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games of chance, and they have been in all countries from the rudest to the most civilized, there must have been some numerical estimate formed of the probability of certain events, by which the stakes and the expectations of the gamesters must have been regulated. The principle just stated, must therefore with more or less distinctness have been long recognized; but nothing like a system of reasoning founded on it is to be found before the time of Fermat and Pascal. Huygens was the next after these two illustrious men who treated of this matter in a treatise, De Ratiociniis in Ludo Aleæ. Several other mathematicians, Huddes and De Witt in Holland, Halley in England, applied the same calculus to the probabilities of human life, and the latter published the first tables relative to that object. James Bernoulli, about the same time, proposed and resolved many problems concerning probabilities, and composed the treatise entitled Ars Conjectandi, which was not published till 1713, some years after his death. This work is worthy of the high reputation of the author, who treats in it of the probability which a succession of the same events, at any time, gives of its continuance; and he was the first to demonstrate a proposition concerning the indefinite multiplication of casual events, to which we shall again have occasion to advert. Monmort published an estimable work on the same subject, Essai sur les Jeux de
Hasard; and Demoivre followed with his treatise On the Doctrine of Chances, which first appeared in the Philosophical Transactions for 1711, but was afterwards published in three editions, which the author successively improved. This work is the first that mentioned the theory of recurring series, a subject of such importance in algebra, and connected with so many of the discoveries which have since been made in the calculus of Finite Differences. Laplace does great justice to it, and has entered into an analysis of the part that relates to series. Demoivre gives a demonstration of the theorem of Bernoulli, just referred to, which, in a series of events, serves to connect the future and the past. Several other mathematicians, and particularly Lagrange, have been attracted by the results which this theory offered, and by the difficulty of the problems it suggested, which seemed in many respects to require a new application of analysis. The last who has treated of it, is our author himself, in a large work in quarto, entitled Théorie Analytique des Probabilités, published at Paris in 1812. The essay now under review, is an abstract of this last, containing an account of the more important conclusions deduced in it, together with many general and profound remarks on the principles of the calculus, and their application to the researches of philosophy, as well as to the affairs of life.
The analytical work contains some valuable improvements in this branch of the mathematics. We have adverted to the use made by Demoivre, in his work on Chances, of the series, called Recurring, in which the coefficient of each term is formed in the same manner from the coefficients of a certain number of the preceding terms. The generalization of this property led Laplace to consider all those series in which the coefficients are formed by substituting the exponents, every where, in the same formula; or where, in every term, the coefficient is the same function of the exponent. A series of this kind being supposed, a function of the variable quantity may be found, from the developement of which the series may be derived; and this function is what Laplace calls the Generating Function (Fonction Génératrice) of the coefficients in the supposed series, or rather of the function in which all those series are included. This gives rise to a new branch of analysis, the calculus of Generating Functions, the principles of which he first explained in the Memoirs of the Academy of Sciences for 1779. From these series, by applying the method of finite and partial differences, he has extracted results that throw great light on the Doctrine of Chances, and readily afford demonstrations of many propositions that cannot but with the greatest difficulty be proved by any other means. It must not seem surprising that the
Doctrine of Series is thus intimately connected with the Theory of Probabilities; for it should be remembered, that the first considerable improvement in that theory came from the same quarter. The numbers of combinations that can be formed of a given number of things, taking them two and two, three and three, &c. are given by the successive coefficients of a binomial raised to the power denoted by the number of things in question. Such combinations are evidently much concerned in the laws of chance; and Bernoulli deduced from them a great number of conclusions concerning those laws. Demoivre went farther than Bernoulli, and Laplace much farther than either; but to give any adequate idea of the analytical methods which he has employed, is not to be expected in an abstract like the present. For a general view of the analytical methods applied to the calculation of probabilities, we may refer the reader to the conclusion of the Essai Philosophique, p. 90, &c., and to the beginning of the Théorie Analytique. To a passage in the latter, however, we cannot but advert, and with much less satisfaction than we have generally felt in pointing out any of the remarks of this celebrated writer to the attention of our readers. "Il paraît que Fermat, le véritable inventeur du calcul différentiel, a considéré ce calcul comme une dérivation de celui des dif ferences finies," &c. Against the affirmation that
Fermat is the real inventor of the differential calculus, we must enter a strong and solemn protestation. The age in which that discovery was made, has been unanimous in ascribing the honour of it either to Newton or Leibnitz; or, as seems to us much the fairest and most probable opinion, to both; that is, to each independently of the other, the priority in respect of time being somewhat on the side of the English mathematician. The writers of the history of the mathematical sciences have given their suffrages to the same effect; -Montucla, for instance, who has treated the subject with great impartiality, and Bossut, with no prejudices certainly in favour of the English philosopher. In the great controversy, to which this invention gave rise, all the claims were likely to be well considered; and the ultimate and fair decision, in which all sides seem to have acquiesced, is that which has just been mentioned. It ought to be on good grounds, that a decision, passed by such competent judges, and that has now been in force for a hundred years, should all at once be reversed. Fermat has strong claims undoubtedly on the gratitude of posterity; and we do not believe that there exists, either among the productions of ancient or modern science, a work of the same size with his Opera Varia, that contains so many traits of ori ginal invention. He had certainly approached very near to the differential or fluxionary calculus,