 there are in fact thirty-six equally possible cases1 among which one single case gives two aces4 ,generally the probability that a simple event in the same circumstances will occur consecutively a given number of times is equal to the probability of this simple event raised to the power indicated by this number4 ,having thus the successive powers of a fraction less than unity diminishing without ceasing1 an event which depends upon a series of very great probabilities may become extremely improbable4 ,suppose then an incident be transmitted to us by twenty witnesses in such manner that the first has transmitted it to the second1 the second to the third1 and so on4 ,suppose again that the probability of each testimony be equal to the fraction #i_/aj2 that of the incident resulting from the testimonies will be less than 4 ,we cannot better compare this diminution of the probability than with the extinction of the light of objects by the interposition of several pieces of glass4 ,a relatively small number of pieces suffices to take away the view of an object that a single piece allows us to perceive in a distinct manner4 ,the historians do not appear to have paid sufficient attention to this degradation of the probability of events when seen across a great number of successive generations2 many historical events reputed as certain would be at least doubtful if they were submitted to this test4 ,in the purely mathematical sciences the most distant consequences participate in the certainty of the principle from which they are derived4 ,in the applications of analysis to physics the results have all the certainty of facts or experiences4 ,but in the moral sciences1 where each inference is deduced from that which precedes it only in a probable manner1 however probable these deductions may be1 the chance of error increases with their number and ultimately surpasses the chance of truth in the consequences very remote from the principle4 ,fourth ,principle4--,when two events depend upon each other1 the probability of the compound event is the product of the probability of the first event and the probability that1 this event having occurred1 the second will occur4 ,thus in the preceding case of the three urns ,a1 ,b1 ,c1 of which two contain only white balls and one contains only black balls1 the probability of drawing a white ball from the urn ,c is 1 since of the three urns only two contain balls of that color4 ,but when a white ball has been drawn from the urn ,c1 the indecision relative to that one of the urns which contain only black balls extends only to the urns ,a and ,b2 the probability of drawing a white ball from the urn ,b is 2 the product of by 1 or 1 is then the probability of drawing two white balls at one time from the urns ,b and ,c4 ,we see by this example the influence of past events upon the probability of future events4 ,for the probability of drawing a white ball from the urn ,b1 which primarily is 1 becomes when a white ball has been drawn from the urn ,c2 it would change to certainty if a black ball had been drawn from the same urn4 ,we will determine this influence by means of the following principle1 which is a corollary of the preceding one4 ,fifth ,principle4--,if we calculate priori the probability of the occurred event and the probability of an event composed of that one and a second one which is expected1 the second probability divided by the first will be the probability of the event expected1 drawn from the observed event4 ,here is presented the question raised by some philosophers touching the influence of the past upon the probability of the future4 ,let us suppose at the play of heads and tails that heads has occurred oftener than tails4 ,by this alone we shall be led to believe that in the constitution of the coin there is a secret cause which favors it4 ,thus in the conduct of life constant happiness is a proof of competency which should induce us to employ preferably happy persons4 ,but if by the unreliability of circumstances we are constantly brought back to a state of absolute indecision1 if1 for example1 we change the coin at each throw at the play of heads and tails1 the past can shed no light upon the future and it would be absurd to take account of it4 ,sixth ,principle4--,each of the causes to which an observed event may be attributed is indicated with just as much likelihood as there is probability that the event will take place1 supposing the event to be constant4 ,the probability of the existence of any one of these causes is then a fraction whose numerator is the probability of the event resulting from this cause and whose denominator is the sum of the similar probabilities relative to all the causes2 if these various causes1 considered priori1 are unequally probable1 it is necessary1 in place of the probability of the event resulting from each cause1 to employ the product of this probability by the possibility of the cause itself4 ,this is the fundamental principle of this branch of the analysis of chances which consists in passing from events to causes4 ,this principle gives the reason why we attribute regular events to a particular cause4 ,some philosophers have thought that these events are less possible than others and that at the play of heads and tails1 for example1 the combination in which heads occurs twenty successive times is less easy in its nature than those where heads and tails are mixed in an irregular manner4 ,but this opinion supposes that past events have an influence on the possibility of future events1 which is not at all admissible4 ,the regular combinations occur more rarely only because they are less numerous4 ,if we seek a cause wherever we perceive symmetry1 it is not that we regard a symmetrical event as less possible than the others1 but1 since this event ought to be the effect of a regular cause or that of chance1 the first of these suppositions is more probable than the second4 ,on a table we see letters arranged in this order1 ,c o n s t a n t i n o p l e1 and we judge that this arrangement is not the result of chance1 not because it is less possible than the others1 for if this word were not employed in any language we should not suspect it came from any particular cause1 but this word being in use among us1 it is incomparably more probable that some person has thus arranged the aforesaid letters than that this arrangement is due to chance4 ,this is the place to define the word extraordinary4 ,we arrange in our thought all possible events in various classes2 and we regard as extraordinary those classes which include a very small number4 ,thus at the play of heads and tails the occurrence of heads a hundred successive times appears to us extraordinary because of the almost infinite number of combinations which may occur in a hundred throws2 and if we divide the combinations into regular series containing an order easy to comprehend1 and into irregular series1 the latter are incomparably more numerous4 ,the drawing of a white ball from an urn which among a million balls contains only one of this color1 the others being black1 would appear to us likewise extraordinary1 because we form only two classes of events relative to the two colors4 ,but the drawing of the number #dgehac1 for example1 from an urn that contains a million numbers seems to us an ordinary event2 because1 comparing individually the numbers with one another without dividing them into classes1 we have no reason to believe that one of them will appear sooner than the others4 ,from what precedes1 we ought generally to conclude that the more extraordinary the event1 the greater the need of its being supported by strong proofs4 ,for those who attest it1 being able to deceive or to have been deceived1 these two causes are as much more probable as the reality of the event is less4 ,we shall see this particularly when we come to speak of the probability of testimony4 ,seventh ,principle4--,the probability of a future event is the sum of the products of the probability of each cause1 drawn from the event observed1 by the probability that1 this cause existing1 the future event will occur4 ,the following example will illustrate this principle4 ,let us imagine an urn which contains only two balls1 each of which may be either white or black4 ,one of these balls is drawn and is put back into the urn before proceeding to a new draw4 ,suppose that in the first two draws white balls have been drawn2 the probability of again drawing a white ball at the third draw is required4 ,only two hypotheses can be made here3 either one of the balls is white and the other black1 or both are white4 ,in the first hypothesis the probability of the event observed is 2 it is unity or certainty in the second4 ,thus in regarding these hypotheses as so many causes1 we shall have for the sixth principle and for their respective probabilities4 ,but if the first hypothesis occurs1 the probability of drawing a white ball at the third draw is 2 it is equal to certainty in the second hypothesis2 multiplying then the last probabilities by those of the corresponding hypotheses1 the sum of the products1 or #i_/aj1 will be the probability of drawing a white ball at the third draw4 ,when the probability of a single event is unknown we may suppose it equal to any value from zero to unity4 ,the probability of each of these hypotheses1 drawn from the event observed1 is1 by the sixth principle1 a fraction whose numerator is the probability of the event in this hypothesis and whose denominator is the sum of the similar probabilities relative to all the hypotheses4 ,thus the probability that the possibility of the event is comprised within given limits is the sum of the fractions comprised within these limits4 ,now if we multiply each fraction by the probability of the future event1 determined in the corresponding hypothesis1 the sum of the products relative to all the hypotheses will be1 by the seventh principle1 the probability of the future event drawn from the event observed4 ,thus we find that an event having occurred successively any number of times1 the probability that it will happen again the next time is equal to this number increased by unity divided by the same number1 increased by two units4 ,placing the most ancient epoch of history at five thousand years ago1 or at #ahbfbac days1 and the sun having risen constantly in the interval at each revolution of twenty-four hours1 it is a bet of #ahbfbad to one that it will rise again to-morrow4 ,but this number is incomparably greater for him who1 recognizing in the totality of phenomena the principal regulator of days and seasons1 sees that nothing at the present moment can arrest the course of it4 ,buffon in his ,political ,arithmetic calculates differently the preceding probability4 ,he supposes that it differs from unity only by a fraction whose numerator is unity and whose denominator is the number #b raised to a power equal to the number of days which have elapsed since the epoch4 ,but the true manner of relating past events with the probability of causes and of future events was unknown to this illustrious writer4 ,,chapter ,,iv4 ,,concerning ,,hope4 ,the probability of events serves to determine the hope or the fear of persons interested in their existence4 ,the word hope has various acceptations2 it expresses generally the advantage of that one who expects a certain benefit in suppositions which are only probable4 ,this advantage in the theory of chance is a product of the sum hoped for by the probability of obtaining it2 it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities4 ,this division is the only equitable one when all strange circumstances are eliminated2 because an equal degree of probability gives an equal right to the sum hoped for4 ,we will call this advantage mathematical hope4 ,eighth ,principle4--,when the advantage depends on several events it is obtained by taking the sum of the products of the probability of each event by the benefit attached to its occurrence4 ,let us apply this principle to some examples4 ,let us suppose that at the play of heads and tails ,paul receives two francs if he throws heads at the first throw and five francs if he throws it only at the second4 ,multiplying two francs by the probability of the first case1 and five francs by the probability of the second case1 the sum of the products1 or two and a quarter francs1 will be ,paul's advantage4 ,it is the sum which he ought to give in advance to that one who has given him this advantage2 for1 in order to maintain the equality of the play1 the throw ought to be equal to the advantage which it procures4 ,if ,paul receives two francs by throwing heads at the first and five francs by throwing it at the second throw1 whether he has thrown it or not at the first1 the probability of throwing heads at the second throw being 1 multiplying two francs and five francs by the sum of these products will give three and one half francs for ,paul's advantage and consequently for his stake at the game4 ,ninth ,principle4--,in a series of probable events of which the ones produce a benefit and the others a loss1 we shall have the advantage which results from it by making a sum of the products of the probability of each favorable event by the benefit which it procures1 and subtracting from this sum that of the products of the probability of each unfavorable event by the loss which is attached to it4 ,if the second sum is greater than the first1 the benefit becomes a loss and hope is changed to fear4 ,consequently we ought always in the conduct of life to make the product of the benefit hoped for1 by its probability1 at least equal to the similar product relative to the loss4 ,but it is necessary1 in order to attain this1 to appreciate exactly the advantages1 the losses1 and their respective probabilities4 ,for this a great accuracy of mind1 a delicate judgment1 and a great experience in affairs is necessary2 it is necessary to know how to guard one's self against prejudices1 illusions of fear or hope1 and erroneous ideas1 ideas of fortune and happiness1 with which the majority of people feed their self-love4 ,the application of the preceding principles to the following question has greatly exercised the geometricians4 ,paul plays at heads and tails with the condition of receiving two francs if he throws heads at the first throw1 four francs if he throws it only at the second throw1 eight francs if he throws it only at the third1 and so on4 ,his stake at the play ought to be1 according to the eighth principle1 equal to the number of throws1 so that if the game continues to infinity the stake ought to be infinite4 ,however1 no reasonable man would wish to risk at this game even a small sum1 for example five francs4 ,whence comes this difference between the result of calculation and the indication of common sense8 ,we soon recognize that it amounts to this3 that the moral advantage which a benefit procures for us is not proportional to this benefit and that it depends upon a thousand circumstances1 often very difficult to define1 but of which the most general and most important is that of fortune4 ,indeed it is apparent that one franc has much greater value for him who possesses only a hundred than for a millionaire4 ,we ought then to distinguish in the hoped-for benefit its absolute from its relative value4 ,but the latter is regulated by the motives which make it desirable1 whereas the first is independent of them4 ,the general principle for appreciating this relative value cannot be given1 but here is one proposed by ,daniel ,bernoulli which will serve in many cases4 ,tenth ,principle4--,the relative value of an infinitely small sum is equal to its absolute value divided by the total benefit of the person interested4 ,this supposes that every one has a certain benefit whose value can never be estimated as zero4 ,indeed even that one who possesses nothing always gives to the product of his labor and to his hopes a value at least equal to that which is absolutely necessary to sustain him4 ,if we apply analysis to the principle just propounded1 we obtain the following rule3 ,let us designate by unity the part of the fortune of an individual1 independent of his expectations4 ,if we determine the different values that this fortune may have by virtue of these expectations and their probabilities1 the product of these values raised respectively to the powers indicated by their probabilities will be the physical fortune which would procure for the individual the same moral advantage which he receives from the part of his fortune taken as unity and from his expectations2 by subtracting unity from the product1 the difference will be the increase of the physical fortune due to expectations3 we will call this increase moral hope4 ,it is easy to see that it coincides with mathematical hope when the fortune taken as unity becomes infinite in reference to the variations which it receives from the expectations4 ,but when these variations are an appreciable part of this unity the two hopes may differ very materially among themselves4 ,this rule conduces to results conformable to the indications of common sense which can by this means be appreciated with some exactitude4 ,thus in the preceding question it is found that if the fortune of ,paul is two hundred francs1 he ought not reasonably to stake more than nine francs4 ,the same rule leads us again to distribute the danger over several parts of a benefit expected rather than to expose the entire benefit to this danger4 ,it results similarly that at the fairest game the loss is always greater than the gain4 ,let us suppose1 for example1 that a player having a fortune of one hundred francs risks fifty at the play of heads and tails2 his fortune after his stake at the play will be reduced to eighty-seven francs1 that is to say1 this last sum would procure for the player the same moral advantage as the state of his fortune after the stake4 ,the play is then disadvantageous even in the case where the stake is equal to the product of the sum hoped for1 by its probability4 ,we can judge by this of the immorality of games in which the sum hoped for is below this product4 ,they subsist only by false reasonings and by the cupidity which they excite and which1 leading the people to sacrifice their necessaries to chimerical hopes whose improbability they are not in condition to appreciate1 are the source of an infinity of evils4 ,the disadvantage of games of chance1 the advantage of not exposing to the same danger the whole benefit that is expected1 and all the similar results indicated by common sense1 subsist1 whatever may be the function of the physical fortune which for each individual expresses his moral fortune4 ,it is enough that the proportion of the increase of this function to the increase of the physical fortune diminishes in the measure that the latter increases4 ,,chapter ,v4 ,,concerning ,,the ,,analytical ,,methods ,,of ,,the ,,calculus ,,of ,,probabilities4 ,the application of the principle which we have just expounded to the various questions of probability requires methods whose investigation has given birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences4 ,if we form the product of the binomials1 unity plus the first letter1 unity plus the second letter1 unity plus the third letter1 and so on up to n letters1 and subtract unity from this developed product1 the result will be the sum of the combination of all these letters taken one by one1 two by two1 three by three1 etc41 each combination having unity for a coefficient4 ,in order to have the number of combinations of these n letters taken s by s times1 we shall observe that if we suppose these letters equal among themselves1 the preceding product will become the nth power of the binomial one plus the first letter2 thus the number of combinations of n letters taken s by s times will be the coefficient of the sth power of the first letter in the development in this binomial2 and this number is obtained by means of the known binomial formula4 ,attention must be paid to the respective situations of the letters in each combination1 observing that if a second letter is joined to the first it may be placed in the first or second position which gives two combinations4 ,if we join to these combinations a third letter1 we can give it in each combination the first1 the second1 and the third rank which forms three combinations relative to each of the two others1 in all six combinations4 ,from this it is easy to conclude that the number of arrangements of which s letters are susceptible is the product of the numbers from unity to s4 ,in order to pay regard to the respective positions of the letters it is necessary then to multiply by this product the number of combinations of n letters s by s times1 which is tantamount to taking away the denominator of the coefficient of the binomial which expresses this number4 ,let us imagine a lottery composed of n numbers1 of which r are drawn at each draw4 ,the probability is demanded of the drawing of s given numbers in one draw4 ,to arrive at this let us form a fraction whose denominator will be the number of all the cases possible or of the combinations of n letters taken r by r times1 and whose numerator will be the number of all the combinations which contain the given s numbers4 ,this last number is evidently that of the combinations of the other numbers taken n less s by n less s times4 ,this fraction will be the required probability1 and we shall easily find that it can be reduced to a fraction whose numerator is the number of combinations of r numbers taken s by s times1 and whose denominator is the number of combinations of n numbers taken similarly s by s times4 ,thus in the lottery of ,france1 formed as is known of #ij numbers of which five are drawn at each draw1 the probability of drawing a given combination is #e_/ij1 or #a_/ah2 the lottery ought then for the equality of the play to give eighteen times the stake4 ,the total number of combinations two by two of the #ij numbers is #djje1 and that of the combinations two by two of #e numbers is #aj4 ,the probability of the drawing of a given pair is then #a_/djje1 and the lottery ought to give four hundred and a half times the stake2 it ought to give #aagdh times for a given tray1 #eaajch times for a quaternary1 and #dcidibfh times for a quint4 ,the lottery is far from giving the player these advantages4 ,suppose in an urn a white balls1 b black balls1 and after having drawn a ball it is put back into the urn2 the probability is asked that in n number of draws m white balls and n - m black balls will be drawn4 ,it is clear that the number of cases that may occur at each drawing is a b4 ,each case of the second drawing being able to combine with all the cases of the first1 the number of possible cases in two drawings is the square of the binomial a b4 ,in the development of this square1 the square of a expresses the number of cases in which a white ball is twice drawn1 the double product of a by b expresses the number of cases in which a white ball and a black ball are drawn4 ,finally1 the square of b expresses the number of cases in which two black balls are drawn4 ,continuing thus1 we see generally that the nth power of the binomial a b expresses the number of all the cases possible in n draws2 and that in the development of this power the term multiplied by the mth power of a expresses the number of cases in which m white balls and n - m black balls may be drawn4 ,dividing then this term by the entire power of the binomial1 we shall have the probability of drawing m white balls and n - m black balls4 ,the ratio of the numbers a and a b being the probability of drawing one white ball at one draw2 and the ratio of the numbers b and a b being the probability of drawing one black ball2 if we call these probabilities p and q1 the probability of drawing m white balls in n draws will be the term multiplied by the mth power of p in the development of the nth power of the binomial p q2 we may see that the sum p q is unity4 ,this remarkable property of the binomial is very useful in the theory of probabilities4 ,but the most general and direct method of resolving questions of probability consists in making them depend upon equations of differences4 ,comparing the successive conditions of the function which expresses the probability when we increase the variables by their respective differences1 the proposed question often furnishes a very simple proportion between the conditions4 ,this proportion is what is called equation of ordinary or partial differentials2 ordinary when there is only one variable1 partial when there are several4 ,let us consider some examples of this4 ,three players of supposed equal ability play together on the following conditions3 that one of the first two players who beats his adversary plays the third1 and if he beats him the game is finished4 ,if he is beaten1 the victor plays against the second until one of the players has defeated consecutively the two others1 which ends the game4 ,the probability is demanded that the game will be finished in a certain number n of plays4 ,let us find the probability that it will end precisely at the nth play4 ,for that the player who wins ought to enter the game at the play n - #a and win it thus at the following play4 ,but if in place of winning the play n - #a he should be beaten by his adversary who had just beaten the other player1 the game would end at this play4 ,thus the probability that one of the players will enter the game at the play n - #a and will win it is equal to the probability that the game will end precisely with this play2 and as this player ought to win the following play in order that the game may be finished at the nth play1 the probability of this last case will be only one half of the preceding one4 ,this probability is evidently a function of the number n2 this function is then equal to the half of the same function when n is diminished by unity4 ,this equality forms one of those equations called ordinary finite differential equations4 ,we may easily determine by its use the probability that the game will end precisely at a certain play4 ,it is evident that the play cannot end sooner than at the second play2 and for this it is necessary that that one of the first two players who has beaten his adversary should beat at the second play the third player2 the probability that the game will end at this play is 4 ,hence by virtue of the preceding equation we conclude that the successive probabilities of the end of the game are for the third play1 for the fourth play1 and so on2 and in general raised to the power n - #a for the nth play4 ,the sum of all these powers of is unity less the last of these powers2 it is the probability that the game will end at the latest in n plays4 ,let us consider again the first problem more difficult which may be solved by probabilities and which ,pascal proposed to ,fermat to solve4 ,two players1 ,a and ,b1 of equal skill play together on the conditions that the one who first shall beat the other a given number of times shall win the game and shall take the sum of the stakes at the game2 after some throws the players agree to quit without having finished the game3 we ask in what manner the sum ought to be divided between them4 ,it is evident that the parts ought to be proportional to the respective probabilities of winning the game4 ,the question is reduced then to the determination of these probabilities4 ,they depend evidently upon the number of points which each player lacks of having attained the given number4 ,hence the probability of ,a is a function of the two numbers which we will call indices4 ,if the two players should agree to play one throw more 7an agreement which does not change their condition1 provided that after this new throw the division is always made proportionally to the new probabilities of winning the game71 then either ,a would win this throw and in that case the number of points which he lacks would be diminished by unity1 or the player ,b would win it and in that case the number of points lacking to this last player would be less by unity4 ,but the probability of each of these cases is 2 the function sought is then equal to one half of this function in which we diminish by unity the first index plus the half of the same function in which the second variable is diminished by unity4 ,this equality is one of those equations called equations of partial differentials4 ,we are able to determine by its use the probabilities of ,a by dividing the smallest numbers1 and by observing that the probability or the function which expresses it is equal to unity when the player ,a does not lack a single point1 or when the first index is zero1 and that this function becomes zero with the second index4 ,supposing thus that the player ,a lacks only one point1 we find that his probability is 1 1 1 etc41 according as ,b lacks one point1 two1 three1 etc4 ,generally it is then unity less the power of 1 equal to the number of points which ,b lacks4 ,we will suppose then that the player ,a lacks two points and his probability will be found equal to 1 1 #aa_/af1 etc41 according as ,b lacks one point1 two points1 three points1 etc4 ,we will suppose again that the player ,a lacks three points1 and so on4 ,this manner of obtaining the successive values of a quantity by means of its equation of differences is long and laborious4 ,the geometricians have sought methods to obtain the general function of indices that satisfies this equation1 so that for any particular case we need only to substitute in this function the corresponding values of the indices4 ,let us consider this subject in a general way4 ,for this purpose let us conceive a series of terms arranged along a horizontal line so that each of them is derived from the preceding one according to a given law4 ,let us suppose this law expressed by an equation among several consecutive terms and their index1 or the number which indicates the rank that they occupy in the series4 ,this equation ,i call the equation of finite differences by a single index4 ,the order or the degree of this equation is the difference of rank of its two extreme terms4 ,we are able by its use to determine successively the terms of the series and to continue it indefinitely2 but for that it is necessary to know a number of terms of the series equal to the degree of the equation4 ,these terms are the arbitrary constants of the expression of the general term of the series or of the integral of the equation of differences4 ,let us imagine now below the terms of the preceding series a second series of terms arranged horizontally2 let us imagine again below the terms of the second series a third horizontal series1 and so on to infinity2 and let us suppose the terms of all these series connected by a general equation among several consecutive terms1 taken as much in the horizontal as in the vertical sense1 and the numbers which indicate their rank in the two senses4 ,this equation is called the equation of partial finite differences by two indices4 ,let us imagine in the same way below the plan of the preceding series a second plan of similar series1 whose terms should be placed respectively below those of the first plan2 let us imagine again below this second plan a third plan of similar series1 and so on to infinity2 let us suppose all the terms of these series connected by an equation among several consecutive terms taken in the sense of length1 width1 and depth1 and the three numbers which indicate their rank in these three senses4 ,this equation ,i call the equation of partial finite differences by three indices4 ,finally1 considering the matter in an abstract way and independently of the dimensions of space1 let us imagine generally a system of magnitudes1 which should be functions of a certain number of indices1 and let us suppose among these magnitudes1 their relative differences to these indices and the indices themselves1 as many equations as there are magnitudes2 these equations will be partial finite differences by a certain number of indices4 ,we are able by their use to determine successively these magnitudes4 ,but in the same manner as the equation by a single index requires for it that we know a certain number of terms of the series1 so the equation by two indices requires that we know one or several lines of series whose general terms should be expressed each by an arbitrary function of one of the indices4 ,similarly the equation by three indices requires that we know one or several plans of series1 the general terms of which should be expressed each by an arbitrary function of two indices1 and so on4 ,in all these cases we shall be able by successive eliminations to determine a certain term of the series4 ,but all the equations among which we eliminate being comprised in the same system of equations1 all the expressions of the successive terms which we obtain by these eliminations ought to be comprised in one general expression1 a function of the indices which determine the rank of the term4 ,this expression is the integral of the proposed equation of differences1 and the search for it is the object of integral calculus4 ,taylor is the first who in his work entitled ,metodus incrementorum has considered linear equations of finite differences4 ,he gives the manner of integrating those of the first order with a coefficient and a last term1 functions of the index4 ,in truth the relations of the terms of the arithmetical and geometrical progressions which have always been taken into consideration are the simplest cases of linear equations of differences2 but they had not been considered from this point of view4 ,it was one of those which1 attaching themselves to general theories1 lead to these theories and are consequently veritable discoveries4 ,about the same time ,moivre was considering under the name of recurring series the equations of finite differences of a certain order having a constant coefficient4 ,he succeeded in integrating them in a very ingenious manner4 ,as it is always interesting to follow the progress of inventors1 ,i shall expound the method of ,moivre by applying it to a recurring series whose relation among three consecutive terms is given4 ,first he considers the relation among the consecutive terms of a geometrical progression or the equation of two terms which expresses it4 ,referring it to terms less than unity1 he multiplies it in this state by a constant factor and subtracts the product from the first equation4 ,thus he obtains an equation among three consecutive terms of the geometrical progression4 ,moivre considers next a second progression whose ratio of terms is the same factor which he has just used4 ,he diminishes similarly by unity the index of the terms of the equation of this new progression4 ,in this condition he multiplies it by the ratio of the terms of the first progression1 and he subtracts the product from the equation of the second progression1 which gives him among three consecutive terms of this progression a relation entirely similar to that which he has found for the first progression4 ,then he observes that if one adds term by term the two progressions1 the same ratio exists among any three of these consecutive terms4 ,he compares the coefficients of this ratio to those of the relation of the terms of the proposed recurrent series1 and he finds for determining the ratios of the two geometrical progressions an equation of the second degree1 whose roots are these ratios4 ,thus ,moivre decomposes the recurrent series into two geometrical progressions1 each multiplied by an arbitrary constant which he determines by means of the first two terms of the recurrent series4 ,this ingenious process is in fact the one that d',alembert has since employed for the integration of linear equations of infinitely small differences with constant coefficients1 and ,lagrange has transformed into similar equations of finite differences4 ,finally1 ,i have considered the linear equations of partial finite differences1 first under the name of recurro-recurrent series and afterwards under their own name4 ,the most general and simplest manner of integrating all these equations appears to me that which ,i have based upon the consideration of discriminant functions1 the idea of which is here given4 ,if we conceive a function ,v of a variable t developed according to the powers of this variable1 the coefficient of any one of these powers will be a function of the exponent or index of this power1 which index ,i shall call x4 ,v is what ,i call the discriminant function of this coefficient or of the function of the index4 ,now if we multiply the series of the development of ,v by a function of the same variable1 such1 for example1 as unity plus two times this variable1 the product will be a new discriminant function in which the coefficient of the power x of the variable t will be equal to the coefficient of the same power in ,v plus twice the coefficient of the power less unity4 ,thus the function of the index x in the product will be equal to the function of the index x in ,v plus twice the same function in which the index is diminished by unity4 ,this function of the index x is thus a derivative of the function of the same index in the development of ,v1 a function which ,i shall call the primitive function of the index4 ,let us designate the derivative function by the letter ,alembert placed before the primitive function4 ,the derivation indicated by this letter will depend upon the multiplier of ,v1 which we will call ,t and which we will suppose developed like ,v by the ratio to the powers of the variable t4 ,if we multiply anew by ,t the product of ,v by ,t1 which is equivalent to multiplying ,v by ,t1 we shall form a third discriminant function1 in which the coefficient of the xth power of t will be a derivative similar to the corresponding coefficient of the preceding product2 it may be expressed by the same character placed before the preceding derivative1 and then this character will be written twice before the primitive function of x4 ,but in place of writing it thus twice we give it #b for an exponent4 ,continuing thus1 we see generally that if we multiply ,v by the nth power of ,t1 we shall have the coefficient of the xth power of t in the product of ,v by the nth power of ,t by placing before the primitive function the character with n for an exponent4 ,let us suppose1 for example1 that ,t be unity divided by t2 then in the product of ,v by ,t the coefficient of the xth power of t will be the coefficient of the power greater by unity in ,v2 this coefficient in the product of ,v by the nth power of ,t will then be the primitive function in which x is augmented by n units4 ,let us consider now a new function ,z of t1 developed like ,v and ,t according to the powers of t2 let us designate by the character placed before the primitive function the coefficient of the xth power of t in the product of ,v by ,z2 this coefficient in the product of ,v by the nth power of ,z will be expressed by the character affected by the exponent n and placed before the primitive function of x4 ,if1 for example1 ,z is equal to unity divided by t less one1 the coefficient of the xth power of t in the product of ,v by ,z will be the coefficient of the x #a power of t in ,v less the coefficient of the xth power4 ,it will be then the finite difference of the primitive function of the index x4 ,then the character indicates a finite difference of the primitive function in the case where the index varies by unity2 and the nth power of this character placed before the primitive function will indicate the finite nth difference of this function4 ,if we suppose that ,t be unity divided by t1 we shall have ,t equal to the binomial ,z #a4 ,the product of ,v by the nth power of ,t will then be equal to the product of ,v by the nth power of the binomial ,z #a4 ,developing this power in the ratio of the powers of ,z1 the product of ,v by the various terms of this development will be the discriminant functions of these same terms in which we substitute in place of the powers of ,z the corresponding finite differences of the primitive function of the index4 ,now the product of ,v by the nth power of ,t is the primitive function in which the index x is augmented by n units2 repassing from the discriminant functions to their coefficients1 we shall have this primitive function thus augmented equal to the development of the nth power of the binomial ,z #a1 provided that in this development we substitute in place of the powers of ,z the corresponding differences of the primitive function and that we multiply the independent term of these powers by the primitive function4 ,we shall thus obtain the primitive function whose index is augmented by any number n by means of its differences4 ,supposing that ,t and ,z always have the preceding values1 we shall have ,z equal to the binomial ,t - #a2 the product of ,v by the nth power of ,z will then be equal to the product of ,v by the development of the nth power of the binomial ,t - #a4 ,repassing from the discriminant functions to their coefficients as has just been done1 we shall have the nth difference of the primitive function expressed by the development of the nth power of the binomial ,t - #a1 in which we substitute for the powers of ,t this same function whose index is augmented by the exponent of the power1 and for the independent term of t1 which is unity1 the primitive function1 which gives this difference by means of the consecutive terms of this function4 ,placing before the primitive function expressing the derivative of this function1 which multiplies the x power of t in the product of ,v by ,t1 and expressing the same derivative in the product of ,v by ,z1 we are led by that which precedes to this general result3 whatever may be the function of the variable t represented by ,t and ,z1 we may1 in the development of all the identical equations susceptible of being formed among these functions1 substitute the characters and in place of ,t and ,z1 provided that we write the primitive function of the index in series with the powers and with the products of the powers of the characters1 and that we multiply by this function the independent terms of these characters4 ,we are able by means of this general result to transform any certain power of a difference of the primitive function of the index x1 in which x varies by unity1 into a series of differences of the same function in which x varies by a certain number of units and reciprocally4 ,let us suppose that ,t be the i power of unity divided by t - #a1 and that ,z be always unity divided by t - #a2 then the coefficient of the x power of t in the product of ,v by ,t will be the coefficient of the x i power of t in ,v less the coefficient of the x power of t2 it will then be the finite difference of the primitive function of the index x in which we vary this index by the number i4 ,it is easy to see that ,t is equal to the difference between the i power of the binomial ,z #a and unity4 ,the nth power of ,t is equal to the nth power of this difference4 ,if in this equality we substitute in place of ,t and ,z the characters and 1 and after the development we place at the end of each term the primitive function of the index x1 we shall have the nth difference of this function in which x varies by i units expressed by a series of differences of the same function in which x varies by unity4 ,this series is only a transformation of the difference which it expresses and which is identical with it2 but it is in similar transformations that the power of analysis resides4 ,the generality of analysis permits us to suppose in this expression that n is negative4 ,then the negative powers of and indicate the integrals4 ,indeed the nth difference of the primitive function having for a discriminant function the product of ,v by the nth power of the binomial one divided by t less unity1 the primitive function which is the nth integral of this difference has for a discriminant function that of the same difference multiplied by the nth power taken less than the binomial one divided by t minus one1 a power to which the same power of the character corresponds2 this power indicates then an integral of the same order1 the index x varying by unity2 and the negative powers of indicate equally the integrals x varying by i units4 ,we see1 thus1 in the clearest and simplest manner the rationality of the analysis observed among the positive powers and differences1 and among the negative powers and the integrals4 ,if the function indicated by placed before the primitive function is zero1 we shall have an equation of finite differences1 and ,v will be the discriminant function of its integral4 ,in order to obtain this discriminant function we shall observe that in the product of ,v by ,t all the powers of t ought to disappear except the powers inferior to the order of the equation of differences2 ,v is then equal to a fraction whose denominator is ,t and whose numerator is a polynomial in which the highest power of t is less by unity than the order of the equation of differences4 ,the arbitrary coefficients of the various powers of t in this polynomial1 including the power zero1 will be determined by as many values of the primitive function of the index when we make successively x equal to zero1 to one1 to two1 etc4 ,when the equation of differences is given we determine ,t by putting all its terms in the first member and zero in the second2 by substituting in the first member unity in place of the function which has the largest index2 the first power of t in place of the primitive function in which this index is diminished by unity2 the second power of t for the primitive function where this index is diminished by two units1 and so on4 ,the coefficient of the xth power of t in the development of the preceding expression of ,v will be the primitive function of x or the integral of the equation of finite differences4 ,analysis furnishes for this development various means1 among which we may choose that one which is most suitable for the question proposed2 this is an advantage of this method of integration4 ,let us conceive now that ,v be a function of the two variables t and t developed according to the powers and products of these variables2 the coefficient of any product of the powers x and x of t and t will be a function of the exponents or indices x and x of these powers2 this function ,i shall call the primitive function of which ,v is the discriminant function4 ,let us multiply ,v by a function ,t of the two variables t and t developed like ,v in ratio of the powers and the products of these variables2 the product will be the discriminant function of a derivative of the primitive function2 if ,t1 for example1 is equal to the variable t plus the variable t minus two1 this derivative will be the primitive function of which we diminish by unity the index x plus this same primitive function of which we diminish by unity the index x less two times the primitive function4 ,designating whatever ,t may be by the character placed before the primitive function1 this derivative1 the product of ,v by the nth power of ,t1 will be the discriminant function of the derivative of the primitive function before which one places the nth power of the character 4 ,hence result the theorems analogous to those which are relative to functions of a single variable4 ,suppose the function indicated by the character be zero2 one will have an equation of partial differences4 ,if1 for example1 we make as before ,t equal to the variable t plus the variable t - #b1 we have zero equal to the primitive function of which we diminish by unity the index x plus the same function of which we diminish by unity the index x minus two times the primitive function4 ,the discriminant function ,v of the primitive function or of the integral of this equation ought then to be such that its product by ,t does not include at all the products of t by t2 but ,v may include separately the powers of t and those of t1 that is to say1 an arbitrary function of t and an arbitrary function of t2 ,v is then a fraction whose numerator is the sum of these two arbitrary functions and whose denominator is ,t4 ,the coefficient of the product of the xth power of t by the x power of t in the development of this fraction will then be the integral of the preceding equation of partial differences4 ,this method of integrating this kind of equations seems to me the simplest and the easiest by the employment of the various analytical processes for the development of rational fractions4 ,more ample details in this matter would be scarcely understood without the aid of calculus4 ,considering equations of infinitely small partial differences as equations of finite partial differences in which nothing is neglected1 we are able to throw light upon the obscure points of their calculus1 which have been the subject of great discussions among geometricians4 ,it is thus that ,i have demonstrated the possibility of introducing discontinued functions in their integrals1 provided that the discontinuity takes place only for the differentials of the order of these equations or of a superior order4 ,the transcendent results of calculus are1 like all the abstractions of the understanding1 general signs whose true meaning may be ascertained only by repassing by metaphysical analysis to the elementary ideas which have led to them2 this often presents great difficulties1 for the human mind tries still less to transport itself into the future than to retire within itself4 ,the comparison of infinitely small differences with finite differences is able similarly to shed great light upon the metaphysics of infinitesimal calculus4 ,it is easily proven that the finite nth difference of a function in which the increase of the variable is ,e being divided by the nth power of ,e1 the quotient reduced in series by ratio to the powers of the increase ,e is formed by a first term independent of ,e4 ,in the measure that ,e diminishes1 the series approaches more and more this first term from which it can differ only by quantities less than any assignable magnitude4 ,this term is then the limit of the series and expresses in differential calculus the infinitely small nth difference of the function divided by the nth power of the infinitely small increase4 ,considering from this point of view the infinitely small differences1 we see that the various operations of differential calculus amount to comparing separately in the development of identical expressions the finite terms or those independent of the increments of the variables which are regarded as infinitely small2 this is rigorously exact1 these increments being indeterminant4 ,thus differential calculus has all the exactitude of other algebraic operations4 ,the same exactitude is found in the applications of differential calculus to geometry and mechanics4 ,if we imagine a curve cut by a secant at two adjacent points1 naming ,e the interval of the ordinates of these two points1 ,e will be the increment of the abscissa from the first to the second ordinate4 ,it is easy to see that the corresponding increment of the ordinate will be the product of ,e by the first ordinate divided by its subsecant2 augmenting then in this equation of the curve the first ordinate by this increment1 we shall have the equation relative to the second ordinate4 ,the difference of these two equations will be a third equation which1 developed by the ratio of the powers of ,e and divided by ,e1 will have its first term independent of ,e1 which will be the limit of this development4 ,this term1 equal to zero1 will give then the limit of the subsecants1 a limit which is evidently the subtangent4 ,this singularly happy method of obtaining the subtangent is due to ,fermat1 who has extended it to transcendent curves4 ,this great geometrician expresses by the character ,e the increment of the abscissa2 and considering only the first power of this increment1 he determines exactly as we do by differential calculus the subtangents of the curves1 their points of inflection1 the maxima and minima of their ordinates1 and in general those of rational functions4 ,we see likewise by his beautiful solution of the problem of the refraction of light inserted in the ,collection of the ,letters of ,descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers4 ,fermat should be regarded1 then1 as the true discoverer of ,differential ,calculus4 ,newton has since rendered this calculus more analytical in his ,method of ,fluxions1 and simplified and generalized the processes by his beautiful theorem of the binomial4 ,finally1 about the same time ,leibnitz has enriched differential calculus by a notation which1 by indicating the passage from the finite to the infinitely small1 adds to the advantage of expressing the general results of calculus that of giving the first approximate values of the differences and of the sums of the quantities2 this notation is adapted of itself to the calculus of partial differentials4 ,we are often led to expressions which contain so many terms and factors that the numerical substitutions are impracticable4 ,this takes place in questions of probability when we consider a great number of events4 ,meanwhile it is necessary to have the numerical value of the formul in order to know with what probability the results are indicated1 which the events develop by multiplication4 ,it is necessary especially to have the law according to which this probability continually approaches certainty1 which it will finally attain if the number of events were infinite4 ,in order to obtain this law ,i considered that the definite integrals of differentials multiplied by the factors raised to great powers would give by integration the formul composed of a great number of terms and factors4 ,this remark brought me to the idea of transforming into similar integrals the complicated expressions of analysis and the integrals of the equation of differences4 ,i fulfilled this condition by a method which gives at the same time the function comprised under the integral sign and the limits of the integration4 ,it offers this remarkable thing1 that the function is the same discriminant function of the expressions and the proposed equations2 this attaches this method to the theory of discriminant functions of which it is thus the complement4 ,further1 it would only be a question of reducing the definite integral to a converging series4 ,this ,i have obtained by a process which makes the series converge with as much more rapidity as the formula which it represents is more complicated1 so that it is more exact as it becomes more necessary4 ,frequently the series has for a factor the square root of the ratio of the circumference to the diameter2 sometimes it depends upon other transcendents whose number is infinite4 ,an important remark which pertains to great generality of analysis1 and which permits us to extend this method to formul and to equations of difference which the theory of probability presents most frequently1 is that the series to which one comes by supposing the limits of the definite integrals to be real and positive take place equally in the case where the equation which determines these limits has only negative or imaginary roots4 ,these passages from the positive to the negative and from the real to the imaginary1 of which ,i first have made use1 have led me further to the values of many singular definite integrals1 which ,i have accordingly demonstrated directly4 ,we may then consider these passages as a means of discovery parallel to induction and analogy long employed by geometricians1 at first with an extreme reserve1 afterwards with entire confidence1 since a great number of examples has justified its use4 ,in the mean time it is always necessary to confirm by direct demonstrations the results obtained by these divers means4 ,i have named the ensemble of the preceding methods the ,calculus of ,discriminant ,functions2 this calculus serves as a basis for the work which ,i have published under the title of the ,analytical ,theory of ,probabilities4 ,it is connected with the simple idea of indicating the repeated multiplications of a quantity by itself or its entire and positive powers by writing toward the top of the letter which expresses it the numbers which mark the degrees of these powers4 ,this notation1 employed by ,descartes in his ,geometry and generally adopted since the publication of this important work1 is a little thing1 especially when compared with the theory of curves and variable functions by which this great geometrician has established the foundations of modern calculus4 ,but the language of analysis1 most perfect of all1 being in itself a powerful instrument of discoveries1 its notations1 especially when they are necessary and happily conceived1 are so many germs of new calculi4 ,this is rendered appreciable by this example4 ,wallis1 who in his work entitled ,arithmetica ,infinitorum1 one of those which have most contributed to the progress of analysis1 has interested himself especially in following the thread of induction and analogy1 considered that if one divides the exponent of a letter by two1 three1 etc41 the quotient will be accordingly the ,cartesian notation1 and when division is possible the exponent of the square1 cube1 etc41 root of the quantity which represents the letter raised to the dividend exponent4 ,extending by analogy this result to the case where division is impossible1 he considered a quantity raised to a fractional exponent as the root of the degree indicated by the denominator of this fraction--namely1 of the quantity raised to a power indicated by the numerator4 ,he observed then that1 according to the ,cartesian notation1 the multiplication of two powers of the same letter amounts to adding their exponents1 and that their division amounts to subtracting the exponents of the power of the divisor from that of the power of the dividend1 when the second of these exponents is greater than the first4 ,wallis extended this result to the case where the first exponent is equal to or greater than the second1 which makes the difference zero or negative4 ,he supposed then that a negative exponent indicates unity divided by the quantity raised to the same exponent taken positively4 ,these remarks led him to integrate generally the monomial differentials1 whence he inferred the definite integrals of a particular kind of binomial differentials whose exponent is a positive integral number4 ,the observation then of the law of the numbers which express these integrals1 a series of interpolations and happy inductions where one perceives the germ of the calculus of definite integrals which has so much exercised geometricians and which is one of the fundaments of my new ,theory of ,probabilities1 gave him the ratio of the area of the circle to the square of its diameter expressed by an infinite product1 which1 when one stops it1 confines this ratio to limits more and more converging2 this is one of the most singular results in analysis4 ,but it is remarkable that ,wallis1 who had so well considered the fractional exponents of radical powers1 should have continued to note these powers as had been done before him4 ,newton in his ,letters to ,oldembourg1 if ,i am not mistaken1 was the first to employ the notation of these powers by fractional exponents4 ,comparing by the way of induction1 of which ,wallis had made such a beautiful use1 the exponents of the powers of the binomial with the coefficients of the terms of its development in the case where this exponent is integral and positive1 he determined the law of these coefficients and extended it by analogy to fractional and negative powers4 ,these various results1 based upon the notation of ,descartes1 show his influence on the progress of analysis4 ,it has still the advantage of giving the simplest and fairest idea of logarithms1 which are indeed only the exponents of a magnitude whose successive powers1 increasing by infinitely small degrees1 can represent all numbers4 ,but the most important extension that this notation has received is that of variable exponents1 which constitutes exponential calculus1 one of the most fruitful branches of modern analysis4 ,leibnitz was the first to indicate the transcendents by variable exponents1 and thereby he has completed the system of elements of which a finite function can be composed2 for every finite explicit function of a variable may be reduced in the last analysis to simple magnitudes1 combined by the method of addition1 subtraction1 multiplication1 and division and raised to constant or variable powers4 ,the roots of the equations formed from these elements are the implicit functions of the variable4 ,it is thus that a variable has for a logarithm the exponent of the power which is equal to it in the series of the powers of the number whose hyperbolic logarithm is unity1 and the logarithm of a variable of it is an implicit function4 ,leibnitz thought to give to his differential character the same exponents as to magnitudes2 but then in place of indicating the repeated multiplications of the same magnitude these exponents indicate the repeated differentiations of the same function4 ,this new extension of the ,cartesian notation led ,leibnitz to the analogy of positive powers with the differentials1 and the negative powers with the integrals4 ,lagrange has followed this singular analogy in all its developments2 and by series of inductions which may be regarded as one of the most beautiful applications which have ever been made of the method of induction he has arrived at general formul which are as curious as useful on the transformations of differences and of integrals the ones into the others when the variables have divers finite increments and when these increments are infinitely small4 ,but he has not given the demonstrations of it which appear to him difficult4 ,the theory of discriminant functions extends the ,cartesian notations to some of its characters2 it shows with proof the analogy of the powers and operations indicated by these characters2 so that it may still be regarded as the exponential calculus of characters4 ,all that concerns the series and the integration of equations of differences springs from it with an extreme facility4 ,,part ,,ii4 ,,applications ,,of ,,the ,,calculus ,,of ,,probabilities4 ,,chapter ,,vi4 ,,games ,,of ,,chance4 ,the combinations which games present were the object of the first investigations of probabilities4 ,in an infinite variety of these combinations many of them lend themselves readily to calculus2 others require more difficult calculi2 and the difficulties increasing in the measure that the combinations become more complicated1 the desire to surmount them and curiosity have excited geometricians to perfect more and more this kind of analysis4 ,it has been seen already that the benefits of a lottery are easily determined by the theory of combinations4 ,but it is more difficult to know in how many draws one can bet one against one1 for example that all the numbers will be drawn1 n being the number of numbers1 r that of the numbers drawn at each draw1 and i the unknown number of draws4 ,the expression of the probability of drawing all the numbers depends upon the nth finite difference of the i power of a product of r consecutive numbers4 ,when the number n is considerable the search for the value of i which renders this probability equal to becomes impossible at least unless this difference is converted into a very converging series4 ,this is easily done by the method here below indicated by the approximations of functions of very large numbers4 ,it is found thus since the lottery is composed of ten thousand numbers1 one of which is drawn at each draw1 that there is a disadvantage in betting one against one that all the numbers will be drawn in #iegfg draws and an advantage in making the same bet for #iegfh draws4 ,in the lottery of ,france this bet is disadvantageous for #he draws and advantageous for #hf draws4 ,let us consider again two players1 ,a and ,b1 playing together at heads and tails in such a manner that at each throw if heads turns up ,a gives one counter to ,b1 who gives him one if tails turns up2 the number of counters of ,b is limited1 while that of ,a is unlimited1 and the game is to end only when ,b shall have no more counters4 ,we ask in how many throws one should bet one to one that the game will end4 ,the expression of the probability that the game will end in an i number of throws is given by a series which comprises a great number of terms and factors if the number of counters of ,b is considerable2 the search for the value of the unknown i which renders this series would then be impossible if we did not reduce the same to a very convergent series4 ,in applying to it the method of which we have just spoken1 we find a very simple expression for the unknown from which it results that if1 for example1 ,b has a hundred counters1 it is a bet of a little less than one against one that the game will end in #bcghj throws1 and a bet of a little more than one against one that it will end in #bcgha throws4 ,these two examples added to those we have already given are sufficient to shows how the problems of games have contributed to the perfection of analysis4 ,,chapter ,,vii4 ,,concerning ,,the ,,unknown ,,inequalities ,,which ,,may ,,exist ,,among ,,chances ,,which ,,are ,,supposed ,,equal4 ,inequalities of this kind have upon the results of the calculation of probabilities a sensible influence which deserves particular attention4 ,let us take the game of heads and tails1 and let us suppose that it is equally easy to throw the one or the other side of the coin4 ,then the probability of throwing heads at the first throw is and that of throwing it twice in succession is 4 ,but if there exist in the coin an inequality which causes one of the faces to appear rather than the other without knowing which side is favored by this inequality1 the probability of throwing heads at the first throw will always 2 because of our ignorance of which face is favored by the inequality the probability of the simple event is increased if this inequality is favorable to it1 just so much is it diminished if the inequality is contrary to it4 ,but in this same ignorance the probability of throwing heads twice in succession is increased4 ,indeed this probability is that of throwing heads at the first throw multiplied by the probability that having thrown it at the first throw it will be thrown at the second2 but its happening at the first throw is a reason for belief that the inequality of the coin favors it2 the unknown inequality increases1 then1 the probability of throwing heads at the second throw2 it consequently increases the product of these two probabilities4 ,in order to submit this matter to calculus let us suppose that this inequality increases by a twentieth the probability of the simple event which it favors4 ,if this event is heads1 its probability will be plus #a_/bj1 or #aa_/bj1 and the probability of throwing it twice in succession will be the square of #aa_/bj1 or #aba_/djj4 ,if the favored event is tails1 the probability of heads1 will be minus #a_/bj1 or #i_/bj1 and the probability of throwing it twice in succession will be #ha_/djj4 ,since we have at first no reason for believing that the inequality favors one of these events rather than the other1 it is clear that in order to have the probability of the compound event heads heads it is necessary to add the two preceding probabilities and take the half of their sum1 which gives #aja_/djj for this probability1 which exceeds by #a_/djj or by the square of the favor #a_/bj that the inequality adds to the possibilities of the event which it favors4 ,the probability of throwing tails tails is similarly #aja_/djj1 but the probability of throwing heads tails or tails heads is each #ii_/djj2 for the sum of these four probabilities ought to equal certainty or unity4 ,we find thus generally that the constant and unknown causes which favor simple events which are judged equally possible always increase the probability of the repetition of the same simple event4 ,in an even number of throws heads and tails ought both to happen either an even number of times or odd number of times4 ,the probability of each of these cases is if the possibilities of the two faces are equal2 but if there is between them an unknown inequality1 this inequality is always favorable to the first case4 ,two players whose skill is supposed to be equal play on the conditions that at each throw that one who loses gives a counter to his adversary1 and that the game continues until one of the players has no more counters4 ,the calculation of the probabilities shows us that for the equality of the play the throws of the players ought to be an inverse ratio to their counters4 ,but if there is between the players a small unknown inequality1 it favors that one of the players who has the smallest number of counters4 ,his probability of winning the game increases if the players agree to double or triple their counters2 and it will be or the same as the probability of the other player in the case where the number of their counters should become infinite1 preserving always the same ratio4 ,one may correct the influence of these unknown inequalities by submitting them themselves to the chances of hazard4 ,thus at the play of heads and tails1 if one has a second coin which is thrown each time with the first and one agrees to name constantly heads the face turned up by the second coin1 the probability of throwing heads twice in succession with the first coin will approach much nearer than in the case of a single coin4 ,in this last case the difference is the square of the small increment of possibility that the unknown inequality gives to the face of the first coin which it favors2 in the other case this difference is the quadruple product of this square by the corresponding square relative to the second coin4 ,let there be thrown into an urn a hundred numbers from #a to #ajj in the order of numeration1 and after having shaken the urn in order to mix the numbers one is drawn2 it is clear that if the mixing has been well done the probabilities of the drawing of the numbers will be the same4 ,but if we fear that there is among them small differences dependent upon the order according to which the numbers have been thrown into the urn1 we shall diminish considerably these differences by throwing into a second urn the numbers according to the order of their drawing from the first urn1 and by shaking then this second urn in order to mix the numbers4 ,a third urn1 a fourth urn1 etc41 would diminish more and more these differences already inappreciable in the second urn4 ,,chapter ,,viii4 ,,concerning ,,the ,,laws ,,of ,,probability ,,which ,,result ,,from ,,the ,,indefinite ,,multiplication ,,of ,,events4 ,amid the variable and unknown causes which we comprehend under the name of chance1 and which render uncertain and irregular the march of events1 we see appearing1 in the measure that they multiply1 a striking regularity which seems to hold to a design and which has been considered as a proof of ,providence4 ,but in reflecting upon this we soon recognize that this regularity is only the development of the respective possibilities of simple events which ought to present themselves more often when they are more probable4 ,let us imagine1 for example1 an urn which contains white balls and black balls2 and let us suppose that each time a ball is drawn it is put back into the urn before proceeding to a new draw4 ,the ratio of the number of the white balls drawn to the number of black balls drawn will be most often very irregular in the first drawings2 but the variable causes of this irregularity produce effects alternately favorable and unfavorable to the regular march of events which destroy each other mutually in the totality of a great number of draws1 allowing us to perceive more and more the ratio of white balls to the black balls contained in the urn1 or the respective possibilities of drawing a white ball or black ball at each draw4 ,from this results the following theorem4 ,the probability that the ratio of the number of white balls drawn to the total number of balls drawn does not deviate beyond a given interval from the ratio of the number of white balls to the total number of balls contained in the urn1 approaches indefinitely to certainty by the indefinite multiplication of events1 however small this interval4 ,this theorem indicated by common sense was difficult to demonstrate by analysis4 ,accordingly the illustrious geometrician ,jacques ,bernoulli1 who first has occupied himself with it1 attaches great importance to the demonstrations he has given4 ,the calculus of discriminant functions applied to this matter not only demonstrates with facility this theorem1 but still more it gives the probability that the ratio of the events observed deviates only in certain limits from the true ratio of their respective possibilities4 ,one may draw from the preceding theorem this consequence which ought to be regarded as a general law1 namely1 that the ratios of the acts of nature are very nearly constant when these acts are considered in great number4 ,thus in spite of the variety of years the sum of the productions during a considerable number of years is sensibly the same2 so that man by useful foresight is able to provide against the irregularity of the seasons by spreading out equally over all the seasons the goods which nature distributes in an unequal manner4 ,i do not except from the above law results due to moral causes4 ,the ratio of annual births to the population1 and that of marriages to births1 show only small variations2 at ,paris the number of annual births is almost the same1 and ,i have heard it said at the post-office in ordinary seasons the number of letters thrown aside on account of defective addresses changes little each year2 this has likewise been observed at ,london4 ,it follows again from this theorem that in a series of events indefinitely prolonged the action of regular and constant causes ought to prevail in the long run over that of irregular causes4 ,it is this which renders the gains of the lotteries just as certain as the products of agriculture2 the chances which they reserve assure them a benefit in the totality of a great number of throws4 ,thus favorable and numerous chances being constantly attached to the observation of the eternal principles of reason1 of justice1 and of humanity which establish and maintain societies1 there is a great advantage in conforming to these principles and of grave inconvenience in departing from them4 ,if one consult histories and his own experience1 one will see all the facts come to the aid of this result of calculus4 ,consider the happy effects of institutions founded upon reason and the natural rights of man among the peoples who have known how to establish and preserve them4 ,consider again the advantages which good faith has procured for the governments who have made it the basis of their conduct and how they have been indemnified for the sacrifices which a scrupulous exactitude in keeping their engagements has cost them4 ,what immense credit at home6 ,what preponderance abroad6 ,on the contrary1 look into what an abyss of misfortunes nations have often been precipitated by the ambition and the perfidy of their chiefs4 ,every time that a great power intoxicated by the love of conquest aspires to universal domination the sentiment of independence produces among the menaced nations a coalition of which it becomes almost always the victim4 ,similarly in the midst of the variable causes which extend or restrain the divers states1 the natural limits acting as constant causes ought to end by prevailing4 ,it is important then to the stability as well as to the happiness of empires not to extend them beyond those limits into which they are led again without cessation by the action of the causes2 just as the waters of the seas raised by violent tempests fall again into their basins by the force of gravity4 ,it is again a result of the calculus of probabilities confirmed by numerous and melancholy experiences4 ,history treated from the point of view of the influence of constant causes would unite to the interest of curiosity that of offering to man most useful lessons4 ,sometimes we attribute the inevitable results of these causes to the accidental circumstances which have produced their action4 ,it is1 for example1 against the nature of things that one people should ever be governed by another when a vast sea or a great distance separates them4 ,it may be affirmed that in the long run this constant cause1 joining itself without ceasing to the variable causes which act in the same way and which the course of time develops1 will end by finding them sufficiently strong to give to a subjugated people its natural independence or to unite it to a powerful state which may be contiguous4 ,in a great number of cases1 and these are the most important of the analysis of hazards1 the possibilities of simple events are unknown and we are forced to search in past events for the indices which can guide us in our conjectures about the causes upon which they depend4 ,in applying the analysis of discriminant functions to the principle elucidated above on the probability of the causes drawn from the events observed1 we are led to the following theorem4 ,when a simple event or one composed of several simple events1 as1 for instance1 in a game1 has been repeated a great number of times the possibilities of the simple events which render most probable that which has been observed are those that observation indicates with the greatest probability2 in the measure that the observed event is repeated this probability increases and would end by amounting to certainty if the numbers of repetitions should become infinite4 ,there are two kinds of approximations3 the one is relative to the limits taken on all sides of the possibilities which give to the past the greatest probability2 the other approximation is related to the probability that these possibilities fall within these limits4 ,the repetition of the compound event increases more and more this probability1 the limits remaining the same2 it reduces more and more the interval of these limits1 the probability remaining the same2 in infinity this interval becomes zero and the probability changes to certainty4 ,if we apply this theorem to the ratio of the births of boys to that of girls observed in the different countries of ,europe1 we find that this ratio1 which is everywhere about equal to that of #bb to #ba1 indicates with an extreme probability a greater facility in the birth of boys4 ,considering further that it is the same at ,naples and at ,st4 ,petersburg1 we shall see that in this regard the influence of climate is without effect4 ,we might then suspect1 contrary to the common belief1 that this predominance of masculine births exists even in the ,orient4 ,i have consequently invited the ,french scholars sent to ,egypt to occupy themselves with this interesting question2 but the difficulty in obtaining exact information about the births has not permitted them to solve it4 ,happily1 ,m4 de ,humboldt has not neglected this matter among the innumerable new things which he has observed and collected in ,america with so much sagacity1 constancy1 and courage4 ,he has found in the tropics the same ratio of the births as we observe in ,paris2 this ought to make us regard the greater number of masculine births as a general law of the human race4 ,the laws which the different kinds of animals follow in this regard seem to me worthy of the attention of naturalists4 ,the fact that the ratio of births of boys to that of girls differs very little from unity even in the great number of the births observed in a place would offer in this regard a result contrary to the general law1 without which we should be right in concluding that this law did not exist4 ,in order to arrive at this result it is necessary to employ great numbers and to be sure that it is indicated by great probability4 ,buffon cites1 for example1 in his ,political ,arithmetic several communities of ,bourgogne where the births of girls have surpassed those of boys4 ,among these communities that of ,carcelle-le-,grignon presents in #bjji births during five years #ajbf girls and #ihc boys4 ,although these numbers are considerable1 they indicate1 however1 only a greater possibility in the births of girls with a probability of #i_/aj1 and this probability1 smaller than that of not throwing heads four times in succession in the game of heads and tails1 is not sufficient to investigate the cause for this anomaly1 which1 according to all probability1 would disappear if one should follow during a century the births in this community4 ,the registers of births1 which are kept with care in order to assure the condition of the citizens1 may serve in determining the population of a great empire without recurring to the enumeration of its inhabitants--a laborious operation and one difficult to make with exactitude4 ,but for this it is necessary to know the ratio of the population to the annual births4 ,the most precise means of obtaining it consists1 first1 in choosing in the empire districts distributed in an almost equal manner over its whole surface1 so as to render the general result independent of local circumstances2 second1 in enumerating with care for a given epoch the inhabitants of several communities in each of these districts2 third1 by determining from the statement of the births during several years which precede and follow this epoch the mean number corresponding to the annual births4 ,this number1 divided by that of the inhabitants1 will give the ratio of the annual births to the population in a manner more and more accurate as the enumeration becomes more considerable4 ,the government1 convinced of the utility of a similar enumeration1 has decided at my request to order its execution4 ,in thirty districts spread out equally over the whole of ,france1 communities have been chosen which would be able to furnish the most exact information4 ,their enumerations have given #bjcgfae individuals as the total number of their inhabitants on the #bc;d of ,september1 #ahjb4 ,the statement of the births in these communities during the years #ahjj1 #ahja1 and #ahjb have given3 ,births4 ,marriages4 ,deaths4 #aajcab boys #dfjcg #ajcfei men #ajebhg girls #iiddc women ,the ratio of the population to annual births is then #bh 7#cebhde_/ajjjjjj72 it is greater than had been estimated up to this time4 ,multiplying the number of annual births in ,france by this ratio1 we shall have the population of this kingdom4 ,but what is the probability that the population thus determined will not deviate from the true population beyond a given limit8 ,resolving this problem and applying to its solution the preceding data1 ,i have found that1 the number of annual births in ,france being supposed to be #ajjjjjj1 which brings the population to #bhcebhde inhabitants1 it is a bet of almost #cjjjjj against #a that the error of this result is not half a million4 ,the ratio of the births of boys to that of girls which the preceding statement offers is that of #bb to #ba2 and the marriages are to the births as #c is to #d4 ,at ,paris the baptisms of children of both sexes vary a little from the ratio of #bb to #ba4 ,since #agde1 the epoch in which one has commenced to distinguish the sexes upon the birth-registers1 up to the end of #aghd1 there have been baptized in this capital #cicchf boys and #cggeee girls4 ,the ratio of the two numbers is almost that of #be to #bd2 it appears then at ,paris that a particular cause approximates an equality of baptisms of the two sexes4 ,if we apply to this matter the calculus of probabilities1 we find that it is a bet of #bch to #a in favor of the existence of this cause1 which is sufficient to authorize the investigation4 ,upon reflection it has appeared to me that the difference observed holds to this1 that the parents in the country and the provinces1 finding some advantage in keeping the boys at home1 have sent to the ,hospital for ,foundlings in ,paris fewer of them relative to the number of girls according to the ratio of births of the two sexes4 ,this is proved by the statement of the registers of this hospital4 ,from the beginning of #agde to the end of #ahji there were entered #afcdii boys and #aeidje girls4 ,the first of these numbers exceeds only by #a_/ch the second1 which it ought to have surpassed at least by #a_/bd4 ,this confirms the existence of the assigned cause1 namely1 that the ratio of births of boys to those of girls is at ,paris that of #bb to #ba1 no attention having been paid to foundlings4 ,the preceding results suppose that we may compare the births to the drawings of balls from an urn which contains an infinite number of white balls and black balls so mixed that at each draw the chances of drawing ought to be the same for each ball2 but it is possible that the variations of the same seasons in different years may have some influence upon the annual ratio of the births of boys to those of girls4 ,the ,bureau of ,longitudes of ,france publishes each year in its annual the tables of the annual movement of the population of the kingdom4 ,the tables already published commence in #ahag2 in that year and in the five following years there were born #bifbcfa boys and #bghaiig girls1 which gives about #af_/ae for the ratio of the births of boys to that of girls4 ,the ratios of each year vary little from this mean result2 the smallest ratio is that of #ahbb1 where it was only #ag_/af2 the greatest is of the year #ahag1 when it was #ae_/ad4 ,these ratios vary appreciably from the ratio of #bb_/ba found above4 ,applying to this deviation the analysis of probabilities in the hypothesis of the comparison of births to the drawings of balls from an urn1 we find that it would be scarcely probable4 ,it appears1 then1 to indicate that this hypothesis1 although closely approximated1 is not rigorously exact4 ,in the number of births which we have just stated there are of natural children #bjjdid boys and #aijfih girls4 ,the ratio of masculine and feminine births was then in this regard #bj_/ai1 smaller than the mean ratio of #af_/ae4 ,this result is in the same sense as that of the births of foundlings2 and it seems to prove that in the class of natural children the births of the two sexes approach more nearly equality than in the class of legitimate children4 ,the difference of the climates from the north to the south of ,france does not appear to influence appreciably the ratio of the births of boys and girls4 ,the thirty most southern districts have given #af_/ae for this ratio1 the same as that of entire ,france4 ,the constancy of the superiority of the births of boys over girls at ,paris and at ,london since they have been observed has appeared to some scholars to be a proof of ,providence1 without which they have thought that the irregular causes which disturb without ceasing the course of events ought several times to have rendered the annual births of girls superior to those of boys4 ,but this proof is a new example of the abuse which has been so often made of final causes which always disappear on a searching examination of the questions when we have the necessary data to solve them4 ,the constancy in question is a result of regular causes which give the superiority to the births of boys and which extend it to the anomalies due to hazard when the number of annual births is considerable4 ,the investigation of the probability that this constancy will maintain itself for a long time belongs to that branch of the analysis of hazards which passes from past events to the probability of future events2 and taking as a basis the births observed from #agde to #aghd1 it is a bet of almost #d against #a that at ,paris the annual births of boys will constantly surpass for a century the births of girls2 there is then no reason to be astonished that this has taken place for a half-century4 ,let us take another example of the development of constant ratios which events present in the measure that they are multiplied4 ,let us imagine a series of urns arranged circularly1 and each containing a very great number of white balls and black balls2 the ratio of white balls to the black in the urns being originally very different and such1 for example1 that one of these urns contains only white balls1 while another contains only black balls4 ,if one draws a ball from the first urn in order to put it into the second1 and1 after having shaken the second urn in order to mix well the new ball with the others1 one draws a ball to put it into the third urn1 and so on to the last urn1 from which is drawn a ball to put into the first1 and if this series is recommenced continually1 the analysis of probability shows us that the ratios of the white balls to the black in these urns will end by being the same and equal to the ratio of the sum of all the white balls to the sum of all the black balls contained in the urns4 ,thus by this regular mode of change the primitive irregularity of these ratios disappears eventually in order to make room for the most simple order4 ,now if among these urns one intercalate new ones in which the ratio of the sum of the white balls to the sum of the black balls which they contain differs from the preceding1 continuing indefinitely in the totality of the urns the drawings which we have just indicated1 the simple order established in the old urns will be at first disturbed1 and the ratios of the white balls to the black balls will become irregular2 but little by little this irregularity will disappear in order to make room for a new order1 which will finally be that of the equality of the ratios of the white balls to the black balls contained in the urns4 ,we may apply these results to all the combinations of nature in which the constant forces by which their elements are animated establish regular modes of action1 suited to bring about in the very heart of chaos systems governed by admirable laws4 ,the phenomena which seem the most dependent upon hazard present1 then1 when multiplied a tendency to approach without ceasing fixed ratios1 in such a manner that if we conceive on all sides of each of these ratios an interval as small as desired1 the probability that the mean result of the observations falls within this interval will end by differing from certainty only by a quantity greater than an assignable magnitude4 ,thus by the calculations of probabilities applied to a great number of observations we may recognize the existence of these ratios4 ,but before seeking the causes it is necessary1 in order not to be led into vain speculations1 to assure ourselves that they are indicated by a probability which does not permit us to regard them as anomalies due to hazard4 ,the theory of discriminant functions gives a very simple expression for this probability1 which is obtained by integrating the product of the differential of the quantity of which the result deduced from a great number of observations varies from the truth by a constant less than unity1 dependent upon the nature of the problem1 and raised to a power whose exponent is the ratio of the square of this variation to the number of observations4 ,the integral taken between the limits given and divided by the same integral1 applied to a positive and negative infinity1 will express the probability that the variation from the truth is comprised between these limits4 ,such is the general law of the probability of results indicated by a great number of observations4 ,,chapter ,,ix4 ,,the ,,application ,,of ,,the ,,calculus ,,of ,,probabilities ,,to ,,natural ,,philosophy4 ,the phenomena of nature are most often enveloped by so many strange circumstances1 and so great a number of disturbing causes mix their influence1 that it is very difficult to recognize them4 ,we may arrive at them only by multiplying the observations or the experiences1 so that the strange effects finally destroy reciprocally each other1 the mean results putting in evidence those phenomena and their divers elements4 ,the more numerous the number of observations and the less they vary among themselves the more their results approach the truth4 ,we fulfil this last condition by the choice of the methods of observations1 by the precision of the instruments1 and by the care which we take to observe closely2 then we determine by the theory of probabilities the most advantageous mean results or those which give the least value of the error4 ,but that is not sufficient2 it is further necessary to appreciate the probability that the errors of these results are comprised in the given limits2 and without this we have only an imperfect knowledge of the degree of exactitude obtained4 ,formul suitable to these matters are then true improvements of the method of sciences1 and it is indeed important to add them to this method4 ,the analysis which they require is the most delicate and the most difficult of the theory of probabilities2 it is one of the principal objects of the work which ,i have published upon this theory1 and in which ,i have arrived at formul of this kind which have the remarkable advantage of being independent of the law of the probability of errors and of including only the quantities given by the observations themselves and their expressions4 ,each observation has for an analytic expression a function of the elements which we wish to determine2 and if these elements are nearly known1 this function becomes a linear function of their corrections4 ,in equating it to the observation itself there is formed an equation of condition4 ,if we have a great number of similar equations1 we combine them in such a manner as to obtain as many final equations as there are elements whose corrections we determine then by resolving these equations4 ,but what is the most advantageous manner of combining equations of condition in order to obtain final equations8 ,what is the law of the probabilities of errors of which the elements are still susceptible that we draw from them8 ,this is made clear to us by the theory of probabilities4 ,the formation of a final equation by means of the equation of condition amounts to multiplying each one of these by an indeterminate factor and by uniting the products2 it is necessary to choose the system of factors which gives the smallest opportunity for error4 ,but it is apparent that if we multiply the possible errors of an element by their respective probabilities1 the most advantageous system will be that in which the sum of these products all1 taken1 positively is a minimum2 for a positive or a negative error ought to be considered as a loss4 ,forming1 then1 this sum of products1 the condition of the minimum will determine the system of factors which it is expedient to adopt1 or the most advantageous system4 ,we find thus that this system is that of the coefficients of the elements in each equation of condition2 so that we form a first final equation by multiplying respectively each equation of condition by its coefficient of the first element and by uniting all these equations thus multiplied4 ,we form a second final equation by employing in the same manner the coefficients of the second element1 and so on4 ,in this manner the elements and the laws of the phenomena obtained in the collection of a great number of observations are developed with the most evidence4 ,the probability of the errors which each element still leaves to be feared is proportional to the number whose hyperbolic logarithm is unity raised to a power equal to the square of the error taken as a minus quantity and multiplied by a constant coefficient which may be considered as the modulus of the probability of the errors2 because1 the error remaining the same1 its probability decreases with rapidity when the former increases2 so that the element obtained weighs1 if ,i may thus speak toward the truth1 as much more as this modulus is greater4 ,i would call for this reason this modulus the weight of the element or of the result4 ,this weight is the greatest possible in the system of factors--the most advantageous2 it is this which gives to this system superiority over others4 ,by a remarkable analogy of this weight with those of bodies compared at their common centre of gravity it results that if the same element is given by divers systems1 composed each of a great number of observations1 the most advantageous1 the mean result of their totality is the sum of the products of each partial result by its weight4 ,moreover1 the total weight of the results of the divers systems is the sum of their partial weights2 so that the probability of the errors of the mean result of their totality is proportional to the number which has unity for an hyperbolic logarithm raised to a power equal to the square of the error taken as minus and multiplied by the sum of the weights4 ,each weight depends in truth upon the law of the probability of error of each system1 and almost always this law is unknown2 but happily ,i have been able to eliminate the factor which contains it by means of the sum of the squares of the variations of the observations in this system from their mean result4 ,it would then be desirable in order to complete our knowledge of the results obtained by the totality of a great number of observations that we write by the side of each result the weight which corresponds to it2 analysis furnishes for this object both general and simple methods4 ,when we have thus obtained the exponential which represents the law of the probability of errors1 we shall have the probability that the error of the result is included within given limits by taking within the limits the integral of the product of this exponential by the differential of the error and multiplying it by the square root of the weight of the result divided by the circumference whose diameter is unity4 ,hence it follows that for the same probability the errors of the results are reciprocal to the square roots of their weights1 which serves to compare their respective precision4 ,in order to apply this method with success it is necessary to vary the circumstances of the observations or the experiences in such a manner as to avoid the constant causes of error4 ,it is necessary that the observations should be numerous1 and that they should be so much the more so as there are more elements to determine2 for the weight of the mean result increases as the number of observations divided by the number of the elements4 ,it is still necessary that the elements follow in these observations a different course2 for if the course of the two elements were exactly the same1 which would render their coefficients proportional in equation of conditions1 these elements would form only a single unknown quantity and it would be impossible to distinguish them by these observations4 ,finally it is necessary that the observations should be precise2 this condition1 the first of all1 increases greatly the weight of the result the expression of which has for a divisor the sum of the squares of the deviations of the observations from this result4 ,with these precautions we shall be able to make use of the preceding method and measure the degree of confidence which the results deduced from a great number of observations merit4 ,the rule which we have just given to conclude equations of condition1 final equations1 amount to rendering a minimum the sum of the squares of the errors of observations2 for each equation of condition becomes exact by substituting in it the observation plus its error2 and if we draw from it the expression of this error1 it is easy to see that the condition of the minimum of the sum of the squares of these expressions gives the rule in question4 ,this rule is the more precise as the observations are more numerous2 but even in the case where their number is small it appears natural to employ the same rule which in all cases offers a simple means of obtaining without groping the corrections which we seek to determine4 ,it serves further to compare the precision of the divers astronomical tables of the same star4 ,these tables may always be supposed as reduced to the same form1 and then they differ only by the epochs1 the mean movements and the coefficients of the arguments2 for if one of them contains a coefficient which is not found in the others1 it is clear that this amounts to supposing zero in them as the coefficient of this argument4 ,if now we rectify these tables by the totality of the good observations1 they would satisfy the condition that the sum of the squares of the errors should be a minimum2 the tables which1 compared to a considerable number of observations1 approach nearest this condition merit then the preference4 ,it is principally in astronomy that the method explained above may be employed with advantage4 ,the astronomical tables owe the truly astonishing exactitude which they have attained to the precision of observations and of theories1 and to the use of equations of conditions which cause to concur a great number of excellent observations in the correction of the same element4 ,but it remains to determine the probability of the errors that this correction leaves still to be feared2 and the method which ,i have just explained enables us to recognize the probability of these errors4 ,in order to give some interesting applications of it ,i have profited by the immense work which ,m4 ,bouvard has just finished on the movements of ,jupiter and ,saturn1 of which he has formed very precise tables4 ,he has discussed with the greatest care the oppositions and quadratures of these two planets observed by ,bradley and by the astronomers who have followed him down to the last years2 he has concluded the corrections of the elements of their movement and their masses compared to that of the sun taken as unity4 ,his calculations give him the mass of ,saturn equal to the #ceabth part of that of the sun4 ,applying to them my formul of probability1 ,i find that it is a bet of #aa1jjj against one that the error of this result is not #a_/ajj of its value1 or that which amounts to almost the same--that after a century of new observations added to the preceding ones1 and examined in the same manner1 the new result will not differ by #a_/ajj from that of ,m4 ,bouvard4 ,this wise astronomer finds again the mass of ,jupiter equal to the #ajgath part of the sun2 and my method of probability gives a bet of #a1jjj1jjj to one that this result is not #a_/ajj in error4 ,this method may be employed again with success in geodetic operations4 ,we determine the length of the great arc on the surface of the earth by triangulation1 which depends upon a base measured with exactitude4 ,but whatever precision may be brought to the measure of the angles1 the inevitable errors can1 by accumulating1 cause the value of the arc concluded from a great number of triangles to deviate appreciably from the truth4 ,we recognize this value1 then1 only imperfectly unless the probability that its error is comprised within given limits can be assigned4 ,the error of a geodetic result is a function of the errors of the angles of each triangle4 ,i have given in the work cited general formul in order to obtain the probability of the values of one or of several linear functions of a great number of partial errors of which we know the law of probability2 we may then by means of these formul determine the probability that the error of a geodetic result is contained within the assigned limits1 whatever may be the law of the probability of partial errors4 ,it is moreover more necessary to render ourselves independent of the law1 since the most simple laws themselves are always infinitely less probable1 seeing the infinite number of those which may exist in nature4 ,but the unknown law of partial errors introduces into the formul an indeterminant which does not permit of reducing them to numbers unless we are able to eliminate it4 ,we have seen that in astronomical questions1 where each observation furnishes an equation of condition for obtaining the elements1 we eliminate this determinant by means of the sum of the squares of the remainders when the most probable values of the elements have been substituted in each equation4 ,geodetic questions not offering similar equations1 it is necessary to seek another means of elimination4 ,the quantity by which the sum of the angles of each observed triangle surpasses two right angles plus the spherical excess furnishes this means4 ,thus we replace by the sum of the squares of these quantities the sum of the squares of the remainders of the equations of condition2 and we may assign in numbers the probability that the error of the final result of a series of geodetic operations will not exceed a given quantity4 ,but what is the most advantageous manner of dividing among the three angles of each triangle the observed sum of their errors8 ,the analysis of probabilities renders it apparent that each angle ought to be diminished by a third of this sum1 provided that the weight of a geodetic result be the greatest possible1 which renders the same error less probable4 ,there is then a great advantage in observing the three angles of each triangle and of correcting them as we have just said4 ,simple common sense indicates this advantage2 but the calculation of probabilities alone is able to appreciate it and to render apparent that by this correction it becomes the greatest possible4 ,in order to assure oneself of the exactitude of the value of a great arc which rests upon a base measured at one of its extremities one measures a second base toward the other extremity2 and one concludes from one of these bases the length of the other4 ,if this length varies very little from the observation1 there is all reason to believe that the chain of triangles which unites these bases is very nearly exact and likewise the value of the large arc which results from it4 ,one corrects1 then1 this value by modifying the angles of the triangles in such a manner that the base is calculated according to the bases measured4 ,but this may be done in an infinity of ways1 among which is preferred that of which the geodetic result has the greatest weight1 inasmuch as the same error becomes less probable4 ,the analysis of probabilities gives formul for obtaining directly the most advantageous correction which results from the measurements of the several bases and the laws of probability which the multiplicity of the bases makes--laws which become very rapidly decreasing by this multiplicity4 ,generally the errors of the results deduced from a great number of observations are the linear functions of the partial errors of each observation4 ,the coefficients of these functions depend upon the nature of the problem and upon the process followed in order to obtain the results4 ,the most advantageous process is evidently that in which the same error in the results is less probable than according to any other process4 ,the application of the calculus of probabilities to natural philosophy consists1 then1 in determining analytically the probability of the values of these functions and in choosing their indeterminant coefficients in such a manner that the law of this probability should be most rapidly descending4 ,eliminating1 then1 from the formul by the data of the question the factor which is introduced by the almost always unknown law of the probability of partial errors1 we may be able to evaluate numerically the probability that the errors of the results do not exceed a given quantity4 ,we shall thus have all that may be desired touching the results deduced from a great number of observations4 ,very approximate results may be obtained by other considerations4 ,suppose1 for example1 that one has a thousand and one observations of the same quantity2 the arithmetical mean of all these observations is the result given by the most advantageous method4 ,but one would be able to choose the result according to the condition that the sum of the variations from each partial value all taken positively should be a minimum4 ,it appears indeed natural to regard as very approximate the result which satisfies this condition4 ,it is easy to see that if one disposes the values given by the observations according to the order of magnitude1 the value which will occupy the mean will fulfil the preceding condition1 and calculus renders it apparent that in the case of an infinite number of observations it would coincide with the truth2 but the result given by the most advantageous method is still preferable4 ,we see by that which precedes that the theory of probabilities leaves nothing arbitrary in the manner of distributing the errors of the observations2 it gives for this distribution the most advantageous formul which diminishes as much as possible the errors to be feared in the results4 ,the consideration of probabilities can serve to distinguish the small irregularities of the celestial movements enveloped in the errors of observations1 and to repass to the cause of the anomalies observed in these movements4 ,in comparing all the observations it was ,ticho-,brah who recognized the necessity of applying to the moon an equation of time different from that which had been applied to the sun and to the planets4 ,it was similarly the totality of a great number of observations which made ,mayer recognize that the coefficient of the inequality of the precession ought to be diminished a little for the moon4 ,but since this diminution1 although confirmed and even augmented by ,mason1 did not appear to result from universal gravitation1 the majority of astronomers neglect it in their calculations4 ,having submitted to the calculation of probabilities a considerable number of lunar observations chosen for this purpose and which ,m4 ,bouvard consented to examine at my request1 it appeared to me to be indicated with so strong a probability that ,i believed the cause of it ought to be investigated4 ,i soon saw that it would be only the ellipticity of the terrestrial spheroid1 neglected up to that time in the theory of the lunar movement as being able to produce only imperceptible terms4 ,i concluded that these terms became perceptible by the successive integrations of differential equations4 ,i determined then those terms by a particular analysis1 and ,i discovered first the inequality of the lunar movement in latitude which is proportional to the sine of the longitude of the moon1 which no astronomer before had suspected4 ,i recognized then by means of this inequality that another exists in the lunar movement in longitude which produces the diminution observed by ,mayer in the equation of the precession applicable to the moon4 ,the quantity of this diminution and the coefficient of the preceding inequality in latitude are very appropriate to fix the oblateness of the earth4 ,having communicated my researches to ,m4 ,burg1 who was occupied at that time in perfecting the tables of the moon by the comparison of all the good observations1 ,i requested him to determine with a particular care these two quantities4 ,by a very remarkable agreement the values which he has found give to the earth the same oblateness1 #a_/cje1 which differs little from the mean derived from the measurements of the degrees of the meridian and the pendulum2 but those regarded from the point of view of the influence of the errors of the observations and of the perturbing causes in these measurements1 did not appear to me exactly determined by these lunar inequalities4 ,it was again by the consideration of probabilities that ,i recognized the cause of the secular equation of the moon4 ,the modern observations of this star compared to the ancient eclipses had indicated to astronomers an acceleration in the lunar movement2 but the geometricians1 and particularly ,lagrange1 having vainly sought in the perturbations which this movement experienced the terms upon which this acceleration depends1 reject it4 ,an attentive examination of the ancient and modern observations and of the intermediary eclipses observed by the ,arabians convinced me that it was indicated with a great probability4 ,i took up again then from this point of view the lunar theory1 and ,i recognized that the secular equation of the moon is due to the action of the sun upon this satellite1 combined with the secular variation of the eccentricity of the terrestrial orb2 this brought me to the discovery of the secular equations of the movements of the nodes and of the perigees of the lunar orbit1 which equations had not been even suspected by astronomers4 ,the very remarkable agreement of this theory with all the ancient and modern observations has brought it to a very high degree of evidence4 ,the calculus of probabilities has led me similarly to the cause of the great irregularities of ,jupiter and ,saturn4 ,comparing modern observations with ancient1 ,halley found an acceleration in the movement of ,jupiter and a retardation in that of ,saturn4 ,in order to conciliate the observations he reduced the movements to two secular equations of contrary signs and increasing as the squares of the times passed since #agjj4 ,euler and ,lagrange submitted to analysis the alterations which the mutual attraction of these two planets ought to produce in these movements4 ,they found in doing this the secular equations2 but their results were so different that one of the two at least ought to be erroneous4 ,i determined then to take up again this important problem of celestial mechanics1 and ,i recognized the invariability of the mean planetary movements1 which nullified the secular equations introduced by ,halley in the tables of ,jupiter and ,saturn4 ,thus there remain1 in order to explain the great irregularity of these planets1 only the attractions of the comets to which many astronomers had effective recourse1 or the existence of an irregularity over a long period produced in the movements of the two planets by their reciprocal action and affected by contrary signs for each of them4 ,a theorem which ,i found in regard to the inequalities of this kind rendered this inequality very probable4 ,according to this theorem1 if the movement of ,jupiter is accelerated1 that of ,saturn is retarded1 which has already conformed to what ,halley had noticed2 moreover1 the acceleration of ,jupiter resulting from the same theorem is to the retardation of ,saturn very nearly in the ratio of the secular equations proposed by ,halley4 ,considering the mean movements of ,jupiter and ,saturn ,i was enabled easily to recognize that two times that of ,jupiter differed only by a very small quantity from five times that of ,saturn4 ,the period of an irregularity which would have for an argument this difference would be about nine centuries4 ,indeed its coefficient would be of the order of the cubes of the eccentricities of the orbits2 but ,i knew that by virtue of successive integrations it acquired for divisor the square of the very small multiplier of the time in the argument of this inequality which is able to give it a great value2 the existence of this inequality appeared to me then very probable4 ,the following observation increased then its probability4 ,supposing its argument zero toward the epoch of the observations of ,ticho-,brah1 ,i saw that ,halley ought to have found by the comparison of modern with ancient observations the alterations which he had indicated2 while the comparison of the modern observations among themselves ought to offer contrary alterations similar to those which ,lambert had concluded from this comparison4 ,i did not then hesitate at all to undertake this long and tedious calculation necessary to assure myself of this inequality4 ,it was entirely confirmed by the result of this calculation1 which moreover made me recognize a great number of other inequalities of which the totality has inclined the tables of ,jupiter and ,saturn to the precision of the same observations4 ,it was again by means of the calculus of probabilities that ,i recognized the remarkable law of the mean movements of the three first satellites of ,jupiter1 according to which the mean longitude of the first minus three times that of the second plus two times that of the third is rigorously equal to the half-circumference4 ,the approximation with which the mean movements of these stars satisfy this law since their discovery indicates its existence with an extreme probability4 ,i sought then the cause of it in their mutual action4 ,the searching examination of this action convinced me that it was sufficient if in the beginning the ratios of their mean movements had approached this law within certain limits1 because their mutual action had established and maintained it rigorously4 ,thus these three bodies will balance one another eternally in space according to the preceding law unless strange causes1 such as comets1 should change suddenly their movements about ,jupiter4 ,accordingly it is seen how necessary it is to be attentive to the indications of nature when they are the result of a great number of observations1 although in other respects they may be inexplicable by known means4 ,the extreme difficulty of problems relative to the system of the world has forced geometricians to recur to the approximation which always leaves room for the fear that the quantities neglected may have an appreciable influence4 ,when they have been warned of this influence by the observations1 they have recurred to their analysis2 in rectifying it they have always found the cause of the anomalies observed2 they have determined the laws and often they have anticipated the observations in discovering the inequalities which it had not yet indicated4 ,thus one may say that nature itself has concurred in the analytical perfection of the theories based upon the principle of universal gravity2 and this is to my mind one of the strongest proofs of the truth of this admirable principle4 ,in the cases which ,i have just considered the analytical solution of the question has changed the probability of the causes into certainty4 ,but most often this solution is impossible and it remains only to augment more and more this probability4 ,in the midst of numerous and incalculable modifications which the action of the causes receives then from strange circumstances these causes conserve always with the effects observed the proper ratios to make them recognizable and to verify their existence4 ,determining these ratios and comparing them with a great number of observations if one finds that they constantly satisfy it1 the probability of the causes may increase to the point of equalling that of facts in regard to which there is no doubt4 ,the investigation of these ratios of causes to their effects is not less useful in natural philosophy than the direct solution of problems whether it be to verify the reality of these causes or to determine the laws from their effects2 since it may be employed in a great number of questions whose direct solution is not possible1 it replaces it in the most advantageous manner4 ,i shall discuss here the application which ,i have made of it to one of the most interesting phenomena of nature1 the flow and the ebb of the sea4 ,pliny has given of this phenomenon a description remarkable for its exactitude1 and in it one sees that the ancients had observed that the tides of each month are greatest toward the syzygies and smallest toward the quadratures2 that they are higher in the perigees than in the apogees of the moon1 and higher in the equinoxes than in the solstices4 ,they concluded from this that this phenomenon is due to the action of the sun and moon upon the sea4 ,in the preface of his work ,de ,stella ,martis ,kepler admits a tendency of the waters of the sea toward the moon2 but1 ignorant of the law of this tendency1 he was able to give on this subject only a probable idea4 ,newton converted into certainty the probability of this idea by attaching it to his great principle of universal gravity4 ,he gave the exact expression of the attractive forces which produced the flood and the ebb of the sea2 and in order to determine the effects he supposed that the sea takes at each instant the position of equilibrium which is agreeable to these forces4 ,he explained in this manner the principal phenomena of the tides2 but it followed from this theory that in our ports the two tides of the same day would be very unequal if the sun and the moon should have a great declination4 ,at ,brest1 for example1 the evening tide would be in the syzygies of the solstices about eight times greater than the morning tide1 which is certainly contrary to the observations which prove that these two tides are very nearly equal4 ,this result from the ,newtonian theory might hold to the supposition that the sea is agreeable at each instant to a position of equilibrium1 a supposition which is not at all admissible4 ,but the investigation of the true figure of the sea presents great difficulties4 ,aided by the discoveries which the geometricians had just made in the theory of the movement of fluids and in the calculus of partial differences1 ,i undertook this investigation1 and ,i gave the differential equations of the movement of the sea by supposing that it covers the entire earth4 ,in drawing thus near to nature ,i had the satisfaction of seeing that my results approached the observations1 especially in regard to the little difference which exists in our ports between the two tides of the solstitial syzygies of the same day4 ,i found that they would be equal if the sea had everywhere the same depth2 ,i found further that in giving to this depth convenient values one was able to augment the height of the tides in a port conformably to the observations4 ,but these investigations1 in spite of their generality1 did not satisfy at all the great differences which even adjacent ports present in this regard and which prove the influence of local circumstances4 ,the impossibility of knowing these circumstances and the irregularity of the basin of the seas and that of integrating the equations of partial differences which are relative has compelled me to make up the deficiency by the method ,i have indicated above4 ,i then endeavored to determine the greatest ratios possible among the forces which affect all the molecules of the sea1 and their effects observable in our ports4 ,for this ,i made use of the following principle1 which may be applied to many other phenomena4 ,the state of the system of a body in which the primitive conditions of the movement have disappeared by the resistances which this movement meets is periodic as the forces which animate it4 ,combining this principle with that of the coexistence of very small oscillations1 ,i have found an expression of the height of the tides whose arbitraries contain the effect of local circumstances of each port and are reduced to the smallest number possible2 it is only necessary to compare it to a great number of observations4 ,upon the invitation of the ,academy of ,sciences1 observations were made at the beginning of the last century at ,brest upon the tides1 which were continued during six consecutive years4 ,the situation of this port is very favorable to this sort of observations2 it communicates with the sea by a canal which empties into a vast roadstead at the far end of which the port has been constructed4 ,the irregularities of the sea extend thus only to a small degree into the port1 just as the oscillations which the irregular movement of a vessel produces in a barometer are diminished by a throttling made in the tube of this instrument4 ,moreover1 the tides being considerable at ,brest1 the accidental variations caused by the winds are only feeble2 likewise we notice in the observations of these tides1 however little we multiply them1 a great regularity which induced me to propose to the government to order in this port a new series of observations of the tides1 continued during a period of the movement of the nodes of the lunar orbit4 ,this has been done4 ,the observations began ,june #a1 #ahjf2 and since this time they have been made every day without interruption4 ,i am indebted to the indefatigable zeal of ,m4 ,bouvard1 for all that interests astronomy1 the immense calculations which the comparison of my analysis with the observations has demanded4 ,there have been used about six thousand observations1 made during the year #ahjg and the fifteen years following4 ,it results from this comparison that my formul represent with a remarkable precision all the varieties of the tides relative to the digression of the moon1 from the sun1 to the declination of these stars1 to their distances from the earth1 and to the laws of variation at the maximum and minimum of each of these elements4 ,there results from this accord a probability that the flow and the ebb of the sea is due to the attraction of the sun and moon1 so approaching certainty that it ought to leave room for no reasonable doubt4 ,it changes into certainty when we consider that this attraction is derived from the law of universal gravity demonstrated by all the celestial phenomena4 ,the action of the moon upon the sea is more than double that of the sun4 ,newton and his successors in the development of this action have paid attention only to the terms divided by the cube of the distance from the moon to the earth1 judging that the effects due to the following terms ought to be inappreciable4 ,but the calculation of probabilities makes it clear to us that the smallest effects of regular causes may manifest themselves in the results of a great number of observations arranged in the order most suitable to indicate them4 ,this calculation again determines their probability and up to what point it is necessary to multiply the observations to make it very great4 ,applying it to the numerous observations discussed by ,m4 ,bouvard ,i recognized that at ,brest the action of the moon upon the sea is greater in the full moons than in the new moons1 and greater when the moon is austral than when it is boreal--phenomena which can result only from the terms of the lunar action divided by the fourth power of the distance from the moon to the earth4 ,to arrive at the ocean the action of the sun and the moon traverses the atmosphere1 which ought consequently to feel its influence and to be subjected to movements similar to those of the sea4 ,these movements produce in the barometer periodic oscillations4 ,analysis has made it clear to me that they are inappreciable in our climates4 ,but as local circumstances increase considerably the tides in our ports1 ,i have inquired again if similar circumstances have made appreciable these oscillations of the barometer4 ,for this ,i have made use of the meteorological observations which have been made every day for many years at the royal observatory4 ,the heights of the barometer and of the thermometer are observed there at nine o'clock in the morning1 at noon1 at three o'clock in the afternoon1 and at eleven o'clock in the evening4 ,m4 ,bouvard has indeed wished to take up the consideration of observations of the eight years elapsed from ,october #a1 #ahae1 to ,october #a1 #ahbc1 on the registers4 ,in disposing the observations in the manner most suitable to indicate the lunar atmospheric flood at ,paris1 ,i find only one eighteenth of a millimeter for the extent of the corresponding oscillation of the barometer4 ,it is this especially which has made us feel the necessity of a method for determining the probability of a result1 and without this method one is forced to present as the laws of nature the results of irregular causes which has often happened in meteorology4 ,this method applied to the preceding result shows the uncertainty of it in spite of the great number of observations employed1 which it would be necessary to increase tenfold in order to obtain a result sufficiently probable4 ,the principle which serves as a basis for my theory of the tides may be extended to all the effects of hazard to which variable causes are joined according to regular laws4 ,the action of these causes produces in the mean results of a great number of effects varieties which follow the same laws and which one may recognize by the analysis of probabilities4 ,in the measure which these effects are multiplied those varieties are manifested with an ever-increasing probability1 which would approach certainty if the number of the effects of the results should become infinite4 ,this theorem is analogous to that which ,i have already developed upon the action of constant causes4 ,every time1 then1 that a cause whose progress is regular can have influence upon a kind of events1 we may seek to discover its influence by multiplying the observations and arranging them in the most suitable order to indicate it4 ,when this influence appears to manifest itself the analysis of probabilities determines the probability of its existence and that of its intensity2 thus the variation of the temperature from day to night modifying the pressure of the atmosphere and consequently the height of the barometer1 it is natural to think that the multiplied observations of these heights ought to show the influence of the solar heat4 ,indeed there has long been recognized at the equator1 where this influence appears to be greatest1 a small diurnal variation in the height of the barometer of which the maximum occurs about nine o'clock in the morning and the minimum about three o'clock in the afternoon4 ,a second maximum occurs about eleven o'clock in the evening and a second minimum about four o'clock in the morning4 ,the oscillations of the night are less than those of the day1 the extent of which is about two millimeters4 ,the inconstancy of our climate has not taken this variation from our observers1 although it may be less appreciable than in the tropics4 ,m4 ,ramond has recognized and determined it at ,clermont1 the chief place of the district of ,puy-de-,dme1 by a series of precise observations made during several years2 he has even found that it is smaller in the months of winter than in other months4 ,the numerous observations which ,i have discussed in order to estimate the influence of attractions of the sun and the moon upon the barometric heights at ,paris have served me in determining their diurnal variation4 ,comparing the heights at nine o'clock in the morning with those of the same days at three o'clock in the afternoon1 this variation is manifested with so much evidence that its mean value each month has been constantly positive for each of the seventy-two months from ,january #a1 #ahag1 to ,january #a1 #ahbc2 its mean value in these seventy-two months has been almost 4#h of a millimeter1 a little less than at ,clermont and much less than at the equator4 ,i have recognized that the mean result of the diurnal variations of the barometer from #i o'clock ,a4,m4 to #c ,p4,m4 has been only 4#edbh millimeter in the three months of ,november1 ,december1 ,january1 and that it has risen to #a4jefc millimeters in the three following months1 which coincides with the observations of ,m4 ,ramond4 ,the other months offer nothing similar4 ,in order to apply to these phenomena the calculation of these probabilities1 ,i commenced by determining the law of the probability of the anomalies of the diurnal variation due to hazard4 ,applying it then to the observations of this phenomenon1 ,i found that it was a bet of more than #cjj1jjj against one that a regular cause produced it4 ,i do not seek to determine this cause2 ,i content myself with stating its existence4 ,the period of the diurnal variation regulated by the solar day indicates evidently that this variation is due to the action of the sun4 ,the extreme smallness of the attractive action of the sun upon the atmosphere is proved by the smallness of the effects due to the united attractions of the sun and the moon4 ,it is then by the action of its heat that the sun produces the diurnal variation of the barometer2 but it is impossible to subject to calculus the effects of its action on the height of the barometer and upon the winds4 ,the diurnal variation of the magnetic needle is certainly a result of the action of the sun4 ,but does this star act here as in the diurnal variation of the barometer by its heat or by its influence upon electricity and upon magnetism1 or finally by the union of these influences8 ,a long series of observations made in different countries will enable us to apprehend this4 ,one of the most remarkable phenomena of the system of the world is that of all the movements of rotation and of revolution of the planets and the satellites in the sense of the rotation of the sun and about in the same plane of its equator4 ,a phenomenon so remarkable is not the effect of hazard3 it indicates a general cause which has determined all its movements4 ,in order to obtain the probability with which this cause is indicated we shall observe that the planetary system1 such as we know it to-day1 is composed of eleven planets and of eighteen satellites at least1 if we attribute with ,herschel six satellites to the planet ,uranus4 ,the movements of the rotation of the sun1 of six planets1 of the moon1 of the satellites of ,jupiter1 of the ring of ,saturn1 and of one of its satellites have been recognized4 ,these movements form with those of revolution a totality of forty-three movements directed in the same sense2 but one finds by the analysis of probabilities that it is a bet of more than #djjjjjjjjjjjj against one that this disposition is not the result of hazard2 this forms a probability indeed superior to that of historical events in regard to which no doubt exists4 ,we ought then to believe at least with equal confidence that a primitive cause has directed the planetary movements1 especially if we consider that the inclination of the greatest number of these movements at the solar equator is very small4 ,another equally remarkable phenomenon of the solar system is the small degree of the eccentricity of the orbs of the planets and the satellites1 while those of the comets are very elongated1 the orbs of the system not offering any intermediate shades between a great and a small eccentricity4 ,we are again forced to recognize here the effect of a regular cause2 chance has certainly not given an almost circular form to the orbits of all the planets and their satellites2 it is then that the cause which has determined the movements of these bodies has rendered them almost circular4 ,it is necessary1 again1 that the great eccentricities of the orbits of the comets should result from the existence of this cause without its having influenced the direction of their movements2 for it is found that there are almost as many retrograde comets as direct comets1 and that the mean inclination of all their orbits to the ecliptic approaches very nearly half a right angle1 as it ought to be if the bodies had been thrown at hazard4 ,whatever may be the nature of the cause in question1 since it has produced or directed the movement of the planets1 it is necessary that it should have embraced all the bodies and considered all the distances which separate them1 it can have been only a fluid of an immense extension4 ,therefore in order to have given them in the same sense an almost circular movement about the sun it is necessary that this fluid should have surrounded this star as an atmosphere4 ,the consideration of the planetary movements leads us then to think that by virtue of an excessive heat the atmosphere of the sun was originally extended beyond the orbits of all the planets1 and that it has contracted gradually to its present limits4 ,in the primitive state where we imagine the sun it resembled the nebul that the telescope shows us composed of a nucleus more or less brilliant surrounded by a nebula which1 condensing at the surface1 ought to transform it some day into a star4 ,if one conceives by analogy all the stars formed in this manner1 one can imagine their anterior state of nebulosity itself preceded by other stars in which the nebulous matter was more and more diffuse1 the nucleus being less and less luminous and dense4 ,going back1 then1 as far as possible1 one would arrive at a nebulosity so diffuse that one would be able scarcely to suspect its existence4 ,such is indeed the first state of the nebul which ,herschel observed with particular care by means of his powerful telescopes1 and in which he has followed the progress of condensation1 not in a single one1 these stages not becoming appreciable to us except after centuries1 but in their totality1 just about as one can in a vast forest follow the increase of the trees by the individuals of the divers ages which the forest contains4 ,he has observed from the beginning nebulous matter spread out in divers masses in the different parts of the heavens1 of which it occupies a great extent4 ,he has seen in some of these masses this matter slightly condensed about one or several faintly luminous nebul4 ,in the other nebul these nuclei shine1 moreover1 in proportion to the nebulosity which surrounds them4 ,the atmospheres of each nucleus becoming separated by an ulterior condensation1 there result the multifold nebul formed of brilliant nuclei very adjacent and surrounded each by an atmosphere2 sometimes the nebulous matter1 by condensing in a uniform manner1 has produced the nebul which are called planetary4 ,finally a greater degree of condensation transforms all these nebul into stars4 ,the nebul classed according to this philosophic view indicate with an extreme probability their future transformation into stars and the anterior state of nebulosity of existing stars4 ,the following considerations come to the aid of proofs drawn from these analogies4 ,for a long time the particular disposition of certain stars visible to the naked eye has struck the attention of philosophical observers4 ,mitchel has already remarked how improbable it is that the stars of the ,pleiades1 for example1 should have been confined in the narrow space which contain them by the chances of hazard alone1 and he has concluded from this that this group of stars and the similar groups that the heaven presents us are the results of a primitive cause or of a general law of nature4 ,these groups are a necessary result of the condensation of the nebul at several nuclei2 it is apparent that the nebulous matter being attracted continuously by the divers nuclei1 they ought to form in time a group of stars equal to that of the ,pleiades4 ,the condensation of the nebul at two nuclei forms similarly very adjacent stars1 revolving the one about the other1 equal to those whose respective movements ,herschel has already considered4 ,such are1 further1 the #fast of the ,swan and its following one in which ,bessel has just recognized particular movements so considerable and so little different that the proximity of these stars to one another and their movement about the common centre of gravity ought to leave no doubt4 ,thus one descends by degrees from the condensation of nebulous matter to the consideration of the sun surrounded formerly by a vast atmosphere1 a consideration to which one repasses1 as has been seen1 by the examination of the phenomena of the solar system4 ,a case so remarkable gives to the existence of this anterior state of the sun a probability strongly approaching certainty4 ,but how has the solar atmosphere determined the movements of rotation and revolution of the planets and the satellites8 ,if these bodies had penetrated deeply the atmosphere its resistance would have caused them to fall upon the sun2 one is then led to believe with much probability that the planets have been formed at the successive limits of the solar atmosphere which1 contracting by the cold1 ought to have abandoned in the plane of its equator zones of vapors which the mutual attraction of their molecules has changed into divers spheroids4 ,the satellites have been similarly formed by the atmospheres of their respective planets4 ,i have developed at length in my ,exposition of the ,system of the ,world this hypothesis1 which appears to me to satisfy all the phenomena which this system presents us4 ,i shall content myself here with considering that the angular velocity of rotation of the sun and the planets being accelerated by the successive condensation of their atmospheres at their surfaces1 it ought to surpass the angular velocity of revolution of the nearest bodies which revolve about them4 ,observation has indeed confirmed this with regard to the planets and satellites1 and even in ratio to the ring of ,saturn1 the duration of whose revolution is 4#dch days1 while the duration of the rotation of ,saturn is 4#dbg days4 ,in this hypothesis the comets are strangers to the planetary system4 ,in attaching their formation to that of the nebul they may be regarded as small nebul at the nuclei1 wandering from systems to solar systems1 and formed by the condensation of the nebulous matter spread out in such great profusion in the universe4 ,the comets would be thus1 in relation to our system1 as the aerolites are relatively to the ,earth1 to which they would appear strangers4 ,when these stars become visible to us they offer so perfect resemblance to the nebul that they are often confounded with them2 and it is only by their movement1 or by the knowledge of all the nebul confined to that part of the heavens where they appear1 that we succeed in distinguishing them4 ,this supposition explains in a happy manner the great extension which the heads and tails of comets take in the measure that they approach the sun1 and the extreme rarity of these tails which1 in spite of their immense depth1 do not weaken at all appreciably the light of the stars which we look across4 ,when the little nebul come into that part of space where the attraction of the sun is predominant1 and which we shall call the sphere of activity of this star1 it forces them to describe elliptic or hyperbolic orbits4 ,but their speed being equally possible in all directions they ought to move indifferently in all the senses and under all inclinations of the elliptic1 which is conformable to that which has been observed4 ,the great eccentricity of the cometary orbits results again from the preceding hypothesis4 ,indeed if these orbits are elliptical they are very elongated1 since their great axes are at least equal to the radius of the sphere of activity of the sun4 ,but these orbits may be hyperbolic2 and if the axes of these hyperbol are not very large in proportion to the mean distance from the sun to the earth1 the movement of the comets which describe them will appear sensibly hyperbolic4 ,however1 of the hundred comets of which we already have the elements1 not one has appeared certainly to move in an hyperbola2 it is necessary1 then1 that the chances which give an appreciable hyperbola should be extremely rare in proportion to the contrary chances4 ,the comets are so small that1 in order to become visible1 their perihelion distance ought to be inconsiderable4 ,up to the present this distance has surpassed only twice the diameter of the terrestrial orbit1 and most often it has been below the radius of this orbit4 ,it is conceived that1 in order to approach so near the sun1 their speed at the moment of their entrance into its sphere of activity ought to have a magnitude and a direction confined within narrow limits4 ,in determining by the analysis of probabilities the ratio of the chances which1 in these limits1 give an appreciable hyperbola1 to the chances which give an orbit which may be confounded with a parabola1 ,i have found that it is a bet of at least #fjjj against one that a nebula which penetrates into the activity of the sun in such a manner as to be observed will describe either a very elongated ellipse or an hyperbola4 ,by the magnitude of its axis1 the latter will be appreciably confounded with a parabola in the part which is observed2 it is then not surprising that1 up to this time1 hyperbolic movements have not been recognized4 ,the attraction of the planets1 and1 perhaps further1 the resistance of the ethereal centres1 ought to have changed many cometary orbits in the ellipses whose great axis is less than the radius of the sphere of activity of the sun1 which augments the chances of the elliptical orbits4 ,we may believe that this change has taken place with the comet of #agei1 and with the comet whose duration is only twelve hundred days1 and which will reappear without ceasing in this short interval1 unless the evaporation which it meets at each of its returns to the perihelion ends by rendering it invisible4 ,we are able further1 by the analysis of probabilities1 to verify the existence or the influence of certain causes whose action is believed to exist upon organized beings4 ,of all the instruments that we are able to employ in order to recognize the imperceptible agents of nature the most sensitive are the nerves1 especially when particular causes increase their sensibility4 ,it is by their aid that the feeble electricity which the contact of two heterogeneous metals develops has been discovered2 this has opened a vast field to the researches of physicists and chemists4 ,the singular phenomena which results from extreme sensibility of the nerves in some individuals have given birth to divers opinions about the existence of a new agent which has been named animal magnetism1 about the action on ordinary magnetism1 and about the influence of the sun and moon in some nervous affections1 and finally1 about the impressions which the proximity of metals or of running water makes felt4 ,it is natural to think that the action of these causes is very feeble1 and that it may be easily disturbed by accidental circumstances2 thus because in some cases it is not manifested at all its existence ought not to be denied4 ,we are so far from recognizing all the agents of nature and their divers modes of action that it would be unphilosophical to deny the phenomena solely because they are inexplicable in the present state of our knowledge4 ,but we ought to examine them with an attention as much the more scrupulous as it appears the more difficult to admit them2 and it is here that the calculation of probabilities becomes indispensable in determining to just what point it is necessary to multiply the observations or the experiences in order to obtain in favor of the agents which they indicate1 a probability superior to the reasons which can be obtained elsewhere for not admitting them4 ,the calculation of probabilities can make appreciable the advantages and the inconveniences of the methods employed in the speculative sciences4 ,thus in order to recognize the best of the treatments in use in the healing of a malady1 it is sufficient to test each of them on an equal number of patients1 making all the conditions exactly similar2 the superiority of the most advantageous treatment will manifest itself more and more in the measure that the number is increased2 and the calculation will make apparent the corresponding probability of its advantage and the ratio according to which it is superior to the others4 ,,chapter ,x4 ,,application ,,of ,,the ,,calculus ,,of ,,probabilities ,,to ,,the ,,moral ,,sciences4 ,we have just seen the advantages of the analysis of probabilities in the investigation of the laws of natural phenomena whose causes are unknown or so complicated that their results cannot be submitted to calculus4 ,this is the case of nearly all subjects of the moral sciences4 ,so many unforeseen causes1 either hidden or inappreciable1 influence human institutions that it is impossible to judge priori the results4 ,the series of events which time brings about develops these results and indicates the means of remedying those that are harmful4 ,wise laws have often been made in this regard2 but because we had neglected to conserve the motives many have been abrogated as useless1 and the fact that vexatious experiences have made the need felt anew ought to have restablished them4 ,it is very important to keep in each branch of the public administration an exact register of the results which the various means used have produced1 and which are so many experiences made on a large scale by governments4 ,let us apply to the political and moral sciences the method founded upon observation and upon calculus1 the method which has served us so well in the natural sciences4 ,let us not offer in the least a useless and often dangerous resistance to the inevitable effects of the progress of knowledge2 but let us change only with an extreme circumspection our institutions and the usages to which we have already so long conformed4 ,we should know well by the experience of the past the difficulties which they present2 but we are ignorant of the extent of the evils which their change can produce4 ,in this ignorance the theory of probability directs us to avoid all change2 especially is it necessary to avoid the sudden changes which in the moral world as well as in the physical world never operate without a great loss of vital force4 ,already the calculus of probabilities has been applied with success to several subjects of the moral sciences4 ,i shall present here the principal results4 ,,chapter ,,xi4 ,,concerning ,,the ,,probabilities ,,of ,,testimonies4 ,the majority of our opinions being founded on the probability of proofs it is indeed important to submit it to calculus4 ,things it is true often become impossible by the difficulty of appreciating the veracity of witnesses and by the great number of circumstances which accompany the deeds they attest2 but one is able in several cases to resolve the problems which have much analogy with the questions which are proposed and whose solutions may be regarded as suitable approximations to guide and to defend us against the errors and the dangers of false reasoning to which we are exposed4 ,an approximation of this kind1 when it is well made1 is always preferable to the most specious reasonings4 ,let us try then to give some general rules for obtaining it4 ,a single number has been drawn from an urn which contains a thousand of them4 ,a witness to this drawing announces that number #gi is drawn2 one asks the probability of drawing this number4 ,let us suppose that experience has made known that this witness deceives one time in ten1 so that the probability of his testimony is #a_/aj4 ,here the event observed is the witness attesting that number #gi is drawn4 ,this event may result from the two following hypotheses1 namely3 that the witness utters the truth or that he deceives4 ,following the principle that has been expounded on the probability of causes drawn from events observed it is necessary first to determine priori the probability of the event in each hypothesis4 ,in the first1 the probability that the witness will announce number #gi is the probability itself of the drawing of this number1 that is to say1 #a_/ajjj4 ,it is necessary to multiply it by the probability #f_/aj of the veracity of the witness2 one will have then #i_/ajjjj for the probability of the event observed in this hypothesis4 ,if the witness deceives1 number #gi is not drawn1 and the probability of this case is #iii_/ajjj4 ,but to announce the drawing of this number the witness has to choose it among the #iii numbers not drawn2 and as he is supposed to have no motive of preference for the ones rather than the others1 the probability that he will choose number #gi is #a_/iii2 multiplying1 then1 this probability by the preceding one1 we shall have #a_/ajjj for the probability that the witness will announce number #gi in the second hypothesis4 ,it is necessary again to multiply this probability by #a_/aj of the hypothesis itself1 which gives #a_/ajjjj for the probability of the event relative to this hypothesis4 ,now if we form a fraction whose numerator is the probability relative to the first hypothesis1 and whose denominator is the sum of the probabilities relative to the two hypotheses1 we shall have1 by the sixth principle1 the probability of the first hypothesis1 and this probability will be #i_/aj2 that is to say1 the veracity itself of the witness4 ,this is likewise the probability of the drawing of number #gi4 ,the probability of the falsehood of the witness and of the failure of drawing this number is #a_/aj4 ,if the witness1 wishing to deceive1 has some interest in choosing number #gi among the numbers not drawn1--if he judges1 for example1 that having placed upon this number a considerable stake1 the announcement of its drawing will increase his credit1 the probability that he will choose this number will no longer be as at first1 #a_/iii1 it will then be 1 1 etc41 according to the interest that he will have in announcing its drawing4 ,supposing it to be #a_/i1 it will be necessary to multiply by this fraction the probability #iii_/ajjj in order to get in the hypothesis of the falsehood the probability of the event observed1 which it is necessary still to multiply by #a_/aj1 which gives #aaa_/ajjjj for the probability of the event in the second hypothesis4 ,then the probability of the first hypothesis1 or of the drawing of number #gi1 is reduced by the preceding rule to #i_/abj4 ,it is then very much decreased by the consideration of the interest which the witness may have in announcing the drawing of number #gi4 ,in truth this same interest increases the probability #i_/aj that the witness will speak the truth if number #gi is drawn4 ,but this probability cannot exceed unity or #aj_/aj2 thus the probability of the drawing of number #gi will not surpass #aj_/aba4 ,common sense tells us that this interest ought to inspire distrust1 but calculus appreciates the influence of it4 ,the probability priori of the number announced by the witness is unity divided by the number of the numbers in the urn2 it is changed by virtue of the proof into the veracity itself of the witness2 it may then be decreased by the proof4 ,if1 for example1 the urn contains only two numbers1 which gives for the probability priori of the drawing of number #a1 and if the veracity of a witness who announces it is #d_/aj1 this drawing becomes less probable4 ,indeed it is apparent1 since the witness has then more inclination towards a falsehood than towards the truth1 that his testimony ought to decrease the probability of the fact attested every time that this probability equals or surpasses 4 ,but if there are three numbers in the urn the probability priori of the drawing of number #a is increased by the affirmation of a witness whose veracity surpasses 4 ,suppose now that the urn contains #iii black balls and one white ball1 and that one ball having been drawn a witness of the drawing announces that this ball is white4 ,the probability of the event observed1 determined priori in the first hypothesis1 will be here1 as in the preceding question1 equal to #i_/ajjjj4 ,but in the hypothesis where the witness deceives1 the white ball is not drawn and the probability of this case is #iii_/ajjj4 ,it is necessary to multiply it by the probability #a_/aj of the falsehood1 which gives #iii_/ajjjj for the probability of the event observed relative to the second hypothesis4 ,this probability was only #a_/ajjjj in the preceding question2 this great difference results from this--that a black ball having been drawn the witness who wishes to deceive has no choice at all to make among the #iii balls not drawn in order to announce the drawing of a white ball4 ,now if one forms two fractions whose numerators are the probabilities relative to each hypothesis1 and whose common denominator is the sum of these probabilities1 one will have #i_/ajjh for the probability of the first hypothesis and of the drawing of a white ball1 and #iii_/ajjh for the probability of the second hypothesis and of the drawing of a black ball4 ,this last probability strongly approaches certainty2 it would approach it much nearer and would become #iiiiii_/ajjjjjh if the urn contained a million balls of which one was white1 the drawing of a white ball becoming then much more extraordinary4 ,we see thus how the probability of the falsehood increases in the measure that the deed becomes more extraordinary4 ,we have supposed up to this time that the witness was not mistaken at all2 but if one admits1 however1 the chance of his error the extraordinary incident becomes more improbable4 ,then in place of the two hypotheses one will have the four following ones1 namely3 that of the witness not deceiving and not being mistaken at all2 that of the witness not deceiving at all and being mistaken2 the hypothesis of the witness deceiving and not being mistaken at all2 finally1 that of the witness deceiving and being mistaken4 ,determining priori in each of these hypotheses the probability of the event observed1 we find by the sixth principle the probability that the fact attested is false equal to a fraction whose numerator is the number of black balls in the urn multiplied by the sum of the probabilities that the witness does not deceive at all and is mistaken1 or that he deceives and is not mistaken1 and whose denominator is this numerator augmented by the sum of the probabilities that the witness does not deceive at all and is not mistaken at all1 or that he deceives and is mistaken at the same time4 ,we see by this that if the number of black balls in the urn is very great1 which renders the drawing of the white ball extraordinary1 the probability that the fact attested is not true approaches most nearly to certainty4 ,applying this conclusion to all extraordinary deeds it results from it that the probability of the error or of the falsehood of the witness becomes as much greater as the fact attested is more extraordinary4 ,some authors have advanced the contrary on this basis that the view of an extraordinary fact being perfectly similar to that of an ordinary fact the same motives ought to lead us to give the witness the same credence when he affirms the one or the other of these facts4 ,simple common sense rejects such a strange assertion2 but the calculus of probabilities1 while confirming the findings of common sense1 appreciates the greatest improbability of testimonies in regard to extraordinary facts4 ,these authors insist and suppose two witnesses equally worthy of belief1 of whom the first attests that he saw an individual dead fifteen days ago whom the second witness affirms to have seen yesterday full of life4 ,the one or the other of these facts offers no improbability4 ,the reservation of the individual is a result of their combination2 but the testimonies do not bring us at all directly to this result1 although the credence which is due these testimonies ought not to be decreased by the fact that the result of their combination is extraordinary4 ,but if the conclusion which results from the combination of the testimonies was impossible one of them would be necessarily false2 but an impossible conclusion is the limit of extraordinary conclusions1 as error is the limit of improbable conclusions2 the value of the testimonies which becomes zero in the case of an impossible conclusion ought then to be very much decreased in that of an extraordinary conclusion4 ,this is indeed confirmed by the calculus of probabilities4 ,in order to make it plain let us consider two urns1 ,a and ,b1 of which the first contains a million white balls and the second a million black balls4 ,one draws from one of these urns a ball1 which he puts back into the other urn1 from which one then draws a ball4 ,two witnesses1 the one of the first drawing1 the other of the second1 attest that the ball which they have seen drawn is white without indicating the urn from which it has been drawn4 ,each testimony taken alone is not improbable2 and it is easy to see that the probability of the fact attested is the veracity itself of the witness4 ,but it follows from the combination of the testimonies that a white ball has been extracted from the urn ,a at the first draw1 and that then placed in the urn ,b it has reappeared at the second draw1 which is very extraordinary2 for this second urn1 containing then one white ball among a million black balls1 the probability of drawing the white ball is #a_/ajjjjja4 ,in order to determine the diminution which results in the probability of the thing announced by the two witnesses we shall notice that the event observed is here the affirmation by each of them that the ball which he has seen extracted is white4 ,let us represent by #i_/aj the probability that he announces the truth1 which can occur in the present case when the witness does not deceive and is not mistaken at all1 and when he deceives and is mistaken at the same time4 ,one may form the four following hypotheses3 #ast4 ,the first and second witness speak the truth4 ,then a white ball has at first been drawn from the urn ,a1 and the probability of this event is 1 since the ball drawn at the first draw may have been drawn either from the one or the other urn4 ,consequently the ball drawn1 placed in the urn ,b1 has reappeared at the second draw2 the probability of this event is #a_/ajjjjja1 the probability of the fact announced is then #a_/bjjjjjb4 ,multiplying it by the product of the probabilities #i_/aj and #i_/aj that the witnesses speak the truth one will have #ha_/bjjjjjbjj for the probability of the event observed in this first hypothesis4 #b;d4 ,the first witness speaks the truth and the second does not1 whether he deceives and is not mistaken or he does not deceive and is mistaken4 ,then a white ball has been drawn from the urn ,a at the first draw1 and the probability of this event is 4 ,then this ball having been placed in the urn ,b a black ball has been drawn from it3 the probability of such drawing is #ajjjjjj_/ajjjjja2 one has then #ajjjjjj_/bjjjjjb for the probability of the compound event4 ,multiplying it by the product of the two probabilities #i_/aj and #a_/aj that the first witness speaks the truth and that the second does not1 one will have #ijjjjjj_/bjjjjjbjj for the probability for the event observed in the second hypothesis4 #c;d4 ,the first witness does not speak the truth and the second announces it4 ,then a black ball has been drawn from the urn ,b at the first drawing1 and after having been placed in the urn ,a a white ball has been drawn from this urn4 ,the probability of the first of these events is and that of the second is #ajjjjjj_/ajjjjja2 the probability of the compound event is then #ajjjjjj_/bjjjjjb4 ,multiplying it by the product of the probabilities #a_/aj and #i_/aj that the first witness does not speak the truth and that the second announces it1 one will have #ijjjjjj_/bjjjjjbjj for the probability of the event observed relative to this hypothesis4 #dth4 ,finally1 neither of the witnesses speaks the truth4 ,then a black ball has been drawn from the urn ,b at the first draw2 then having been placed in the urn ,a it has reappeared at the second drawing3 the probability of this compound event is #a_/bjjjjjb4 ,multiplying it by the product of the probabilities #a_/aj and #a_/aj that each witness does not speak the truth one will have #a_/bjjjjjbjj for the probability of the event observed in this hypothesis4 ,now in order to obtain the probability of the thing announced by the two witnesses1 namely1 that a white ball has been drawn at each draw1 it is necessary to divide the probability corresponding to the first hypothesis by the sum of the probabilities relative to the four hypotheses2 and then one has for this probability #ha_/ahjjjjhb1 an extremely small fraction4 ,if the two witnesses affirm the first1 that a white ball has been drawn from one of the two urns ,a and ,b2 the second that a white ball has been likewise drawn from one of the two urns ,a and ,b1 quite similar to the first ones1 the probability of the thing announced by the two witnesses will be the product of the probabilities of their testimonies1 or #ha_/ajj2 it will then be at least a hundred and eighty thousand times greater than the preceding one4 ,one sees by this how much1 in the first case1 the reappearance at the second draw of the white ball drawn at the first draw1 the extraordinary conclusion of the two testimonies decreases the value of it4 ,we would give no credence to the testimony of a man who should attest to us that in throwing a hundred dice into the air they had all fallen on the same face4 ,if we had ourselves been spectators of this event we should believe our own eyes only after having carefully examined all the circumstances1 and after having brought in the testimonies of other eyes in order to be quite sure that there had been neither hallucination nor deception4 ,but after this examination we should not hesitate to admit it in spite of its extreme improbability2 and no one would be tempted1 in order to explain it1 to recur to a denial of the laws of vision4 ,we ought to conclude from it that the probability of the constancy of the laws of nature is for us greater than this1 that the event in question has not taken place at all--a probability greater than that of the majority of historical facts which we regard as incontestable4 ,one may judge by this the immense weight of testimonies necessary to admit a suspension of natural laws1 and how improper it would be to apply to this case the ordinary rules of criticism4 ,all those who without offering this immensity of testimonies support this when making recitals of events contrary to those laws1 decrease rather than augment the belief which they wish to inspire2 for then those recitals render very probable the error or the falsehood of their authors4 ,but that which diminishes the belief of educated men increases often that of the uneducated1 always greedy for the wonderful4 ,there are things so extraordinary that nothing can balance their improbability4 ,but this1 by the effect of a dominant opinion1 can be weakened to the point of appearing inferior to the probability of the testimonies2 and when this opinion changes an absurd statement admitted unanimously in the century which has given it birth offers to the following centuries only a new proof of the extreme influence of the general opinion upon the more enlightened minds4 ,two great men of the century of ,louis ,,xiv4--,racine and ,pascal--are striking examples of this4 ,it is painful to see with what complaisance ,racine1 this admirable painter of the human heart and the most perfect poet that has ever lived1 reports as miraculous the recovery of ,mlle4 ,perrier1 a niece of ,pascal and a day pupil at the monastery of ,port-,royal2 it is painful to read the reasons by which ,pascal seeks to prove that this miracle should be necessary to religion in order to justify the doctrine of the monks of this abbey1 at that time persecuted by the ,jesuits4 ,the young ,perrier had been afflicted for three years and a half by a lachrymal fistula2 she touched her afflicted eye with a relic which was pretended to be one of the thorns of the crown of the ,saviour and she had faith in instant recovery4 ,some days afterward the physicians and the surgeons attest the recovery1 and they declare that nature and the remedies have had no part in it4 ,this event1 which took place in #afef1 made a great sensation1 and all ,paris rushed1 says ,racine1 to ,port-,royal4 ,the crowd increased from day to day1 and ,god himself seemed to take pleasure in authorizing the devotion of the people by the number of miracles which were performed in this church4 ,at this time miracles and sorcery did not yet appear improbable1 and one did not hesitate at all to attribute to them the singularities of nature which could not be explained otherwise4 ,this manner of viewing extraordinary results is found in the most remarkable works of the century of ,louis ,,xiv42 even in the ,essay on the ,human ,understanding by the philosopher ,locke1 who says1 in speaking of the degree of assent3 ,though the common experience and the ordinary course of things have justly a mighty influence on the minds of men1 to make them give or refuse credit to anything proposed to their belief2 yet there is one case1 wherein the strangeness of the fact lessens not the assent to a fair testimony of it4 ,for where such supernatural events are suitable to ends aimed at by him who has the power to change the course of nature1 there1 under such circumstances1 they may be the fitter to procure belief1 by how much the more they are beyond or contrary to ordinary observation4 ,the true principles of the probability of testimonies having been thus misunderstood by philosophers to whom reason is principally indebted for its progress1 ,i have thought it necessary to present at length the results of calculus upon this important subject4 ,there comes up naturally at this point the discussion of a famous argument of ,pascal1 that ,craig1 an ,english mathematician1 has produced under a geometric form4 ,witnesses declare that they have it from ,divinity that in conforming to a certain thing one will enjoy not one or two but an infinity of happy lives4 ,however feeble the probability of the proofs may be1 provided that it be not infinitely small1 it is clear that the advantage of those who conform to the prescribed thing is infinite since it is the product of this probability and an infinite good2 one ought not to hesitate then to procure for oneself this advantage4 ,this argument is based upon the infinite number of happy lives promised in the name of the ,divinity by the witnesses2 it is necessary then to prescribe them1 precisely because they exaggerate their promises beyond all limits1 a consequence which is repugnant to good sense4 ,also calculus teaches us that this exaggeration itself enfeebles the probability of their testimony to the point of rendering it infinitely small or zero4 ,indeed this case is similar to that of a witness who should announce the drawing of the highest number from an urn filled with a great number of numbers1 one of which has been drawn and who would have a great interest in announcing the drawing of this number4 ,one has already seen how much this interest enfeebles his testimony4 ,in evaluating only at the probability that if the witness deceives he will choose the largest number1 calculus gives the probability of his announcement as smaller than a fraction whose numerator is unity and whose denominator is unity plus the half of the product of the number of the numbers by the probability of falsehood considered priori or independently of the announcement4 ,in order to compare this case to that of the argument of ,pascal it is sufficient to represent by the numbers in the urn all the possible numbers of happy lives which the number of these numbers renders infinite2 and to observe that if the witnesses deceive they have the greatest interest1 in order to accredit their falsehood1 in promising an eternity of happiness4 ,the expression of the probability of their testimony becomes then infinitely small4 ,multiplying it by the infinite number of happy lives promised1 infinity would disappear from the product which expresses the advantage resultant from this promise which destroys the argument of ,pascal4 ,let us consider now the probability of the totality of several testimonies upon an established fact4 ,in order to fix our ideas let us suppose that the fact be the drawing of a number from an urn which contains a hundred of them1 and of which one single number has been drawn4 ,two witnesses of this drawing announce that number #b has been drawn1 and one asks for the resultant probability of the totality of these testimonies4 ,one may form these two hypotheses3 the witnesses speak the truth2 the witnesses deceive4 ,in the first hypothesis the number #b is drawn and the probability of this event is #a_/ajj4 ,it is necessary to multiply it by the product of the veracities of the witnesses1 veracities which we will suppose to be #i_/aj and #g_/aj3 one will have then #fc_/ajjjj for the probability of the event observed in this hypothesis4 ,in the second1 the number #b is not drawn and the probability of this event is #ii_/ajj4 ,but the agreement of the witnesses requires then that in seeking to deceive they both choose the number #b from the #ii numbers not drawn3 the probability of this choice if the witnesses do not have a secret agreement is the product of the fraction #a_/ii by itself2 it becomes necessary then to multiply these two probabilities together1 and by the product of the probabilities #a_/aj and #c_/aj that the witnesses deceive2 one will have thus #a_/ccjjjj for the probability of the event observed in the second hypothesis4 ,now one will have the probability of the fact attested or of the drawing of number #b in dividing the probability relative to the first hypothesis by the sum of the probabilities relative to the two hypotheses2 this probability will be then #bjgi_/bjhj1 and the probability of the failure to draw this number and of the falsehood of the witnesses will be #a_/bjhj4 ,if the urn should contain only the numbers #a and #b one would find in the same manner #ba_/bb for the probability of the drawing of number #b1 and consequently #a_/bb for the probability of the falsehood of the witnesses1 a probability at least ninety-four times larger than the preceding one4 ,one sees by this how much the probability of the falsehood of the witnesses diminishes when the fact which they attest is less probable in itself4 ,indeed one conceives that then the accord of the witnesses1 when they deceive1 becomes more difficult1 at least when they do not have a secret agreement1 which we do not suppose here at all4 ,in the preceding case where the urn contained only two numbers the priori probability of the fact attested is 1 the resultant probability of the testimonies is the product of the veracities of the witnesses divided by this product added to that of the respective probabilities of their falsehood4 ,it now remains for us to consider the influence of time upon the probability of facts transmitted by a traditional chain of witnesses4 ,it is clear that this probability ought to diminish in proportion as the chain is prolonged4 ,if the fact has no probability itself1 such as the drawing of a number from an urn which contains an infinity of them1 that which it acquires by the testimonies decreases according to the continued product of the veracity of the witnesses4 ,if the fact has a probability in itself2 if1 for example1 this fact is the drawing of the number #b from an urn which contains an infinity of them1 and of which it is certain that one has drawn a single number2 that which the traditional chain adds to this probability decreases1 following a continued product of which the first factor is the ratio of the number of numbers in the urn less one to the same number1 and of which each other factor is the veracity of each witness diminished by the ratio of the probability of his falsehood to the number of the numbers in the urn less one2 so that the limit of the probability of the fact is that of this fact considered priori1 or independently of the testimonies1 a probability equal to unity divided by the number of the numbers in the urn4 ,the action of time enfeebles then1 without ceasing1 the probability of historical facts just as it changes the most durable monuments4 ,one can indeed diminish it by multiplying and conserving the testimonies and the monuments which support them4 ,printing offers for this purpose a great means1 unfortunately unknown to the ancients4 ,in spite of the infinite advantages which it procures the physical and moral revolutions by which the surface of this globe will always be agitated will end1 in conjunction with the inevitable effect of time1 by rendering doubtful after thousands of years the historical facts regarded to-day as the most certain4 ,craig has tried to submit to calculus the gradual enfeebling of the proofs of the ,christian religion2 supposing that the world ought to end at the epoch when it will cease to be probable1 he finds that this ought to take place #aded years after the time when he writes4 ,but his analysis is as faulty as his hypothesis upon the duration of the moon is bizarre4 ,,chapter ,,xii4 ,,concerning ,,the ,,selections ,,and ,,the ,,decisions ,,of ,,assemblies4 ,the probability of the decisions of an assembly depends upon the plurality of votes1 the intelligence and the impartiality of the members who compose it4 ,so many passions and particular interests so often add their influence that it is impossible to submit this probability to calculus4 ,there are1 however1 some general results dictated by simple common sense and confirmed by calculus4 ,if1 for example1 the assembly is poorly informed about the subject submitted to its decision1 if this subject requires delicate considerations1 or if the truth on this point is contrary to established prejudices1 so that it would be a bet of more than one against one that each voter will err2 then the decision of the majority will be probably wrong1 and the fear of it will be the better based as the assembly is more numerous4 ,it is important then1 in public affairs1 that assemblies should have to pass upon subjects within reach of the greatest number2 it is important for them that information be generally diffused and that good works founded upon reason and experience should enlighten those who are called to decide the lot of their fellows or to govern them1 and should forewarn them against false ideas and the prejudices of ignorance4 ,scholars have had frequent occasion to remark that first conceptions often deceive and that the truth is not always probable4 ,it is difficult to understand and to define the desire of an assembly in the midst of a variety of opinions of its members4 ,let us attempt to give some rules in regard to this matter by considering the two most ordinary cases3 the election among several candidates1 and that among several propositions relative to the same subject4 ,when an assembly has to choose among several candidates who present themselves for one or for several places of the same kind1 that which appears simplest is to have each voter write upon a ticket the names of all the candidates according to the order of merit that he attributes to them4 ,supposing that he classifies them in good faith1 the inspection of these tickets will give the results of the elections in such a manner that the candidates may be compared among themselves2 so that new elections can give nothing more in this regard4 ,it is a question now to conclude the order of preference which the tickets establish among the candidates4 ,let us imagine that one gives to each voter an urn which contains an infinity of balls by means of which he is able to shade all the degrees of merit of the candidates2 let us conceive again that he draws from his urn a number of balls proportional to the merit of each candidate1 and let us suppose this number written upon a ticket at the side of the name of the candidate4 ,it is clear that by making a sum of all the numbers relative to each candidate upon each ticket1 that one of all the candidates who shall have the largest sum will be the candidate whom the assembly prefers2 and that in general the order of preference of the candidates will be that of the sums relative to each of them4 ,but the tickets do not mark at all the number of balls which each voter gives to the candidates2 they indicate solely that the first has more of them than the second1 the second more than the third1 and so on4 ,in supposing then at first upon a given ticket a certain number of balls all the combinations of the inferior numbers which fulfil the preceding conditions are equally admissible2 and one will have the number of balls relative to each candidate by making a sum of all the numbers which each combination gives him and dividing it by the entire number of combinations4 ,a very simple analysis shows that the numbers which must be written upon each ticket at the side of the last name1 of the one before the last1 etc41 are proportional to the terms of the arithmetical progression #a1 #b1 #c1 etc4 ,writing then thus upon each ticket the terms of this progression1 and adding the terms relative to each candidate upon these tickets1 the divers sums will indicate by their magnitude the order of their preference which ought to be established among the candidates4 ,such is the mode of election which ,the ,theory of ,probabilities indicates4 ,without doubt it would be better if each voter should write upon his ticket the names of the candidates in the order of merit which he attributes to them4 ,but particular interests and many strange considerations of merit would affect this order and place sometimes in the last rank the candidate most formidable to that one whom one prefers1 which gives too great an advantage to the candidates of mediocre merit4 ,likewise experience has caused the abandonment of this mode of election in the societies which had adopted it4 ,the election by the absolute majority of the suffrages unites to the certainty of not admitting any one of the candidates whom this majority rejects1 the advantage of expressing most often the desire of the assembly4 ,it always coincides with the preceding mode when there are only two candidates4 ,indeed it exposes an assembly to the inconvenience of rendering elections interminable4 ,but experience has shown that this inconvenience is nil1 and that the general desire to put an end to elections soon unites the majority of the suffrages upon one of the candidates4 ,the choice among several propositions relative to the same object ought to be subjected1 seemingly1 to the same rules as the election among several candidates4 ,but there exists between the two cases this difference1 namely1 that the merit of a candidate does not exclude that of his competitors2 but if it is necessary to choose among propositions which are contrary1 the truth of the one excludes the truth of the others4 ,let us see how one ought then to view this question4 ,let us give to each voter an urn which contains an infinite number of balls1 and let us suppose that he distributes them upon the divers propositions according to the respective probabilities which he attributes to them4 ,it is clear that the total number of balls expressing certainty1 and the voter being by the hypothesis assured that one of the propositions ought to be true1 he will distribute this number at length upon the propositions4 ,the problem is reduced then to this1 namely1 to determine the combinations in which the balls will be distributed in such a manner that there may be more of them upon the first proposition of the ticket than upon the second1 more upon the second than upon the third1 etc42 to make the sums of all the numbers of balls relative to each proposition in the divers combinations1 and to divide this sum by the number of combinations2 the quotients will be the numbers of balls that one ought to attribute to the propositions upon a certain ticket4 ,one finds by analysis that in going from the last proposition these quotients are among themselves as the following quantities3 first1 unity divided by the number of propositions2 second1 the preceding quantity1 augmented by unity1 divided by the number of propositions less one2 third1 this second quantity1 augmented by unity1 divided by the number of propositions less two1 and so on for the others4 ,one will write then upon each ticket these quantities at the side of the corresponding propositions1 and adding the relative quantities to each proposition upon the divers tickets the sums will indicate by their magnitude the order of preference which the assembly gives to these propositions4 ,let us speak a word about the manner of renewing assemblies which should change in totality in a definite number of years4 ,ought the renewal to be made at one time1 or is it advantageous to divide it among these years8 ,according to the last method the assembly would be formed under the influence of the divers opinions dominant during the time of its renewal2 the opinion which obtained then would be probably the mean of all these opinions4 ,the assembly would receive thus at the time the same advantage that is given to it by the extension of the elections of its members to all parts of the territory which it represents4 ,now if one considers what experience has only too clearly taught1 namely1 that elections are always directed in the greatest degree by dominant opinions1 one will feel how useful it is to temper these opinions1 the ones by the others1 by means of a partial renewal4 ,,chapter ,,xiii4 ,,concerning ,,the ,,probability ,,of ,,the ,,judgments ,,of ,,tribunals4 ,analysis confirms what simple common sense teaches us1 namely1 the correctness of judgments is as much more probable as the judges are more numerous and more enlightened4 ,it is important then that tribunals of appeal should fulfil these two conditions4 ,the tribunals of the first instance standing in closer relation to those amenable offer to the higher tribunal the advantage of a first judgment already probable1 and with which the latter often agree1 be it in compromising or in desisting from their claims4 ,but if the uncertainty of the matter in litigation and its importance determine a litigant to have recourse to the tribunal of appeals1 he ought to find in a greater probability of obtaining an equitable judgment greater security for his fortune and the compensation for the trouble and expense which a new procedure entails4 ,it is this which had no place in the institution of the reciprocal appeal of the tribunals of the district1 an institution thereby very prejudicial to the interest of the citizens4 ,it would be perhaps proper and conformable to the calculus of probabilities to demand a majority of at least two votes in a tribunal of appeal in order to invalidate the sentence of the lower tribunal4 ,one would obtain this result if the tribunal of appeal being composed of an even number of judges the sentence should stand in the case of the equality of votes4 ,i shall consider particularly the judgments in criminal matters4 ,in order to condemn an accused it is necessary without doubt that the judges should have the strongest proofs of his offence4 ,but a moral proof is never more than a probability2 and experience has only too clearly shown the errors of which criminal judgments1 even those which appear to be the most just1 are still susceptible4 ,the impossibility of amending these errors is the strongest argument of the philosophers who have wished to proscribe the penalty of death4 ,we should then be obliged to abstain from judging if it were necessary for us to await mathematical evidence4 ,but the judgment is required by the danger which would result from the impunity of the crime4 ,this judgment reduces itself1 if ,i am not mistaken1 to the solution of the following question3 ,has the proof of the offence of the accused the high degree of probability necessary so that the citizens would have less reason to doubt the errors of the tribunals1 if he is innocent and condemned1 than they would have to fear his new crimes and those of the unfortunate ones who would be emboldened by the example of his impunity if he were guilty and acquitted8 ,the solution of this question depends upon several elements very difficult to ascertain4 ,such is the eminence of danger which would threaten society if the criminal accused should remain unpunished4 ,sometimes this danger is so great that the magistrate sees himself constrained to waive forms wisely established for the protection of innocence4 ,but that which renders almost always this question insoluble is the impossibility of appreciating exactly the probability of the offence and of fixing that which is necessary for the condemnation of the accused4 ,each judge in this respect is forced to rely upon his own judgment4 ,he forms his opinion by comparing the divers testimonies and the circumstances by which the offence is accompanied1 to the results of his reflections and his experiences1 and in this respect a long habitude of interrogating and judging accused persons gives great advantage in ascertaining the truth in the midst of indices often contradictory4 ,the preceding question depends again upon the care taken in the investigation of the offence2 for one demands naturally much stronger proofs for imposing the death penalty than for inflicting a detention of some months4 ,it is a reason for proportioning the care to the offence1 great care taken with an unimportant case inevitably clearing many guilty ones4 ,a law which gives to the judges power of moderating the care in the case of attenuating circumstances is then conformable at the same time to principles of humanity towards the culprit1 and to the interest of society4 ,the product of the probability of the offence by its gravity being the measure of the danger to which the acquittal of the accused can expose society1 one would think that the care taken ought to depend upon this probability4 ,this is done indirectly in the tribunals where one retains for some time the accused against whom there are very strong proofs1 but insufficient to condemn him2 in the hope of acquiring new light one does not place him immediately in the midst of his fellow citizens1 who would not see him again without great alarm4 ,but the arbitrariness of this measure and the abuse which one can make of it have caused its rejection in the countries where one attaches the greatest price to individual liberty4 ,now what is the probability that the decision of a tribunal which can condemn only by a given majority will be just1 that is to say1 conform to the true solution of the question proposed above8 ,this important problem well solved will give the means of comparing among themselves the different tribunals4 ,the majority of a single vote in a numerous tribunal indicates that the affair in question is very doubtful2 the condemnation of the accused would be then contrary to the principles of humanity1 protectors of innocence4 ,the unanimity of the judges would give very strong probability of a just decision2 but in abstaining from it too many guilty ones would be acquitted4 ,it is necessary1 then1 either to limit the number of judges1 if one wishes that they should be unanimous1 or increase the majority necessary for a condemnation1 when the tribunal becomes more numerous4 ,i shall attempt to apply calculus to this subject1 being persuaded that it is always the best guide when one bases it upon the data which common sense suggests to us4 ,the probability that the opinion of each judge is just enters as the principal element into this calculation4 ,if in a tribunal of a thousand and one judges1 five hundred and one are of one opinion1 and five hundred are of the contrary opinion1 it is apparent that the probability of the opinion of each judge surpasses very little 2 for supposing it obviously very large a single vote of difference would be an improbable event4 ,but if the judges are unanimous1 this indicates in the proofs that degree of strength which entails conviction2 the probability of the opinion of each judge is then very near unity or certainty1 provided that the passions or the ordinary prejudices do not affect at the same time all the judges4 ,outside of these cases the ratio of the votes for or against the accused ought alone to determine this probability4 ,i suppose thus that it can vary from to unity1 but that it cannot be below 4 ,if that were not the case the decision of the tribunal would be as insignificant as chance2 it has value only in so far as the opinion of the judge has a greater tendency to truth than to error4 ,it is thus by the ratio of the numbers of votes favorable1 and contrary to the accused1 that ,i determine the probability of this opinion4 ,these data suffice to ascertain the general expression of the probability that the decision of a tribunal judging by a known majority is just4 ,in the tribunals where of eight judges five votes would be necessary for the condemnation of an accused1 the probability of the error to be feared in the justice of the decision would surpass 4 ,if the tribunal should be reduced to six members who are able to condemn only by a plurality of four votes1 the probability of the error to be feared would be below 4 ,there would be then for the accused an advantage in this reduction of the tribunal4 ,in both cases the majority required is the same and is equal to two4 ,thus the majority remaining constant1 the probability of error increases with the number of judges2 this is general whatever may be the majority required1 provided that it remains the same4 ,taking1 then1 for the rule the arithmetical ratio1 the accused finds himself in a position less and less advantageous in the measure that the tribunal becomes more numerous4 ,one might believe that in a tribunal where one might demand a majority of twelve votes1 whatever the number of the judges was1 the votes of the minority1 neutralizing an equal number of votes of the majority1 the twelve remaining votes would represent the unanimity of a jury of twelve members1 required in ,england for the condemnation of an accused2 but one would be greatly mistaken4 ,common sense shows that there is a difference between the decision of a tribunal of two hundred and twelve judges1 of which one hundred and twelve condemn the accused1 while one hundred acquit him1 and that of a tribunal of twelve judges unanimous for condemnation4 ,in the first case the hundred votes favorable to the accused warrant in thinking that the proofs are far from attaining the degree of strength which entails conviction2 in the second case1 the unanimity of the judges leads to the belief that they have attained this degree4 ,but simple common sense does not suffice at all to appreciate the extreme difference of the probability of error in the two cases4 ,it is necessary then to recur to calculus1 and one finds nearly one fifth for the probability of error in the first case1 and only #a_/haib for this probability in the second case1 a probability which is not one thousandth of the first4 ,it is a confirmation of the principle that the arithmetical ratio is unfavorable to the accused when the number of judges increases4 ,on the contrary1 if one takes for a rule the geometrical ratio1 the probability of the error of the decision diminishes when the number of judges increases4 ,for example1 in the tribunals which can condemn only by a plurality of two thirds of the votes1 the probability of the error to be feared is nearly one fourth if the number of the judges is six2 it is below #a_/g if this number is increased to twelve4 ,thus one ought to be governed neither by the arithmetical ratio nor by the geometrical ratio if one wishes that the probability of error should never be above nor below a given fraction4 ,but what fraction ought to be determined upon8 ,it is here that the arbitrariness begins and the tribunals offer in this regard the greatest variety4 ,in the special tribunals where five of the eight votes suffice for the condemnation of the accused1 the probability of the error to be feared in regard to justice of the judgment is #fe_/bef1 or more than 4 ,the magnitude of this fraction is dreadful2 but that which ought to reassure us a little is the consideration that most frequently the judge who acquits an accused does not regard him as innocent2 he pronounces solely that it is not attained by proofs sufficient for condemnation4 ,one is especially reassured by the pity which nature has placed in the heart of man and which disposes the mind to see only with reluctance a culprit in the accused submitted to his judgment4 ,this sentiment1 more active in those who have not the habitude of criminal judgments1 compensates for the inconveniences attached to the inexperience of the jurors4 ,in a jury of twelve members1 if the plurality demanded for the condemnation is eight of twelve votes1 the probability of the error to be feared #ajic_/haib1 or a little more than one eighth1 it is almost #a_/bb if this plurality consists of nine votes4 ,in the case of unanimity the probability of the error to be feared is #a_/haib1 that is to say1 more than a thousand times less than in our juries4 ,this supposes that the unanimity results only from proofs favorable or contrary to the accused2 but motives that are entirely strange1 ought oftentimes to concur in producing it1 when it is imposed upon the jury as a necessary condition of its judgment4 ,then its decisions depending upon the temperament1 the character1 the habits of the jurors1 and the circumstances in which they are placed1 they are sometimes contrary to the decisions which the majority of the jury would have made if they had listened only to the proofs2 this seems to me to be a great fault of this manner of judging4 ,the probability of the decision is too feeble in our juries1 and ,i think that in order to give a sufficient guarantee to innocence1 one ought to demand at least a plurality of nine votes in twelve4 ,,chapter ,,xiv4 ,,concerning ,,tables ,,of ,,mortality1 ,,and ,,of ,,mean ,,durations ,,of ,,life1 ,,of ,,marriages1 ,,and ,,of ,,associations4 ,the manner of preparing tables of mortality is very simple4 ,one takes in the civil registers a great number of individuals whose birth and death are indicated4 ,one determines how many of these individuals have died in the first year of their age1 how many in the second year1 and so on4 ,it is concluded from these the number of individuals living at the commencement of each year1 and this number is written in the table at the side of that which indicates the year4 ,thus one writes at the side of zero the number of births2 at the side of the year #a the number of infants who have attained one year2 at the side of the year #b the number of infants who have attained two years1 and so on for the rest4 ,but since in the first two years of life the mortality is very great1 it is necessary for the sake of greater exactitude to indicate in this first age the number of survivors at the end of each half year4 ,if we divide the sum of the years of the life of all the individuals inscribed in a table of mortality by the number of these individuals we shall have the mean duration of life which corresponds to this table4 ,for this1 we will multiply by a half year the number of deaths in the first year1 a number equal to the difference of the numbers of individuals inscribed at the side of the years #j and #a4 ,their mortality being distributed over the entire year the mean duration of their life is only a half year4 ,we will multiply by a year and a half the number of deaths in the second year2 by two years and a half the number of deaths in the third year2 and so on4 ,the sum of these products divided by the number of births will be the mean duration of life4 ,it is easy to conclude from this that we will obtain this duration1 by making the sum of the numbers inscribed in the table at the side of each year1 dividing it by the number of births and subtracting one half from the quotient1 the year being taken as unity4 ,the mean duration of life that remains1 starting from any age1 is determined in the same manner1 working upon the number of individuals who have arrived at this age1 as has just been done with the number of births4 ,but it is not at the moment of birth that the mean duration of life is the greatest2 it is when one has escaped the dangers of infancy and it is then about forty-three years4 ,the probability of arriving at a certain age1 starting from a given age is equal to the ratio of the two numbers of individuals indicated in the table at these two ages4 ,the precision of these results demands that for the formation of tables we should employ a very great number of births4 ,analysis gives then very simple formul for appreciating the probability that the numbers indicated in these tables will vary from the truth only within narrow limits4 ,we see by these formul that the interval of the limits diminishes and that the probability increases in proportion as we take into consideration more births2 so that the tables would represent exactly the true law of mortality if the number of births employed were infinite4 ,a table of mortality is then a table of the probability of human life4 ,the ratio of the individuals inscribed at the side of each year to the number of births is the probability that a new birth will attain this year4 ,as we estimate the value of hope by making a sum of the products of each benefit hoped for1 by the probability of obtaining it1 so we can equally evaluate the mean duration of life by adding the products of each year by half the sum of the probabilities of attaining the commencement and the end of it1 which leads to the result found above4 ,but this manner of viewing the mean duration of life has the advantage of showing that in a stationary population1 that is to say1 such that the number of births equals that of deaths1 the mean duration of life is the ratio itself of the population to the annual births2 for the population being supposed stationary1 the number of individuals of an age comprised between two consecutive years of the table is equal to the number of annual births1 multiplied by half the sum of the probabilities of attaining these years2 the sum of all these products will be then the entire population4 ,now it is easy to see that this sum1 divided by the number of annual births1 coincides with the mean duration of life as we have just defined it4 ,it is easy by means of a table of mortality to form the corresponding table of the population supposed to be stationary4 ,for this we take the arithmetical means of the numbers of the table of mortality corresponding to the ages zero and one year1 one and two years1 two and three years1 etc4 ,the sum of all these means is the entire population2 it is written at the side of the age zero4 ,there is subtracted from this sum the first mean and the remainder is the number of individuals of one year and upwards2 it is written at the side of the year #a4 ,there is subtracted from this first remainder the second mean2 this second remainder is the number of individuals of two years and upwards2 it is written at the side of the year #b1 and so on4 ,so many variable causes influence mortality that the tables which represent it ought to be changed according to place and time4 ,the divers states of life offer in this regard appreciable differences relative to the fatigues and the dangers inseparable from each state and of which it is indispensable to keep account in the calculations founded upon the duration of life4 ,but these differences have not been sufficiently observed4 ,some day they will be and then will be known what sacrifice of life each profession demands and one will profit by this knowledge to diminish the dangers4 ,the greater or less salubrity of the soil1 its elevation1 its temperature1 the customs of the inhabitants1 and the operations of governments have a considerable influence upon mortality4 ,but it is always necessary to precede the investigation of the cause of the differences observed by that of the probability with which this cause is indicated4 ,thus the ratio of the population to annual births1 which one has seen raised in ,france to twenty-eight and one third1 is not equal to twenty-five in the ancient duchy of ,milan4 ,these ratios1 both established upon a great number of births1 do not permit of calling into question the existence among the ,milanese of a special cause of mortality1 which it is of moment for the government of our country to investigate and remove4 ,the ratio of the population to the births would increase again if we could diminish and remove certain dangerous and widely spread maladies4 ,this has happily been done for the smallpox1 at first by the inoculation of this disease1 then in a manner much more advantageous1 by the inoculation of vaccine1 the inestimable discovery of ,jenner1 who has thereby become one of the greatest benefactors of humanity4 ,the smallpox has this in particular1 namely1 that the same individual is not twice affected by it1 or at least such cases are so rare that they may be abstracted from the calculation4 ,this malady1 from which few escaped before the discovery of vaccine1 is often fatal and causes the death of one seventh of those whom it attacks4 ,sometimes it is mild1 and experience has taught that it can be given this latter character by inoculating it upon healthy persons1 prepared for it by a proper diet and in a favorable season4 ,then the ratio of the individuals who die to the inoculated ones is not one three hundredth4 ,this great advantage of inoculation1 joined to those of not altering the appearance and of preserving from the grievous consequences which the natural smallpox often brings1 caused it to be adopted by a great number of persons4 ,the practice was strongly recommended1 but it was strongly combated1 as is nearly always the case in things subject to inconvenience4 ,in the midst of this dispute ,daniel ,bernoulli proposed to submit to the calculus of probabilities the influence of inoculation upon the mean duration of life4 ,since precise data of the mortality produced by the smallpox at the various ages of life were lacking1 he supposed that the danger of having this malady and that of dying of it are the same at every age4 ,by means of these suppositions he succeeded by a delicate analysis in converting an ordinary table of mortality into that which would be used if smallpox did not exist1 or if it caused the death of only a very small number of those affected1 and he concludes from it that inoculation would augment by three years at least the mean duration of life1 which appeared to him beyond doubt the advantage of this operation4 ,d',alembert attacked the analysis of ,bernoulli3 at first in regard to the uncertainty of his two hypotheses1 then in regard to its insufficiency in this1 that no comparison was made of the immediate danger1 although very small1 of dying of inoculation1 to the very great but very remote danger of succumbing to natural smallpox4 ,this consideration1 which disappears when one considers a great number of individuals1 is for this reason immaterial for governments and the advantages of inoculation for them still remain2 but it is of great weight for the father of a family who must fear1 in having his children inoculated1 to see that one perish whom he holds most dear and to be the cause of it4 ,many parents were restrained by this fear1 which the discovery of vaccine has happily dissipated4 ,by one of those mysteries which nature offers to us so frequently1 vaccine is a preventive of smallpox just as certain as variolar virus1 and there is no danger at all2 it does not expose to any malady and demands only very little care4 ,therefore the practice of it has spread quickly2 and to render it universal it remains only to overcome the natural inertia of the people1 against which it is necessary to strive continually1 even when it is a question of their dearest interests4 ,the simplest means of calculating the advantage which the extinction of a malady would produce consists in determining by observation the number of individuals of a given age who die of it each year and subtracting this number from the number of deaths at the same age4 ,the ratio of the difference to the total number of individuals of the given age would be the probability of dying in the year at this age if the malady did not exist4 ,making1 then1 a sum of these probabilities from birth up to any given age1 and subtracting this sum from unity1 the remainder will be the probability of living to that age corresponding to the extinction of the malady4 ,the series of these probabilities will be the table of mortality relative to this hypothesis1 and we may conclude from it1 by what precedes1 the mean duration of life4 ,it is thus that ,duvilard has found that the increase of the mean duration of life1 due to inoculation with vaccine1 is three years at the least4 ,an increase so considerable would produce a very great increase in the population if the latter1 for other reasons1 were not restrained by the relative diminution of subsistences4 ,it is principally by the lack of subsistences that the progressive march of the population is arrested4 ,in all kinds of animals and vegetables1 nature tends without ceasing to augment the number of individuals until they are on a level of the means of subsistence4 ,in the human race moral causes have a great influence upon the population4 ,if easy clearings of the forest can furnish an abundant nourishment for new generations1 the certainty of being able to support a numerous family encourages marriages and renders them more productive4 ,upon the same soil the population and the births ought to increase at the same time simultaneously in geometric progression4 ,but when clearings become more difficult and more rare then the increase of population diminishes2 it approaches continually the variable state of subsistences1 making oscillations about it just as a pendulum whose periodicity is retarded by changing the point of suspension1 oscillates about this point by virtue of its own weight4 ,it is difficult to evaluate the maximum increase of the population2 it appears after observations that in favorable circumstances the population of the human race would be doubled every fifteen years4 ,we estimate that in ,north ,america the period of this doubling is twenty-two years4 ,in this state of things1 the population1 births1 marriages1 mortality1 all increase according to the same geometric progression of which we have the constant ratio of consecutive terms by the observation of annual births at two epochs4 ,by means of a table of mortality representing the probabilities of human life1 we may determine the duration of marriages4 ,supposing in order to simplify the matter that the mortality is the same for the two sexes1 we shall obtain the probability that the marriage will subsist one year1 or two1 or three1 etc41 by forming a series of fractions whose common denominator is the product of the two numbers of the table corresponding to the ages of the consorts1 and whose numerators are the successive products of the numbers corresponding to these ages augmented by one1 by two1 by three1 etc41 years4 ,the sum of these fractions augmented by one half will be the mean duration of marriage1 the year being taken as unity4 ,it is easy to extend the same rule to the mean duration of an association formed of three or of a greater number of individuals4 ,,chapter ,,xv4 ,,concerning ,,the ,,benefits ,,of ,,institutions ,,which ,,depend ,,upon ,,the ,,probability ,,of ,,events4 ,let us recall here what has been said in speaking of hope4 ,it has been seen that in order to obtain the advantage which results from several simple events1 of which the ones produce a benefit and the others a loss1 it is necessary to add the products of the probability of each favorable event by the benefit which it procures1 and subtract from their sum that of the products of the probability of each unfavorable event by the loss which is attached to it4 ,but whatever may be the advantage expressed by the difference of these sums1 a single event composed of these simple events does not guarantee against the fear of experiencing a loss4 ,one imagines that this fear ought to decrease when one multiplies the compound event4 ,the analysis of probabilities leads to this general theorem4 ,by the repetition of an advantageous event1 simple or compound1 the real benefit becomes more and more probable and increases without ceasing2 it becomes certain in the hypothesis of an infinite number of repetitions2 and dividing it by this number the quotient or the mean benefit of each event is the mathematical hope itself or the advantage relative to the event4 ,it is the same with a loss which becomes certain in the long run1 however small the disadvantage of the event may be4 ,this theorem upon benefits and losses is analogous to those which we have already given upon the ratios which are indicated by the indefinite repetition of events simple or compound2 and1 like them1 it proves that regularity ends by establishing itself even in the things which are most subordinated to that which we name hazard4 ,when the events are in great number1 analysis gives another very simple expression of the probability that the benefit will be comprised within determined limits4 ,this is the expression which enters again into the general law of probability given above in speaking of the probabilities which result from the indefinite multiplication of events4 ,the stability of institutions which are based upon probabilities depends upon the truth of the preceding theorem4 ,but in order that it may be applied to them it is necessary that those institutions should multiply these advantageous events for the sake of numerous things4 ,there have been based upon the probabilities of human life divers institutions1 such as life annuities and tontines4 ,the most general and the most simple method of calculating the benefits and the expenses of these institutions consists in reducing these to actual amounts4 ,the annual interest of unity is that which is called the rate of interest4 ,at the end of each year an amount acquires for a factor unity plus the rate of interest2 it increases then according to a geometrical progression of which this factor is the ratio4 ,thus in the course of time it becomes immense4 ,if1 for example1 the rate of interest is #a_/bj or five per cent1 the capital doubles very nearly in fourteen years1 quadruples in twenty-nine years1 and in less than three centuries it becomes two million times larger4 ,an increase so prodigious has given birth to the idea of making use of it in order to pay off the public debt4 ,one forms for this purpose a sinking fund to which is devoted an annual fund employed for the redemption of public bills and without ceasing increased by the interest of the bills redeemed4 ,it is clear that in the long run this fund will absorb a great part of the national debt4 ,if1 when the needs of the ,state make a loan necessary1 a part of this loan is devoted to the increasing of the annual sinking fund1 the variation of public bills will be less2 the confidence of the lenders and the probability of retiring without loss of capital loaned when one desires will be augmented and will render the conditions of the loan less onerous4 ,favorable experiences have fully confirmed these advantages4 ,but the fidelity in engagements and the stability1 so necessary to the success of such institutions1 can be guaranteed only by a government in which the legislative power is divided among several independent powers4 ,the confidence which the necessary coperation of these powers inspires1 doubles the strength of the ,state1 and the sovereign himself gains then in legal power more than he loses in arbitrary power4 ,it results from that which precedes that the actual capital equivalent to a sum which is to be paid only after a certain number of years is equal to this sum multiplied by the probability that it will be paid at that time and divided by unity augmented by the rate of interest and raised to a power expressed by the number of these years4 ,it is easy to apply this principle to life annuities upon one or several persons1 and to savings banks1 and to assurance societies of any nature4 ,suppose that one proposes to form a table of life annuities according to a given table of mortality4 ,a life annuity payable at the end of five years1 for example1 and reduced to an actual amount is1 by this principle1 equal to the product of the two following quantities1 namely1 the annuity divided by the fifth power of unity augmented by the rate of interest and the probability of paying it4 ,this probability is the inverse ratio of the number of individuals inscribed in the table opposite to the age of that one who settles the annuity to the number inscribed opposite to this age augmented by five years4 ,forming1 then1 a series of fractions whose denominators are the products of the number of persons indicated in the table of mortality as living at the age of that one who settles the annuity1 by the successive powers of unity augmented by the rate of interest1 and whose numerators are the products of the annuity by the number of persons living at the same age augmented successively by one year1 by two years1 etc41 the sum of these fractions will be the amount required for the life annuity at that age4 ,let us suppose that a person wishes by means of a life annuity to assure to his heirs an amount payable at the end of the year of his death4 ,in order to determine the value of this annuity1 one may imagine that the person borrows in life at a bank this capital and that he places it at perpetual interest in the same bank4 ,it is clear that this same capital will be due by the bank to his heirs at the end of the year of his death2 but he will have paid each year only the excess of the life interest over the perpetual interest4 ,the table of life annuities will then show that which the person ought to pay annually to the bank in order to assure this capital after his death4 ,maritime assurance1 that against fire and storms1 and generally all the institutions of this kind1 are computed on the same principles4 ,a merchant having vessels at sea wishes to assure their value and that of their cargoes against the dangers that they may run2 in order to do this1 he gives a sum to a company which becomes responsible to him for the estimated value of his cargoes and his vessels4 ,the ratio of this value to the sum which ought to be given for the price of the assurance depends upon the dangers to which the vessels are exposed and can be appreciated only by numerous observations upon the fate of vessels which have sailed from port for the same destination4 ,if the persons assured should give to the assurance company only the sum indicated by the calculus of probabilities1 this company would not be able to provide for the expenses of its institution2 it is necessary then that they should pay a sum much greater than the cost of such insurance4 ,what then is their advantage8 ,it is here that the consideration of the moral disadvantage attached to an uncertainty becomes necessary4 ,one conceives that the fairest game becomes1 as has already been seen1 disadvantageous1 because the player exchanges a certain stake for an uncertain benefit2 assurance by which one exchanges the uncertain for the certain ought to be advantageous4 ,it is indeed this which results from the rule which we have given above for determining moral hope and by which one sees moreover how far the sacrifice may extend which ought to be made to the assurance company by reserving always a moral advantage4 ,this company can then in procuring this advantage itself make a great benefit1 if the number of the assured persons is very large1 a condition necessary to its continued existence4 ,then its benefits become certain and the mathematical and moral hopes coincide2 for analysis leads to this general theorem1 namely1 that if the expectations are very numerous the two hopes approach each other without ceasing and end by coinciding in the case of an infinite number4 ,we have said in speaking of mathematical and moral hopes that there is a moral advantage in distributing the risks of a benefit which one expects over several of its parts4 ,thus in order to send a sum of money to a distant part it is much better to send it on several vessels than to expose it on one4 ,this one does by means of mutual assurances4 ,if two persons1 each having the same sum upon two different vessels which have sailed from the same port to the same destination1 agree to divide equally all the money which may arrive1 it is clear that by this agreement each of them divides equally between the two vessels the sum which he expects4 ,indeed this kind of assurance always leaves uncertainty as to the loss which one may fear4 ,but this uncertainty diminishes in proportion as the number of policy-holders increases2 the moral advantage increases more and more and ends by coinciding with the mathematical advantage1 its natural limit4 ,this renders the association of mutual assurances when it is very numerous more advantageous to the assured ones than the companies of assurance which1 in proportion to the benefit that they give1 give a moral advantage always inferior to the mathematical advantage4 ,but the surveillance of their administration can balance the advantage of the mutual assurances4 ,all these results are1 as has already been seen1 independent of the law which expresses the moral advantage4 ,one may look upon a free people as a great association whose members secure mutually their properties by supporting proportionally the charges of this guaranty4 ,the confederation of several peoples would give to them advantages analogous to those which each individual enjoys in the society4 ,a congress of their representatives would discuss objects of a utility common to all and without doubt the system of weights1 measures1 and moneys proposed by the ,french scientists would be adopted in this congress as one of the things most useful to commercial relations4 ,among the institutions founded upon the probabilities of human life the better ones are those in which1 by means of a light sacrifice of his revenue1 one assures his existence and that of his family for a time when one ought to fear to be unable to satisfy their needs4 ,as far as games are immoral1 so far these institutions are advantageous to customs by favoring the strongest bents of our nature4 ,the government ought then to encourage them and respect them in the vicissitudes of public fortune2 since the hopes which they present look toward a distant future1 they are able to prosper only when sheltered from all inquietude during their existence4 ,it is an advantage that the institution of a representative government assures them4 ,let us say a word about loans4 ,it is clear that in order to borrow perpetually it is necessary to pay each year the product of the capital by the rate of interest4 ,but one may wish to discharge this principal in equal payments made during a definite number of years1 payments which are called annuities and whose value is obtained in this manner4 ,each annuity in order to be reduced at the actual moment ought to be divided by a power of unity augmented by the rate of interest equal to the number of years after which this annuity ought to be paid4 ,forming then a geometric progression whose first term is the annuity divided by unity augmented by the rate of interest1 and whose last term is this annuity divided by the same quantity raised to a power equal to the number of years during which the payment should have been made1 the sum of this progression will be equivalent to the capital borrowed1 which will determine the value of the annuity4 ,a sinking fund is at bottom only a means of converting into annuities a perpetual rent with the sole difference that in the case of a loan by annuities the interest is supposed constant1 while the interest of funds acquired by the sinking fund is variable4 ,if it were the same in both cases1 the annuity corresponding to the funds acquired would be formed by these funds and from this annuity the ,state contributes annually to the sinking fund4 ,if one wishes to make a life loan it will be observed that the tables of life annuities give the capital required to constitute a life annuity at any age1 a simple proportion will give the rent which one ought to pay to the individual from whom the capital is borrowed4 ,from these principles all the possible kinds of loans may be calculated4 ,the principles which we have just expounded concerning the benefits and the losses of institutions may serve to determine the mean result of any number of observations already made1 when one wishes to regard the deviations of the results corresponding to divers observations4 ,let us designate by x the correction of the least result and by x augmented successively by q1 q1 q1 etc41 the corrections of the following results4 ,let us name e1 e1 e1 etc41 the errors of the observations whose law of probability we will suppose known4 ,each observation being a function of the result1 it is easy to see that by supposing the correction x of this result to be very small1 the error e of the first observation will be equal to the product of x by a determined coefficient4 ,likewise the error e of the second observation will be the product of the sum q plus x1 by a determined coefficient1 and so on4 ,the probability of the error e being given by a known function1 it will be expressed by the same function of the first of the preceding products4 ,the probability of e will be expressed by the same function of the second of these products1 and so on of the others4 ,the probability of the simultaneous existence of the errors e1 e1 e1 etc41 will be then proportional to the product of these divers functions1 a product which will be a function of x4 ,this being granted1 if one conceives a curve whose abscissa is x1 and whose corresponding ordinate is this product1 this curve will represent the probability of the divers values of x1 whose limits will be determined by the limits of the errors e1 e1 e1 etc4 ,now let us designate by ,x the abscissa which it is necessary to choose2 ,x diminished by x will be the error which would be committed if the abscissa x were the true correction4 ,this error1 multiplied by the probability of x or by the corresponding ordinate of the curve1 will be the product of the loss by its probability1 regarding1 as one should1 this error as a loss attached to the choice ,x4 ,multiplying this product by the differential of x the integral taken from the first extremity of the curve to ,x will be the disadvantage of ,x resulting from the values of x inferior to ,x4 ,for the values of x superior to ,x1 x less ,x would be the error of ,x if x were the true correction2 the integral of the product of x by the corresponding ordinate of the curve and by the differential of x will be then the disadvantage of ,x resulting from the values x superior to x1 this integral being taken from x equal to ,x up to the last extremity of the curve4 ,adding this disadvantage to the preceding one1 the sum will be the disadvantage attached to the choice of ,x4 ,this choice ought to be determined by the condition that this disadvantage be a minimum2 and a very simple calculation shows that for this1 ,x ought to be the abscissa whose ordinate divides the curve into two equal parts1 so that it is thus probable that the true value of x falls on neither the one side nor the other of ,x4 ,celebrated geometricians have chosen for ,x the most probable value of x and consequently that which corresponds to the largest ordinate of the curve2 but the preceding value appears to me evidently that which the theory of probability indicates4 ,,chapter ,,xvi4 ,,concerning ,,illusions ,,in ,,the ,,estimation ,,of ,,probabilities4 ,the mind has its illusions as the sense of sight2 and in the same manner that the sense of feeling corrects the latter1 reflection and calculation correct the former4 ,probability based upon a daily experience1 or exaggerated by fear and by hope1 strikes us more than a superior probability but it is only a simple result of calculus4 ,thus we do not fear in return for small advantages to expose our life to dangers much less improbable than the drawing of a quint in the lottery of ,france2 and yet no one would wish to procure for himself the same advantages with the certainty of losing his life if this quint should be drawn4 ,our passions1 our prejudices1 and dominating opinions1 by exaggerating the probabilities which are favorable to them and by attenuating the contrary probabilities1 are the abundant sources of dangerous illusions4 ,present evils and the cause which produced them effect us much more than the remembrance of evils produced by the contrary cause2 they prevent us from appreciating with justice the inconveniences of the ones and the others1 and the probability of the proper means to guard ourselves against them4 ,it is this which leads alternately to despotism and to anarchy the people who are driven from the state of repose to which they never return except after long and cruel agitations4 ,this vivid impression which we receive from the presence of events1 and which allows us scarcely to remark the contrary events observed by others1 is a principal cause of error against which one cannot sufficiently guard himself4 ,it is principally at games of chance that a multitude of illusions support hope and sustain it against unfavorable chances4 ,the majority of those who play at lotteries do not know how many chances are to their advantage1 how many are contrary to them4 ,they see only the possibility by a small stake of gaining a considerable sum1 and the projects which their imagination brings forth1 exaggerate to their eyes the probability of obtaining it2 the poor man especially1 excited by the desire of a better fate1 risks at play his necessities by clinging to the most unfavorable combinations which promise him a great benefit4 ,all would be without doubt surprised by the immense number of stakes lost if they could know of them2 but one takes care on the contrary to give to the winnings a great publicity1 which becomes a new cause of excitement for this funereal play4 ,when a number in the lottery of ,france has not been drawn for a long time the crowd is eager to cover it with stakes4 ,they judge since the number has not been drawn for a long time that it ought at the next drawing to be drawn in preference to others4 ,so common an error appears to me to rest upon an illusion by which one is carried back involuntarily to the origin of events4 ,it is1 for example1 very improbable that at the play of heads and tails one will throw heads ten times in succession4 ,this improbability which strikes us indeed when it has happened nine times1 leads us to believe that at the tenth throw tails will be thrown4 ,but the past indicating in the coin a greater propensity for heads than for tails renders the first of the events more probable than the second2 it increases as one has seen the probability of throwing heads at the following throw4 ,a similar illusion persuades many people that one can certainly win in a lottery by placing each time upon the same number1 until it is drawn1 a stake whose product surpasses the sum of all the stakes4 ,but even when similar speculations would not often be stopped by the impossibility of sustaining them they would not diminish the mathematical disadvantage of speculators and they would increase their moral disadvantage1 since at each drawing they would risk a very large part of their fortune4 ,i have seen men1 ardently desirous of having a son1 who could learn only with anxiety of the births of boys in the month when they expected to become fathers4 ,imagining that the ratio of these births to those of girls ought to be the same at the end of each month1 they judged that the boys already born would render more probable the births next of girls4 ,thus the extraction of a white ball from an urn which contains a limited number of white balls and of black balls increases the probability of extracting a black ball at the following drawing4 ,but this ceases to take place when the number of balls in the urn is unlimited1 as one must suppose in order to compare this case with that of births4 ,if1 in the course of a month1 there were born many more boys than girls1 one might suspect that toward the time of their conception a general cause had favored masculine conception1 which would render more probable the birth next of a boy4 ,the irregular events of nature are not exactly comparable to the drawing of the numbers of a lottery in which all the numbers are mixed at each drawing in such a manner as to render the chances of their drawing perfectly equal4 ,the frequency of one of these events seems to indicate a cause slightly favoring it1 which increases the probability of its next return1 and its repetition prolonged for a long time1 such as a long series of rainy days1 may develop unknown causes for its change2 so that at each expected event we are not1 as at each drawing of a lottery1 led back to the same state of indecision in regard to what ought to happen4 ,but in proportion as the observation of these events is multiplied1 the comparison of their results with those of lotteries becomes more exact4 ,by an illusion contrary to the preceding ones one seeks in the past drawings of the lottery of ,france the numbers most often drawn1 in order to form combinations upon which one thinks to place the stake to advantage4 ,but when the manner in which the mixing of the numbers in this lottery is considered1 the past ought to have no influence upon the future4 ,the very frequent drawings of a number are only the anomalies of chance2 ,i have submitted several of them to calculation and have constantly found that they are included within the limits which the supposition of an equal possibility of the drawing of all the numbers allows us to admit without improbability4 ,in a long series of events of the same kind the single chances of hazard ought sometimes to offer the singular veins of good luck or bad luck which the majority of players do not fail to attribute to a kind of fatality4 ,it happens often in games which depend at the same time upon hazard and upon the competency of the players1 that that one who loses1 troubled by his loss1 seeks to repair it by hazardous throws which he would shun in another situation2 thus he aggravates his own ill luck and prolongs its duration4 ,it is then that prudence becomes necessary and that it is of importance to convince oneself that the moral disadvantage attached to unfavorable chances is increased by the ill luck itself4 ,the opinion that man has long been placed in the centre of the universe1 considering himself the special object of the cares of nature1 leads each individual to make himself the centre of a more or less extended sphere and to believe that hazard has preference for him4 ,sustained by this belief1 players often risk considerable sums at games when they know that the chances are unfavorable4 ,in the conduct of life a similar opinion may sometimes have advantages2 but most often it leads to disastrous enterprises4 ,here as everywhere illusions are dangerous and truth alone is generally useful4 ,one of the great advantages of the calculus of probabilities is to teach us to distrust first opinions4 ,as we recognize that they often deceive when they may be submitted to calculus1 we ought to conclude that in other matters confidence should be given only after extreme circumspection4 ,let us prove this by example4 ,an urn contains four balls1 black and white1 but which are not all of the same color4 ,one of these balls has been drawn whose color is white and which has been put back in the urn in order to proceed again to similar drawings4 ,one demands the probability of extracting only black balls in the four following drawings4 ,if the white and black were in equal number this probability would be the fourth power of the probability of extracting a black ball at each drawing2 it would be then #a_/af4 ,but the extraction of a white ball at the first drawing indicates a superiority in the number of white balls in the urn2 for if one supposes in the urn three white balls and one black the probability of extracting a white ball is 2 it is #b_/d if one supposes two white balls and two black2 finally it is reduced to if one supposes three black balls and one white4 ,following the principle of the probability of causes drawn from events the probabilities of these three suppositions are among themselves as the quantities 1 #b_/d1 2 they are consequently equal to #c_/f1 #b_/f1 4 ,it is thus a bet of #e against #a that the number of black balls is inferior1 or at the most equal1 to that of the white4 ,it seems then that after the extraction of a white ball at the first drawing1 the probability of extracting successively four black balls ought to be less than in the case of the equality of the colors or smaller than one sixteenth4 ,however1 it is not1 and it is found by a very simple calculation that this probability is greater than one fourteenth4 ,indeed it would be the fourth power of 1 of #b_/d1 and of in the first1 the second1 and the third of the preceding suppositions concerning the colors of the balls in the urn4 ,multiplying respectively each power by the probability of the corresponding supposition1 or by #c_/f1 #b_/f1 and 1 the sum of the products will be the probability of extracting successively four black balls4 ,one has thus for this probability #bi_/chd1 a fraction greater than #a_/ad4 ,this paradox is explained by considering that the indication of the superiority of white balls over the black ones at the first drawing does not exclude at all the superiority of the black balls over the white ones1 a superiority which excludes the supposition of the equality of the colors4 ,but this superiority1 though but slightly probable1 ought to render the probability of drawing successively a given number of black balls greater than in this supposition if the number is considerable2 and one has just seen that this commences when the given number is equal to four4 ,let us consider again an urn which contains several white and black balls4 ,let us suppose at first that there is only one white ball and one black4 ,it is then an even bet that a white ball will be extracted in one drawing4 ,but it seems for the equality of the bet that one who bets on extracting the white ball ought to have two drawings if the urn contains two black and one white1 three drawings if it contains three black and one white1 and so on2 it is supposed that after each drawing the extracted ball is placed again in the urn4 ,we are convinced easily that this first idea is erroneous4 ,indeed in the case of two black and one white ball1 the probability of extracting two black in two drawings is the second power of or #d_/i2 but this probability added to that of drawing a white ball in two drawings is certainty or unity1 since it is certain that two black balls or at least one white ball ought to be drawn2 the probability in this last case is then #e_/i1 a fraction greater than 4 ,there would still be a greater advantage in the bet of drawing one white ball in five draws when the urn contains five black and one white ball2 this bet is even advantageous in four drawings2 it returns then to that of throwing six in four throws with a single die4 ,the ,chevalier de ,mer1 who caused the invention of the calculus of probabilities by encouraging his friend ,pascal1 the great geometrician1 to occupy himself with it1 said to him that he had found error in the numbers by this ratio4 ,if we undertake to make six with one die there is an advantage in undertaking it in four throws1 as #fga to #fbe4 ,if we undertake to make two sixes with two dice1 there is a disadvantage in undertaking in #bd throws4 ,at least #bd is to #cf1 the number of the faces of the two dice1 as #d is to #f1 the number of faces of one die4 ,this was1 wrote ,pascal to ,fermat1 his great scandal which caused him to say boldly that the propositions were not constant and that arithmetic was demented4444 ,he has a very good mind1 but he is not a geometrician1 which is1 as you know1 a great fault4 ,the ,chevalier de ,mer1 deceived by a false analogy1 thought that in the case of the equality of bets the number of throws ought to increase in proportion to the number of all the chances possible1 which is not exact1 but which approaches exactness as this number becomes larger4 ,one has endeavored to explain the superiority of the births of boys over those of girls by the general desire of fathers to have a son who would perpetuate the name4 ,thus by imagining an urn filled with an infinity of white and black balls in equal number1 and supposing a great number of persons each of whom draws a ball from this urn and continues with the intention of stopping when he shall have extracted a white ball1 one has believed that this intention ought to render the number of white balls extracted superior to that of the black ones4 ,indeed this intention gives necessarily after all the drawings a number of white balls equal to that of persons1 and it is possible that these drawings would never lead a black ball4 ,but it is easy to see that this first notion is only an illusion2 for if one conceives that in the first drawing all the persons draw at once a ball from the urn1 it is evident that their intention can have no influence upon the color of the balls which ought to appear at this drawing4 ,its unique effect will be to exclude from the second drawing the persons who shall have drawn a white one at the first4 ,it is likewise apparent that the intention of the persons who shall take part in the new drawing will have no influence upon the color of the balls which shall be drawn1 and that it will be the same at the following drawings4 ,this intention will have no influence then upon the color of the balls extracted in the totality of drawings2 it will1 however1 cause more or fewer to participate at each drawing4 ,the ratio of the white balls extracted to the black ones will differ thus very little from unity4 ,it follows that the number of persons being supposed very large1 if observation gives between the colors extracted a ratio which differs sensibly from unity1 it is very probable that the same difference is found between unity and the ratio of the white balls to the black contained in the urn4 ,i count again among illusions the application which ,liebnitz and ,daniel ,bernoulli have made of the calculus of probabilities to the summation of series4 ,if one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable1 in a series prescribed by the ratio to the powers of this variable1 it is easy to see that in supposing the variable equal to unity the fraction becomes 1 and the series becomes plus one1 minus one1 plus one1 minus one1 etc4 ,in adding the first two terms1 the second two1 and so on1 the series is transformed into another of which each term is zero4 ,grandi1 an ,italian ,jesuit1 concluded from this the possibility of the creation2 because the series being always 1 he saw this fraction spring from an infinity of zeros or from nothing4 ,it was thus that ,liebnitz believed he saw the image of creation in his binary arithmetic where he employed only the two characters1 unity and zero4 ,he imagined1 since ,god can be represented by unity and nothing by zero1 that the ,supreme ,being had drawn from nothing all beings1 as unity with zero expresses all the numbers in this system of arithmetic4 ,this idea was so pleasing to ,liebnitz that he communicated it to the ,jesuit ,grimaldi1 president of the tribunal of mathematics in ,china1 in the hope that this emblem of creation would convert to ,christianity the emperor there who particularly loved the sciences4 ,i report this incident only to show to what extent the prejudices of infancy can mislead the greatest men4 ,liebnitz1 always led by a singular and very loose metaphysics1 considered that the series plus one1 minus one1 plus one1 etc41 becomes unity or zero according as one stops at a number of terms odd or even2 and as in infinity there is no reason to prefer the even number to the odd1 one ought following the rules of probability1 to take the half of the results relative to these two kinds of numbers1 and which are zero and unity1 which gives for the value of the series4 ,daniel ,bernoulli has since extended this reasoning to the summation of series formed from periodic terms4 ,but all these series have no values properly speaking2 they get them only in the case where their terms are multiplied by the successive powers of a variable less than unity4 ,then these series are always convergent1 however small one supposes the difference of the variable from unity2 and it is easy to demonstrate that the values assigned by ,bernoulli1 by virtue of the rule of probabilities1 are the same values of the generative fraction of the series1 when one supposes in these fractions the variable equal to unity4 ,these values are again the limits which the series approach more and more1 in proportion as the variable approaches unity4 ,but when the variable is exactly equal to unity the series cease to be convergent2 they have values only as far as one arrests them4 ,the remarkable ratio of this application of the calculus of probabilities with the limits of the values of periodic series supposes that the terms of these series are multiplied by all the consecutive powers of the variable4 ,but this series may result from the development of an infinity of different fractions in which this did not occur4 ,thus the series plus one1 minus one1 plus one1 etc41 may spring from the development of a fraction whose numerator is unity plus the variable1 and whose denominator is this numerator augmented by the square of the variable4 ,supposing the variable equal to unity1 this development changes1 in the series proposed1 and the generative fraction becomes equal to 2 the rules of probabilities would give then a false result1 which proves how dangerous it would be to employ similar reasoning1 especially in the mathematical sciences1 which ought to be especially distinguished by the rigor of their operations4 ,we are led naturally to believe that the order according to which we see things renewed upon the earth has existed from all times and will continue always4 ,indeed if the present state of the universe were exactly similar to the anterior state which has produced it1 it would give birth in its turn to a similar state2 the succession of these states would then be eternal4 ,i have found by the application of analysis to the law of universal gravity that the movement of rotation and of revolution of the planets and satellites1 and the position of the orbits and of their equators are subjected only to periodic inequalities4 ,in comparing with ancient eclipses the theory of the secular equation of the moon ,i have found that since ,hipparchus the duration of the day has not varied by the hundredth of a second1 and that the mean temperature of the earth has not diminished the one-hundredth of a degree4 ,thus the stability of actual order appears established at the same time by theory and by observations4 ,but this order is effected by divers causes which an attentive examination reveals1 and which it is impossible to submit to calculus4 ,the actions of the ocean1 of the atmosphere1 and of meteors1 of earthquakes1 and the eruptions of volcanoes1 agitate continually the surface of the earth and ought to effect in the long run great changes4 ,the temperature of climates1 the volume of the atmosphere1 and the proportion of the gases which constitute it1 may vary in an inappreciable manner4 ,the instruments and the means suitable to determine these variations being new1 observation has been unable up to this time to teach us anything in this regard4 ,but it is hardly probable that the causes which absorb and renew the gases constituting the air maintain exactly their respective proportions4 ,a long series of centuries will show the alterations which are experienced by all these elements so essential to the conservation of organized beings4 ,although historical monuments do not go back to a very great antiquity they offer us nevertheless sufficiently great changes which have come about by the slow and continued action of natural agents4 ,searching in the bowels of the earth one discovers numerous dbris of former nature1 entirely different from the present4 ,moreover1 if the entire earth was in the beginning fluid1 as everything appears to indicate1 one imagines that in passing from that state to the one which it has now1 its surface ought to have experienced prodigious changes4 ,the heavens itself in spite of the order of its movements1 is not unchangeable4 ,the resistance of light and of other ethereal fluids1 and the attraction of the stars ought1 after a great number of centuries1 to alter considerably the planetary movements4 ,the variations already observed in the stars and in the form of the nebul give us a presentiment of those which time will develop in the system of these great bodies4 ,one may represent the successive states of the universe by a curve1 of which time would be the abscissa and of which the ordinates are the divers states4 ,scarcely knowing an element of this curve we are far from being able to go back to its origin2 and if in order to satisfy the imagination1 always restless from our ignorance of the cause of the phenomena which interest it1 one ventures some conjectures it is wise to present them only with extreme reserve4 ,there exists in the estimation of probabilities a kind of illusions1 which depending especially upon the laws of the intellectual organization demands1 in order to secure oneself against them1 a profound examination of these laws4 ,the desire to penetrate into the future and the ratios of some remarkable events1 to the predictions of astrologers1 of diviners and soothsayers1 to presentiments and dreams1 to the numbers and the days reputed lucky or unlucky1 have given birth to a multitude of prejudices still very widespread4 ,one does not reflect upon the great number of non-coincidences which have made no impression or which are unknown4 ,however1 it is necessary to be acquainted with them in order to appreciate the probability of the causes to which the coincidences are attributed4 ,this knowledge would confirm without doubt that which reason tells us in regard to these prejudices4 ,thus the philosopher of antiquity to whom is shown in a temple1 in order to exalt the power of the god who is adored there1 the ex veto of all those who after having invoked it were saved from shipwreck1 presents an incident consonant with the calculus of probabilities1 observing that he does not see inscribed the names of those who1 in spite of this invocation1 have perished4 ,cicero has refuted all these prejudices with much reason and eloquence in his ,treatise on ,divination1 which he ends by a passage which ,i shall cite2 for one loves to find again among the ancients the thunderbolts of reason1 which1 after having dissipated all the prejudices by its light1 shall become the sole foundation of human institutions4 ,it is necessary1 says the ,roman orator1 to reject divination by dreams and all similar prejudices4 ,widespread superstition has subjugated the majority of minds and has taken possession of the feebleness of men4 ,it is this we have expounded in our books upon the nature of the gods and especially in this work1 persuaded that we shall render a service to others and to ourselves if we succeed in destroying superstition4 ,however 7and ,i desire especially in this regard my thought be well comprehended71 in destroying superstition ,i am far from wishing to disturb religion4 ,wisdom enjoins us to maintain the institutions and the ceremonies of our ancestors1 touching the cult of the gods4 ,moreover1 the beauty of the universe and the order of celestial things force us to recognize some superior nature which ought to be remarked and admired by the human race4 ,but as far as it is proper to propagate religion1 which is joined to the knowledge of nature1 so far it is necessary to work toward the extirpation of superstition1 for it torments one1 importunes one1 and pursues one continually and in all places4 ,if one consult a diviner or a soothsayer1 if one immolates a victim1 if one regards the flight of a bird1 if one encounters a ,chaldean or an aruspex1 if it lightens1 if it thunders1 if the thunderbolt strikes1 finally1 if there is born or is manifested a kind of prodigy1 things one of which ought often to happen1 then superstition dominates and leaves no repose4 ,sleep itself1 this refuge of mortals in their troubles and their labors1 becomes by it a new source of inquietude and fear4 ,all these prejudices and the terrors which they inspire are connected with physiological causes which continue sometimes to operate strongly after reason has disabused us of them4 ,but the repetition of acts contrary to these prejudices can always destroy them4 ,,chapter ,,xvii4 ,,concerning ,,the ,,various ,,means ,,of ,,approaching ,,certainty4 ,induction1 analogy1 hypotheses founded upon facts and rectified continually by new observations1 a happy tact given by nature and strengthened by numerous comparisons of its indications with experience1 such are the principal means for arriving at truth4 ,if one considers a series of objects of the same nature one perceives among them and in their changes ratios which manifest themselves more and more in proportion as the series is prolonged1 and which1 extending and generalizing continually1 lead finally to the principle from which they were derived4 ,but these ratios are enveloped by so many strange circumstances that it requires great sagacity to disentangle them and to recur to this principle3 it is in this that the true genius of sciences consists4 ,analysis and natural philosophy owe their most important discoveries to this fruitful means1 which is called induction4 ,newton was indebted to it for his theorem of the binomial and the principle of universal gravity4 ,it is difficult to appreciate the probability of the results of induction1 which is based upon this that the simplest ratios are the most common2 this is verified in the formul of analysis and is found again in natural phenomena1 in crystallization1 and in chemical combinations4 ,this simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results of a small number of immutable laws4 ,yet induction1 in leading to the discovery of the general principles of the sciences1 does not suffice to establish them absolutely4 ,it is always necessary to confirm them by demonstrations or by decisive experiences2 for the history of the sciences shows us that induction has sometimes led to inexact results4 ,i shall cite1 for example1 a theorem of ,fermat in regard to prime numbers4 ,this great geometrician1 who had meditated profoundly upon this theorem1 sought a formula which1 containing only prime numbers1 gave directly a prime number greater than any other number assignable4 ,induction led him to think that two1 raised to a power which was itself a power of two1 formed with unity a prime number4 ,thus1 two raised to the square plus one1 forms the prime number five2 two raised to the second power of two1 or sixteen1 forms with one the prime number seventeen4 ,he found that this was still true for the eighth and the sixteenth power of two augmented by unity2 and this induction1 based upon several arithmetical considerations1 caused him to regard this result as general4 ,however1 he avowed that he had not demonstrated it4 ,indeed1 ,euler recognized that this does not hold for the thirty-second power of two1 which1 augmented by unity1 gives #d1bid1ifg1big1 a number divisible by #fda4 ,we judge by induction that if various events1 movements1 for example1 appear constantly and have been long connected by a simple ratio1 they will continue to be subjected to it2 and we conclude from this1 by the theory of probabilities1 that this ratio is due1 not to hazard1 but to a regular cause4 ,thus the equality of the movements of the rotation and the revolution of the moon2 that of the movements of the nodes of the orbit and of the lunar equator1 and the coincidence of these nodes2 the singular ratio of the movements of the first three satellites of ,jupiter1 according to which the mean longitude of the first satellite1 less three times that of the second1 plus two times that of the third1 is equal to two right angles2 the equality of the interval of the tides to that of the passage of the moon to the meridian2 the return of the greatest tides with the syzygies1 and of the smallest with the quadratures2 all these things1 which have been maintained since they were first observed1 indicate with an extreme probability1 the existence of constant causes which geometricians have happily succeeded in attaching to the law of universal gravity1 and the knowledge of which renders certain the perpetuity of these ratios4 ,the chancellor ,bacon1 the eloquent promoter of the true philosophical method1 has made a very strange misuse of induction in order to prove the immobility of the earth4 ,he reasons thus in the ,novum ,organum1 his finest work3 ,the movement of the stars from the orient to the occident increases in swiftness1 in proportion to their distance from the earth4 ,this movement is swiftest with the stars2 it slackens a little with ,saturn1 a little more with ,jupiter1 and so on to the moon and the highest comets4 ,it is still perceptible in the atmosphere1 especially between the tropics1 on account of the great circles which the molecules of the air describe there2 finally1 it is almost inappreciable with the ocean2 it is then nil for the earth4 ,but this induction proves only that ,saturn1 and the stars which are inferior to it1 have their own movements1 contrary to the real or apparent movement which sweeps the whole celestial sphere from the orient to the occident1 and that these movements appear slower with the more remote stars1 which is conformable to the laws of optics4 ,bacon ought to have been struck by the inconceivable swiftness which the stars require in order to accomplish their diurnal revolution1 if the earth is immovable1 and by the extreme simplicity with which its rotation explains how bodies so distant1 the ones from the others1 as the stars1 the sun1 the planets1 and the moon1 all seem subjected to this revolution4 ,as to the ocean and to the atmosphere1 he ought not to compare their movement with that of the stars which are detached from the earth2 but since the air and the sea make part of the terrestrial globe1 they ought to participate in its movement or in its repose4 ,it is singular that ,bacon1 carried to great prospects by his genius1 was not won over by the majestic idea which the ,copernican system of the universe offers4 ,he was able1 however1 to find in favor of that system1 strong analogies in the discoveries of ,galileo1 which were continued by him4 ,he has given for the search after truth the precept1 but not the example4 ,but by insisting1 with all the force of reason and of eloquence1 upon the necessity of abandoning the insignificant subtleties of the school1 in order to apply oneself to observations and to experiences1 and by indicating the true method of ascending to the general causes of phenomena1 this great philosopher contributed to the immense strides which the human mind made in the grand century in which he terminated his career4 ,analogy is based upon the probability1 that similar things have causes of the same kind and produce the same effects4 ,this probability increase as the similitude becomes more perfect4 ,thus we judge without doubt that beings provided with the same organs1 doing the same things1 experience the same sensations1 and are moved by the same desires4 ,the probability that the animals which resemble us have sensations analogous to ours1 although a little inferior to that which is relative to individuals of our species1 is still exceedingly great2 and it has required all the influence of religious prejudices to make us think with some philosophers that animals are mere automatons4 ,the probability of the existence of feeling decreases in the same proportion as the similitude of the organs with ours diminishes1 but it is always very great1 even with insects4 ,in seeing those of the same species execute very complicated things exactly in the same manner from generation to generation1 and without having learned them1 one is led to believe that they act by a kind of affinity analogous to that which brings together the molecules of crystals1 but which1 together with the sensation attached to all animal organization1 produces1 with the regularity of chemical combinations1 combinations that are much more singular2 one might1 perhaps1 name this mingling of elective affinities and sensations animal affinity4 ,although there exists a great analogy between the organization of plants and that of animals1 it does not seem to me sufficient to extend to vegetables the sense of feeling2 but nothing authorizes us in denying it to them4 ,since the sun brings forth1 by the beneficent action of its light and of its heat1 the animals and plants which cover the earth1 we judge by analogy that it produces similar effects upon the other planets2 for it is not natural to think that the cause whose activity we see developed in so many ways should be sterile upon so great a planet as ,jupiter1 which1 like the terrestrial globe1 has its days1 its nights1 and its years1 and upon which observations indicate changes which suppose very active forces4 ,yet this would be giving too great an extension to analogy to conclude from it the similitude of the inhabitants of the planets and of the earth4 ,man1 made for the temperature which he enjoys1 and for the element which he breathes1 would not be able1 according to all appearance1 to live upon the other planets4 ,but ought there not to be an infinity of organization relative to the various constitutions of the globes of this universe8 ,if the single difference of the elements and of the climates make so much variety in terrestrial productions1 how much greater the difference ought to be among those of the various planets and of their satellites6 ,the most active imagination can form no idea of it2 but their existence is very probable4 ,we are led by a strong analogy to regard the stars as so many suns endowed1 like ours1 with an attractive power proportional to the mass and reciprocal to the square of the distances2 for this power being demonstrated for all the bodies of the solar system1 and for their smallest molecules1 it appears to appertain to all matter4 ,already the movements of the small stars1 which have been called double1 on account of their being binary1 appear to indicate it2 a century at most of precise observations1 by verifying their movements of revolution1 the ones about the others1 will place beyond doubt their reciprocal attractions4 ,the analogy which leads us to make each star the centre of a planetary system is far less strong than the preceding one2 but it acquires probability by the hypothesis which has been proposed in regard to the formation of the stars and of the sun2 for in this hypothesis each star1 having been like the sun1 primitively environed by a vast atmosphere1 it is natural to attribute to this atmosphere the same effects as to the solar atmosphere1 and to suppose that it has produced1 in condensing1 planets and satellites4 ,a great number of discoveries in the sciences is due to analogy4 ,i shall cite as one of the most remarkable1 the discovery of atmospheric electricity1 to which one has been led by the analogy of electric phenomena with the effects of thunder4 ,the surest method which can guide us in the search for truth1 consists in rising by induction from phenomena to laws and from laws to forces4 ,laws are the ratios which connect particular phenomena together3 when they have shown the general principle of the forces from which they are derived1 one verifies it either by direct experiences1 when this is possible1 or by examination if it agrees with known phenomena2 and if by a rigorous analysis we see them proceed from this principle1 even in their small details1 and if1 moreover1 they are quite varied and very numerous1 then science acquires the highest degree of certainty and of perfection that it is able to attain4 ,such1 astronomy has become by the discovery of universal gravity4 ,the history of the sciences shows that the slow and laborious path of induction has not always been that of inventors4 ,the imagination1 impatient to arrive at the causes1 takes pleasure in creating hypotheses1 and often it changes the facts in order to adapt them to its work2 then the hypotheses are dangerous4 ,but when one regards them only as the means of connecting the phenomena in order to discover the laws2 when1 by refusing to attribute them to a reality1 one rectifies them continually by new observations1 they are able to lead to the veritable causes1 or at least put us in a position to conclude from the phenomena observed those which given circumstances ought to produce4 ,if we should try all the hypotheses which can be formed in regard to the cause of phenomena we should arrive1 by a process of exclusion1 at the true one4 ,this means has been employed with success2 sometimes we have arrived at several hypotheses which explain equally well all the facts known1 and among which scholars are divided1 until decisive observations have made known the true one4 ,then it is interesting1 for the history of the human mind1 to return to these hypotheses1 to see how they succeed in explaining a great number of facts1 and to investigate the changes which they ought to undergo in order to agree with the history of nature4 ,it is thus that the system of ,ptolemy1 which is only the realization of celestial appearances1 is transformed into the hypothesis of the movement of the planets about the sun1 by rendering equal and parallel to the solar orbit the circles and the epicycles which he causes to be described annually1 and the magnitude of which he leaves undetermined4 ,it suffices1 then1 in order to change this hypothesis into the true system of the world1 to transport the apparent movement of the sun in a sense contrary to the earth4 ,it is almost always impossible to submit to calculus the probability of the results obtained by these various means2 this is true likewise for historical facts4 ,but the totality of the phenomena explained1 or of the testimonies1 is sometimes such that without being able to appreciate the probability we cannot reasonably permit ourselves any doubt in regard to them4 ,in the other cases it is prudent to admit them only with great reserve4 ,,chapter ,,xviii4 ,,historical ,,notice ,,concerning ,,the ,,calculus ,,of ,,probabilities4 ,long ago were determined1 in the simplest games1 the ratios of the chances which are favorable or unfavorable to the players2 the stakes and the bets were regulated according to these ratios4 ,but no one before ,pascal and ,fermat had given the principles and the methods for submitting this subject to calculus1 and no one had solved the rather complicated questions of this kind4 ,it is1 then1 to these two great geometricians that we must refer the first elements of the science of probabilities1 the discovery of which can be ranked among the remarkable things which have rendered illustrious the seventeenth century--the century which has done the greatest honor to the human mind4 ,the principal problem which they solved by different methods1 consists1 as we have seen1 in distributing equitably the stake among the players1 who are supposed to be equally skilful and who agree to stop the game before it is finished1 the condition of play being that1 in order to win the game1 one must gain a given number of points different for each of the players4 ,it is clear that the distribution should be made proportionally to the respective probabilities of the players of winning this game1 the probabilities depending upon the numbers of points which are still lacking4 ,the method of ,pascal is very ingenious1 and is at bottom only the equation of partial differences of this problem applied in determining the successive probabilities of the players1 by going from the smallest numbers to the following ones4 ,this method is limited to the case of two players2 that of ,fermat1 based upon combinations1 applies to any number of players4 ,pascal believed at first that it was1 like his own1 restricted to two players2 this brought about between them a discussion1 at the conclusion of which ,pascal recognized the generality of the method of ,fermat4 ,huygens united the divers problems which had already been solved and added new ones in a little treatise1 the first that has appeared on this subject and which has the title ,de ,ratiociniis in ludo ale4 ,several geometricians have occupied themselves with the subject since3 ,hudde1 the great pensionary1 ,witt in ,holland1 and ,halley in ,england1 applied calculus to the probabilities of human life1 and ,halley published in this field the first table of mortality4 ,about the same time ,jacques ,bernoulli proposed to geometricians various problems of probability1 of which he afterwards gave solutions4 ,finally he composed his beautiful work entitled ,ars conjectandi1 which appeared seven years after his death1 which occurred in #agjf4 ,the science of probabilities is more profoundly investigated in this work than in that of ,huygens4 ,the author gives a general theory of combinations and series1 and applies it to several difficult questions concerning hazards4 ,this work is still remarkable on account of the justice and the cleverness of view1 the employment of the formula of the binomial in this kind of questions1 and by the demonstration of this theorem1 namely1 that in multiplying indefinitely the observations and the experiences1 the ratio of the events of different natures approaches that of their respective probabilities in the limits whose interval becomes more and more narrow in proportion as they are multiplied1 and become less than any assignable quantity4 ,this theorem is very useful for obtaining by observations the laws and the causes of phenomena4 ,bernoulli attaches1 with reason1 a great importance to his demonstration1 upon which he has said to have meditated for twenty years4 ,in the interval1 from the death of ,jacques ,bernoulli to the publication of his work1 ,montmort and ,moivre produced two treatises upon the calculus of probabilities4 ,that of ,montmort has the title ,essai sur les ,jeux de hasard2 it contains numerous applications of this calculus to various games4 ,the author has added in the second edition some letters in which ,nicolas ,bernoulli gives the ingenious solutions of several difficult problems4 ,the treatise of ,moivre1 later than that of ,montmort1 appeared at first in the ,transactions philosophiques of the year #agaa4 ,then the author published it separately1 and he has improved it successively in three editions4 ,this work is principally based upon the formula of the binomial and the problems which it contains have1 like their solutions1 a grand generality4 ,but its distinguishing feature is the theory of recurrent series and their use in this subject4 ,this theory is the integration of linear equations of finite differences with constant coefficients1 which ,moivre made in a very happy manner4 ,in his work1 ,moivre has taken up again the theory of ,jacques ,bernoulli in regard to the probability of results determined by a great number of observations4 ,he does not content himself with showing1 as ,bernoulli does1 that the ratio of the events which ought to occur approaches without ceasing that of their respective probabilities2 but he gives besides an elegant and simple expression of the probability that the difference of these two ratios is contained within the given limits4 ,for this purpose he determines the ratio of the greatest term of the development of a very high power of the binomial to the sum of all its terms1 and the hyperbolic logarithm of the excess of this term above the terms adjacent to it4 ,the greatest term being then the product of a considerable number of factors1 his numerical calculus becomes impracticable4 ,in order to obtain it by a convergent approximation1 ,moivre makes use of a theorem of ,stirling in regard to the mean term of the binomial raised to a high power1 a remarkable theorem1 especially in this1 that it introduces the square root of the ratio of the circumference to the radius in an expression which seemingly ought to be irrelevant to this transcendent4 ,moreover1 ,moivre was greatly struck by this result1 which ,stirling had deduced from the expression of the circumference in infinite products2 ,wallis had arrived at this expression by a singular analysis which contains the germ of the very curious and useful theory of definite integrals4 ,many scholars1 among whom one ought to name ,deparcieux1 ,kersseboom1 ,wargentin1 ,dupr de ,saint-,maure1 ,simpson1 ,sussmilch1 ,messne1 ,moheau1 ,price1 ,bailey1 and ,duvillard1 have collected a great amount of precise data in regard to population1 births1 marriages1 and mortality4 ,they have given formul and tables relative to life annuities1 tontines1 assurances1 etc4 ,but in this short notice ,i can only indicate these useful works in order to adhere to original ideas4 ,of this number special mention is due to the mathematical and moral hopes and to the ingenious principle which ,daniel ,bernoulli has given for submitting the latter to analysis4 ,such is again the happy application which he has made of the calculus of probabilities to inoculation4 ,one ought especially to include1 in the number of these original ideas1 direct consideration of the possibility of events drawn from events observed4 ,jacques ,bernoulli and ,moivre supposed these possibilities known1 and they sought the probability that the result of future experiences will more and more nearly represent them4 ,bayes1 in the ,transactions philosophiques of the year #agfc1 sought directly the probability that the possibilities indicated by past experiences are comprised within given limits2 and he has arrived at this in a refined and very ingenious manner1 although a little perplexing4 ,this subject is connected with the theory of the probability of causes and future events1 concluded from events observed4 ,some years later ,i expounded the principles of this theory with a remark as to the influence of the inequalities which may exist among the chances which are supposed to be equal4 ,although it is not known which of the simple events these inequalities favor1 nevertheless this ignorance itself often increases the probability of compound events4 ,in generalizing analysis and the problems concerning probabilities1 ,i was led to the calculus of partial finite differences1 which ,lagrange has since treated by a very simple method1 elegant applications of which he has used in this kind of problems4 ,the theory of generative functions which ,i published about the same time includes these subjects among those it embraces1 and is adapted of itself and with the greatest generality to the most difficult questions of probability4 ,it determines again1 by very convergent approximations1 the values of the functions composed of a great number of terms and factors2 and in showing that the square root of the ratio of the circumference to the radius enters most frequently into these values1 it shows that an infinity of other transcendents may be introduced4 ,testimonies1 votes1 and the decisions of electoral and deliberative assemblies1 and the judgments of tribunals1 have been submitted likewise to the calculus of probabilities4 ,so many passions1 divers interests1 and circumstances complicate the questions relative to the subjects1 that they are almost always insoluble4 ,but the solution of very simple problems which have a great analogy with them1 may often shed upon difficult and important questions great light1 which the surety of calculus renders always preferable to the most specious reasonings4 ,one of the most interesting applications of the calculus of probabilities concerns the mean values which must be chosen among the results of observations4 ,many geometricians have studied the subject1 and ,lagrange has published in the ,mmoires de ,turin a beautiful method for determining these mean values when the law of the errors of the observations is known4 ,i have given for the same purpose a method based upon a singular contrivance which may be employed with advantage in other questions of analysis2 and this1 by permitting indefinite extension in the whole course of a long calculation of the functions which ought to be limited by the nature of the problem1 indicates the modifications which each term of the final result ought to receive by virtue of these limitations4 ,it has already been seen that each observation furnishes an equation of condition of the first degree1 which may always be disposed of in such a manner that all its terms be in the first member1 the second being zero4 ,the use of these equations is one of the principal causes of the great precision of our astronomical tables1 because an immense number of excellent observations has thus been made to concur in determining their elements4 ,when there is only one element to be determined ,ctes prescribed that the equations of condition should be prepared in such a manner that the coefficient of the unknown element be positive in each of them2 and that all these equations should be added in order to form a final equation1 whence is derived the value of this element4 ,the rule of ,ctes was followed by all calculators1 but since he failed to determine several elements1 there was no fixed rule for combining the equations of condition in such a manner as to obtain the necessary final equations2 but one chose for each element the observations most suitable to determine it4 ,it was in order to obviate these gropings that ,legendre and ,gauss concluded to add the squares of the first members of the equations of condition1 and to render the sum a minimum1 by varying each unknown element2 by this means is obtained directly as many final equations as there are elements4 ,but do the values determined by these equations merit the preference over all those which may be obtained by other means8 ,this question1 the calculus of probabilities alone was able to answer4 ,i applied it1 then1 to this subject1 and obtained by a delicate analysis a rule which includes the preceding method1 and which adds to the advantage of giving1 by a regular process1 the desired elements that of obtaining them with the greatest show of evidence from the totality of observations1 and of determining the values which leave only the smallest possible errors to be feared4 ,however1 we have only an imperfect knowledge of the results obtained1 as long as the law of the errors of which they are susceptible is unknown2 we must be able to assign the probability that these errors are contained within given limits1 which amounts to determining that which ,i have called the weight of a result4 ,analysis leads to general and simple formul for this purpose4 ,i have applied this analysis to the results of geodetic observations4 ,the general problem consists in determining the probabilities that the values of one or of several linear functions of the errors of a very great number of observations are contained within any limits4 ,the law of the possibility of the errors of observations introduces into the expressions of these probabilities a constant1 whose value seems to require the knowledge of this law1 which is almost always unknown4 ,happily this constant can be determined from the observations4 ,in the investigation of astronomical elements it is given by the sum of the squares of the differences between each observation and the calculated one4 ,the errors equally probable being proportional to the square root of this sum1 one can1 by the comparison of these squares1 appreciate the relative exactitude of the different tables of the same star4 ,in geodetic operations these squares are replaced by the squares of the errors of the sums observed of the three angles of each triangle4 ,the comparison of the squares of these errors will enable us to judge of the relative precision of the instruments with which the angles have been measured4 ,by this comparison is seen the advantage of the repeating circle over the instruments which it has replaced in geodesy4 ,there often exists in the observations many sources of errors3 thus the positions of the stars being determined by means of the meridian telescope and of the circle1 both susceptible of errors whose law of probability ought not to be supposed the same1 the elements that are deduced from these positions are affected by these errors4 ,the equations of condition1 which are made to obtain these elements1 contain the errors of each instrument and they have various coefficients4 ,the most advantageous system of factors by which these equations ought to be multiplied respectively1 in order to obtain1 by the union of the products1 as many final equations as there are elements to be determined1 is no longer that of the coefficients of the elements in each equation of condition4 ,the analysis which ,i have used leads easily1 whatever the number of the sources of error may be1 to the system of factors which gives the most advantageous results1 or those in which the same error is less probable than in any other system4 ,the same analysis determines the laws of probability of the errors of these results4 ,these formul contain as many unknown constants as there are sources of error1 and they depend upon the laws of probability of these errors4 ,it has been seen that1 in the case of a single source1 this constant can be determined by forming the sum of the squares of the residuals of each equation of condition1 when the values found for these elements have been substituted4 ,a similar process generally gives values of these constants1 whatever their number may be1 which completes the application of the calculus of probabilities to the results of observations4 ,i ought to make here an important remark4 ,the small uncertainty that the observations1 when they are not numerous1 leave in regard to the values of the constants of which ,i have just spoken1 renders a little uncertain the probabilities determined by analysis4 ,but it almost always suffices to know if the probability1 that the errors of the results obtained are comprised within narrow limits1 approaches closely to unity2 and when it is not1 it suffices to know up to what point the observations should be multiplied1 in order to obtain a probability such that no reasonable doubt remains in regard to the correctness of the results4 ,the analytic formul of probabilities satisfy perfectly this requirement2 and in this connection they may be viewed as the necessary complement of the sciences1 based upon a totality of observations susceptible of error4 ,they are likewise indispensable in solving a great number of problems in the natural and moral sciences4 ,the regular causes of phenomena are most frequently either unknown1 or too complicated to be submitted to calculus2 again1 their action is often disturbed by accidental and irregular causes2 but its impression always remains in the events produced by all these causes1 and it leads to modifications which only a long series of observations can determine4 ,the analysis of probabilities develops these modifications2 it assigns the probability of their causes and it indicates the means of continually increasing this probability4 ,thus in the midst of the irregular causes which disturb the atmosphere1 the periodic changes of solar heat1 from day to night1 and from winter to summer1 produce in the pressure of this great fluid mass and in the corresponding height of the barometer1 the diurnal and annual oscillations2 and numerous barometric observations have revealed the former with a probability at least equal to that of the facts which we regard as certain4 ,thus it is again that the series of historical events shows us the constant action of the great principles of ethics in the midst of the passions and the various interests which disturb societies in every way4 ,it is remarkable that a science1 which commenced with the consideration of games of chance1 should be elevated to the rank of the most important subjects of human knowledge4 ,i have collected all these methods in my ,thorie analytique des ,probabilits1 in which ,i have proposed to expound in the most general manner the principles and the analysis of the calculus of probabilities1 likewise the solutions of the most interesting and most difficult problems which calculus presents4 ,it is seen in this essay that the theory of probabilities is at bottom only common sense reduced to calculus2 it makes us appreciate with exactitude that which exact minds feel by a sort of instinct without being able ofttimes to give a reason for it4 ,it leaves no arbitrariness in the choice of opinions and sides to be taken2 and by its use can always be determined the most advantageous choice4 ,thereby it supplements most happily the ignorance and the weakness of the human mind4 ,if we consider the analytical methods to which this theory has given birth2 the truth of the principles which serve as a basis2 the fine and delicate logic which their employment in the solution of problems requires2 the establishments of public utility which rest upon it2 the extension which it has received and which it can still receive by its application to the most important questions of natural philosophy and the moral science2 if we consider again that1 even in the things which cannot be submitted to calculus1 it gives the surest hints which can guide us in our judgments1 and that it teaches us to avoid the illusions which ofttimes confuse us1 then we shall see that there is no science more worthy of our meditations1 and that no more useful one could be incorporated in the system of public instruction4