But Galileo's attention was called to the subject of Chance in another form. From his letters we learn that in his day the Florentine gentlemen, instead of employing their time in attention to ladies, or in the stables, or in excessive gaming, were accustomed to improve themselves by learned conversation in cultivated society. In one of their meetings the following question was proposed: a horse is really worth a 100 crowns; one person estimated it at io crowns, another at a 1000-which of the two made the more extravagant estimate? Among the persons who were consulted was Galileo; he pronounced the two estimates to be equally extravagant, because the ratio of a 1000 to a 100 is the same as the ratio of 100 to 10. On the other hand, a priest named Nozzolini, who was also consulted, pronounced the higher estimate to be more extravagant than the other, because the excess of a 1000 above a 100 is greater than that of a 100 above 10. It appears that Galileo had the same notion as Nozzolini when the question was first proposed to him, but afterwards changed his mind.

It was in 1654 that the subject was destined to receive a greater development. The Chevalier de Méré applied to Pascal for a solution of two problems, for which he was unable to find answers. The one was to ascertain in how many throws one might bet with advantage that two sixes would be thrown with two dice; the other to find a rule for dividing the stakes between two players-who were desirous of breaking off an unfinished game-in exact proportion to their relative fortune at the time, and to their chances of winning the remaining stakes. Pascal considered all the possible combinations that could be formed by the simultaneous throw of two dice, and of all the possible changes that might occur in a game of cards, interrupted at any point, and what number of them were in favour of the event for which his solution was required. He then computed the number of cases in which two sixes could be thrown with two dice, and the number of changes which in the actual state of the game of cards would secure to each player, separately, the whole or any part of the stakes, and thus arrived by proportion at the required result. Simple as this method seems (continues Mr. Samuel Brown), it was the first attempt to employ mathematics in such subjects; at least the first that, being closely followed up, led directly to the great discoveries that ensued. Boole, in his Laws of Thought, says this was the first of a long series of problems destined to call into existence new methods in mathematical analysis, and to render valuable service to the practical concerns of life.

Fermat, a magistrate of the Parl. of Toulouse, a mathematician of great repute in his day, was a friend of Pascal, one with whom he corresponded daily on the subject of his studies, and to whom he freely communicated his doubts and his discoveries. He forwarded to him the solution he had arrived at. The original correspondence is now lost; but it appears clear that in his solution he had merely replied to the questions put to him; and however ingenious and minute the investigation, it did not lead to ready solutions of other questions of a similar kind. Copies of the correspondence will be found in the works of the respective authors.

It was Fermat who generalized the solution, and found a rule not merely for ascertaining the value of each player's expectation in the particular case referred to, but at any moment of interrupting the game between any number of players. This was the next step, and by far the most important one, in the science of Prob. Without it the attempt of Pascal might have remained, like some previous problems and speculations by Galileo and Cardan, in obscurity till a much later period.

The correspondence of Pascal and Fermat was not generally made known at this time, though Pascal (as is shown by a letter to one of the learned societies in Paris in 1654) appears to have entertained the thought of introducing his discovery to the world. He evidently appreciated the importance of it; but about this time he met with the accident which led him to retire altogether from his scientific studies, and devote himself to those religious pursuits of which his celebrated Provincial Letters were the fruit. Fermat appears to have been indifferent to his discovery, and but little progress was made for nearly half a century.

Huygens, a celebrated geometrician, on the mere rumour of the questions submitted to Pascal, wrote a treatise in Dutch, which was afterwards translated into Latin by Schootens, and pub. by the latter in a work which appeared in 1658. This was the first systematic treatise which appeared on the Doctrine of Chances. It contained an analysis of the various questions which had been solved by Pascal and Fermat, and at the end five new questions were proposed; the solutions of which, simple as they may now appear, were then attended with considerable difficulty. The analysis of two of them was in fact given for the first time by Montmort half a century after their pub.-Galloway.

In the treatment of the subject these great men had already in effect passed beyond the immediate range of the original inquiry, and were rapidly developing the Theory of Probabilities. Prof. Todhunter indeed remarks: "The practice of games of chance must at all times have directed attention to some of the elementary considerations of the Theory of Prob." A still further stage of progress was near at hand. [1671.]

In 1663 was pub. a treatise De Ludo Alea by Cardan. This was included in the collected works of that author, then for the first time pub. [Cardan died in 1576]. It contains much miscellaneous matter connected with gambling, such as descriptions of 32

VOL. I.

games, and an account of the precautions necessary to be employed in order to guard against adversaries disposed to cheat. The discussions relating to Chance form but a small portion of the treatise, which may be best described as the Gambler's Manual.-Todhunter.

In 1671 the Grand Pensionary De Wit came upon the scene. This great man, celebrated alike as a statesman and a mathematician of the highest repute, who had already pub., in 1650, a work on Curves, to which Condorcet refers in terms of eulogy, conceived the design of applying the doctrines of probabilities to the valuation of human life in the question of Government annuities. The result of his labours we have already given in some detail in our hist. of ANNU. ON LIVES.

We must confine ourselves in the remainder of the present art. as much as possible to the Doctrine and Laws of Chance. The Theory of Prob. as developed from the same will be treated of in detail under PROBABILITY; while the application of the Science of Prob. to the contingencies of human life will be treated of under LIFE CONTINGENCIES. In 1692 John Arbuthnot, M.D., pub. a work, Of the Laws of Chance; or, a Method of Calculating the Hazards of Game Plainly Demonstrated. This was prob. the first work pub. in England on the subject. In the same year there appeared a trans. of Huygens' tract into English, accompanied by an Essay on the Laws of Chance, which is supposed by some to have been written by Motte, stated to have been the then Sec. of the Royal So.; but Prof. Todhunter attributes it to Arbuthnot. In this essay are some remarks relative to the advantage of the banker in the game of Pharaon. [See 1738.]

We have shown in our account of the BRESLAU TABLE OF MORT., pub. 1693, how Dr. Halley applied the Doctrine of Chance to the solution of the problems first presented to him by the study of his newly-formed T.

In 1693 there was also pub. in vol. xvii. of Phil. Trans., An Arithmetical Paradox Concerning the Chances of Lotteries, by the Hon. Francis Roberts, F.R.S. [LOTTERIES.] In 1699 John Craig, a Scotch mathematician, pub. a remarkable tract, of which we shall have to speak more at large hereafter. Its main feature was the application of mathematical calculations to the credibility of Gospel history; and he predicted the termination of the Christian religion at a date determined by the Doctrine of Chances! About this date Nicolas Bernouilli pub. a thesis, De Arte Conjectandi in Jure, of which we do not find any detailed account. It is mentioned by Montmort.

In 1708 Pierre Redmond de Montmort pub. his Essai d'Analyse sur les Jeux de Hazards, of which we shall have to speak more at large under date 1714, when the 2nd ed. appeared. Todhunter says of this work of 1708 that, "with the courage of Columbus, he revealed a new world to mathematicians." He adds that much which Montmort had included in his chapter on Combinations would now be considered to belong rather to the chapter on Chances. There were numerous examples about drawing cards and throwing dice. In 1709 M. Barbeyrac pub., Traité du feu, one of the objects of which appears to have been to show that religion and morality do not prohibit the use of games in general, or of games of chance in particular. Montmort refers to this book, which he says he had lately received from Paris. He said it was un livre de morale. He praises the author, but considers him to be wrong sometimes in his calculations, and gives an example. Nicolas Bernouilli, in reply, says that the author of the book is M. Barbeyrac ; he agrees with Montmort in his general opinion respecting the book; but in the example in question he thinks Barbeyrac right, and Montmort wrong.

In 1710 De Moivre submitted to the Royal So. a paper, On the Doctrine of Chances, and the same was pub. in the Phil. Trans. for that year. This paper was afterwards expanded, and pub. in book form. [See 1718.]

In 1713 the Ars Conjectandi of James Bernouilli was pub. under the circumstances we have already explained. [BERNOUILLI, JAMES.] The author solved four out of the five problems which Huygens had placed at the end of his treatise. The last of the five problems which Huygens left to be solved was the most remarkable of all. It is the first example on the Duration of Play, a subject which afterwards exercised the highest powers of De Moivre, Lagrange, and Laplace. James Bernouilli solved the problem, and added, without a demonstration, the result for a more general problem, of which that of Huygens was a particular case. "Perhaps (says Todhunter) the most valuable contribution to the subject which this part of the work contains, is a method of solving problems in chance which James Bernouilli speaks of as his own, and which he frequently uses." Finally, "We may observe that Bernouilli seems to have found-as most who have studied the subject of chances have also found that it was extremely easy to fall into mistakes, especially by attempting to reason without strict calculation."

In 1714 the 2nd ed. of Montmort's essay appeared [1st ed., 1708]. It was much more bulky than the first ed. He makes some judicious obs. on the foolish and superstitious notions which were prevalent among persons devoted to games of chance, and proposes to check these by showing, not only to such persons, but to men in general, that there are rules in chance, and that for want of knowing these rules mistakes are made which entail adverse results; and these results men impute to destiny instead of their own ignorance. The work is divided into four parts. The Ist contains the theory of combinations; the 2nd discusses certain games of chance depending on cards; the 3rd discusses certain games of chance depending on dice; the 4th part contains the solution

of various problems in chances, including five problems proposed by Huygens. Todhunter concludes an exhaustive criticism of this ed. as follows:

Montmort's work, on the whole, must be considered highly creditable to his acuteness, perseverance, and energy. The courage is to be commended which led him to labour in a field hitherto so little cultivated, and his example served to stimulate his more distinguished successor. De Moivre was certainly far superior in mathematical power to Montmort, and enjoyed the great advantage of a long life, extending to more than twice the duration of that of his predecessor; on the other hand, the fortunate circumstances of Montmort's position gave him that abundant leisure which De Moivre in exile and poverty must have found it impossible to secure.

De Moivre spoke in very high terms of Montmort's work, and said that therein he had given "many proofs of his singular genius and extraordinary capacity."

In 1714 also M. Barbeyrac pub. in Amsterdam a discourse, Sur la Nature du Sort. In the same year Nicolas Bernouilli transmitted to the Royal So. a problem in the doctrine of chances, which was pub. in the Phil. Trans.

In 1714 also appeared a work, Christiani Hugenii Libellus de Ratiociniis in Ludo Alea; or, the Value of all Chances in Games of Fortune; Cards, Dice, Wagers, Lotteries, etc., Mathematically Demonstrated. Lond.: Printed by S. Keimer for T. Woodward, near the Temple Gate, in Fleet St., 1714. This was a trans. of Huygens' treatise [1658], by W. Browne. In his adv. to the reader, Browne refers to a trans. of Huygens' treatise which had been made by Arbuthnot; he also notices the labours of Montmort and De Moivre. In 1718 De Moivre pub., in book form, The Doctrine of Chances; or, a Method of Calculating the Prob. of Events at Play. This, as we have said, was an expansion of his paper of 1710. A 2nd ed. of this work was pub. in 1738; 3rd ed., 1756 (after the author's death). The author says in his preface:

'Tis now about 7 years since I gave a specimen in the Phil. Trans. of what I now more largely treat of in this book. The occasion of my then undertaking this subject was chiefly owing to the desire and encouragement of the Hon. Francis Robartes, Esq. (now Earl of Radnor), who, upon occasion of a French tract, called L'Analyse des Jeux de Hazards, which had lately been pub., was pleased to propose to me some problems of much greater difficulty than any he had found in that book; which having solved to his satisfaction, he engaged me to methodize those problems, and to lay down the rules which had led me to their solution. After I had proceeded thus far, it was enjoined me by the Royal So. to communicate to them what I had discovered on this subject; and thereupon it was ordered to be put in the Trans., not so much as a matter relating to play, but as containing some general speculations not unworthy to be considered by the lovers of truth.

Many important results were here first pub. by De Moivre, although it is true that these results already existed in manuscript in the Ars Conjectandi, and the correspondence between Montmort and the Bernouillis.-Todhunter.

In the Hist. of the Academy of Paris for 1728 [pub. 1730], there is a notice respecting some results obtained by Mairan-Sur le Jeu de Pair ou Non. The art. is not by Mairan. In the 9th vol. of Actorum Eruditorum . Supplementa, pub. in Leipzig in 1729, there is a memoir: Johannis Rizzetti Ludorum Scientia, sive Artis Conjectandi elementa ad alias applicata, from which it appears that Daniel Bernouilli had a controversy with Rizzetti and Riccati relating to some problems in chances. It led to nothing new, and chiefly turned upon the proper definition of "expectation.'

In the Hist. of the Academy of Paris for 1730 [pub. 1732], there is a memoir by M. Nicole, entitled: Examen et Résolution de quelques Questions sur les feux. In the same vol. is another memoir by Nicole. But, in each case, Montmort and De Moivre had already covered the same ground.

In the St. Petersburg Memoirs (vol. 5) for 1730-31, there is an interesting paper by Daniel Bernouilli on the relative values of the expectations of individuals who engage in play, or stake sums on contingent benefits, when regard is had to the difference of their fortunes; a consideration which in many cases it is necessary to take into account; for it is obvious that the value of a sum of money to an individual depends not merely on its absolute amount, but also on his previous wealth. On this principle Bernouilli has founded a theory of moral expectation, which admits of numerous and important applications to the ordinary affairs of life.-Galloway.

In 1733 the Compte de Buffon communicated to the Academy of Sciences in Paris the solution of some problems in chances. [See 1777.]

In 1738 there was pub. another ed. of the trans. of Huygens, spoken of under date 1692, with the following title: Of the Laws of Chance; or, a Method of Calculation of the Hazards of Game, plainly demonstrated, and applied to Games at present most in use; which may be easily extended to the most intricate cases of Chance imaginable. The 4th ed. revis'd by John Ham. By whom is added a Demonstration of the Gain of the Banker in any circumstance of the Game call'd Pharaon; and how to determine the Odds at the Ace of Hearts, or Fair Chance; with the Arithmetical Solution of some Questions relating to Lotteries; and a few Remarks upon Hazard and Backgammon. London: Printed for B. Motte and C. Bathurst, at the Middle Temple Gate in Fleet-street, M. DCC. XXXVIII. This second part, which is here attributed to John Ham, Todhunter believes to have been taken in greater part from De Moivre, who however is not named in the work.

In 1738 also there was pub. 2nd ed. of De Moivre's Doctrine of Chances, "fuller, clearer, and more correct than the first," by admission of the author. [See 1718.]

In 1739 there was pub. in Florence, "Par Mr. D. M.," a work, Calcul du feu appellé par les François le trente-et-quaranté, et que l'on nomme à Florence le trente-et-un.":

The problem is solved by examining all cases which can occur, and counting up the number of ways. The operation is most laborious, and the work is perhaps the most conspicuous example of misdirected industry which the literature of games of chance can furnish.- Todhunter.

In 1740 Mr. Thomas Simpson pub., The Nature and Laws of Chance, containing among other Particulars [see below]. The whole after a New, General, and Conspicuous Manner, and Illustrated with a great variety of Examples. The part of the title just

omitted is as follows:

(1) The solutions of several abstruse and important problems. (2) The doctrine of combinations and permutations clearly deduced. (3) A new and comprehensive problem of great use in discovering the advantage or loss in lotteries, raffles, etc. (4) A curious and extensive problem on the duration of play. (5) Problems for determining the prob, of winning at bowls, coits, cards, etc. (6) A problem for finding the trials wherein it may be undertaken that a proposed event shall happen or fail a given number of times. (7) A problem to find the chance for a given number of points on a given number of dice. (8) Full and clear investigations of two problems, added at the end of Mr. De Moivre's last ed., one of them allowed by that great man to be the most useful on the subject, but their demonstrations there omitted. (9) Two new methods for summing of series.

This work engaged some attention. Simpson implies in his preface that his design was to produce an introduction to the subject less expensive and less abstruse than De Moivre's work; and in fact Simpson's work may be considered as an abridgment of De Moivre's. Simpson's problems are nearly all taken from De Moivre, and the mode of treatment is substantially the same. The very small amount of new matter which is contributed by a writer of such high power as Simpson shows how closely De Moivre had examined the subject, as far as it was accessible to the mathematical resources of the period.-Todhunter. In the Hist. of the Berlin Academy for 1751 [pub. 1753] there appeared Euler's first memoir, entitled Calcul de la Probabilité dans le Jeu de Rencontre. The problem discussed is that which is called the game of Treize, which had previously been treated of by Montmort and Nicolas Bernouilli, and more simple results than those now given had been obtained by the latter. [See also 1764.]

In the 2nd vol. of Dodson's Mathematical Repository, dated 1753, there are some problems on Chances, which, however, present nothing new or important.

In 1754 Edmund Hoyle pub. An Essay towards making the Doctrine of Chances easy, to those who understand Vulgar Arithmetic only, &c., &c., &c. The preface says:

In order to put play upon the most equal foot, in this treatise you have practical rules and examples, plainly expressed in words at length, whereby all various cases and the odds of games of any kind, may be easily resolved, without any knowledge of algebra or logarithms; by which the most unskilful person in betting his money is put upon an equal foot with those who have applied themselves to this study for many years. He also gave: "A short table of the powers of two, showing the odds of winning or losing any number of games upon an equality of chance."

In 1754 D'Alembert contributed to the Encyclopédie [Paris] an art., Croix ou Pile, wherein he proposes to find the chance of throwing head in the course of two throws with a coin. He deals with questions of play also. [See 1761.]

The result of all De Moivre's laboured and continuous investigation of the subject was embodied in this proposition-which we believe only appeared in the last ed. of his work, 1756, revised just before and pub. after his death-" That although chance produces irregularities, still the odds will be infinitely great, that in process of time, those irregularities will bear no proportion to the recurrency of that order which naturally results from ORIGINAL DESIGN."

In 1757 there was pub. in Padua a quarto vol., Dell' Azione del Caso nelle Invenzioni, e dell'influsso degli Åstri ne' Corpi Terrestri Dissertazioni due. The first dissertation is on the influence of Chance in inventions. It recognizes this influence, and gives various examples. The second is on the influence of the celestial bodies on men, animals, and plants; and is intended to show that there is no influence produced in the sense in which astrologers understand such influence. The author expressed more belief in the squaring of the circle than in the Newtonian theory of gravitation then recently propounded.

In 1758 Mr. Samuel Clark pub., The Laws of Chance; a Mathematical Investigation of the Prob. arising from any proposed Circumstances of Play, applied to the Solution of a great variety of Problems relating to Cards, Bowls, Dice, Lotteries, etc. The work is written in a very plain and simple style. It is almost entirely based upon De Moivre and Simpson; but it does not contain anything new or important.

In the 2nd vol. of his Opuscules Mathématiques, pub. 1761, D'Alembert says, in reference to chance and probability, that we must distinguish between what is metaphysically possible and what is physically possible. In the first class are included all those things of which the existence is not absurd; in the second are included only those things not too extraordinary to occur in the ordinary course of events. It is metaphysically possible to throw two sixes with two dice a hundred times running; but it is physically impossible, because it never has happened, and never will happen. He applied this principle in various forms.

In vol. v. of the Acta Helvetica, 1762, there is a memoir by M. Mallet, entitled, Recherches sur les avantages de trois Feueurs qui font entr'eux une Poule au trictrac ou à un autre Feu quelconque. The problem had been treated by De Moivre, but was now extended by Mallet.

In the Phil. Trans. for 1763 [pub. 1764] there is the following memoir: An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price in a Letter to John Canton, A.M., F.R.S. This letter commences: "Dear Sir, I now send you an essay which I have found among the papers of our deceased friend Mr. Bayes, and which, in my opinion, has great merit, and well deserves to be preserved." The essay of Bayes follows the introductory letter. It begins with a brief demonstration of the general laws of the Theory of Prob., and then establishes his theorem. Dr. Price added: An Appendix containing an application of the foregoing rules to some particular case.

The Phil. Trans. for 1764 [pub. 1765] contains a memoir: A Demonstration of the second Rule in the Essay towards the solution of a Problem in the Doctrine of Chances, pub. in the Phil. Trans., vol. liii., communicated by the Rev. Mr. Richard Price in a Letter to Mr. John Canton, M.A., F.R.S. This memoir contains Bayes' demonstration of his principal rule for approximation; and some investigations by Dr. Price which also relate to the subject of approximation.

In the Hist. of the Berlin Academy for 1764 [pub. 1766] was a further memoir by Euler: Sur l'avantage du Banquier au Jeu de Pharaon. Euler here merely solves the problems which had been already solved by Montmort and N. Bernouilli, to whom, however, he makes no reference [see 1751]. In the same vol. were two other memoirs on the same problem by Beguelin.

In the Hist. of the Berlin Academy for 1768 [pub. 1770] there is a contribution from John Bernouilli: Mémoire sur un Problème de la Doctrine du Hazard. It requires no special comment here. [MARRIAGE.]

In 1772 Mr. Fenn pub. Calculations and Formula for determining the advantages or disadvantages of Gamesters.

In the Mémoires par divers Savans [vol. vi.], 1774, there are two memoirs by Laplace. The first: Mémoire sur les suites récurro-récurrentes et sur leurs usages dans la théorie des hazards. The author considers three problems. The first is the problem of the duration of play, supposing two players of unequal skill and unequal capital. Todhunter says that Laplace rather shows how the problem may be solved than actually solves it. He begins with the case of equal skill and equal capital, and then passes on to the case of unequal skill. He does not actually discuss the case of unequal cap., but intimates that there will be no obstacle except the length of the process. The second memoir is on Prob. [PROBABILITIES.]

In the Select Memoirs of the Academy of Berlin for 1775 [pub. 1777] there is a paper by Lagrange which treats of the Theory of Chances. The 5th problem therein relates to the duration of play in the case in which one player has unlimited capital. This subject had been previously treated by De Moivre.

In 1776 was pub. anon. Reflections on Gaming, Annuities, and Usurious Contracts; and in the same year some Essays on Mathematical Subjects, by W. Emerson, including one on the Laws of Chance. "Emerson's work would be most dangerous for a beginner, and quite useless for a more advanced student.”—Todhunter.

In 1777 Buffon pub. his Essai d'Arithmétique Morale. Struck by the remarkable speculations of Daniel Bernouilli, on the subject of the relative gain or loss at play on the private fortunes of the players, he commenced to write in 1760, and pub. in 1777, that most eloquent and impressive attack on the immorality, the danger, and the vicious principle of government gaming tables, and contributed perhaps more than any other to their gradual extinction in some countries, and the disrepute to which they have justly fallen in others. "The Doctrine of Chances (says Mr. Brown) has seldom been converted to a more honourable or benevolent purpose, and never with greater eloquence and force." The following is one of Buffon's ingenious arguments: He supposes two players of equal fortune, and that each stakes half of his fortune. He says that the player who wins will increase his fortune by a third, and the player who loses will diminish his by a half; and as half is greater than a third, there is more to fear from loss than to hope from gain. Other writers, as we have seen, have touched upon this same consideration. [MATHEMATICAL EXPECTATION.] [MONEY.]

In the Select Berlin Memoirs for 1780 [pub. 1782], there is one by Herr Prevost, Sur les principes de la Théorie des gains fortuits. This simply consists of an attempted criticism of the elementary principles laid down by James Bernouilli, Huygens, and De Moivre. In the Select Memoirs of the Berlin Academy for 1784 [pub. 1786] there is one by D'Anieres, entitled Réflexions sur les Jeux de Hazard. The memoir is not mathematical; it alludes to the fact that games of hazard are prohibited by governments, and shows that there are different kinds of such games, namely, those in which a man may ruin his fortune, and those which cannot produce more than a trifling loss in any case.-Todhunter. In 1787 W. Painter pub., Guide to the Lottery; or, the Laws of Chance. In the same year John Johnson pub., A Complete Abridgment of the Laws respecting Gaming and Usury, with Adjudged Cases.

In 1792 there was pub. in Paris, in connexion with the Encyclopédie Méthodique, a special vol. entitled Dictionnaire des Jeux, faisant suite au Tome III. des Mathématiques. There are no mathematical investigations, but in several cases the numerical values of the chances are given. The work does not appear to have been completed.